Short-time Lyapunov exponent analysis
Vastano, J. A.
1990-01-01
A new technique for analyzing complicated fluid flows in numerical simulations has been successfully tested. The analysis uses short time Lyapunov exponent contributions and the associated Lyapunov perturbation fields. A direct simulation of the Taylor-Couette flow just past the onset of chaos demonstrated that this new technique marks important times during the system evolution and identifies the important flow features at those times. This new technique will now be applied to a 'minimal' turbulent channel.
Short-time Lyapunov exponent analysis
Vastano, J. A.
1990-01-01
A new technique for analyzing complicated fluid flows in numerical simulations has been successfully tested. The analysis uses short time Lyapunov exponent contributions and the associated Lyapunov perturbation fields. A direct simulation of the Taylor-Couette flow just past the onset of chaos demonstrated that this new technique marks important times during the system evolution and identifies the important flow features at those times. This new technique will now be applied to a 'minimal' turbulent channel.
Crauel, Hans; Eckmann, Jean-Pierre
1991-01-01
Since the predecessor to this volume (LNM 1186, Eds. L. Arnold, V. Wihstutz)appeared in 1986, significant progress has been made in the theory and applications of Lyapunov exponents - one of the key concepts of dynamical systems - and in particular, pronounced shifts towards nonlinear and infinite-dimensional systems and engineering applications are observable. This volume opens with an introductory survey article (Arnold/Crauel) followed by 26 original (fully refereed) research papers, some of which have in part survey character. From the Contents: L. Arnold, H. Crauel: Random Dynamical Systems.- I.Ya. Goldscheid: Lyapunov exponents and asymptotic behaviour of the product of random matrices.- Y. Peres: Analytic dependence of Lyapunov exponents on transition probabilities.- O. Knill: The upper Lyapunov exponent of Sl (2, R) cocycles:Discontinuity and the problem of positivity.- Yu.D. Latushkin, A.M. Stepin: Linear skew-product flows and semigroups of weighted composition operators.- P. Baxendale: Invariant me...
Analysis of human standing balance by largest lyapunov exponent.
Liu, Kun; Wang, Hongrui; Xiao, Jinzhuang; Taha, Zahari
2015-01-01
The purpose of this research is to analyse the relationship between nonlinear dynamic character and individuals' standing balance by the largest Lyapunov exponent, which is regarded as a metric for assessing standing balance. According to previous study, the largest Lyapunov exponent from centre of pressure time series could not well quantify the human balance ability. In this research, two improvements were made. Firstly, an external stimulus was applied to feet in the form of continuous horizontal sinusoidal motion by a moving platform. Secondly, a multiaccelerometer subsystem was adopted. Twenty healthy volunteers participated in this experiment. A new metric, coordinated largest Lyapunov exponent was proposed, which reflected the relationship of body segments by integrating multidimensional largest Lyapunov exponent values. By using this metric in actual standing performance under sinusoidal stimulus, an obvious relationship between the new metric and the actual balance ability was found in the majority of the subjects. These results show that the sinusoidal stimulus can make human balance characteristics more obvious, which is beneficial to assess balance, and balance is determined by the ability of coordinating all body segments.
He, Jianbin; Yu, Simin; Cai, Jianping
2016-12-01
Lyapunov exponent is an important index for describing chaotic systems behavior, and the largest Lyapunov exponent can be used to determine whether a system is chaotic or not. For discrete-time dynamical systems, the Lyapunov exponents are calculated by an eigenvalue method. In theory, according to eigenvalue method, the more accurate calculations of Lyapunov exponent can be obtained with the increment of iterations, and the limits also exist. However, due to the finite precision of computer and other reasons, the results will be numeric overflow, unrecognized, or inaccurate, which can be stated as follows: (1) The iterations cannot be too large, otherwise, the simulation result will appear as an error message of NaN or Inf; (2) If the error message of NaN or Inf does not appear, then with the increment of iterations, all Lyapunov exponents will get close to the largest Lyapunov exponent, which leads to inaccurate calculation results; (3) From the viewpoint of numerical calculation, obviously, if the iterations are too small, then the results are also inaccurate. Based on the analysis of Lyapunov-exponent calculation in discrete-time systems, this paper investigates two improved algorithms via QR orthogonal decomposition and SVD orthogonal decomposition approaches so as to solve the above-mentioned problems. Finally, some examples are given to illustrate the feasibility and effectiveness of the improved algorithms.
Entanglement production and Lyapunov exponents
Hackl, Lucas; Bianchi, Eugenio; Yokomizo, Nelson
2017-01-01
Squeezed vacua play a prominent role in quantum field theory in curved spacetime. Instabilities and resonances that arise from the coupling in the field to the background geometry, result in a large squeezing of the vacuum. In this talk, I discuss the relation between squeezing and Lyapunov exponents of the system. In particular, I derive a new formula for the rate of growth of the entanglement entropy expressed as the sum of the Lyapunov exponents. Examples of such a linear production regime can be found during inflation and in the preheating phase directly after inflation.
Principal component cluster analysis of ECG time series based on Lyapunov exponent spectrum
Institute of Scientific and Technical Information of China (English)
WANG Nai; RUAN Jiong
2004-01-01
In this paper we propose an approach of principal component cluster analysis based on Lyapunov exponent spectrum (LES) to analyze the ECG time series. Analysis results of 22 sample-files of ECG from the MIT-BIH database confirmed the validity of our approach. Another technique named improved teacher selecting student (TSS) algorithm is presented to analyze unknown samples by means of some known ones, which is of better accuracy. This technique combines the advantages of both statistical and nonlinear dynamical methods and is shown to be significant to the analysis of nonlinear ECG time series.
Wang, Zhikang; Lou, Haifang; Sun, Jianzhong
2011-07-01
Attempting to use nonlinear spatiotemporal Lyapunov exponent to characterize fMRI brain functional connectivity of anxiety disease patients, we adopted the methods of nonlinear spatiotemporal Lyapunov exponent and linear correlation coefficients to analyses fMRI datum of 11 anxiety disease patients and 11 healthy volunteers, respectively. The results show that there are significant normalized variance exponent (NVE) differences in Inferior Frontal Gyrus (rIFG) and Medial Frontal Gyrus (MFG) between the two groups (PLyapunov exponent method had higher sensitivity than the correlation coefficient method in the characterization of functional connectivity; Anxiety disease patients have abnormal functional connectivity in rIFG and MFG during our experiment.
Analysis of the Emergence in Swarm Model Based on Largest Lyapunov Exponent
Directory of Open Access Journals (Sweden)
Yu Wu
2011-01-01
Full Text Available Emergent behaviors of collective intelligence systems, exemplified by swarm model, have attracted broad interests in recent years. However, current research mostly stops at observational interpretations and qualitative descriptions of emergent phenomena and is essentially short of quantitative analysis and evaluation. In this paper, we conduct a quantitative study on the emergence of swarm model by using chaos analysis of complex dynamic systems. This helps to achieve a more exact understanding of emergent phenomena. In particular, we evaluate the emergent behaviors of swarm model quantitatively by using the chaos and stability analysis of swarm model based on largest Lyapunov exponent. It is concluded that swarm model is at the edge of chaos when emergence occurs, and whether chaotic or stable at the beginning, swarm model will converge to stability with the elapse of time along with interactions among agents.
Directory of Open Access Journals (Sweden)
Rui Wang
2014-01-01
Full Text Available A modified multiple structural changes model is built to test structural breaks of the financial system based on calculating the largest Lyapunov exponents of the financial time series. Afterwards, the Lorenz system is used as a simulation example to inspect the new model. As the Lorenz system has strong nonlinearity, the verification results show that the new model has good capability in both finding the breakpoint and revealing the changes in nonlinear characteristics of the time series. The empirical study based on the model used daily data from the S&P 500 stock index during the global financial crisis from 2005 to 2012. The results provide four breakpoints of the period, which divide the contagion into four stages: stationary, local outbreak, global outbreak, and recovery period. An additional significant result is the obvious chaos characteristic difference in the largest Lyapunov exponents and the standard deviation at various stages, particularly at the local outbreak stage.
Multiscale analysis of biological data by scale-dependent lyapunov exponent.
Gao, Jianbo; Hu, Jing; Tung, Wen-Wen; Blasch, Erik
2011-01-01
Physiological signals often are highly non-stationary (i.e., mean and variance change with time) and multiscaled (i.e., dependent on the spatial or temporal interval lengths). They may exhibit different behaviors, such as non-linearity, sensitive dependence on small disturbances, long memory, and extreme variations. Such data have been accumulating in all areas of health sciences and rapid analysis can serve quality testing, physician assessment, and patient diagnosis. To support patient care, it is very desirable to characterize the different signal behaviors on a wide range of scales simultaneously. The Scale-Dependent Lyapunov Exponent (SDLE) is capable of such a fundamental task. In particular, SDLE can readily characterize all known types of signal data, including deterministic chaos, noisy chaos, random 1/f(α) processes, stochastic limit cycles, among others. SDLE also has some unique capabilities that are not shared by other methods, such as detecting fractal structures from non-stationary data and detecting intermittent chaos. In this article, we describe SDLE in such a way that it can be readily understood and implemented by non-mathematically oriented researchers, develop a SDLE-based consistent, unifying theory for the multiscale analysis, and demonstrate the power of SDLE on analysis of heart-rate variability (HRV) data to detect congestive heart failure and analysis of electroencephalography (EEG) data to detect seizures.
Wang, Qiqi; Rigas, Georgios; Esclapez, Lucas; Magri, Luca; Blonigan, Patrick
2016-11-01
Bluff body flows are of fundamental importance to many engineering applications involving massive flow separation and in particular the transport industry. Coherent flow structures emanating in the wake of three-dimensional bluff bodies, such as cars, trucks and lorries, are directly linked to increased aerodynamic drag, noise and structural fatigue. For low Reynolds laminar and transitional regimes, hydrodynamic stability theory has aided the understanding and prediction of the unstable dynamics. In the same framework, sensitivity analysis provides the means for efficient and optimal control, provided the unstable modes can be accurately predicted. However, these methodologies are limited to laminar regimes where only a few unstable modes manifest. Here we extend the stability analysis to low-dimensional chaotic regimes by computing the Lyapunov covariant vectors and their associated Lyapunov exponents. We compare them to eigenvectors and eigenvalues computed in traditional hydrodynamic stability analysis. Computing Lyapunov covariant vectors and Lyapunov exponents also enables the extension of sensitivity analysis to chaotic flows via the shadowing method. We compare the computed shadowing sensitivities to traditional sensitivity analysis. These Lyapunov based methodologies do not rely on mean flow assumptions, and are mathematically rigorous for calculating sensitivities of fully unsteady flow simulations.
Sun, Yuming; Wu, Christine Qiong
2012-12-01
Balancing control is important for biped standing. In spite of large efforts, it is very difficult to design balancing control strategies satisfying three requirements simultaneously: maintaining postural stability, improving energy efficiency and satisfying the constraints between the biped feet and the ground. In this article, a proportional-derivative (PD) controller is proposed for a standing biped, which is simplified as a two-link inverted pendulum with one additional rigid foot-link. The genetic algorithm (GA) is used to search for the control gain meeting all three requirements. The stability analysis of such a deterministic biped control system is carried out using the concept of Lyapunov exponents (LEs), based on which, the system stability, where the disturbance comes from the initial states, and the structural stability, where the disturbance comes from the PD gains, are examined quantitively in terms of stability region. This article contributes to the biped balancing control, more significantly, the method shown in the studied case of biped provides a general framework of systematic stability analysis for certain deterministic nonlinear dynamical systems.
Upper quantum Lyapunov exponent and parametric oscillators
Jauslin, H. R.; Sapin, O.; Guérin, S.; Wreszinski, W. F.
2004-11-01
We introduce a definition of upper Lyapunov exponent for quantum systems in the Heisenberg representation, and apply it to parametric quantum oscillators. We provide a simple proof that the upper quantum Lyapunov exponent ranges from zero to a positive value, as the parameters range from the classical system's region of stability to the instability region. It is also proved that in the instability region the parametric quantum oscillator satisfies the discrete quantum Anosov relations defined by Emch, Narnhofer, Sewell, and Thirring.
Lyapunov exponents for continuous random transformations
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
In this paper, the concept of Lyapunov exponent is generalized to random transformations that are not necessarily differentiable. For a class of random repellers and of random hyperbolic sets obtained via small perturbations of deterministic ones respectively, the new exponents are shown to coincide with the classical ones.
Finite-time Lyapunov exponent-based analysis for compressible flows
González, D. R.; Speth, R. L.; Gaitonde, D. V.; Lewis, M. J.
2016-08-01
The finite-time Lyapunov exponent (FTLE) technique has shown substantial success in analyzing incompressible flows by capturing the dynamics of coherent structures. Recent applications include river and ocean flow patterns, respiratory tract dynamics, and bio-inspired propulsors. In the present work, we extend FTLE to the compressible flow regime so that coherent structures, which travel at convective speeds, can be associated with waves traveling at acoustic speeds. This is particularly helpful in the study of jet acoustics. We first show that with a suitable choice of integration time interval, FTLE can extract wave dynamics from the velocity field. The integration time thus acts as a pseudo-filter separating coherent structures from waves. Results are confirmed by examining forward and backward FTLE coefficients for several simple, well-known acoustic fields. Next, we use this analysis to identify events associated with intermittency in jet noise pressure probe data. Although intermittent events are known to be dominant causes of jet noise, their direct source in the turbulent jet flow has remained unexplained. To this end, a Large-Eddy Simulation of a Mach 0.9 jet is subjected to FTLE to simultaneously examine, and thus expose, the causal relationship between coherent structures and the corresponding acoustic waves. Results show that intermittent events are associated with entrainment in the initial roll up region and emissive events downstream of the potential-core collapse. Instantaneous acoustic disturbances are observed to be primarily induced near the collapse of the potential core and continue propagating towards the far-field at the experimentally observed, approximately 30° angle relative to the jet axis.
Finite-time Lyapunov exponent-based analysis for compressible flows.
González, D R; Speth, R L; Gaitonde, D V; Lewis, M J
2016-08-01
The finite-time Lyapunov exponent (FTLE) technique has shown substantial success in analyzing incompressible flows by capturing the dynamics of coherent structures. Recent applications include river and ocean flow patterns, respiratory tract dynamics, and bio-inspired propulsors. In the present work, we extend FTLE to the compressible flow regime so that coherent structures, which travel at convective speeds, can be associated with waves traveling at acoustic speeds. This is particularly helpful in the study of jet acoustics. We first show that with a suitable choice of integration time interval, FTLE can extract wave dynamics from the velocity field. The integration time thus acts as a pseudo-filter separating coherent structures from waves. Results are confirmed by examining forward and backward FTLE coefficients for several simple, well-known acoustic fields. Next, we use this analysis to identify events associated with intermittency in jet noise pressure probe data. Although intermittent events are known to be dominant causes of jet noise, their direct source in the turbulent jet flow has remained unexplained. To this end, a Large-Eddy Simulation of a Mach 0.9 jet is subjected to FTLE to simultaneously examine, and thus expose, the causal relationship between coherent structures and the corresponding acoustic waves. Results show that intermittent events are associated with entrainment in the initial roll up region and emissive events downstream of the potential-core collapse. Instantaneous acoustic disturbances are observed to be primarily induced near the collapse of the potential core and continue propagating towards the far-field at the experimentally observed, approximately 30° angle relative to the jet axis.
Smith, Beth A.; Stergiou, Nicholas; Ulrich, Beverly D.
2010-01-01
In previous studies we found that while preadolescents with Down syndrome (DS) produce higher amounts of variability (Smith et al., 2007) and larger Lyapunov exponent (LyE) values (indicating more instability) during walking than peers with typical development (TD) (Buzzi & Ulrich, 2004), they also partition more of this into goal-equivalent variability (UCM//), that can be exploited to increase options for success when perturbed (Black et al., 2007). Here we use nonlinear methods to examine the patterns that characterize gait variability as it emerges, in toddlers with TD and with DS, rather than after years of practice. We calculated Lyapunov exponent (LyE) values to assess stability of leg trajectories. We also tested the use of 3 algorithms for surrogation analysis to investigate mathematical periodicity of toddlers’ strides. Results show that toddlers’ LyE values were not different between groups or with practice and strides of both groups become more periodic with practice. PMID:20237407
Short-time Lyapunov exponent analysis and the transition to chaos in Taylor-Couette flow
Vastano, John A.; Moser, Robert D.
1991-01-01
The physical mechanism driving the weakly chaotic Taylor-Couette flow is investigated using the short-time Liapunov exponent analysis. In this procedure, the transition from quasi-periodicity to chaos is studied using direct numerical 3D simulations of axially periodic Taylor-Couette flow, and a partial Liapunov exponent spectrum for the flow is computed by simultaneously advancing the full solution and a set of perturbations. It is shown that the short-time Liapunov exponent analysis yields more information on the exponents and dimension than that obtained from the common Liapunov exponent calculations. Results show that the chaotic state studied here is caused by a Kelvin-Helmholtz-type instability of the outflow boundary jet of Taylor vortices.
Short-time Lyapunov exponent analysis and the transition to chaos in Taylor-Couette flow
Vastano, John A.; Moser, Robert D.
1991-01-01
The physical mechanism driving the weakly chaotic Taylor-Couette flow is investigated using the short-time Liapunov exponent analysis. In this procedure, the transition from quasi-periodicity to chaos is studied using direct numerical 3D simulations of axially periodic Taylor-Couette flow, and a partial Liapunov exponent spectrum for the flow is computed by simultaneously advancing the full solution and a set of perturbations. It is shown that the short-time Liapunov exponent analysis yields more information on the exponents and dimension than that obtained from the common Liapunov exponent calculations. Results show that the chaotic state studied here is caused by a Kelvin-Helmholtz-type instability of the outflow boundary jet of Taylor vortices.
Lyapunov exponents computation for hybrid neurons.
Bizzarri, Federico; Brambilla, Angelo; Gajani, Giancarlo Storti
2013-10-01
Lyapunov exponents are a basic and powerful tool to characterise the long-term behaviour of dynamical systems. The computation of Lyapunov exponents for continuous time dynamical systems is straightforward whenever they are ruled by vector fields that are sufficiently smooth to admit a variational model. Hybrid neurons do not belong to this wide class of systems since they are intrinsically non-smooth owing to the impact and sometimes switching model used to describe the integrate-and-fire (I&F) mechanism. In this paper we show how a variational model can be defined also for this class of neurons by resorting to saltation matrices. This extension allows the computation of Lyapunov exponent spectrum of hybrid neurons and of networks made up of them through a standard numerical approach even in the case of neurons firing synchronously.
Convex Optimization methods for computing the Lyapunov Exponent of matrices
Protasov, Vladimir Yu
2012-01-01
We introduce a new approach to evaluate the largest Lyapunov exponent of a family of nonnegative matrices. The method is based on using special positive homogeneous functionals on $R^{d}_+,$ which gives iterative lower and upper bounds for the Lyapunov exponent. They improve previously known bounds and converge to the real value. The rate of convergence is estimated and the efficiency of the algorithm is demonstrated on several problems from applications (in functional analysis, combinatorics, and lan- guage theory) and on numerical examples with randomly generated matrices. The method computes the Lyapunov exponent with a prescribed accuracy in relatively high dimensions (up to 60). We generalize this approach to all matrices, not necessar- ily nonnegative, derive a new universal upper bound for the Lyapunov exponent, and show that such a lower bound, in general, does not exist.
Lyapunov Exponents and Covariant Vectors for Turbulent Flow Simulations
Blonigan, Patrick; Murman, Scott; Fernandez, Pablo; Wang, Qiqi
2016-11-01
As computational power increases, engineers are beginning to use scale-resolving turbulent flow simulations for applications in which jets, wakes, and separation dominate. However, the chaotic dynamics exhibited by scale-resolving simulations poses problems for the conventional sensitivity analysis and stability analysis approaches that are vital for design and control. Lyapunov analysis is used to study the chaotic behavior of dynamical systems, including flow simulations. Lyapunov exponents are the growth or a decay rate of specific flow field perturbations called the Lyapunov covariant vectors. Recently, the authors have used Lyapunov analysis to study the breakdown in conventional sensitivity analysis and the cost of new shadowing-based sensitivity analysis. The current work reviews Lyapunov analysis and presents new results for a DNS of turbulent channel flow, wall-modeled channel flow, and a DNS of a low pressure turbine blade. Additionally, the implications of these Lyapunov analyses for computing sensitivities of these flow simulations will be discussed.
Lyapunov exponent in quantum mechanics A phase-space approach
Man'ko, V I
2000-01-01
Using the symplectic tomography map, both for the probability distributionsin classical phase space and for the Wigner functions of its quantumcounterpart, we discuss a notion of Lyapunov exponent for quantum dynamics.Because the marginal distributions, obtained by the tomography map, are alwayswell defined probabilities, the correspondence between classical and quantumnotions is very clear. Then we also obtain the corresponding expressions inHilbert space. Some examples are worked out. Classical and quantum exponentsare seen to coincide for local and non-local time-dependent quadraticpotentials. For non-quadratic potentials classical and quantum exponents aredifferent and some insight is obtained on the taming effect of quantummechanics on classical chaos. A detailed analysis is made for the standard map.Providing an unambiguous extension of the notion of Lyapunov exponent toquantum mechnics, the method that is developed is also computationallyefficient in obtaining analytical results for the Lyapunov expone...
Zeren, Tamer; Özbek, Mustafa; Kutlu, Necip; Akilli, Mahmut
2016-01-05
Pneumocardiography (PNCG) is the recording method of cardiac-induced tracheal air flow and pressure pulsations in the respiratory airways. PNCG signals reflect both the lung and heart actions and could be accurately recorded in spontaneously breathing anesthetized rats. Nonlinear analysis methods, including the Lyapunov exponent, can be used to explain the biological dynamics of systems such as the cardiorespiratory system. In this study, we recorded tracheal air flow signals, including PNCG signals, from 3 representative anesthetized rats and analyzed the nonlinear behavior of these complex signals using Lyapunov exponents. Lyapunov exponents may also be used to determine the normal and pathological structure of biological systems. If the signals have at least one positive Lyapunov exponent, the signals reflect chaotic activity, as seen in PNCG signals in rats; the largest Lyapunov exponents of the signals of the healthy rats were greater than zero in this study. A method was proposed to determine the diagnostic and prognostic values of the cardiorespiratory system of rats using the arrangement of the PNCG and Lyapunov exponents, which may be monitored as vitality indicators.
Lyapunov exponents for infinite dimensional dynamical systems
Mhuiris, Nessan Mac Giolla
1987-01-01
Classically it was held that solutions to deterministic partial differential equations (i.e., ones with smooth coefficients and boundary data) could become random only through one mechanism, namely by the activation of more and more of the infinite number of degrees of freedom that are available to such a system. It is only recently that researchers have come to suspect that many infinite dimensional nonlinear systems may in fact possess finite dimensional chaotic attractors. Lyapunov exponents provide a tool for probing the nature of these attractors. This paper examines how these exponents might be measured for infinite dimensional systems.
Clustering and synchronization with positive Lyapunov exponents
Mendes, R V
1998-01-01
Clustering and correlation effects are frequently observed in chaotic systems in situations where, because of the positivity of the Lyapunov exponents, no dimension reduction is to be expected. In this paper, using a globally coupled network of Bernoulli units, one finds a general mechanism by which strong correlations and slow structures are obtained at the synchronization edge. A structure index is defined, which diverges at the transition points. Some conclusions are drawn concerning the construction of an ergodic theory of self-organization.
Diverging Fluctuations of the Lyapunov Exponents.
Pazó, Diego; López, Juan M; Politi, Antonio
2016-07-15
We show that in generic one-dimensional Hamiltonian lattices the diffusion coefficient of the maximum Lyapunov exponent diverges in the thermodynamic limit. We trace this back to the long-range correlations associated with the evolution of the hydrodynamic modes. In the case of normal heat transport, the divergence is even stronger, leading to the breakdown of the usual single-function Family-Vicsek scaling ansatz. A similar scenario is expected to arise in the evolution of rough interfaces in the presence of suitably correlated background noise.
Yang, Hong-liu; Radons, Günter; Kantz, Holger
2012-12-14
The estimation of Lyapunov exponents from time series suffers from the appearance of spurious Lyapunov exponents due to the necessary embedding procedure. Separating true from spurious exponents poses a fundamental problem which is not yet solved satisfactorily. We show, in this Letter, analytically and numerically that covariant Lyapunov vectors associated with true exponents lie in the tangent space of the reconstructed attractor. Therefore, we use the angle between the covariant Lyapunov vectors and the tangent space of the reconstructed attractor to identify the true Lyapunov exponents. The usefulness of our method, also for noisy situations, is demonstrated by applications to data from model systems and a NMR laser experiment.
Lyapunov exponents for a Duffing oscillator
Zeni, Andrea R.; Gallas, Jason A. C.
With the help of a parallel computer we perform a systematic computation of Lyapunov exponents for a Duffing oscillator driven externally by a force proportional to cos( t). In contrast to the familiar situation in discrete-time systems where one finds “windows” of regularity embedded in intervals of chaos, we find the continuous-time Duffing oscillator to contain a quite regular epetition of relatively self-similar “islands of chaos” (i.e. regions characterized by positive exponents) embedded in large “seas of regularity” (negative exponents). We also investigate the effect of driving the oscillator with a Jacobian elliptic function cn( t, m). For m = 0 one has cn( t, 0) ≡ cos( t), the usual trigonometric pumping. For m = 1 one has cn( t, 1) ≡ sech( t), a hyperbolic pumping. When 0 displace the islands of chaos in parameter space. Thus, Jacobian pumping provides a possible way of “cleaning chaos” in regions of the parameter space for periodically driven systems.
Stability analysis and quasinormal modes of Reissner–Nordstrøm space-time via Lyapunov exponent
Indian Academy of Sciences (India)
PRADHAN PARTHAPRATIM
2016-07-01
We explicitly derive the proper-time (τ ) principal Lyapunov exponent (λp) and coordinate-time (t ) principal Lyapunov exponent $(\\lambda_c)$ for Reissner–Nordstrøm (RN) black hole (BH). We also compute their ratio. For RN space-time, it is shown that the ratio is $(\\lambda_{p}/\\lambda_{c}) = r_{0}/\\sqrt{r^{2}0 − 3Mr_{0} + 2Q^{2}}$ for time-like circulargeodesics and for Schwarzschild BH, it is $(\\lambda_{p}/\\lambda_{c}) = \\sqrt{r_{0}}/\\sqrt{r_{0} − 3M}. We further show that their ratio $\\lambda_{p}/\\lambda_{c}$ may vary from orbit to orbit. For instance, for Schwarzschild BH at the innermost stable circular orbit (ISCO), the ratio is $(\\lambda_{p}/\\lambda_{c})_{|rISCO}=6M = \\sqrt{2}$ and at marginally bound circular orbit (MBCO) the ratio is calculated to be $(\\lambda_{p}/\\lambda_{c})|_{rmb}=4M = 2$. Similarly, for extremal RN BH, the ratio at ISCO is $(\\lambda_{p}/\\lambda_{c})|_{rISCO}=4M = 2\\sqrt{2}/\\sqrt{3}$. We also further analyse the geodesic stability via this exponent. By evaluating the Lyapunov exponent, it is shown that in the eikonal limit, the real and imaginary parts of the quasinormal modes of RN BH is given by the frequency and instability time-scale of the unstable null circular geodesics.
Lyapunov exponent diagrams of a 4-dimensional Chua system.
Stegemann, Cristiane; Albuquerque, Holokx A; Rubinger, Rero M; Rech, Paulo C
2011-09-01
We report numerical results on the existence of periodic structures embedded in chaotic and hyperchaotic regions on the Lyapunov exponent diagrams of a 4-dimensional Chua system. The model was obtained from the 3-dimensional Chua system by the introduction of a feedback controller. Both the largest and the second largest Lyapunov exponents were considered in our colorful Lyapunov exponent diagrams, and allowed us to characterize periodic structures and regions of chaos and hyperchaos. The shrimp-shaped periodic structures appear to be malformed on some of Lyapunov exponent diagrams, and they present two different bifurcation scenarios to chaos when passing the boundaries of itself, namely via period-doubling and crisis. Hyperchaos-chaos transition can also be observed on the Lyapunov exponent diagrams for the second largest exponent.
Statistics of Lyapunov exponent spectrum in randomly coupled Kuramoto oscillators.
Patra, Soumen K; Ghosh, Anandamohan
2016-03-01
Characterization of spatiotemporal dynamics of coupled oscillatory systems can be done by computing the Lyapunov exponents. We study the spatiotemporal dynamics of randomly coupled network of Kuramoto oscillators and find that the spectral statistics obtained from the Lyapunov exponent spectrum show interesting sensitivity to the coupling matrix. Our results indicate that in the weak coupling limit the gap distribution of the Lyapunov spectrum is Poissonian, while in the limit of strong coupling the gap distribution shows level repulsion. Moreover, the oscillators settle to an inhomogeneous oscillatory state, and it is also possible to infer the random network properties from the Lyapunov exponent spectrum.
The Lyapunov exponents of the Van der Pol oscillator
Grasman, J.; Verhulst, F.; Shih, S.D.
2005-01-01
Lyapunov exponents characterize the dynamics of a system near its attractor. For the Van der Pol oscillator these are quantities for which an approximation should be at hand. Similar to the asymptotic approximation of amplitude and period, expressions are derived for the non-zero Lyapunov exponent
Calculating Lyapunov Exponents: Applying Products and Evaluating Integrals
McCartney, Mark
2010-01-01
Two common examples of one-dimensional maps (the tent map and the logistic map) are generalized to cases where they have more than one control parameter. In the case of the tent map, this still allows the global Lyapunov exponent to be found analytically, and permits various properties of the resulting global Lyapunov exponents to be investigated…
Calculating Lyapunov Exponents: Applying Products and Evaluating Integrals
McCartney, Mark
2010-01-01
Two common examples of one-dimensional maps (the tent map and the logistic map) are generalized to cases where they have more than one control parameter. In the case of the tent map, this still allows the global Lyapunov exponent to be found analytically, and permits various properties of the resulting global Lyapunov exponents to be investigated…
Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations
Kuznetsov, N. V.; Alexeeva, T. A.; Leonov, G. A.
2014-01-01
Nowadays the Lyapunov exponents and Lyapunov dimension have become so widespread and common that they are often used without references to the rigorous definitions or pioneering works. It may lead to a confusion since there are at least two well-known definitions, which are used in computations: the upper bounds of the exponential growth rate of the norms of linearized system solutions (Lyapunov characteristic exponents, LCEs) and the upper bounds of the exponential growth rate of the singula...
Generalized Lyapunov exponent as a unified characterization of dynamical instabilities.
Akimoto, Takuma; Nakagawa, Masaki; Shinkai, Soya; Aizawa, Yoji
2015-01-01
The Lyapunov exponent characterizes an exponential growth rate of the difference of nearby orbits. A positive Lyapunov exponent (exponential dynamical instability) is a manifestation of chaos. Here, we propose the Lyapunov pair, which is based on the generalized Lyapunov exponent, as a unified characterization of nonexponential and exponential dynamical instabilities in one-dimensional maps. Chaos is classified into three different types, i.e., superexponential, exponential, and subexponential chaos. Using one-dimensional maps, we demonstrate superexponential and subexponential chaos and quantify the dynamical instabilities by the Lyapunov pair. In subexponential chaos, we show superweak chaos, which means that the growth of the difference of nearby orbits is slower than a stretched exponential growth. The scaling of the growth is analytically studied by a recently developed theory of a continuous accumulation process, which is related to infinite ergodic theory.
Lyapunov exponent of chaos generated by acousto-optic modulators with feedback
Ghosh, Anjan K.; Verma, Pramode
2011-01-01
Generation of chaos from acousto-optic modulators with an electronic feedback has been studied for several years. Such chaotic signals have an important application in providing secure encryption in free-space optical communication systems. Lyapunov exponent is an important parameter for analysis of chaos generated by a nonlinear system. The Lyapunov exponent of an acousto-optic system is determined and calculated in this paper to understand the dependence of the chaotic response on the system parameters such as bias, feedback gain, input intensity and initial condition exciting the cell. Analysis of chaos using Lyapunov exponent is consistent with bifurcation analysis and is useful in encrypting data signals.
Pseudo-Lyapunov exponents and predictability of Hodgkin-Huxley neuronal network dynamics.
Sun, Yi; Zhou, Douglas; Rangan, Aaditya V; Cai, David
2010-04-01
We present a numerical analysis of the dynamics of all-to-all coupled Hodgkin-Huxley (HH) neuronal networks with Poisson spike inputs. It is important to point out that, since the dynamical vector of the system contains discontinuous variables, we propose a so-called pseudo-Lyapunov exponent adapted from the classical definition using only continuous dynamical variables, and apply it in our numerical investigation. The numerical results of the largest Lyapunov exponent using this new definition are consistent with the dynamical regimes of the network. Three typical dynamical regimes-asynchronous, chaotic and synchronous, are found as the synaptic coupling strength increases from weak to strong. We use the pseudo-Lyapunov exponent and the power spectrum analysis of voltage traces to characterize the types of the network behavior. In the nonchaotic (asynchronous or synchronous) dynamical regimes, i.e., the weak or strong coupling limits, the pseudo-Lyapunov exponent is negative and there is a good numerical convergence of the solution in the trajectory-wise sense by using our numerical methods. Consequently, in these regimes the evolution of neuronal networks is reliable. For the chaotic dynamical regime with an intermediate strong coupling, the pseudo-Lyapunov exponent is positive, and there is no numerical convergence of the solution and only statistical quantifications of the numerical results are reliable. Finally, we present numerical evidence that the value of pseudo-Lyapunov exponent coincides with that of the standard Lyapunov exponent for systems we have been able to examine.
The Lyapunov exponents of C~1 hyperbolic systems
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
Let f be a C 1 diffeomorphisim of smooth Riemannian manifold and preserve a hyperbolic ergodic measure μ. We prove that if the Osledec splitting is dominated, then the Lyapunov exponents of μ can be approximated by the exponents of atomic measures on hyperbolic periodic orbits.
Scaling of Lyapunov Exponents in Homogeneous, Isotropic DNS
Fitzsimmons, Nicholas; Malaya, Nicholas; Moser, Robert
2013-11-01
Lyapunov exponents measure the rate of separation of initially infinitesimally close trajectories in a chaotic system. Using the exponents, we are able to probe the chaotic nature of homogeneous isotropic turbulence and study the instabilities of the chaotic field. The exponents are measured by calculating the instantaneous growth rate of a linear disturbance, evolved with the linearized Navier-Stokes equation, at each time step. In this talk, we examine these exponents in the context of homogeneous isotropic turbulence with two goals: 1) to investigate the scaling of the exponents with respect to the parameters of forced homogeneous isotropic turbulence, and 2) to characterize the instabilities that lead to chaos in turbulence. Specifically, we explore the scaling of the Lyapunov exponents with respect to the Reynolds number and with respect to the ratio of the integral length scale and the computational domain size.
Branicki, Michal
2009-01-01
We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of "transport barriers" in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields and associated Lagra...
OBSERVING LYAPUNOV EXPONENTS OF INFINITE-DIMENSIONAL DYNAMICAL SYSTEMS.
Ott, William; Rivas, Mauricio A; West, James
2015-12-01
Can Lyapunov exponents of infinite-dimensional dynamical systems be observed by projecting the dynamics into ℝ (N) using a 'typical' nonlinear projection map? We answer this question affirmatively by developing embedding theorems for compact invariant sets associated with C(1) maps on Hilbert spaces. Examples of such discrete-time dynamical systems include time-T maps and Poincaré return maps generated by the solution semigroups of evolution partial differential equations. We make every effort to place hypotheses on the projected dynamics rather than on the underlying infinite-dimensional dynamical system. In so doing, we adopt an empirical approach and formulate checkable conditions under which a Lyapunov exponent computed from experimental data will be a Lyapunov exponent of the infinite-dimensional dynamical system under study (provided the nonlinear projection map producing the data is typical in the sense of prevalence).
Characterizing weak chaos using time series of Lyapunov exponents.
da Silva, R M; Manchein, C; Beims, M W; Altmann, E G
2015-06-01
We investigate chaos in mixed-phase-space Hamiltonian systems using time series of the finite-time Lyapunov exponents. The methodology we propose uses the number of Lyapunov exponents close to zero to define regimes of ordered (stickiness), semiordered (or semichaotic), and strongly chaotic motion. The dynamics is then investigated looking at the consecutive time spent in each regime, the transition between different regimes, and the regions in the phase space associated to them. Applying our methodology to a chain of coupled standard maps we obtain (i) that it allows for an improved numerical characterization of stickiness in high-dimensional Hamiltonian systems, when compared to the previous analyses based on the distribution of recurrence times; (ii) that the transition probabilities between different regimes are determined by the phase-space volume associated to the corresponding regions; and (iii) the dependence of the Lyapunov exponents with the coupling strength.
Lyapunov exponents for synchronous 12-lead ECG signals
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
The Lyapunov exponents of synchronous 12-lead ECG signals have been investigated for the first time using a multi-sensor (electrode) technique. The results show that the Lyapunov exponents computed from different locations on the body surface are not the same, but have a distribution characteristic for the ECG signals recorded from coronary artery disease (CAD) patients with sinus rhythms and for signals from healthy older people. The maximum Lyapunov exponent L1 of all signals is positive. While all the others are negative, so the ECG signal has chaotic characteristics. With the same leads, L1 of CAD patients is less than that of healthy people, so the CAD patients and healthy people can be classified by L1, L1 therefore has potential values in the diagnosis of heart disease.
Lyapunov exponents of stochastic systems—from micro to macro
Laffargue, Tanguy; Tailleur, Julien; van Wijland, Frédéric
2016-03-01
Lyapunov exponents of dynamical systems are defined from the rates of divergence of nearby trajectories. For stochastic systems, one typically assumes that these trajectories are generated under the ‘same noise realization’. The purpose of this work is to critically examine what this expression means. For Brownian particles, we consider two natural interpretations of the noise: intrinsic to the particles or stemming from the fluctuations of the environment. We show how they lead to different distributions of the largest Lyapunov exponent as well as different fluctuating hydrodynamics for the collective density field. We discuss, both at microscopic and macroscopic levels, the limits in which these noise prescriptions become equivalent. We close this paper by providing an estimate of the largest Lyapunov exponent and of its fluctuations for interacting particles evolving with Dean-Kawasaki dynamics.
On the bound of the Lyapunov exponents for the fractional differential systems.
Li, Changpin; Gong, Ziqing; Qian, Deliang; Chen, YangQuan
2010-03-01
In recent years, fractional(-order) differential equations have attracted increasing interests due to their applications in modeling anomalous diffusion, time dependent materials and processes with long range dependence, allometric scaling laws, and complex networks. Although an autonomous system cannot define a dynamical system in the sense of semigroup because of the memory property determined by the fractional derivative, we can still use the Lyapunov exponents to discuss its dynamical evolution. In this paper, we first define the Lyapunov exponents for fractional differential systems then estimate the bound of the corresponding Lyapunov exponents. For linear fractional differential system, the bounds of its Lyapunov exponents are conveniently derived which can be regarded as an example for the theoretical results established in this paper. Numerical example is also included which supports the theoretical analysis.
Lyapunov exponents for multi-parameter tent and logistic maps.
McCartney, Mark
2011-12-01
The behaviour of logistic and tent maps is studied in cases where the control parameter is dependent on iteration number. Analytic results for global Lyapunov exponent are presented in the case of the tent map and numerical results are presented in the case of the logistic map. In the case of a tent map with N control parameters, the fraction of parameter space for which the global Lyapunov exponent is positive is calculated. The case of bi-parameter maps of period N are investigated.
Do Finite-Size Lyapunov Exponents detect coherent structures?
Karrasch, Daniel; Haller, George
2013-12-01
Ridges of the Finite-Size Lyapunov Exponent (FSLE) field have been used as indicators of hyperbolic Lagrangian Coherent Structures (LCSs). A rigorous mathematical link between the FSLE and LCSs, however, has been missing. Here, we prove that an FSLE ridge satisfying certain conditions does signal a nearby ridge of some Finite-Time Lyapunov Exponent (FTLE) field, which in turn indicates a hyperbolic LCS under further conditions. Other FSLE ridges violating our conditions, however, are seen to be false positives for LCSs. We also find further limitations of the FSLE in Lagrangian coherence detection, including ill-posedness, artificial jump-discontinuities, and sensitivity with respect to the computational time step.
Directory of Open Access Journals (Sweden)
M. Branicki
2010-01-01
Full Text Available We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of transport barriers in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE fields and associated Lagrangian coherent structures have been the main tools for characterising transport barriers in the time-aperiodic situation. In this paper we consider a variety of examples, some with explicit solutions, that illustrate in a concrete manner the issues and phenomena that arise in the setting of finite-time dynamical systems. Of particular significance for geophysical applications is the notion of flow transition which occurs when finite-time hyperbolicity is lost or gained. The phenomena discovered and analysed in our examples point the way to a variety of directions for rigorous mathematical research in this rapidly developing and important area of dynamical systems theory.
Characterizing heart rate variability by scale-dependent Lyapunov exponent
Hu, Jing; Gao, Jianbo; Tung, Wen-wen
2009-06-01
Previous studies on heart rate variability (HRV) using chaos theory, fractal scaling analysis, and many other methods, while fruitful in many aspects, have produced much confusion in the literature. Especially the issue of whether normal HRV is chaotic or stochastic remains highly controversial. Here, we employ a new multiscale complexity measure, the scale-dependent Lyapunov exponent (SDLE), to characterize HRV. SDLE has been shown to readily characterize major models of complex time series including deterministic chaos, noisy chaos, stochastic oscillations, random 1/f processes, random Levy processes, and complex time series with multiple scaling behaviors. Here we use SDLE to characterize the relative importance of nonlinear, chaotic, and stochastic dynamics in HRV of healthy, congestive heart failure, and atrial fibrillation subjects. We show that while HRV data of all these three types are mostly stochastic, the stochasticity is different among the three groups.
Lyapunov exponents and particle dispersion in drift wave turbulence
DEFF Research Database (Denmark)
Pedersen, T.S.; Michelsen, Poul; Juul Rasmussen, J.
1996-01-01
The Hasegawa-Wakatani model equations for resistive drift waves are solved numerically for a range of values of the coupling due to the parallel electron motion. The largest Lyapunov exponent, lambda(1), is calculated to quantify the unpredictability of the turbulent flow and compared to other...
Mizuta, Keisuke; Tokita, Takashi; Ito, Yatsuji; Aoki, Mitsuhiro; Kuze, Bunya
2009-12-01
In the present study, we investigated the body sway in patients with unilateral vestibular dysfunction by the largest Lyapunov exponents using a chaotic time series analysis. The largest Lyapunov exponent is regarded as a parameter indexing an orbital instability. Subjects consisted of 55 normal healthy subjects, 11 patients diagnosed as having vestibular neuritis (VN), 6 patients diagnosed as having sudden deafness (SD) with vertigo, 23 patients diagnosed as having Meniere disease (MD), 11 patients diagnosed as having benign paroxysmal positional vertigo (BPPV) and 14 patients diagnosed as having other vestibular disorders. Using a stabilometer, the sway of the body center of gravity in an upright standing position was recorded with eyes open and closed for 60 seconds under each condition. From the time series data obtained, the largest Lyapunov exponents were calculated using a chaos analysis program. In normal healthy subjects and patients with unilateral vestibular dysfunction, the largest Lyapunov exponents on right-left sway were larger than those on forward-backward sway with eyes open and closed. The largest Lyapunov exponents in patients with unilateral vestibular dysfunction on forward-backward sway with eyes closed were significantly larger than those in normal healthy subjects. A few patients with the instability of standing posture judged from conventional analysis (area of sway, locus length per time) showed higher values of the LLE. We investigated the variation of the values of the largest Lyapunov exponents in patients with unilateral vestibular dysfunction at each stage during recovery from their vestibular damage. The largest Lyapunov exponents at the early stage with stable standing posture were significantly higher than those at the late stable stage with stable standing posture. Some patients at the very early stage had lower values of the largest Lyapunov exponents. We speculate that the orbital instability indicated by the values of the
Lyapunov exponent for aging process in induction motor
Bayram, Duygu; Ünnü, Sezen Yıdırım; Şeker, Serhat
2012-09-01
Nonlinear systems like electrical circuits and systems, mechanics, optics and even incidents in nature may pass through various bifurcations and steady states like equilibrium point, periodic, quasi-periodic, chaotic states. Although chaotic phenomena are widely observed in physical systems, it can not be predicted because of the nature of the system. On the other hand, it is known that, chaos is strictly dependent on initial conditions of the system [1-3]. There are several methods in order to define the chaos. Phase portraits, Poincaré maps, Lyapunov Exponents are the most common techniques. Lyapunov Exponents are the theoretical indicator of the chaos, named after the Russian mathematician Aleksandr Lyapunov (1857-1918). Lyapunov Exponents stand for the average exponential divergence or convergence of nearby system states, meaning estimating the quantitive measure of the chaotic attractor. Negative numbers of the exponents stand for a stable system whereas zero stands for quasi-periodic systems. On the other hand, at least if one of the exponents is positive, this situation is an indicator of the chaos. For estimating the exponents, the system should be modeled by differential equation but even in that case mathematical calculation of Lyapunov Exponents are not very practical and evaluation of these values requires a long signal duration [4-7]. For experimental data sets, it is not always possible to acquire the differential equations. There are several different methods in literature for determining the Lyapunov Exponents of the system [4, 5]. Induction motors are the most important tools for many industrial processes because they are cheap, robust, efficient and reliable. In order to have healthy processes in industrial applications, the conditions of the machines should be monitored and the different working conditions should be addressed correctly. To the best of our knowledge, researches related to Lyapunov exponents and electrical motors are mostly
From Lyapunov modes to their exponents for hard disk systems.
Chung, Tony; Truant, Daniel; Morriss, Gary P
2010-06-01
We demonstrate the preservation of the Lyapunov modes in a system of hard disks by the underlying tangent space dynamics. This result is exact for the Zero modes and correct to order ϵ for the Transverse and Longitudinal-Momentum modes, where ϵ is linear in the mode number. For sufficiently large mode numbers, the ϵ terms become significant and the dynamics no longer preserves the mode structure. We propose a modified Gram-Schmidt procedure based on orthogonality with respect to the center zero space that produces the exact numerical mode. This Gram-Schmidt procedure can also exploit the orthogonality between conjugate modes and their symplectic structure in order to find a simple relation that determines the Lyapunov exponent from the Lyapunov mode. This involves a reclassification of the modes into either direction preserving or form preserving. These analytic methods assume a knowledge of the ordering of the modes within the Lyapunov spectrum, but gives both predictive power for the values of the exponents from the modes and describes the modes in greater detail than was previously achievable. Thus the modes and the exponents contain the same information.
Integral expressions of Lyapunov exponents for autonomous ordinary differential systems
Institute of Scientific and Technical Information of China (English)
DAI XiongPing
2009-01-01
In the paper,the author addresses the Lyapunov characteristic spectrum of an ergodic autonomous ordinary differential system on a complete riemannian manifold of finite dimension such as the d-dimensional euclidean space Rd,not necessarily compact,by Liaowise spectral theorems that give integral expressions of Lyapunov exponents.In the context of smooth linear skew-product flows with Polish driving systems,the results are still valid.This paper seems to be an interesting contribution to the stability theory of ordinary differential systems with non-compact phase spaces.
Integral expressions of Lyapunov exponents for autonomous ordinary differential systems
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
In the paper, the author addresses the Lyapunov characteristic spectrum of an ergodic autonomous ordinary differential system on a complete riemannian manifold of finite dimension such as the d-dimensional euclidean space Rd, not necessarily compact, by Liaowise spectral theorems that give integral expressions of Lyapunov exponents. In the context of smooth linear skew-product flows with Polish driving systems, the results are still valid. This paper seems to be an interesting contribution to the stability theory of ordinary differential systems with non-compact phase spaces.
Lyapunov exponents for one-dimensional aperiodic photonic bandgap structures
Kissel, Glen J.
2011-10-01
Existing in the "gray area" between perfectly periodic and purely randomized photonic bandgap structures are the socalled aperoidic structures whose layers are chosen according to some deterministic rule. We consider here a onedimensional photonic bandgap structure, a quarter-wave stack, with the layer thickness of one of the bilayers subject to being either thin or thick according to five deterministic sequence rules and binary random selection. To produce these aperiodic structures we examine the following sequences: Fibonacci, Thue-Morse, Period doubling, Rudin-Shapiro, as well as the triadic Cantor sequence. We model these structures numerically with a long chain (approximately 5,000,000) of transfer matrices, and then use the reliable algorithm of Wolf to calculate the (upper) Lyapunov exponent for the long product of matrices. The Lyapunov exponent is the statistically well-behaved variable used to characterize the Anderson localization effect (exponential confinement) when the layers are randomized, so its calculation allows us to more precisely compare the purely randomized structure with its aperiodic counterparts. It is found that the aperiodic photonic systems show much fine structure in their Lyapunov exponents as a function of frequency, and, in a number of cases, the exponents are quite obviously fractal.
On finite-size Lyapunov exponents in multiscale systems
Mitchell, Lewis
2012-01-01
We study the effect of regime switches on finite size Lyapunov exponents (FSLEs) in determining the error growth rates and predictability of multiscale systems. We consider a dynamical system involving slow and fast regimes and switches between them. The surprising result is that due to the presence of regimes the error growth rate can be a non-monotonic function of initial error amplitude. In particular, troughs in the large scales of FSLE spectra is shown to be a signature of slow regimes, whereas fast regimes are shown to cause large peaks in the spectra where error growth rates far exceed those estimated from the maximal Lyapunov exponent. We present analytical results explaining these signatures and corroborate them with numerical simulations. We show further that these peaks disappear in stochastic parametrizations of the fast chaotic processes, and the associated FSLE spectra reveal that large scale predictability properties of the full deterministic model are well approximated whereas small scale feat...
Behavior of the Lyapunov Exponent and Phase Transition in Nuclei
Institute of Scientific and Technical Information of China (English)
WANG Nan; WU Xi-Zhen; LI Zhu-Xia; WANG Ning; ZHUO Yi-Zhong; SUN Xiu-Quan
2000-01-01
Based on the quantum molecular dynamics model, we investigate the dynamical behaviors of the excited nuclear system to simulate the latter stage of heavy ion reactions, which associate with a liquid-gas phase transition. We try to search a microscopic way to describe the phase transition in realnuclei. The Lyapunov exponent is employed and examined for our purpose. We find out that the Lyapunov exponent is one of good microscopic quantities to describe the phase transition in hot nuclei. Coulomb potential and the finite size effect may give a strong influence on the critical temperature. However, the collision term plays a minor role in the process of the liquid-gas phase transition in finite systems.
Lyapunov exponents a tool to explore complex dynamics
Pikovsky, Arkady
2016-01-01
Lyapunov exponents lie at the heart of chaos theory, and are widely used in studies of complex dynamics. Utilising a pragmatic, physical approach, this self-contained book provides a comprehensive description of the concept. Beginning with the basic properties and numerical methods, it then guides readers through to the most recent advances in applications to complex systems. Practical algorithms are thoroughly reviewed and their performance is discussed, while a broad set of examples illustrate the wide range of potential applications. The description of various numerical and analytical techniques for the computation of Lyapunov exponents offers an extensive array of tools for the characterization of phenomena such as synchronization, weak and global chaos in low and high-dimensional set-ups, and localization. This text equips readers with all the investigative expertise needed to fully explore the dynamical properties of complex systems, making it ideal for both graduate students and experienced researchers...
Geometry of dynamics, Lyapunov exponents and phase transitions
Caiani, L; Clementi, C; Pettini, M; Caiani, Lando; Casetti, Lapo; Clementi, Cecilia; Pettini, Marco
1997-01-01
The Hamiltonian dynamics of classical planar Heisenberg model is numerically investigated in two and three dimensions. By considering the dynamics as a geodesic flow on a suitable Riemannian manifold, it is possible to analytically estimate the largest Lyapunov exponent in terms of some curvature fluctuations. The agreement between numerical and analytical values for Lyapunov exponents is very good in a wide range of temperatures. Moreover, in the three dimensional case, in correspondence with the second order phase transition, the curvature fluctuations exibit a singular behaviour which is reproduced in an abstract geometric model suggesting that the phase transition might correspond to a change in the topology of the manifold whose geodesics are the motions of the system.
Riemannian theory of Hamiltonian chaos and Lyapunov exponents
Casetti, L; Pettini, M; Casetti, Lapo; Clementi, Cecilia; Pettini, Marco
1996-01-01
This paper deals with the problem of analytically computing the largest Lyapunov exponent for many degrees of freedom Hamiltonian systems. This aim is succesfully reached within a theoretical framework that makes use of a geometrization of newtonian dynamics in the language of Riemannian geometry. A new point of view about the origin of chaos in these systems is obtained independently of homoclinic intersections. Chaos is here related to curvature fluctuations of the manifolds whose geodesics are natural motions and is described by means of Jacobi equation for geodesic spread. Under general conditions ane effective stability equation is derived; an analytic formula for the growth-rate of its solutions is worked out and applied to the Fermi-Pasta-Ulam beta model and to a chain of coupled rotators. An excellent agreement is found the theoretical prediction and the values of the Lyapunov exponent obtained by numerical simulations for both models.
A local Echo State Property through the largest Lyapunov exponent.
Wainrib, Gilles; Galtier, Mathieu N
2016-04-01
Echo State Networks are efficient time-series predictors, which highly depend on the value of the spectral radius of the reservoir connectivity matrix. Based on recent results on the mean field theory of driven random recurrent neural networks, enabling the computation of the largest Lyapunov exponent of an ESN, we develop a cheap algorithm to establish a local and operational version of the Echo State Property.
Bohmian quantum mechanical and classical Lyapunov exponents for kicked rotor
Energy Technology Data Exchange (ETDEWEB)
Zheng Yindong [Department of Physics, University of North Texas, Denton, TX 76203-1427 (United States); Kobe, Donald H. [Department of Physics, University of North Texas, Denton, TX 76203-1427 (United States)], E-mail: kobe@unt.edu
2008-04-15
Using de Broglie-Bohm approach to quantum theory, we show that the kicked rotor at quantum resonance exhibits quantum chaos for the control parameter K above a threshold. Lyapunov exponents are calculated from the method of Benettin et al. for bounded systems for both the quantum and classical kicked rotor. In the chaotic regime we find stability regions for control parameters equal to even and odd multiples of {pi}, but the quantum regions are only remnants of the classical ones.
[A Standing Balance Evaluation Method Based on Largest Lyapunov Exponent].
Liu, Kun; Wang, Hongrui; Xiao, Jinzhuang; Zhao, Qing
2015-12-01
In order to evaluate the ability of human standing balance scientifically, we in this study proposed a new evaluation method based on the chaos nonlinear analysis theory. In this method, a sinusoidal acceleration stimulus in forward/backward direction was forced under the subjects' feet, which was supplied by a motion platform. In addition, three acceleration sensors, which were fixed to the shoulder, hip and knee of each subject, were applied to capture the balance adjustment dynamic data. Through reconstructing the system phase space, we calculated the largest Lyapunov exponent (LLE) of the dynamic data of subjects' different segments, then used the sum of the squares of the difference between each LLE (SSDLLE) as the balance capabilities evaluation index. Finally, 20 subjects' indexes were calculated, and compared with evaluation results of existing methods. The results showed that the SSDLLE were more in line with the subjects' performance during the experiment, and it could measure the body's balance ability to some extent. Moreover, the results also illustrated that balance level was determined by the coordinate ability of various joints, and there might be more balance control strategy in the process of maintaining balance.
Phase space reconstruction and estimation of the largest Lyapunov exponent for gait kinematic data
Energy Technology Data Exchange (ETDEWEB)
Josiński, Henryk [Silesian University of Technology, Akademicka 16, 44-100 Gliwice (Poland); Świtoński, Adam [Polish-Japanese Institute of Information Technology, Aleja Legionów 2, 41-902 Bytom (Poland); Silesian University of Technology, Akademicka 16, 44-100 Gliwice (Poland); Michalczuk, Agnieszka; Wojciechowski, Konrad [Polish-Japanese Institute of Information Technology, Aleja Legionów 2, 41-902 Bytom (Poland)
2015-03-10
The authors describe an example of application of nonlinear time series analysis directed at identifying the presence of deterministic chaos in human motion data by means of the largest Lyapunov exponent. The method was previously verified on the basis of a time series constructed from the numerical solutions of both the Lorenz and the Rössler nonlinear dynamical systems.
Universal scaling of Lyapunov-exponent fluctuations in space-time chaos.
Pazó, Diego; López, Juan M; Politi, Antonio
2013-06-01
Finite-time Lyapunov exponents of generic chaotic dynamical systems fluctuate in time. These fluctuations are due to the different degree of stability across the accessible phase space. A recent numerical study of spatially extended systems has revealed that the diffusion coefficient D of the Lyapunov exponents (LEs) exhibits a nontrivial scaling behavior, D(L)~L(-γ), with the system size L. Here, we show that the wandering exponent γ can be expressed in terms of the roughening exponents associated with the corresponding "Lyapunov surface." Our theoretical predictions are supported by the numerical analysis of several spatially extended systems. In particular, we find that the wandering exponent of the first LE is universal: in view of the known relationship with the Kardar-Parisi-Zhang equation, γ can be expressed in terms of known critical exponents. Furthermore, our simulations reveal that the bulk of the spectrum exhibits a clearly different behavior and suggest that it belongs to a possibly unique universality class, which has, however, yet to be identified.
Extracting Lyapunov exponents from the echo dynamics of Bose-Einstein condensates on a lattice
Tarkhov, Andrei E.; Wimberger, Sandro; Fine, Boris V.
2017-08-01
We propose theoretically an experimentally realizable method to demonstrate the Lyapunov instability and to extract the value of the largest Lyapunov exponent for a chaotic many-particle interacting system. The proposal focuses specifically on a lattice of coupled Bose-Einstein condensates in the classical regime describable by the discrete Gross-Pitaevskii equation. We suggest to use imperfect time reversal of the system's dynamics known as the Loschmidt echo, which can be realized experimentally by reversing the sign of the Hamiltonian of the system. The routine involves tracking and then subtracting the noise of virtually any observable quantity before and after the time reversal. We support the theoretical analysis by direct numerical simulations demonstrating that the largest Lyapunov exponent can indeed be extracted from the Loschmidt echo routine. We also discuss possible values of experimental parameters required for implementing this proposal.
Nonlinear local Lyapunov exponent and atmospheric predictability research
Institute of Scientific and Technical Information of China (English)
CHEN; Baohua; LI; Jianping; DING; Ruiqiang
2006-01-01
Because atmosphere itself is a nonlinear system and there exist some problems using the linearized equations to study the initial error growth, in this paper we try to use the error nonlinear growth theory to discuss its evolution, based on which we first put forward a new concept: nonlinear local Lyapunov exponent. It is quite different from the classic Lyapunov exponent because it may characterize the finite time error local average growth and its value depends on the initial condition,initial error, variables, evolution time, temporal and spatial scales. Based on its definition and the atmospheric features, we provide a reasonable algorithm to the exponent for the experimental data,obtain the atmospheric initial error growth in finite time and gain the maximal prediction time. Lastly,taking 500 hPa height field as example, we discuss the application of the nonlinear local Lyapunov exponent in the study of atmospheric predictability and get some reliable results: atmospheric predictability has a distinct spatial structure. Overall, predictability shows a zonal distribution. Prediction time achieves the maximum over tropics, the second near the regions of Antarctic, it is also longer next to the Arctic and in subtropics and the mid-latitude the predictability is lowest. Particularly speaking, the average prediction time near the equation is 12 days and the maximum is located in the tropical Indian, Indonesia and the neighborhood, tropical eastern Pacific Ocean, on these regions the prediction time is about two weeks. Antarctic has a higher predictability than the neighboring latitudes and the prediction time is about 9 days. This feature is more obvious on Southern Hemispheric summer. In Arctic, the predictability is also higher than the one over mid-high latitudes but it is not pronounced as in Antarctic. Mid-high latitude of both Hemispheres (30°S-60°S, 30°-60°N) have the lowest predictability and the mean prediction time is just 3-4 d. In addition
Local Lyapunov exponents sublimiting growth rates of linear random differential equations
Siegert, Wolfgang
2009-01-01
Establishing a new concept of local Lyapunov exponents the author brings together two separate theories, namely Lyapunov exponents and the theory of large deviations. Specifically, a linear differential system is considered which is controlled by a stochastic process that during a suitable noise-intensity-dependent time is trapped near one of its so-called metastable states. The local Lyapunov exponent is then introduced as the exponential growth rate of the linear system on this time scale. Unlike classical Lyapunov exponents, which involve a limit as time increases to infinity in a fixed system, here the system itself changes as the noise intensity converges, too.
A Simple Method for the Computation of the COnditional Lyapunov Exponents
Institute of Scientific and Technical Information of China (English)
无
1999-01-01
An handy method of calculating the conditional Lyapunov exponents is put forward.Lyapunov exponents of differential dynamical systems and the conditional Lyapunov exponents can be acquired easily with the method.The method has been successfully used in kinds of synchronization ,such as continuous driving synchronization,impulsive(sporadic)driving synchronization,intermittently driving synchronization.The conditional Lyapunov exponents obtained with our method can give the largest and the best time interval for impulsive synchronization that can hardly be settled in other ways.
A Hyperchaotic Attractor with Multiple Positive Lyapunov Exponents
Institute of Scientific and Technical Information of China (English)
HU Guo-Si
2009-01-01
There are many hyperchaotic systems,but few systems can generate hyperchaotic attractors with more than three PLEs(positive Lyapunov exponents).A new hyperchaotic system,constructed by adding an approximate time-delay state feedback to a five-dimensional hyperchaotic system,is presented.With the increasing number of phase-shift units used in this system,the number of PLEs also steadily increases.Hyperchaotic attractors with 25 PLEs can be generated by this system with 32 phase-shift units.The sum of the PLEs will reach the maximum value when 23 phase-shift units are used.A simple electronic circuit,consisting of 16 operational amplifiers and two analogy multipliers,is presented for confirming hyperchaos of order 5,i.e.,with 5 PLEs.
GPU and APU computations of Finite Time Lyapunov Exponent fields
Conti, Christian; Rossinelli, Diego; Koumoutsakos, Petros
2012-03-01
We present GPU and APU accelerated computations of Finite-Time Lyapunov Exponent (FTLE) fields. The calculation of FTLEs is a computationally intensive process, as in order to obtain the sharp ridges associated with the Lagrangian Coherent Structures an extensive resampling of the flow field is required. The computational performance of this resampling is limited by the memory bandwidth of the underlying computer architecture. The present technique harnesses data-parallel execution of many-core architectures and relies on fast and accurate evaluations of moment conserving functions for the mesh to particle interpolations. We demonstrate how the computation of FTLEs can be efficiently performed on a GPU and on an APU through OpenCL and we report over one order of magnitude improvements over multi-threaded executions in FTLE computations of bluff body flows.
Lyapunov exponents of linear cocycles continuity via large deviations
Duarte, Pedro
2016-01-01
The aim of this monograph is to present a general method of proving continuity of Lyapunov exponents of linear cocycles. The method uses an inductive procedure based on a general, geometric version of the Avalanche Principle. The main assumption required by this method is the availability of appropriate large deviation type estimates for quantities related to the iterates of the base and fiber dynamics associated with the linear cocycle. We establish such estimates for various models of random and quasi-periodic cocycles. Our method has its origins in a paper of M. Goldstein and W. Schlag. Our present work expands upon their approach in both depth and breadth. We conclude this monograph with a list of related open problems, some of which may be treated using a similar approach.
Refining and classifying finite-time Lyapunov exponent ridges
Allshouse, Michael R
2015-01-01
While more rigorous and sophisticated methods for identifying Lagrangian based coherent structures exist, the finite-time Lyapunov exponent (FTLE) field remains a straightforward and popular method for gaining some insight into transport by complex, time-dependent two-dimensional flows. In light of its enduring appeal, and in support of good practice, we begin by investigating the effects of discretization and noise on two numerical approaches for calculating the FTLE field. A practical method to extract and refine FTLE ridges in two-dimensional flows, which builds on previous methods, is then presented. Seeking to better ascertain the role of an FTLE ridge in flow transport, we adapt an existing classification scheme and provide a thorough treatment of the challenges of classifying the types of deformation represented by an FTLE ridge. As a practical demonstration, the methods are applied to an ocean surface velocity field data set generated by a numerical model.
Computation of entropy and Lyapunov exponent by a shift transform.
Matsuoka, Chihiro; Hiraide, Koichi
2015-10-01
We present a novel computational method to estimate the topological entropy and Lyapunov exponent of nonlinear maps using a shift transform. Unlike the computation of periodic orbits or the symbolic dynamical approach by the Markov partition, the method presented here does not require any special techniques in computational and mathematical fields to calculate these quantities. In spite of its simplicity, our method can accurately capture not only the chaotic region but also the non-chaotic region (window region) such that it is important physically but the (Lebesgue) measure zero and usually hard to calculate or observe. Furthermore, it is shown that the Kolmogorov-Sinai entropy of the Sinai-Ruelle-Bowen measure (the physical measure) coincides with the topological entropy.
Computation of entropy and Lyapunov exponent by a shift transform
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Matsuoka, Chihiro, E-mail: matsuoka.chihiro.mm@ehime-u.ac.jp [Department of Physics, Graduate School of Science and Technology, Ehime University, Matsuyama, Ehime 790-8577 (Japan); Hiraide, Koichi [Department of Mathematics, Graduate School of Science and Technology, Ehime University, Matsuyama, Ehime 790-8577 (Japan)
2015-10-15
We present a novel computational method to estimate the topological entropy and Lyapunov exponent of nonlinear maps using a shift transform. Unlike the computation of periodic orbits or the symbolic dynamical approach by the Markov partition, the method presented here does not require any special techniques in computational and mathematical fields to calculate these quantities. In spite of its simplicity, our method can accurately capture not only the chaotic region but also the non-chaotic region (window region) such that it is important physically but the (Lebesgue) measure zero and usually hard to calculate or observe. Furthermore, it is shown that the Kolmogorov-Sinai entropy of the Sinai-Ruelle-Bowen measure (the physical measure) coincides with the topological entropy.
Symmetry properties of orthogonal and covariant Lyapunov vectors and their exponents
Posch, Harald A.
2013-06-01
Lyapunov exponents are indicators for the chaotic properties of a classical dynamical system. They are most naturally defined in terms of the time evolution of a set of so-called covariant vectors, co-moving with the linearized flow in tangent space. Taking a simple spring pendulum and the Hénon-Heiles system as examples, we demonstrate the consequences of symplectic symmetry and of time-reversal invariance for such vectors, and study the transformation between different parameterizations of the flow. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to applications’.
Predictability of large-scale atmospheric motions: Lyapunov exponents and error dynamics.
Vannitsem, Stéphane
2017-03-01
The deterministic equations describing the dynamics of the atmosphere (and of the climate system) are known to display the property of sensitivity to initial conditions. In the ergodic theory of chaos, this property is usually quantified by computing the Lyapunov exponents. In this review, these quantifiers computed in a hierarchy of atmospheric models (coupled or not to an ocean) are analyzed, together with their local counterparts known as the local or finite-time Lyapunov exponents. It is shown in particular that the variability of the local Lyapunov exponents (corresponding to the dominant Lyapunov exponent) decreases when the model resolution increases. The dynamics of (finite-amplitude) initial condition errors in these models is also reviewed, and in general found to display a complicated growth far from the asymptotic estimates provided by the Lyapunov exponents. The implications of these results for operational (high resolution) atmospheric and climate modelling are also discussed.
Dünki, Rudolf M.
2000-11-01
Limited predictability is one of the remarkable features of deterministic chaos and this feature may be quantized in terms of Lyapunov exponents. Accordingly, Lyapunov-exponent estimates may be expected to follow in a natural way from forecast algorithms. Exploring this idea, we propose a method estimating the largest Lyapunov exponent from a time series which uses the behavior of so-called simplex forecasts. The method considers the estimation of properties of the distribution of local simplex expansion coefficients. These are also used for the definition of error bars for the Lyapunov-exponent estimates and allows for selective forecasts with improved prediction accuracy. We demonstrate these concepts on standard test examples and three realistic applications to time series concerning largest Lyapunov-exponent estimation of an experimentally obtained hyperchaotic NMR signal, brain state differentiation, and stock-market prediction.
Entropy, Lyapunov Exponents and Escape Rates in Open Systems
Demers, Mark; Young, Lai-Sang
2011-01-01
We study the relation between escape rates and pressure in general dynamical systems with holes, where pressure is defined to be the difference between entropy and the sum of positive Lyapunov exponents. Central to the discussion is the formulation of a class of invariant measures supported on the survivor set over which we take the supremum to measure the pressure. Upper bounds for escape rates are proved for general diffeomorphisms of manifolds, possibly with singularities, for arbitrary holes and natural initial distributions including Lebesgue and SRB measures. Lower bounds do not hold in such generality, but for systems admitting Markov tower extensions with spectral gaps, we prove the equality of the escape rate with the absolute value of the pressure and the existence of an invariant measure realizing the escape rate, i.e. we prove a full variational principle. As an application of our results, we prove a variational principle for the billiard map associated with a planar Lorentz gas of finite horizon ...
Regularized semiclassical limits: Linear flows with infinite Lyapunov exponents
Athanassoulis, Agissilaos
2016-08-30
Semiclassical asymptotics for Schrödinger equations with non-smooth potentials give rise to ill-posed formal semiclassical limits. These problems have attracted a lot of attention in the last few years, as a proxy for the treatment of eigenvalue crossings, i.e. general systems. It has recently been shown that the semiclassical limit for conical singularities is in fact well-posed, as long as the Wigner measure (WM) stays away from singular saddle points. In this work we develop a family of refined semiclassical estimates, and use them to derive regularized transport equations for saddle points with infinite Lyapunov exponents, extending the aforementioned recent results. In the process we answer a related question posed by P.L. Lions and T. Paul in 1993. If we consider more singular potentials, our rigorous estimates break down. To investigate whether conical saddle points, such as -|x|, admit a regularized transport asymptotic approximation, we employ a numerical solver based on posteriori error control. Thus rigorous upper bounds for the asymptotic error in concrete problems are generated. In particular, specific phenomena which render invalid any regularized transport for -|x| are identified and quantified. In that sense our rigorous results are sharp. Finally, we use our findings to formulate a precise conjecture for the condition under which conical saddle points admit a regularized transport solution for the WM. © 2016 International Press.
Backward Finite-Time Lyapunov Exponents in Inertial Flows.
Gunther, Tobias; Theisel, Holger
2017-01-01
Inertial particles are finite-sized objects that are carried by fluid flows and in contrast to massless tracer particles they are subject to inertia effects. In unsteady flows, the dynamics of tracer particles have been extensively studied by the extraction of Lagrangian coherent structures (LCS), such as hyperbolic LCS as ridges of the Finite-Time Lyapunov Exponent (FTLE). The extension of the rich LCS framework to inertial particles is currently a hot topic in the CFD literature and is actively under research. Recently, backward FTLE on tracer particles has been shown to correlate with the preferential particle settling of small inertial particles. For larger particles, inertial trajectories may deviate strongly from (massless) tracer trajectories, and thus for a better agreement, backward FTLE should be computed on inertial trajectories directly. Inertial backward integration, however, has not been possible until the recent introduction of the influence curve concept, which - given an observation and an initial velocity - allows to recover all sources of inertial particles as tangent curves of a derived vector field. In this paper, we show that FTLE on the influence curve vector field is in agreement with preferential particle settling and more importantly it is not only valid for small (near-tracer) particles. We further generalize the influence curve concept to general equations of motion in unsteady spatio-velocity phase spaces, which enables backward integration with more general equations of motion. Applying the influence curve concept to tracer particles in the spatio-velocity domain emits streaklines in massless flows as tangent curves of the influence curve vector field. We demonstrate the correlation between inertial backward FTLE and the preferential particle settling in a number of unsteady vector fields.
Joint Statistics of Finite Time Lyapunov Exponents in Isotropic Turbulence
Johnson, Perry; Meneveau, Charles
2014-11-01
Recently, the notion of Lagrangian Coherent Structures (LCS) has gained attention as a tool for qualitative visualization of flow features. LCS visualize repelling and attracting manifolds marked by local ridges in the field of maximal and minimal finite-time Lyapunov exponents (FTLE), respectively. To provide a quantitative characterization of FTLEs, the statistical theory of large deviations can be used based on the so-called Cramér function. To obtain the Cramér function from data, we use both the method based on measuring moments and measuring histograms (with finite-size correction). We generalize the formalism to characterize the joint distributions of the two independent FTLEs in 3D. The ``joint Cramér function of turbulence'' is measured from the Johns Hopkins Turbulence Databases (JHTDB) isotropic simulation at Reλ = 433 and results are compared with those computed using only the symmetric part of the velocity gradient tensor, as well as with those of instantaneous strain-rate eigenvalues. We also extend the large-deviation theory to study the statistics of the ratio of FTLEs. When using only the strain contribution of the velocity gradient, the maximal FTLE nearly doubles in magnitude and the most likely ratio of FTLEs changes from 4:1:-5 to 8:3:-11, highlighting the role of rotation in de-correlating the fluid deformations along particle paths. Supported by NSF Graduate Fellowship (DGE-1232825), a JHU graduate Fellowship, and NSF Grant CMMI-0941530. CM thanks Prof. Luca Biferale for useful discussions on the subject.
Energy Technology Data Exchange (ETDEWEB)
Morawetz, K
1999-07-01
Within the frame of kinetic theory a response function is derived for finite Fermi systems which includes dissipation in relaxation time approximation and a contribution from additional chaotic processes characterized by the largest Lyapunov exponent. A generalized local density approximation is presented including the effect of many particle relaxation and the additional chaotic scattering. For small Lyapunov exponents relative to the product of wave vector and Fermi time. Therefore the transport coefficients can be connected with the largest positive Lyapunov exponent in the same way as known the transport theory in relaxation time approximation. (author)
Finite-time Lyapunov exponents in time-delayed nonlinear dynamical systems.
Kanno, Kazutaka; Uchida, Atsushi
2014-03-01
We introduce a method for the calculation of finite-time Lyapunov exponents in time-delayed nonlinear dynamical systems. We apply the method to the Mackey-Glass model with time-delayed feedback. We investigate the standard deviation of the probability distribution of the finite-time Lyapunov exponents when the finite time or the delay time is changed. It is found that the standard deviation decreases in a power-law scaling with the exponent ∼0.5 as the finite time or the delay time is increased. Similar results are obtained for the finite-time Lyapunov spectrum.
Lyapunov exponents for small aspect ratio Rayleigh-Bénard convection.
Scheel, J D; Cross, M C
2006-12-01
Leading order Lyapunov exponents and their corresponding eigenvectors have been computed numerically for small aspect ratio, three-dimensional Rayleigh-Benard convection cells with no-slip boundary conditions. The parameters are the same as those used by Ahlers and Behringer [Phys. Rev. Lett. 40, 712 (1978)] and Gollub and Benson [J. Fluid Mech. 100, 449 (1980)] in their work on a periodic time dependence in Rayleigh-Benard convection cells. Our work confirms that the dynamics in these cells truly are chaotic as defined by a positive Lyapunov exponent. The time evolution of the leading order Lyapunov eigenvector in the chaotic regime will also be discussed. In addition we study the contributions to the leading order Lyapunov exponent for both time periodic and aperiodic states and find that while repeated dynamical events such as dislocation creation/annihilation and roll compression do contribute to the short time Lyapunov exponent dynamics, they do not contribute to the long time Lyapunov exponent. We find instead that nonrepeated events provide the most significant contribution to the long time leading order Lyapunov exponent.
Using Lyapunov exponents to predict the onset of chaos in nonlinear oscillators.
Ryabov, Vladimir B
2002-07-01
An analytic technique for predicting the emergence of chaotic instability in nonlinear nonautonomous dissipative oscillators is proposed. The method is based on the Lyapunov-type stability analysis of an arbitrary phase trajectory and the standard procedure of calculating the Lyapunov characteristic exponents. The concept of temporally local Lyapunov exponents is then utilized for specifying the area in the phase space where any trajectory is asymptotically stable, and, therefore, the existence of chaotic attractors is impossible. The procedure of linear coordinate transform optimizing the linear part of the vector field is developed for the purpose of maximizing the stability area in the vicinity of a stable fixed point. By considering the inverse conditions of asymptotic stability, this approach allows formulating a necessary condition for chaotic motion in a broad class of nonlinear oscillatory systems, including many cases of practical interest. The examples of externally excited one- and two-well Duffing oscillators and a planar pendulum demonstrate efficiency of the proposed method, as well as a good agreement of the theoretical predictions with the results of numerical experiments. The comparison of the proposed method with Melnikov's criterion shows a potential advantage of using the former one at high values of dissipation parameter and/or multifrequency type of excitation in dynamical systems.
Fazanaro, Filipe I; Soriano, Diogo C; Suyama, Ricardo; Attux, Romis; Madrid, Marconi K; de Oliveira, José Raimundo
2013-06-01
The present work aims to apply a recently proposed method for estimating Lyapunov exponents to characterize-with the aid of the metric entropy and the fractal dimension-the degree of information and the topological structure associated with multiscroll attractors. In particular, the employed methodology offers the possibility of obtaining the whole Lyapunov spectrum directly from the state equations without employing any linearization procedure or time series-based analysis. As a main result, the predictability and the complexity associated with the phase trajectory were quantified as the number of scrolls are progressively increased for a particular piecewise linear model. In general, it is shown here that the trajectory tends to increase its complexity and unpredictability following an exponential behaviour with the addition of scrolls towards to an upper bound limit, except for some degenerated situations where a non-uniform grid of scrolls is attained. Moreover, the approach employed here also provides an easy way for estimating the finite time Lyapunov exponents of the dynamics and, consequently, the Lagrangian coherent structures for the vector field. These structures are particularly important to understand the stretching/folding behaviour underlying the chaotic multiscroll structure and can provide a better insight of phase space partition and exploration as new scrolls are progressively added to the attractor.
Gustavsson, K
2013-01-01
We calculate the Lyapunov exponents describing spatial clustering of particles advected in one- and two-dimensional random velocity fields at finite Kubo number Ku (a dimensionless parameter characterising the correlation time of the velocity field). In one dimension we obtain accurate results up to Ku ~ 1 by resummation of a perturbation expansion in Ku. At large Kubo numbers we compute the Lyapunov exponent by taking into account the fact that the particles follow the minima of the potential function corresponding to the velocity field. In two dimensions we compute the first four non-vanishing terms in the small-Ku expansion of the Lyapunov exponents. For large Kubo numbers we estimate the Lyapunov exponents by assuming that the particles sample stagnation points of the velocity field with det A > 0 and Tr A < 0 where A is the matrix of flow-velocity gradients.
Short-Term Forecasting of Urban Water Consumption Based on the Largest Lyapunov Exponent
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
An approach for short-term forecasting of municipal water consumption was presented based on the largest Lyapunov exponent of chaos theory. The chaotic characteristics of time series of urban water consumption were examined by means of the largest Lyapunov exponent and correlation dimension. By using the largest Lyapunov exponent a short-term forecasting model for urban water consumption was developed, which was compared with the artificial neural network (ANN) approach in a case study. The result indicates that the model based on the largest Lyapunov exponent has higher prediction precision and forecasting stability than the ANN method, and its forecasting mean relative error is 9.6% within its maximum predictable time scale while it is 60.6% beyond the scale.
Application of local Lyapunov exponents to maneuver design and navigation in the three-body problem
Anderson, Rodney L.; Lo, Martin W.; Born, George H.
2003-01-01
Dynamical systems theory has recently been employed to design trajectories within the three-body problem for several missions. This research has applied one stability technique, the calculation of local Lyapunov exponents, to such trajectories. Local Lyapunov exponents give an indication of the effects that perturbations or maneuvers will have on trajectories over a specified time. A numerical comparison of local Lyapunov exponents was first made with the distance random perturbations traveled from a nominal trajectory, and the local Lyapunov exponents were found to correspond well with the perturbations that caused the greatest deviation from the nominal. This would allow them to be used as an indicator of the points where it would be important to reduce navigation uncertainties.
Application of local Lyapunov exponents to maneuver design and navigation in the three-body problem
Anderson, Rodney L.; Lo, Martin W.; Born, George H.
2003-01-01
Dynamical systems theory has recently been employed to design trajectories within the three-body problem for several missions. This research has applied one stability technique, the calculation of local Lyapunov exponents, to such trajectories. Local Lyapunov exponents give an indication of the effects that perturbations or maneuvers will have on trajectories over a specified time. A numerical comparison of local Lyapunov exponents was first made with the distance random perturbations traveled from a nominal trajectory, and the local Lyapunov exponents were found to correspond well with the perturbations that caused the greatest deviation from the nominal. This would allow them to be used as an indicator of the points where it would be important to reduce navigation uncertainties.
Zeta function for the Lyapunov exponent of a product of random matrices
Energy Technology Data Exchange (ETDEWEB)
Mainieri, R. (Neils Bohr Institute, Blegdamsvej 17, Copenhagen O, 2100 (Denmark) Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States))
1992-03-30
A cycle expansion for the Lyapunov exponent of a product of random matrices is derived. The formula is nonperturbative and numerically effective, which allows the Lyapunov exponent to be computed to high accuracy. In particular, the free energy and heat capacity are computed for the one-dimensional Ising model with quenched disorder. The formula is derived by using a Bernoulli dynamical system to mimic the randomness.
Lyapunov Exponents and Kolmogorov-Sinai Entropy for the Lorentz Gas at Low Densities
van Beijeren, Henk; Dorfman, J. R.
1995-05-01
The Lyapunov exponents and the Kolmogorov-Sinai (KS) entropy for a two-dimensional Lorentz gas at low densities are defined for general nonequilibrium states and calculated with the use of a Lorentz-Boltzmann type equation. In equilibrium the density dependence of these quantities, predicted by Krylov, is recovered and explicit expressions are obtained. The relationship between KS entropy, Lyapunov exponents, and diffusion coefficients, developed by Gaspard and Nicolis, is generalized to a wide class of nonequilibrium states.
Lyapunov exponent evaluation of a digital watermarking scheme proven to be secure
Bahi, Jacques M; Guyeux, Christophe
2012-01-01
In our previous researches, a new digital watermarking scheme based on chaotic iterations has been introduced. This scheme was both stego-secure and topologically secure. The stego-security is to face an attacker in the "watermark only attack" category, whereas the topological security concerns other categories of attacks. Its Lyapunov exponent is evaluated here, to quantify the chaos generated by this scheme. Keywords : Lyapunov exponent; Information hiding; Security; Chaotic iterations; Digital Watermarking.
Symmetry of Lyapunov exponents in bifurcation structures of one-dimensional maps.
Shimada, Yutaka; Takagi, Emiko; Ikeguchi, Tohru
2016-12-01
We observe a symmetry of Lyapunov exponents in bifurcation structures of one-dimensional maps in which there exists a pair of parameter values in a dynamical system such that two dynamical systems with these paired parameter values have the same Lyapunov exponent. We show that this is a consequence of the presence of an invariant transformation from a dynamical system with one of the two paired parameter values to that with another parameter value, which does not change natures of dynamical systems.
Critical behavior of the Lyapunov exponent in type-III intermittency
Energy Technology Data Exchange (ETDEWEB)
Alvarez-Llamoza, O. [Departamento de Fisica, FACYT, Universidad de Carabobo, Valencia (Venezuela); Centro de Fisica Fundamental, Grupo de Caos y Sistemas Complejos, Universidad de Los Andes, Merida 5251, Merida (Venezuela)], E-mail: llamoza@ula.ve; Cosenza, M.G. [Centro de Fisica Fundamental, Grupo de Caos y Sistemas Complejos, Universidad de Los Andes, Merida 5251, Merida (Venezuela); Ponce, G.A. [Departamento de Fisica, Universidad Nacional Autonoma de Honduras (Honduras); Departamento de Ciencias Naturales, Universidad Pedagogica Nacional Francisco Morazan, Tegucigalpa (Honduras)
2008-04-15
The critical behavior of the Lyapunov exponent near the transition to robust chaos via type-III intermittency is determined for a family of one-dimensional singular maps. Critical boundaries separating the region of robust chaos from the region where stable fixed points exist are calculated on the parameter space of the system. A critical exponent {beta} expressing the scaling of the Lyapunov exponent is calculated along the critical curve corresponding to the type-III intermittent transition to chaos. It is found that {beta} varies on the interval 0 {<=} {beta} < 1/2 as a function of the order of the singularity of the map. This contrasts with earlier predictions for the scaling behavior of the Lyapunov exponent in type-III intermittency. The variation of the critical exponent {beta} implies a continuous change in the nature of the transition to chaos via type-III intermittency, from a second-order, continuous transition to a first-order, discontinuous transition.
Lyapunov exponent of magnetospheric activity from AL time series
Vassiliadis, D.; Sharma, A. S.; Papadopoulos, K.
1991-01-01
A correlation dimension analysis of the AE index indicates that the magnetosphere behaves as a low-dimensional chaotic system with a dimension close to 4. Similar techniques are used to determine if the system's behavior is due to an intrinsic sensitivity to initial conditions and thus is truly chaotic. The quantity used to measure the sensitivity to initial conditions is the Liapunov exponent. Its calculation for AL shows that it is nonzero (0.11 + or - 0.05/min). This gives the exponential rate at which initially similar configurations of the magnetosphere evolve into completely different states. Also, predictions of deterministic nonlinear models are expected to deviate from the observed behavior at the same rate.
Lyapunov exponent of magnetospheric activity from AL time series
Vassiliadis, D.; Sharma, A. S.; Papadopoulos, K.
1991-01-01
A correlation dimension analysis of the AE index indicates that the magnetosphere behaves as a low-dimensional chaotic system with a dimension close to 4. Similar techniques are used to determine if the system's behavior is due to an intrinsic sensitivity to initial conditions and thus is truly chaotic. The quantity used to measure the sensitivity to initial conditions is the Liapunov exponent. Its calculation for AL shows that it is nonzero (0.11 + or - 0.05/min). This gives the exponential rate at which initially similar configurations of the magnetosphere evolve into completely different states. Also, predictions of deterministic nonlinear models are expected to deviate from the observed behavior at the same rate.
Study on the expression of systematic Lyapunov exponent based on UPOs
Institute of Scientific and Technical Information of China (English)
岳毅宏; 韩文秀; 程国平
2004-01-01
The natural measure of a certain area in phase space is defined firstly. On the basis of natural measure, the expression of Lyapunov exponent based on unstable periodic orbits (UPOs) of chaotic systems is deduced from theoretical aspect. Then, by means of the inherent relation between UPOs and systematic Lyapunov exponent, the transitional mechanism and route of chaotic systems from low-dimensional chaos to high-dimensional chaos are explained. In the end,a novel method for computing systematic Lyapunov exponents based on UPOs is proposed. Its computing procedure is also summarized. The chaotic system described by Henon map is taken as example. Through calculating the Lypunov exponents of this system, validity of the suggested method is verified.
Lyapunov exponents and phase diagrams reveal multi-factorial control over TRAIL-induced apoptosis
Aldridge, Bree B; Gaudet, Suzanne; Lauffenburger, Douglas A; Sorger, Peter K
2011-01-01
Receptor-mediated apoptosis proceeds via two pathways: one requiring only a cascade of initiator and effector caspases (type I behavior) and the second requiring an initiator–effector caspase cascade and mitochondrial outer membrane permeabilization (type II behavior). Here, we investigate factors controlling type I versus II phenotypes by performing Lyapunov exponent analysis of an ODE-based model of cell death. The resulting phase diagrams predict that the ratio of XIAP to pro-caspase-3 concentrations plays a key regulatory role: type I behavior predominates when the ratio is low and type II behavior when the ratio is high. Cell-to-cell variability in phenotype is observed when the ratio is close to the type I versus II boundary. By positioning multiple tumor cell lines on the phase diagram we confirm these predictions. We also extend phase space analysis to mutations affecting the rate of caspase-3 ubiquitylation by XIAP, predicting and showing that such mutations abolish all-or-none control over activation of effector caspases. Thus, phase diagrams derived from Lyapunov exponent analysis represent a means to study multi-factorial control over a complex biochemical pathway. PMID:22108795
Lyapunov exponents and phase diagrams reveal multi-factorial control over TRAIL-induced apoptosis.
Aldridge, Bree B; Gaudet, Suzanne; Lauffenburger, Douglas A; Sorger, Peter K
2011-11-22
Receptor-mediated apoptosis proceeds via two pathways: one requiring only a cascade of initiator and effector caspases (type I behavior) and the second requiring an initiator-effector caspase cascade and mitochondrial outer membrane permeabilization (type II behavior). Here, we investigate factors controlling type I versus II phenotypes by performing Lyapunov exponent analysis of an ODE-based model of cell death. The resulting phase diagrams predict that the ratio of XIAP to pro-caspase-3 concentrations plays a key regulatory role: type I behavior predominates when the ratio is low and type II behavior when the ratio is high. Cell-to-cell variability in phenotype is observed when the ratio is close to the type I versus II boundary. By positioning multiple tumor cell lines on the phase diagram we confirm these predictions. We also extend phase space analysis to mutations affecting the rate of caspase-3 ubiquitylation by XIAP, predicting and showing that such mutations abolish all-or-none control over activation of effector caspases. Thus, phase diagrams derived from Lyapunov exponent analysis represent a means to study multi-factorial control over a complex biochemical pathway.
Effect of parameter calculation in direct estimation of the Lyapunov exponent in short time series
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A. M. López Jiménez
2002-01-01
Full Text Available The literature about non-linear dynamics offers a few recommendations, which sometimes are divergent, about the criteria to be used in order to select the optimal calculus parameters in the estimation of Lyapunov exponents by direct methods. These few recommendations are circumscribed to the analysis of chaotic systems. We have found no recommendation for the estimation of λ starting from the time series of classic systems. The reason for this is the interest in distinguishing variability due to a chaotic behavior of determinist dynamic systems of variability caused by white noise or linear stochastic processes, and less in the identification of non-linear terms from the analysis of time series. In this study we have centered in the dependence of the Lyapunov exponent, obtained by means of direct estimation, of the initial distance and the time evolution. We have used generated series of chaotic systems and generated series of classic systems with varying complexity. To generate the series we have used the logistic map.
Truant, Daniel P; Morriss, Gary P
2014-11-01
The covariant Lyapunov analysis is generalized to systems attached to deterministic thermal reservoirs that create a heat current across the system and perturb it away from equilibrium. The change in the Lyapunov exponents as a function of heat current is described and explained. Both the nonequilibrium backward and covariant hydrodynamic Lyapunov modes are analyzed and compared. The movement of the converged angle between the hydrodynamic stable and unstable conjugate manifolds with the free flight time of the dynamics is accurately predicted for any nonequilibrium system simply as a function of their exponent. The nonequilibrium positive and negative LP mode frequencies are found to be asymmetrical, causing the negative mode to oscillate between the two functional forms of each mode in the positive conjugate mode pair. This in turn leads to the angular distributions between the conjugate modes to oscillate symmetrically about π/2 at a rate given by the difference between the positive and negative mode frequencies.
Geometrical constraints on finite-time Lyapunov exponents in two and three dimensions.
Thiffeault, Jean-Luc; Boozer, Allen H.
2001-03-01
Constraints are found on the spatial variation of finite-time Lyapunov exponents of two- and three-dimensional systems of ordinary differential equations. In a chaotic system, finite-time Lyapunov exponents describe the average rate of separation, along characteristic directions, of neighboring trajectories. The solution of the equations is a coordinate transformation that takes initial conditions (the Lagrangian coordinates) to the state of the system at a later time (the Eulerian coordinates). This coordinate transformation naturally defines a metric tensor, from which the Lyapunov exponents and characteristic directions are obtained. By requiring that the Riemann curvature tensor vanish for the metric tensor (a basic result of differential geometry in a flat space), differential constraints relating the finite-time Lyapunov exponents to the characteristic directions are derived. These constraints are realized with exponential accuracy in time. A consequence of the relations is that the finite-time Lyapunov exponents are locally small in regions where the curvature of the stable manifold is large, which has implications for the efficiency of chaotic mixing in the advection-diffusion equation. The constraints also modify previous estimates of the asymptotic growth rates of quantities in the dynamo problem, such as the magnitude of the induced current. (c) 2001 American Institute of Physics.
The Multivariate Largest Lyapunov Exponent as an Age-Related Metric of Quiet Standing Balance
Directory of Open Access Journals (Sweden)
Kun Liu
2015-01-01
Full Text Available The largest Lyapunov exponent has been researched as a metric of the balance ability during human quiet standing. However, the sensitivity and accuracy of this measurement method are not good enough for clinical use. The present research proposes a metric of the human body’s standing balance ability based on the multivariate largest Lyapunov exponent which can quantify the human standing balance. The dynamic multivariate time series of ankle, knee, and hip were measured by multiple electrical goniometers. Thirty-six normal people of different ages participated in the test. With acquired data, the multivariate largest Lyapunov exponent was calculated. Finally, the results of the proposed approach were analysed and compared with the traditional method, for which the largest Lyapunov exponent and power spectral density from the centre of pressure were also calculated. The following conclusions can be obtained. The multivariate largest Lyapunov exponent has a higher degree of differentiation in differentiating balance in eyes-closed conditions. The MLLE value reflects the overall coordination between multisegment movements. Individuals of different ages can be distinguished by their MLLE values. The standing stability of human is reduced with the increment of age.
The Multivariate Largest Lyapunov Exponent as an Age-Related Metric of Quiet Standing Balance.
Liu, Kun; Wang, Hongrui; Xiao, Jinzhuang
2015-01-01
The largest Lyapunov exponent has been researched as a metric of the balance ability during human quiet standing. However, the sensitivity and accuracy of this measurement method are not good enough for clinical use. The present research proposes a metric of the human body's standing balance ability based on the multivariate largest Lyapunov exponent which can quantify the human standing balance. The dynamic multivariate time series of ankle, knee, and hip were measured by multiple electrical goniometers. Thirty-six normal people of different ages participated in the test. With acquired data, the multivariate largest Lyapunov exponent was calculated. Finally, the results of the proposed approach were analysed and compared with the traditional method, for which the largest Lyapunov exponent and power spectral density from the centre of pressure were also calculated. The following conclusions can be obtained. The multivariate largest Lyapunov exponent has a higher degree of differentiation in differentiating balance in eyes-closed conditions. The MLLE value reflects the overall coordination between multisegment movements. Individuals of different ages can be distinguished by their MLLE values. The standing stability of human is reduced with the increment of age.
Ding, Ruiqiang; Li, Jianping; Li, Baosheng
2017-09-01
For an n-dimensional chaotic system, we extend the definition of the nonlinear local Lyapunov exponent (NLLE) from one- to n-dimensional spectra, and present a method for computing the NLLE spectrum. The method is tested on three chaotic systems with different complexity. The results indicate that the NLLE spectrum realistically characterizes the growth rates of initial error vectors along different directions from the linear to nonlinear phases of error growth. This represents an improvement over the traditional Lyapunov exponent spectrum, which only characterizes the error growth rates during the linear phase of error growth. In addition, because the NLLE spectrum can effectively separate the slowly and rapidly growing perturbations, it is shown to be more suitable for estimating the predictability of chaotic systems, as compared to the traditional Lyapunov exponent spectrum.
Institute of Scientific and Technical Information of China (English)
ALI M.; SAHA L.M.
2005-01-01
A chaotic dynamical system is characterized by a positive averaged exponential separation of two neighboring trajectories over a chaotic attractor. Knowledge of the Largest Lyapunov Exponent λ1 of a dynamical system over a bounded attractor is necessary and sufficient for determining whether it is chaotic (λ1＞0) or not (λ1≤0). We intended in this work to elaborate the connection between Local Lyapunov Exponents and the Largest Lyapunov Exponent where an altemative method to calculate λ1has emerged. Finally, we investigated some characteristics of the fixed points and periodic orbits embedded within a chaotic attractor which led to the conclusion of the existence of chaotic attractors that may not embed in any fixed point or periodic orbit within it.
New prediction of chaotic time series based on local Lyapunov exponent
Institute of Scientific and Technical Information of China (English)
Zhang Yong
2013-01-01
A new method of predicting chaotic time series is presented based on a local Lyapunov exponent,by quantitatively measuring the exponential rate of separation or attraction of two infinitely close trajectories in state space.After reconstructing state space from one-dimensional chaotic time series,neighboring multiple-state vectors of the predicting point are selected to deduce the prediction formula by using the definition of the local Lyapunov exponent.Numerical simulations are carried out to test its effectiveness and verify its higher precision over two older methods.The effects of the number of referential state vectors and added noise on forecasting accuracy are also studied numerically.
Lyapunov exponent corresponding to enslaved phase dynamics: Estimation from time series.
Moskalenko, Olga I; Koronovskii, Alexey A; Hramov, Alexander E
2015-07-01
A method for the estimation of the Lyapunov exponent corresponding to enslaved phase dynamics from time series has been proposed. It is valid for both nonautonomous systems demonstrating periodic dynamics in the presence of noise and coupled chaotic oscillators and allows us to estimate precisely enough the value of this Lyapunov exponent in the supercritical region of the control parameters. The main results are illustrated with the help of the examples of the noised circle map, the nonautonomous Van der Pole oscillator in the presence of noise, and coupled chaotic Rössler systems.
Institute of Scientific and Technical Information of China (English)
Shihui Fu; Qi Wang
2006-01-01
Using the properties of chaos synchronization. the method for estimating the largest Lyapunov exponent in a multibody system with dry friction is presented in this paper. The Lagrange equations with multipliers of the systems are given in matrix form. which is adequate for numerical calculation. The approach for calculating the generalized velocity and acceleration of the slider is given to determine slipping or sticking of the slider in the systems. For slip-slip and stick-slip multibody systems, their largest Lyapunov exponents are calculated to characterize their dynamics.
Structured scale dependence in the Lyapunov exponent of a Boolean chaotic map.
Cohen, Seth D
2015-04-01
We report on structures in a scale-dependent Lyapunov exponent of an experimental chaotic map that arise due to discontinuities in the map. The chaos is realized in an autonomous Boolean network, which is constructed using asynchronous logic gates to form a map operator that outputs an unclocked pulse-train of varying widths. The map operator executes pulse-width stretching and folding and the operator's output is fed back to its input to continuously iterate the map. Using a simple model, we show that the structured scale-dependence in the system's Lyapunov exponent is the result of the discrete logic elements in the map operator's stretching function.
An Isomorphism between Lyapunov Exponents and Shannon's Channel Capacity
Energy Technology Data Exchange (ETDEWEB)
Friedland, Gerald [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Metere, Alfredo [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
2017-06-07
We demonstrate that discrete Lyapunov exponents are isomorphic to numeric overflows of the capacity of an arbitrary noiseless and memoryless channel in a Shannon communication model with feedback. The isomorphism allows the understanding of Lyapunov exponents in terms of Information Theory, rather than the traditional definitions in chaos theory. The result also implies alternative approaches to the calculation of related quantities, such as the Kolmogorov Sinai entropy which has been linked to thermodynamic entropy. This work provides a bridge between fundamental physics and information theory. It suggests, among other things, that machine learning and other information theory methods can be employed at the core of physics simulations.
How reliable are Finite-Size Lyapunov Exponents for the assessment of ocean dynamics?
Hernández-Carrasco, Ismael; López, Cristóbal; Turiel, Antonio
2010-01-01
Much of atmospheric and oceanic transport is associated with coherent structures. Lagrangian methods are emerging as optimal tools for their identification and analysis. An important Lagrangian technique which is starting to be widely used in oceanography is that of Finite-Size Lyapunov Exponents (FSLEs). Despite this growing relevance there are still many open questions concerning the reliability of the FSLEs in order to analyse the ocean dynamics. In particular, it is still unclear how robust they are when confronted with real data. In this paper we analyze the effect on this Lagrangian technique of the two most important effects when facing real data, namely noise and dynamics of unsolved scales. Our results, using as a benchmarch data from a primitive numerical model of the Mediterranean Sea, show that even when some dynamics is missed the FSLEs results still give an accurate picture of the oceanic transport properties.
A perturbation method to the tent map based on Lyapunov exponent and its application
Institute of Scientific and Technical Information of China (English)
曹绿晨; 罗玉玲; 丘森辉; 刘俊秀
2015-01-01
Perturbation imposed on a chaos system is an effective way to maintain its chaotic features. A novel parameter perturbation method for the tent map based on the Lyapunov exponent is proposed in this paper. The pseudo-random sequence generated by the tent map is sent to another chaos function—the Chebyshev map for the post processing. If the output value of the Chebyshev map falls into a certain range, it will be sent back to replace the parameter of the tent map. As a result, the parameter of the tent map keeps changing dynamically. The statistical analysis and experimental results prove that the disturbed tent map has a highly random distribution and achieves good cryptographic properties of a pseudo-random sequence. As a result, it weakens the phenomenon of strong correlation caused by the finite precision and effectively compensates for the digital chaos system dynamics degradation.
No ISCOs in Charged Myers Perry Spacetimes by Measuring Lyapunov Exponent
Pradhan, Parthapratim
2015-01-01
By computing coordinate time Lyapunov exponent, we prove that for more than four spacetime dimensions (N ≥ 3), there are no Innermost Stable Circular Orbit (ISCO) in charged Myers Perry blackhole spacetime.Using it, we show that the instability of equatorial circular geodesics, both massive and massless particles for such types of blackhole space-times.
Predicting Traffic Flow in Local Area Networks by the Largest Lyapunov Exponent
Directory of Open Access Journals (Sweden)
Yan Liu
2016-01-01
Full Text Available The dynamics of network traffic are complex and nonlinear, and chaotic behaviors and their prediction, which play an important role in local area networks (LANs, are studied in detail, using the largest Lyapunov exponent. With the introduction of phase space reconstruction based on the time sequence, the high-dimensional traffic is projected onto the low dimension reconstructed phase space, and a reduced dynamic system is obtained from the dynamic system viewpoint. Then, a numerical method for computing the largest Lyapunov exponent of the low-dimensional dynamic system is presented. Further, the longest predictable time, which is related to chaotic behaviors in the system, is studied using the largest Lyapunov exponent, and the Wolf method is used to predict the evolution of the traffic in a local area network by both Dot and Interval predictions, and a reliable result is obtained by the presented method. As the conclusion, the results show that the largest Lyapunov exponent can be used to describe the sensitivity of the trajectory in the reconstructed phase space to the initial values. Moreover, Dot Prediction can effectively predict the flow burst. The numerical simulation also shows that the presented method is feasible and efficient for predicting the complex dynamic behaviors in LAN traffic, especially for congestion and attack in networks, which are the main two complex phenomena behaving as chaos in networks.
Institute of Scientific and Technical Information of China (English)
Cheng Changjun; Fan Xiaojun
2000-01-01
The relation between the Lyapunov exponent spectrun of a periodically excited non-autono mous dynamical system and the Lyapunov exponent spectrum of the corresponding autonomous system is given and the validity of the relation is verified theoretically and computationally. A direct method for calculating the Lyapunov exponent spectrum of non-autonomous dynamical systems is suggested in this paper, which makes it more convenient to calculate the Lyapunov exponent spectrum of the dynamical system periodically excited. Following the defi nition of the Lyapunov dimension D(LA) of the autonomous system, the definition of the Lyapunov dimension Dl of the non-autonomous dynamical system is also given, and the difference be- tween them is the integer 1, namely, D(A)L - DL = 1. For a quasi-poriodically excited dynamical system, similar conclusions are formed.
Directory of Open Access Journals (Sweden)
M Seidi
2016-12-01
Full Text Available Lyapunov exponent method is one of the best tools for investigating the range of stability and the transient behavior of the dynamical systems. In beryllium-moderated and heavy water-moderated reactors, photo-neutron plays an important role in dynamic behavior of the reactor. Therefore, stability analysis for changes in the control parameters of the reactor in order to guarantee safety and control nuclear reactor is important. In this work, the range of stability has been investigated using Lyapunov exponent method in response to step, ramp and sinusoidal external reactivities regarding six groups of delayed neutrons plus nine groups of photo-neutrons. The qualitative results are in good agreement with quantitative results of other works
Quantification of the degree of mixing in chaotic micromixers using finite time Lyapunov exponents
Sarkar, Aniruddha; Harting, Jens
2010-01-01
Chaotic micromixers such as the staggered herringbone mixer developed by Stroock et al. allow efficient mixing of fluids even at low Reynolds number by repeated stretching and folding of the fluid interfaces. The ability of the fluid to mix well depends on the rate at which "chaotic advection" occurs in the mixer. An optimization of mixer geometries based on the quantification of chaotic advection is a non trivial task which is often performed by time consuming and expensive trial and error experiments. In this paper it is shown that the concept of finite-time Lyapunov exponents is a suitable tool to provide a quantitative measure of the chaotic advection of the flow. By performing lattice Boltzmann simulations of the flow inside a mixer geometry, introducing massless and non-interacting tracer particles and following their trajectories the finite time Lyapunov exponents can be calculated. The applicability of the method is demonstrated by optimizing the geometrical structure of the staggered herringbone mixe...
Using largest Lyapunov exponent to confirm the intrinsic stability of boiling water reactors
Energy Technology Data Exchange (ETDEWEB)
Gavilian-Moreno, Carlos [Iberdrola Generacion, S.A., Cofrentes Nuclear Power Plant, Project Engineering Department, Paraje le Plano S/N, Valencia (Spain); Espinosa-Paredes, Gilberto [Area de ingeniera en Recursos Energeticos, Universidad Autonoma Metropolitana-Iztapalapa, Mexico city (Mexico)
2016-04-15
The aim of this paper is the study of instability state of boiling water reactors with a method based in largest Lyapunov exponents (LLEs). Detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the LLE. Lyapunov exponents quantify the exponential divergence of initially close state-space trajectories and estimate the amount of chaos in a system. This method was applied to a set of signals from several nuclear power plant (NPP) reactors under commercial operating conditions that experienced instabilities events, apparently each of a different nature. Laguna Verde and Forsmark NPPs with in-phase instabilities, and Cofrentes NPP with out-of-phases instability. This study presents the results of intrinsic instability in the boiling water reactors of three NPPs. In the analyzed cases the limit cycle was not reached, which implies that the point of equilibrium exerts influence and attraction on system evolution.
Romero, K M F; Parreira, J E; Souza, L A M; Wreszinski, W F
2007-01-01
We study and compare the information loss of a large class of gaussian bipartite systems. It includes the usual Caldeira-Leggett type model as well as Anosov models (parametric oscillators, the inverted oscillator environment, etc), which exhibit instability, one of the most important characteristics of chaotic systems. We establish a rigorous connection between the quantum Lyapunov exponents and coherence loss. We show that in the case of unstable environments, coherence loss is completely determined by the upper quantum Lyapunov exponent, a behavior dramatically different to that of the Caldeira-Leggett type model. For this class of systems we have been able to prove a long standing conjecture that for information loss the complexity of even a few (one) degrees of freedom is far more effective in destroying quantum coherence than stable many-body environments.
Xavier, J C; Strunz, W T; Beims, M W
2015-08-01
We consider the energy flow between a classical one-dimensional harmonic oscillator and a set of N two-dimensional chaotic oscillators, which represents the finite environment. Using linear response theory we obtain an analytical effective equation for the system harmonic oscillator, which includes a frequency dependent dissipation, a shift, and memory effects. The damping rate is expressed in terms of the environment mean Lyapunov exponent. A good agreement is shown by comparing theoretical and numerical results, even for environments with mixed (regular and chaotic) motion. Resonance between system and environment frequencies is shown to be more efficient to generate dissipation than larger mean Lyapunov exponents or a larger number of bath chaotic oscillators.
THE RELATION OF DIMENSION,ENTROPY AND LYAPUNOV EXPONENT IN RANDOM CASE
Institute of Scientific and Technical Information of China (English)
Yun Zhao
2008-01-01
We consider random systems generated by two-sided compositions of random surface diffeomorphisms,together with an ergodic Borel probability measure μ.Let D(μω)be its dimension of the sample measure,then we prove a formula relating D(μω)to the entropy and Lyapunov exponents of the random system,where D(μω)is dimHμω,-/dinBμω,or-/dimBμω.
Institute of Scientific and Technical Information of China (English)
ZHANGYing-xun; WUXi-zhen; LIZhu-xia
2003-01-01
The largest Lyapunov exponent (LLE) has been widely used to measure the levelof chaos of a system and was used to study the “solid-like” to “liquid-like” phase transition. Nuclear multifragmentation has been considered to be associated with a liquid-gas phase transition. Thus, in this paper we want to extend the study to the energy regime that encompasses fragmentation phenomena.
Lyapunov Exponent and Out-of-Time-Ordered Correlator's Growth Rate in a Chaotic System.
Rozenbaum, Efim B; Ganeshan, Sriram; Galitski, Victor
2017-02-24
It was proposed recently that the out-of-time-ordered four-point correlator (OTOC) may serve as a useful characteristic of quantum-chaotic behavior, because, in the semiclassical limit ℏ→0, its rate of exponential growth resembles the classical Lyapunov exponent. Here, we calculate the four-point correlator C(t) for the classical and quantum kicked rotor-a textbook driven chaotic system-and compare its growth rate at initial times with the standard definition of the classical Lyapunov exponent. Using both quantum and classical arguments, we show that the OTOC's growth rate and the Lyapunov exponent are, in general, distinct quantities, corresponding to the logarithm of the phase-space averaged divergence rate of classical trajectories and to the phase-space average of the logarithm, respectively. The difference appears to be more pronounced in the regime of low kicking strength K, where no classical chaos exists globally. In this case, the Lyapunov exponent quickly decreases as K→0, while the OTOC's growth rate may decrease much slower, showing a higher sensitivity to small chaotic islands in the phase space. We also show that the quantum correlator as a function of time exhibits a clear singularity at the Ehrenfest time t_{E}: transitioning from a time-independent value of t^{-1}lnC(t) at tt_{E}. We note that the underlying physics here is the same as in the theory of weak (dynamical) localization [Aleiner and Larkin, Phys. Rev. B 54, 14423 (1996)PRBMDO0163-182910.1103/PhysRevB.54.14423; Tian, Kamenev, and Larkin, Phys. Rev. Lett. 93, 124101 (2004)PRLTAO0031-900710.1103/PhysRevLett.93.124101] and is due to a delay in the onset of quantum interference effects, which occur sharply at a time of the order of the Ehrenfest time.
A Lower Bound on the Lyapunov Exponent for the Generalized Harper's Model
Jitomirskaya, Svetlana; Liu, Wencai
2017-02-01
We obtain a lower bound for the Lyapunov exponent of a family of discrete Schrödinger operators (Hu)_n=u_{n+1}+u_{n-1}+2a_1 cos 2π (θ +nα )u_n+2a_2 cos 4π (θ +nα )u_n, that incorporates both a_1 and a_2, thus going beyond the Herman's bound.
Perturbation theory for Lyapunov exponents of an Anderson model on a strip
Schulz-Baldes, H
2003-01-01
It is proven that the localization length of an Anderson model on a strip of width $L$ is bounded above by $L/\\lambda^2$ for small values of the coupling constant $\\lambda$ of the disordered potential. For this purpose, a new formalism is developed in order to calculate the bottom Lyapunov exponent associated with random products of large symplectic matrices perturbatively in the coupling constant of the randomness.
A Lower Bound on the Lyapunov Exponent for the Generalized Harper's Model
Jitomirskaya, Svetlana; Liu, Wencai
2016-05-01
We obtain a lower bound for the Lyapunov exponent of a family of discrete Schrödinger operators (Hu)_n=u_{n+1}+u_{n-1}+2a_1 cos 2π (θ +nα )u_n+2a_2 cos 4π (θ +nα )u_n , that incorporates both a_1 and a_2, thus going beyond the Herman's bound.
Lyapunov Exponent and Out-of-Time-Ordered Correlator's Growth Rate in a Chaotic System
Rozenbaum, Efim B.; Ganeshan, Sriram; Galitski, Victor
2017-02-01
It was proposed recently that the out-of-time-ordered four-point correlator (OTOC) may serve as a useful characteristic of quantum-chaotic behavior, because, in the semiclassical limit ℏ→0 , its rate of exponential growth resembles the classical Lyapunov exponent. Here, we calculate the four-point correlator C (t ) for the classical and quantum kicked rotor—a textbook driven chaotic system—and compare its growth rate at initial times with the standard definition of the classical Lyapunov exponent. Using both quantum and classical arguments, we show that the OTOC's growth rate and the Lyapunov exponent are, in general, distinct quantities, corresponding to the logarithm of the phase-space averaged divergence rate of classical trajectories and to the phase-space average of the logarithm, respectively. The difference appears to be more pronounced in the regime of low kicking strength K , where no classical chaos exists globally. In this case, the Lyapunov exponent quickly decreases as K →0 , while the OTOC's growth rate may decrease much slower, showing a higher sensitivity to small chaotic islands in the phase space. We also show that the quantum correlator as a function of time exhibits a clear singularity at the Ehrenfest time tE: transitioning from a time-independent value of t-1ln C (t ) at t tE. We note that the underlying physics here is the same as in the theory of weak (dynamical) localization [Aleiner and Larkin, Phys. Rev. B 54, 14423 (1996), 10.1103/PhysRevB.54.14423; Tian, Kamenev, and Larkin, Phys. Rev. Lett. 93, 124101 (2004), 10.1103/PhysRevLett.93.124101] and is due to a delay in the onset of quantum interference effects, which occur sharply at a time of the order of the Ehrenfest time.
Hramov, Alexander E; Maximenko, Vladimir A; Moskalenko, Olga I; 10.1063/1.4740063
2013-01-01
The spectrum of Lyapunov exponents is powerful tool for the analysis of the complex system dynamics. In the general framework of nonlinear dynamical systems a number of the numerical technics have been developed to obtain the spectrum of Lyapunov exponents for the complex temporal behavior of the systems with a few degree of freedom. Unfortunately, these methods can not apply directly to analysis of complex spatio-temporal dynamics in plasma devices which are characterized by the infinite phase space, since they are the spatially extended active media. In the present paper we propose the method for the calculation of the spectrum of the spatial Lyapunov exponents (SLEs) for the spatially extended beam-plasma systems. The calculation technique is applied to the analysis of chaotic spatio-temporal oscillations in three different beam-plasma model: (1) simple plasma Pierce diode, (2) coupled Pierce diodes, and (3) electron-wave system with backward electromagnetic wave. We find an excellent agreement between the...
Experimental Realization of a Multiscroll Chaotic Oscillator with Optimal Maximum Lyapunov Exponent
Directory of Open Access Journals (Sweden)
Esteban Tlelo-Cuautle
2014-01-01
Full Text Available Nowadays, different kinds of experimental realizations of chaotic oscillators have been already presented in the literature. However, those realizations do not consider the value of the maximum Lyapunov exponent, which gives a quantitative measure of the grade of unpredictability of chaotic systems. That way, this paper shows the experimental realization of an optimized multiscroll chaotic oscillator based on saturated function series. First, from the mathematical description having four coefficients (a, b, c, d1, an optimization evolutionary algorithm varies them to maximize the value of the positive Lyapunov exponent. Second, a realization of those optimized coefficients using operational amplifiers is given. Herein a, b, c, d1 are implemented with precision potentiometers to tune up to four decimals of the coefficients having the range between 0.0001 and 1.0000. Finally, experimental results of the phase-space portraits for generating from 2 to 10 scrolls are listed to show that their associated value for the optimal maximum Lyapunov exponent increases by increasing the number of scrolls, thus guaranteeing a more complex chaotic behavior.
Experimental realization of a multiscroll chaotic oscillator with optimal maximum Lyapunov exponent.
Tlelo-Cuautle, Esteban; Pano-Azucena, Ana Dalia; Carbajal-Gomez, Victor Hugo; Sanchez-Sanchez, Mauro
2014-01-01
Nowadays, different kinds of experimental realizations of chaotic oscillators have been already presented in the literature. However, those realizations do not consider the value of the maximum Lyapunov exponent, which gives a quantitative measure of the grade of unpredictability of chaotic systems. That way, this paper shows the experimental realization of an optimized multiscroll chaotic oscillator based on saturated function series. First, from the mathematical description having four coefficients (a, b, c, d1 ), an optimization evolutionary algorithm varies them to maximize the value of the positive Lyapunov exponent. Second, a realization of those optimized coefficients using operational amplifiers is given. Herein a, b, c, d1 are implemented with precision potentiometers to tune up to four decimals of the coefficients having the range between 0.0001 and 1.0000. Finally, experimental results of the phase-space portraits for generating from 2 to 10 scrolls are listed to show that their associated value for the optimal maximum Lyapunov exponent increases by increasing the number of scrolls, thus guaranteeing a more complex chaotic behavior.
Effective power-law dependence of Lyapunov exponents on the central mass in galaxies
Delis, N; Kalapotharakos, C
2015-01-01
Using both numerical and analytical approaches, we demonstrate the existence of an effective power-law relation $L\\propto m^p$ between the mean Lyapunov exponent $L$ of stellar orbits chaotically scattered by a supermassive black hole in the center of a galaxy and the mass parameter $m$, i.e. ratio of the mass of the black hole over the mass of the galaxy. The exponent $p$ is found numerically to obtain values in the range $p \\approx 0.3$--$0.5$. We propose a theoretical interpretation of these exponents, based on estimates of local `stretching numbers', i.e. local Lyapunov exponents at successive transits of the orbits through the black hole's sphere of influence. We thus predict $p=2/3-q$ with $q\\approx 0.1$--$0.2$. Our basic model refers to elliptical galaxy models with a central core. However, we find numerically that an effective power law scaling of $L$ with $m$ holds also in models with central cusp, beyond a mass scale up to which chaos is dominated by the influence of the cusp itself. We finally show...
Presence of nonlinearity in intracranial EEG recordings: detected by Lyapunov exponents
Liu, Chang-Chia; Shiau, Deng-Shan; Chaovalitwongse, W. Art; Pardalos, Panos M.; Sackellares, J. C.
2007-11-01
In this communication, we performed nonlinearity analysis in the EEG signals recorded from patients with temporal lobe epilepsy (TLE). The largest Lyapunov exponent (Lmax) and phase randomization surrogate data technique were employed to form the statistical test. EEG recordings were acquired invasively from three patients in six brain regions (left and right temporal depth, sub-temporal and orbitofrontal) with 28-32 depth electrodes placed in depth and subdural of the brain. All three patients in this study have unilateral epileptic focus region on the right hippocampus(RH). Nonlinearity was detected by comparing the Lmax profiles of the EEG recordings to its surrogates. The nonlinearity was seen in all different states of the patient with the highest found in post-ictal state. Further our results for all patients exhibited higher degree of differences, quantified by paired t-test, in Lmax values between original and its surrogate from EEG signals recorded from epileptic focus regions. The results of this study demonstrated the Lmax is capable to capture spatio-temporal dynamics that may not be able to detect by linear measurements in the intracranial EEG recordings.
A perturbation method to the tent map based on Lyapunov exponent and its application
Cao, Lv-Chen; Luo, Yu-Ling; Qiu, Sen-Hui; Liu, Jun-Xiu
2015-10-01
Perturbation imposed on a chaos system is an effective way to maintain its chaotic features. A novel parameter perturbation method for the tent map based on the Lyapunov exponent is proposed in this paper. The pseudo-random sequence generated by the tent map is sent to another chaos function — the Chebyshev map for the post processing. If the output value of the Chebyshev map falls into a certain range, it will be sent back to replace the parameter of the tent map. As a result, the parameter of the tent map keeps changing dynamically. The statistical analysis and experimental results prove that the disturbed tent map has a highly random distribution and achieves good cryptographic properties of a pseudo-random sequence. As a result, it weakens the phenomenon of strong correlation caused by the finite precision and effectively compensates for the digital chaos system dynamics degradation. Project supported by the Guangxi Provincial Natural Science Foundation, China (Grant No. 2014GXNSFBA118271), the Research Project of Guangxi University, China (Grant No. ZD2014022), the Fund from Guangxi Provincial Key Laboratory of Multi-source Information Mining & Security, China (Grant No. MIMS14-04), the Fund from the Guangxi Provincial Key Laboratory of Wireless Wideband Communication & Signal Processing, China (Grant No. GXKL0614205), the Education Development Foundation and the Doctoral Research Foundation of Guangxi Normal University, the State Scholarship Fund of China Scholarship Council (Grant No. [2014]3012), and the Innovation Project of Guangxi Graduate Education, China (Grant No. YCSZ2015102).
Designing Hyperchaotic Cat Maps With Any Desired Number of Positive Lyapunov Exponents.
Hua, Zhongyun; Yi, Shuang; Zhou, Yicong; Li, Chengqing; Wu, Yue
2017-01-04
Generating chaotic maps with expected dynamics of users is a challenging topic. Utilizing the inherent relation between the Lyapunov exponents (LEs) of the Cat map and its associated Cat matrix, this paper proposes a simple but efficient method to construct an n-dimensional (n-D) hyperchaotic Cat map (HCM) with any desired number of positive LEs. The method first generates two basic n-D Cat matrices iteratively and then constructs the final n-D Cat matrix by performing similarity transformation on one basic n-D Cat matrix by the other. Given any number of positive LEs, it can generate an n-D HCM with desired hyperchaotic complexity. Two illustrative examples of n-D HCMs were constructed to show the effectiveness of the proposed method, and to verify the inherent relation between the LEs and Cat matrix. Theoretical analysis proves that the parameter space of the generated HCM is very large. Performance evaluations show that, compared with existing methods, the proposed method can construct n-D HCMs with lower computation complexity and their outputs demonstrate strong randomness and complex ergodicity.
Valenza, Gaetano; Allegrini, Paolo; Lanatà, Antonio; Scilingo, Enzo Pasquale
2012-01-01
In this work we characterized the non-linear complexity of Heart Rate Variability (HRV) in short time series. The complexity of HRV signal was evaluated during emotional visual elicitation by using Dominant Lyapunov Exponents (DLEs) and Approximate Entropy (ApEn). We adopted a simplified model of emotion derived from the Circumplex Model of Affects (CMAs), in which emotional mechanisms are conceptualized in two dimensions by the terms of valence and arousal. Following CMA model, a set of standardized visual stimuli in terms of arousal and valence gathered from the International Affective Picture System (IAPS) was administered to a group of 35 healthy volunteers. Experimental session consisted of eight sessions alternating neutral images with high arousal content images. Several works can be found in the literature showing a chaotic dynamics of HRV during rest or relax conditions. The outcomes of this work showed a clear switching mechanism between regular and chaotic dynamics when switching from neutral to arousal elicitation. Accordingly, the mean ApEn decreased with statistical significance during arousal elicitation and the DLE became negative. Results showed a clear distinction between the neutral and the arousal elicitation and could be profitably exploited to improve the accuracy of emotion recognition systems based on HRV time series analysis. PMID:22393320
Valenza, Gaetano; Allegrini, Paolo; Lanatà, Antonio; Scilingo, Enzo Pasquale
2012-01-01
In this work we characterized the non-linear complexity of Heart Rate Variability (HRV) in short time series. The complexity of HRV signal was evaluated during emotional visual elicitation by using Dominant Lyapunov Exponents (DLEs) and Approximate Entropy (ApEn). We adopted a simplified model of emotion derived from the Circumplex Model of Affects (CMAs), in which emotional mechanisms are conceptualized in two dimensions by the terms of valence and arousal. Following CMA model, a set of standardized visual stimuli in terms of arousal and valence gathered from the International Affective Picture System (IAPS) was administered to a group of 35 healthy volunteers. Experimental session consisted of eight sessions alternating neutral images with high arousal content images. Several works can be found in the literature showing a chaotic dynamics of HRV during rest or relax conditions. The outcomes of this work showed a clear switching mechanism between regular and chaotic dynamics when switching from neutral to arousal elicitation. Accordingly, the mean ApEn decreased with statistical significance during arousal elicitation and the DLE became negative. Results showed a clear distinction between the neutral and the arousal elicitation and could be profitably exploited to improve the accuracy of emotion recognition systems based on HRV time series analysis.
Valenza, Gaetano; Citi, Luca; Barbieri, Riccardo
2014-01-01
Measures of nonlinearity and complexity, and in particular the study of Lyapunov exponents, have been increasingly used to characterize dynamical properties of a wide range of biological nonlinear systems, including cardiovascular control. In this work, we present a novel methodology able to effectively estimate the Lyapunov spectrum of a series of stochastic events in an instantaneous fashion. The paradigm relies on a novel point-process high-order nonlinear model of the event series dynamics. The long-term information is taken into account by expanding the linear, quadratic, and cubic Wiener-Volterra kernels with the orthonormal Laguerre basis functions. Applications to synthetic data such as the Hénon map and Rössler attractor, as well as two experimental heartbeat interval datasets (i.e., healthy subjects undergoing postural changes and patients with severe cardiac heart failure), focus on estimation and tracking of the Instantaneous Dominant Lyapunov Exponent (IDLE). The novel cardiovascular assessment demonstrates that our method is able to effectively and instantaneously track the nonlinear autonomic control dynamics, allowing for complexity variability estimations.
Control of chaos in permanent magnet synchronous motor by using optimal Lyapunov exponents placement
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Ataei, Mohammad, E-mail: ataei@eng.ui.ac.i [Department of Electrical Engineering, Faculty of Engineering, University of Isfahan, Hezar-Jerib St., Postal Code 8174673441, Isfahan (Iran, Islamic Republic of); Kiyoumarsi, Arash, E-mail: kiyoumarsi@eng.ui.ac.i [Department of Electrical Engineering, Faculty of Engineering, University of Isfahan, Hezar-Jerib St., Postal Code 8174673441, Isfahan (Iran, Islamic Republic of); Ghorbani, Behzad, E-mail: behzad.ghorbani63@gmail.co [Department of Control Engineering, Najafabad Azad University, Najafabad, Isfahan (Iran, Islamic Republic of)
2010-09-13
Permanent Magnet Synchronous Motor (PMSM) experiences chaotic behavior for a certain range of its parameters. In this case, since the performance of the PMSM degrades, the chaos should be eliminated. In this Letter, the control of the undesirable chaos in PMSM using Lyapunov exponents (LEs) placement is proposed that is also improved by choosing optimal locations of the LEs in the sense of predefined cost function. Moreover, in order to provide the physical realization of the method, nonlinear parameter estimator for the system is suggested. Finally, to show the effectiveness of the proposed methodology, the simulation results for applying this control strategy are provided.
A unified approach to finite-time hyperbolicity which extends finite-time Lyapunov exponents
Doan, T. S.; Karrasch, D.; Nguyen, T. Y.; Siegmund, S.
A hyperbolicity notion for linear differential equations x˙=A(t)x, t∈[t-,t+], is defined which unifies different existing notions like finite-time Lyapunov exponents (Haller, 2001, [13], Shadden et al., 2005, [24]), uniform or M-hyperbolicity (Haller, 2001, [13], Berger et al., 2009, [6]) and (t-,(t+-t-))-dichotomy (Rasmussen, 2010, [21]). Its relation to the dichotomy spectrum (Sacker and Sell, 1978, [23], Siegmund, 2002, [26]), D-hyperbolicity (Berger et al., 2009, [6]) and real parts of the eigenvalues (in case A is constant) is described. We prove a spectral theorem and provide an approximation result for the spectral intervals.
Effective Power-Law Dependence of Lyapunov Exponents on the Central Mass in Galaxies
Delis, N.; Efthymiopoulos, C.; Kalapotharakos, C.
2015-01-01
Using both numerical and analytical approaches, we demonstrate the existence of an effective power-law relation L alpha m(sup p) between themean Lyapunov exponent L of stellar orbits chaotically scattered by a supermassive black hole (BH) in the centre of a galaxy and the mass parameter m, i.e. ratio of the mass of the BH over the mass of the galaxy. The exponent p is found numerically to obtain values in the range p approximately equals 0.3-0.5. We propose a theoretical interpretation of these exponents, based on estimates of local 'stretching numbers', i.e. local Lyapunov exponents at successive transits of the orbits through the BH's sphere of influence. We thus predict p = 2/3 - q with q approximately equaling 0.1-0.2. Our basic model refers to elliptical galaxy models with a central core. However, we find numerically that an effective power-law scaling of L with m holds also in models with central cusp, beyond a mass scale up to which chaos is dominated by the influence of the cusp itself. We finally show numerically that an analogous law exists also in disc galaxies with rotating bars. In the latter case, chaotic scattering by the BH affects mainly populations of thick tube-like orbits surrounding some low-order branches of the x(sub 1) family of periodic orbits, as well as its bifurcations at low-order resonances, mainly the inner Lindblad resonance and the 4/1 resonance. Implications of the correlations between L and m to determining the rate of secular evolution of galaxies are discussed.
Effective Power-Law Dependence of Lyapunov Exponents on the Central Mass in Galaxies
Delis, N.; Efthymiopoulos, C.; Kalapotharakos, C.
2015-01-01
Using both numerical and analytical approaches, we demonstrate the existence of an effective power-law relation L alpha m(sup p) between themean Lyapunov exponent L of stellar orbits chaotically scattered by a supermassive black hole (BH) in the centre of a galaxy and the mass parameter m, i.e. ratio of the mass of the BH over the mass of the galaxy. The exponent p is found numerically to obtain values in the range p approximately equals 0.3-0.5. We propose a theoretical interpretation of these exponents, based on estimates of local 'stretching numbers', i.e. local Lyapunov exponents at successive transits of the orbits through the BH's sphere of influence. We thus predict p = 2/3 - q with q approximately equaling 0.1-0.2. Our basic model refers to elliptical galaxy models with a central core. However, we find numerically that an effective power-law scaling of L with m holds also in models with central cusp, beyond a mass scale up to which chaos is dominated by the influence of the cusp itself. We finally show numerically that an analogous law exists also in disc galaxies with rotating bars. In the latter case, chaotic scattering by the BH affects mainly populations of thick tube-like orbits surrounding some low-order branches of the x(sub 1) family of periodic orbits, as well as its bifurcations at low-order resonances, mainly the inner Lindblad resonance and the 4/1 resonance. Implications of the correlations between L and m to determining the rate of secular evolution of galaxies are discussed.
Lyapunov exponents from CHUA's circuit time series using artificial neural networks
Gonzalez, J. Jesus; Espinosa, Ismael E.; Fuentes, Alberto M.
1995-01-01
In this paper we present the general problem of identifying if a nonlinear dynamic system has a chaotic behavior. If the answer is positive the system will be sensitive to small perturbations in the initial conditions which will imply that there is a chaotic attractor in its state space. A particular problem would be that of identifying a chaotic oscillator. We present an example of three well known different chaotic oscillators where we have knowledge of the equations that govern the dynamical systems and from there we can obtain the corresponding time series. In a similar example we assume that we only know the time series and, finally, in another example we have to take measurements in the Chua's circuit to obtain sample points of the time series. With the knowledge about the time series the phase plane portraits are plotted and from them, by visual inspection, it is concluded whether or not the system is chaotic. This method has the problem of uncertainty and subjectivity and for that reason a different approach is needed. A quantitative approach is the computation of the Lyapunov exponents. We describe several methods for obtaining them and apply a little known method of artificial neural networks to the different examples mentioned above. We end the paper discussing the importance of the Lyapunov exponents in the interpretation of the dynamic behavior of biological neurons and biological neural networks.
Babaee, Hessam; Farazmand, Mohamad; Haller, George; Sapsis, Themistoklis P.
2017-06-01
High-dimensional chaotic dynamical systems can exhibit strongly transient features. These are often associated with instabilities that have a finite-time duration. Because of the finite-time character of these transient events, their detection through infinite-time methods, e.g., long term averages, Lyapunov exponents or information about the statistical steady-state, is not possible. Here, we utilize a recently developed framework, the Optimally Time-Dependent (OTD) modes, to extract a time-dependent subspace that spans the modes associated with transient features associated with finite-time instabilities. As the main result, we prove that the OTD modes, under appropriate conditions, converge exponentially fast to the eigendirections of the Cauchy-Green tensor associated with the most intense finite-time instabilities. Based on this observation, we develop a reduced-order method for the computation of finite-time Lyapunov exponents (FTLE) and vectors. In high-dimensional systems, the computational cost of the reduced-order method is orders of magnitude lower than the full FTLE computation. We demonstrate the validity of the theoretical findings on two numerical examples.
Babaee, Hessam; Farazmand, Mohamad; Haller, George; Sapsis, Themistoklis P
2017-06-01
High-dimensional chaotic dynamical systems can exhibit strongly transient features. These are often associated with instabilities that have a finite-time duration. Because of the finite-time character of these transient events, their detection through infinite-time methods, e.g., long term averages, Lyapunov exponents or information about the statistical steady-state, is not possible. Here, we utilize a recently developed framework, the Optimally Time-Dependent (OTD) modes, to extract a time-dependent subspace that spans the modes associated with transient features associated with finite-time instabilities. As the main result, we prove that the OTD modes, under appropriate conditions, converge exponentially fast to the eigendirections of the Cauchy-Green tensor associated with the most intense finite-time instabilities. Based on this observation, we develop a reduced-order method for the computation of finite-time Lyapunov exponents (FTLE) and vectors. In high-dimensional systems, the computational cost of the reduced-order method is orders of magnitude lower than the full FTLE computation. We demonstrate the validity of the theoretical findings on two numerical examples.
Classification of Heart Rate Signals during Meditation using Lyapunov Exponents and Entropy
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Ateke Goshvarpour
2012-03-01
Full Text Available Meditation is commonly perceived as an alternative medicine method of psychological diseases management tool that assist in alleviating depression and anxiety disorders. The purpose of this study is to evaluate the accuracy of different classifiers on the heart rate signals in a specific psychological state. Two types of heart rate time series (before, and during meditation of 25 healthy women are collected in the meditation clinic in Mashhad. Nonlinear features such as Lyapunov Exponents and Entropy were extracted. To evaluate performance of the classifiers, the classification accuracies and mean square error (MSE of the classifiers were examined. Different classifiers were tested and the studies confirmed that for the heart rate signals, Quadratic classifier trained on Lyapunov Exponents and Entropy results in higher classification accuracy. The classification accuracy of the Quadratic classifier is 92.31%. However, the accuracies of Fisher and k-Nearest Neighbor (k-NN classifiers are encouraging. The classification results demonstrate that the dynamical measures are useful parameters which contain comprehensive information about signals and the Quadratic classifier using nonlinear features can be useful in analyzing the heart rate signals in a specific psychological state.
Determining the sub-Lyapunov exponent of delay systems from time series.
Jüngling, Thomas; Soriano, Miguel C; Fischer, Ingo
2015-06-01
For delay systems the sign of the sub-Lyapunov exponent (sub-LE) determines key dynamical properties. This includes the properties of strong and weak chaos and of consistency. Here we present a robust algorithm based on reconstruction of the local linearized equations of motion, which allows for calculating the sub-LE from time series. The algorithm is inspired by a method introduced by Pyragas for a nondelayed drive-response scheme [K. Pyragas, Phys. Rev. E 56, 5183 (1997)]. In the presented extension to delay systems, the delayed feedback takes over the role of the drive, whereas the response of the low-dimensional node leads to the sub-Lyapunov exponent. Our method is based on a low-dimensional representation of the delay system. We introduce the basic algorithm for a discrete scalar map, extend the concept to scalar continuous delay systems, and give an outlook to the case of a full vector-state system, from which only a scalar observable is recorded.
Liu, Xiuling; Du, Haiman; Wang, Guanglei; Zhou, Suiping; Zhang, Hong
2015-10-01
Premature ventricular contraction (PVC) is a common type of abnormal heartbeat. Without early diagnosis and proper treatment, PVC may result in serious harms. Diagnosis of PVC is of great importance in goal-directed treatment and preoperation prognosis. This paper proposes a novel diagnostic method for PVC based on Lyapunov exponents of electrocardiogram (ECG) beats. The methodology consists of preprocessing, feature extraction and classification integrated into the system. PVC beats can be classified and differentiated from other types of abnormal heartbeats by analyzing Lyapunov exponents and training a learning vector quantization (LVQ) neural network. Our algorithm can obtain a good diagnostic result with little features by using single lead ECG data. The sensitivity, positive predictability, and the overall accuracy of the automatic diagnosis of PVC is 90.26%, 92.31%, and 98.90%, respectively. The effectiveness of the new method is validated through extensive tests using data from MIT-BIH database. The experimental results show that the proposed method is efficient and robust.
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Pablo César Rodríguez Gómez
2017-05-01
Full Text Available Context: Because feedback systems are very common and widely used, studies of the structural characteristics under which chaotic behavior is generated have been developed. These can be separated into a nonlinear system and a linear system at least of the third order. Methods such as the descriptive function have been used for analysis. Method: A feedback system is proposed comprising a linear system, a nonlinear system and a delay block, in order to assess his behavior using Lyapunov exponents. It is evaluated with three different linear systems, different delay values and different values for parameters of nonlinear characteristic, aiming to reach chaotic behavior. Results: One hundred experiments were carried out for each of the three linear systems, by changing the value of some parameters, assessing their influence on the dynamics of the system. Contour plots that relate these parameters to the Largest Lyapunov exponent were obtained and analyzed. Conclusions: In spite non-linearity is a condition for the existence of chaos, this does not imply that any nonlinear characteristic generates a chaotic system, it is reflected by the contour plots showing the transitions between chaotic and no chaotic behavior of the feedback system. Language: English
Institute of Scientific and Technical Information of China (English)
Li Qun-Hong; Tan Jie-Yan
2011-01-01
A two-degree-of-freedom vibro-impact system having symmetrical rigid stops and subjected to periodic excitation is investigated in this paper. By introducing local maps between different stages of motion in the whole impact process,the Poincar6 map of the system is constructed. Using the Poincare map and the Gram-Schmidt orthonormalization, a method of calculating the spectrum of Lyapunov exponents of the above vibro-impact system is presented. Then the phase portraits of periodic and chaotic attractors for the system and the corresponding convergence diagrams of the spectrum of Lyapunov exponents are given out through the numerical simulations. To further identify the validity of the aforementioned computation method, the bifurcation diagram of the system with respect to the bifurcation parameter and the corresponding largest Lyapunov exponents are shown.
Computing Finite-Time Lyapunov Exponents with Optimally Time Dependent Reduction
Babaee, Hessam; Farazmand, Mohammad; Sapsis, Themis; Haller, George
2016-11-01
We present a method to compute Finite-Time Lyapunov Exponents (FTLE) of a dynamical system using Optimally Time-Dependent (OTD) reduction recently introduced by H. Babaee and T. P. Sapsis. The OTD modes are a set of finite-dimensional, time-dependent, orthonormal basis {ui (x , t) } |i=1N that capture the directions associated with transient instabilities. The evolution equation of the OTD modes is derived from a minimization principle that optimally approximates the most unstable directions over finite times. To compute the FTLE, we evolve a single OTD mode along with the nonlinear dynamics. We approximate the FTLE from the reduced system obtained from projecting the instantaneous linearized dynamics onto the OTD mode. This results in a significant reduction in the computational cost compared to conventional methods for computing FTLE. We demonstrate the efficiency of our method for double Gyre and ABC flows. ARO project 66710-EG-YIP.
Ota, Y; Ota, Yukihiro; Ohba, Ichiro
2003-01-01
We discuss the quantum--classical correspondence in a specific dissipative chaotic system, Duffing oscillator. We quantize it on the basis of quantum state diffusion (QSD) which is a certain formulation for open quantum systems and an effective tool for analyzing complex problems numerically. We consider a sensitivity to initial conditions, `` pseudo-Lyapunov exponent '', and investigate it in detail, varying Planck constant effectively. We show that in a dissipative system there exists a certain critical stage in which the crossover from classical to quantum behavior occurs. Furthermore, we show that an effect of dissipation suppresses the occurrence of chaos in the quantum region, while it, combined with the periodic external force, plays a crucial role in the chaotic behaviors of classical system.
Mean field theory for Lyapunov exponents and KS entropy in Lorentz lattice gases
Ernst, M H; Nix, R; Jacobs, D; Ernst, M H; Dorfman, J R; Nix, R; Jacobs, D
1994-01-01
automata lattice gases are useful systems for systematically exploring the connections between non-equilibrium statistical mechanics and dynamical systems theory. Here the chaotic properties of a Lorentz lattice gas are studied analytically and by means of computer simulations. The escape-rates, Lyapunov exponents, and KS entropies are estimated for a one- dimensional example using a mean field theory. The results are compared with simulations for a range of densities and scattering parameters of the lattice gas. The computer results show a distribution of values for the dynamical quantities with average values that are in good agreement with the mean field theory and consistent with the escape-rate formalism for the coefficient of diffusion.
On the Validity of the Conjugate Pairing Rule for Lyapunov Exponents
Bonetto, F; Pugh, C
1998-01-01
For Hamiltonian systems subject to an external potential, which in the presence of a thermostat will reach a nonequilibrium stationary state, Dettmann and Morriss proved a strong conjugate pairing rule (SCPR) for pairs of Lyapunov exponents in the case of isokinetic (IK) stationary states which have a given kinetic energy. This SCPR holds for all initial phases of the system, all times t and all numbers of particles N. This proof was generalized by Wojtkovski and Liverani to include hard interparticle potentials. A geometrical reformulation of those results is presented. The present paper proves numerically, using periodic orbits for the Lorentz gas, that SCPR cannot hold for isoenergetic (IE) stationary states, which have a given total internal energy. In that case strong evidence is obtained for CPR to hold for large N and t, where it can be conjectured that the larger N, the smaller t will be. This suffices for statistical mechanics.
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Look, Nicole [Department of Applied Mathematics, University of Colorado Boulder, Boulder, Colorado 80309 (United States); Arellano, Christopher J.; Grabowski, Alena M.; Kram, Rodger [Department of Integrative Physiology, University of Colorado Boulder, Boulder, Colorado 80309 (United States); McDermott, William J. [The Orthopedic Specialty Hospital, Murray, Utah 84107 (United States); Bradley, Elizabeth [Department of Computer Science, University of Colorado Boulder, Boulder, Colorado 80309, USA and Santa Fe Institute, Santa Fe, New Mexico 87501 (United States)
2013-12-15
In this paper, we study dynamic stability during running, focusing on the effects of speed, and the use of a leg prosthesis. We compute and compare the maximal Lyapunov exponents of kinematic time-series data from subjects with and without unilateral transtibial amputations running at a wide range of speeds. We find that the dynamics of the affected leg with the running-specific prosthesis are less stable than the dynamics of the unaffected leg and also less stable than the biological legs of the non-amputee runners. Surprisingly, we find that the center-of-mass dynamics of runners with two intact biological legs are slightly less stable than those of runners with amputations. Our results suggest that while leg asymmetries may be associated with instability, runners may compensate for this effect by increased control of their center-of-mass dynamics.
Ghayoumi Zadeh, Hossein; Haddadnia, Javad; Montazeri, Alimohammad
2016-05-01
The segmentation of cancerous areas in breast images is important for the early detection of disease. Thermal imaging has advantages, such as being non-invasive, non-radiation, passive, quick, painless, inexpensive, and non-contact. Imaging technique is the focus of this research. The proposed model in this paper is a combination of surf and corners that are very resistant. Obtained features are resistant to changes in rotation and revolution then with the help of active contours, this feature has been used for segmenting cancerous areas. Comparing the obtained results from the proposed method and mammogram show that proposed method is Accurate and appropriate. Benign and malignance of segmented areas are detected by Lyapunov exponent. Values obtained include TP=91.31%, FN=8.69%, FP=7.26%. The proposed method can classify those abnormally segmented areas of the breast, to the Benign and malignant cancer.
Detection of the onset of numerical chaotic instabilities by lyapunov exponents
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Alicia Serfaty De Markus
2001-01-01
Full Text Available It is commonly found in the fixed-step numerical integration of nonlinear differential equations that the size of the integration step is opposite related to the numerical stability of the scheme and to the speed of computation. We present a procedure that establishes a criterion to select the largest possible step size before the onset of chaotic numerical instabilities, based upon the observation that computational chaos does not occur in a smooth, continuous way, but rather abruptly, as detected by examining the largest Lyapunov exponent as a function of the step size. For completeness, examination of the bifurcation diagrams with the step reveals the complexity imposed by the algorithmic discretization, showing the robustness of a scheme to numerical instabilities, illustrated here for explicit and implicit Euler schemes. An example of numerical suppression of chaos is also provided.
A method to calculate finite-time Lyapunov exponents for inertial particles in incompressible flows
Garaboa-Paz, D.; Pérez-Muñuzuri, V.
2015-10-01
The present study aims to improve the calculus of finite-time Lyapunov exponents (FTLEs) applied to describe the transport of inertial particles in a fluid flow. To this aim, the deformation tensor is modified to take into account that the stretching rate between particles separated by a certain distance is influenced by the initial velocity of the particles. Thus, the inertial FTLEs (iFTLEs) are defined in terms of the maximum stretching between infinitesimally close trajectories that have different initial velocities. The advantages of this improvement, if compared to the standard method (Shadden et al., 2005), are discussed for the double-gyre flow and the meandering jet flow. The new method allows one to identify the initial velocity that inertial particles must have in order to maximize their dispersion.
Refining finite-time Lyapunov exponent ridges and the challenges of classifying them.
Allshouse, Michael R; Peacock, Thomas
2015-08-01
While more rigorous and sophisticated methods for identifying Lagrangian based coherent structures exist, the finite-time Lyapunov exponent (FTLE) field remains a straightforward and popular method for gaining some insight into transport by complex, time-dependent two-dimensional flows. In light of its enduring appeal, and in support of good practice, we begin by investigating the effects of discretization and noise on two numerical approaches for calculating the FTLE field. A practical method to extract and refine FTLE ridges in two-dimensional flows, which builds on previous methods, is then presented. Seeking to better ascertain the role of a FTLE ridge in flow transport, we adapt an existing classification scheme and provide a thorough treatment of the challenges of classifying the types of deformation represented by a FTLE ridge. As a practical demonstration, the methods are applied to an ocean surface velocity field data set generated by a numerical model.
Positivity of Lyapunov exponents for Anderson-type models on two coupled strings
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Hakim Boumaza
2007-03-01
Full Text Available We study two models of Anderson-type random operators on two deterministically coupled continuous strings. Each model is associated with independent, identically distributed four-by-four symplectic transfer matrices, which describe the asymptotics of solutions. In each case we use a criterion by Gol'dsheid and Margulis (i.e. Zariski denseness of the group generated by the transfer matrices in the group of symplectic matrices to prove positivity of both leading Lyapunov exponents for most energies. In each case this implies almost sure absence of absolutely continuous spectrum (at all energies in the first model and for sufficiently large energies in the second model. The methods used allow for singularly distributed random parameters, including Bernoulli distributions.
Quenched Lyapunov exponent for the parabolic Anderson model in a dynamic random environment
Gärtner, Jürgen; Maillard, Grégory
2010-01-01
We continue our study of the parabolic Anderson equation $\\partial u/\\partial t = \\kappa\\Delta u + \\gamma\\xi u$ for the space-time field $u\\colon\\,\\Z^d\\times [0,\\infty)\\to\\R$, where $\\kappa \\in [0,\\infty)$ is the diffusion constant, $\\Delta$ is the discrete Laplacian, $\\gamma\\in (0,\\infty)$ is the coupling constant, and $\\xi\\colon\\,\\Z^d\\times [0,\\infty)\\to\\R$ is a space-time random environment that drives the equation. The solution of this equation describes the evolution of a "reactant" $u$ under the influence of a "catalyst" $\\xi$, both living on $\\Z^d$. In earlier work we considered three choices for $\\xi$: independent simple random walks, the symmetric exclusion process, and the symmetric voter model, all in equilibrium at a given density. We analyzed the \\emph{annealed} Lyapunov exponents, i.e., the exponential growth rates of the successive moments of $u$ w.r.t.\\ $\\xi$, and showed that these exponents display an interesting dependence on the diffusion constant $\\kappa$, with qualitatively different beha...
The Comparison for Lyapunov Exponents Calculation Methods%关于Lyapunov指数计算方法的比较
Institute of Scientific and Technical Information of China (English)
张海龙; 闵富红; 王恩荣
2012-01-01
针对常用的几种Lyapunov指数数值计算方法,即定义法、正交法、wolf法和小数据量法,以典型的Lorenz系统为例,分别计算Lorenz混沌吸引子的Lyapunov指数谱或者最大Lyapunov指数,比较各种方法的计算精度、计算复杂度,并且对含噪声的混沌时间序列给出Lyapunov指数计算结果,比较各种抗干扰能力.给出了不同计算方法的性能差异、适用场合和选择依据.%In this paper,the several computational methods of Lyapunov exponents are compared,i.e.,the definition method,the orthogonal method,the wolf method and the small data sets.The Lyapunov exponent power and the max-Lyapunov exponent are computed through the above methods for Lorenz system.From the results,the accuracies and the complexity of the above methods are investigated.Furthermore,the max-Lyapunov exponents are also calculated for the chaotic time series including the noise.Finally,numerical results demonstrate that the performances of different computational methods have differences,and some summaries will be presented.
Moura, R. C.; Silva, A. F. C.; Bigarella, E. D. V.; Fazenda, A. L.; Ortega, M. A.
2016-08-01
This paper proposes two important improvements to shock-capturing strategies using a discontinuous Galerkin scheme, namely, accurate shock identification via finite-time Lyapunov exponent (FTLE) operators and efficient shock treatment through a point-implicit discretization of a PDE-based artificial viscosity technique. The advocated approach is based on the FTLE operator, originally developed in the context of dynamical systems theory to identify certain types of coherent structures in a flow. We propose the application of FTLEs in the detection of shock waves and demonstrate the operator's ability to identify strong and weak shocks equally well. The detection algorithm is coupled with a mesh refinement procedure and applied to transonic and supersonic flows. While the proposed strategy can be used potentially with any numerical method, a high-order discontinuous Galerkin solver is used in this study. In this context, two artificial viscosity approaches are employed to regularize the solution near shocks: an element-wise constant viscosity technique and a PDE-based smooth viscosity model. As the latter approach is more sophisticated and preferable for complex problems, a point-implicit discretization in time is proposed to reduce the extra stiffness introduced by the PDE-based technique, making it more competitive in terms of computational cost.
Gauss map and Lyapunov exponents of interacting particles in a billiard
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Manchein, C. [Departamento de Fisica, Universidade Federal do Parana, 81531-990 Curitiba, PR (Brazil); Beims, M.W. [Departamento de Fisica, Universidade Federal do Parana, 81531-990 Curitiba, PR (Brazil); Max Planck Institute for the Physics of Complex Systems, Noethnitzer Strasse 38, D-01187 Dresden (Germany)], E-mail: mbeims@fisica.ufpr.br
2009-03-15
We show that the Lyapunov exponent (LE) of periodic orbits with Lebesgue measure zero from the Gauss map can be used to determine the main qualitative behavior of the LE of a Hamiltonian system. The Hamiltonian system is a one-dimensional box with two particles interacting via a Yukawa potential and does not possess Kolmogorov-Arnold-Moser (KAM) curves. In our case the Gauss map is applied to the mass ratio ({gamma} = m{sub 2}/m{sub 1}) between particles. Besides the main qualitative behavior, some unexpected peaks in the {gamma} dependence of the mean LE and the appearance of 'stickness' in phase space can also be understand via LE from the Gauss map. This shows a nice example of the relation between the 'instability' of the continued fraction representation of a number with the stability of non-periodic curves (no KAM curves) from the physical model. Our results also confirm the intuition that pseudo-integrable systems with more complicated invariant surfaces of the flow (higher genus) should be more unstable under perturbation.
Influence of finite-time Lyapunov exponents on winter precipitation over the Iberian Peninsula
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D. Garaboa-Paz
2017-05-01
Full Text Available Seasonal forecasts have improved during the last decades, mostly due to an increase in understanding of the coupled ocean–atmosphere dynamics, and the development of models able to predict the atmosphere variability. Correlations between different teleconnection patterns and severe weather in different parts of the world are constantly evolving and changing. This paper evaluates the connection between winter precipitation over the Iberian Peninsula and the large-scale tropospheric mixing over the eastern Atlantic Ocean. Finite-time Lyapunov exponents (FTLEs have been calculated from 1979 to 2008 to evaluate this mixing. Our study suggests that significant negative correlations exist between summer FTLE anomalies and winter precipitation over Portugal and Spain. To understand the mechanisms behind this correlation, summer anomalies of the FTLE have also been correlated with other climatic variables such as the sea surface temperature (SST, the sea level pressure (SLP or the geopotential. The East Atlantic (EA teleconnection index correlates with the summer FTLE anomalies, confirming their role as a seasonal predictor for winter precipitation over the Iberian Peninsula.
Finite-Time Lyapunov Exponents and Lagrangian Coherent Structures in Uncertain Unsteady Flows.
Guo, Hanqi; He, Wenbin; Peterka, Tom; Shen, Han-Wei; Collis, Scott; Helmus, Jonathan
2016-02-29
The objective of this paper is to understand transport behavior in uncertain time-varying flow fields by redefining the finite-time Lyapunov exponent (FTLE) and Lagrangian coherent structure (LCS) as stochastic counterparts of their traditional deterministic definitions. Three new concepts are introduced: the distribution of the FTLE (D-FTLE), the FTLE of distributions (FTLE-D), and uncertain LCS (U-LCS). The D-FTLE is the probability density function of FTLE values for every spatiotemporal location, which can be visualized with different statistical measurements. The FTLE-D extends the deterministic FTLE by measuring the divergence of particle distributions. It gives a statistical overview of how transport behaviors vary in neighborhood locations. The U-LCS, the probabilities of finding LCSs over the domain, can be extracted with stochastic ridge finding and density estimation algorithms. We show that our approach produces better results than existing variance-based methods do. Our experiments also show that the combination of D-FTLE, FTLE-D, and U-LCS can help users understand transport behaviors and find separatrices in ensemble simulations of atmospheric processes.
The monomer-dimer problem and moment Lyapunov exponents of homogeneous Gaussian random fields
Vladimirov, Igor G
2012-01-01
We consider an "elastic" version of the statistical mechanical monomer-dimer problem on the n-dimensional integer lattice. Our setting includes the classical "rigid" formulation as a special case and extends it by allowing each dimer to consist of particles at arbitrarily distant sites of the lattice, with the energy of interaction between the particles in a dimer depending on their relative position. We reduce the free energy of the elastic dimer-monomer (EDM) system per lattice site in the thermodynamic limit to the moment Lyapunov exponent (MLE) of a homogeneous Gaussian random field (GRF) whose mean value and covariance function are the Boltzmann factors associated with the monomer energy and dimer potential. In particular, the classical monomer-dimer problem becomes related to the MLE of a moving average GRF. We outline an approach to recursive computation of the partition function for "Manhattan" EDM systems where the dimer potential is a weighted l1-distance and the auxiliary GRF is a Markov random fie...
Statistical properties of the maximum Lyapunov exponent calculated via the divergence rate method.
Franchi, Matteo; Ricci, Leonardo
2014-12-01
The embedding of a time series provides a basic tool to analyze dynamical properties of the underlying chaotic system. To this purpose, the choice of the embedding dimension and lag is crucial. Although several methods have been devised to tackle the issue of the optimal setting of these parameters, a conclusive criterion to make the most appropriate choice is still lacking. An accepted procedure to rank different embedding methods relies on the evaluation of the maximum Lyapunov exponent (MLE) out of embedded time series that are generated by chaotic systems with explicit analytic representation. The MLE is evaluated as the local divergence rate of nearby trajectories. Given a system, embedding methods are ranked according to how close such MLE values are to the true MLE. This is provided by the so-called standard method in a way that exploits the mathematical description of the system and does not require embedding. In this paper we study the dependence of the finite-time MLE evaluated via the divergence rate method on the embedding dimension and lag in the case of time series generated by four systems that are widely used as references in the scientific literature. We develop a completely automatic algorithm that provides the divergence rate and its statistical uncertainty. We show that the uncertainty can provide useful information about the optimal choice of the embedding parameters. In addition, our approach allows us to find which systems provide suitable benchmarks for the comparison and ranking of different embedding methods.
Characteristic distribution of finite-time Lyapunov exponents for chimera states.
Botha, André E
2016-07-04
Our fascination with chimera states stems partially from the somewhat paradoxical, yet fundamental trait of identical, and identically coupled, oscillators to split into spatially separated, coherently and incoherently oscillating groups. While the list of systems for which various types of chimeras have already been detected continues to grow, there is a corresponding increase in the number of mathematical analyses aimed at elucidating the fundamental reasons for this surprising behaviour. Based on the model systems, there are strong indications that chimera states may generally be ubiquitous in naturally occurring systems containing large numbers of coupled oscillators - certain biological systems and high-Tc superconducting materials, for example. In this work we suggest a new way of detecting and characterising chimera states. Specifically, it is shown that the probability densities of finite-time Lyapunov exponents, corresponding to chimera states, have a definite characteristic shape. Such distributions could be used as signatures of chimera states, particularly in systems for which the phases of all the oscillators cannot be measured directly. For such cases, we suggest that chimera states could perhaps be detected by reconstructing the characteristic distribution via standard embedding techniques, thus making it possible to detect chimera states in systems where they could otherwise exist unnoticed.
Lyapunov analysis: from dynamical systems theory to applications
Cencini, Massimo; Ginelli, Francesco
2013-06-01
[17], von Neumann [18], Krylov [19]3 and Asonov and Sinai [20] on ergodic theory. Lyapunov exponents quantify exponential sensitivity to initial conditions and provide direct access to the entropy production in ergodic systems via the Pesin theory [21]. Further advances have been made possible by the introduction of proper physical invariant measures for certain dissipative systems due to Sinai [22], Ruelle [23] and Bowen [24, 25]. However, it was necessary to wait until the end of the 1970s before the independent works of Shimada and Nagashima [26] and Benettin et al [27] introduced the numerical algorithms required to compute Lyapunov exponents beyond the largest one. The availability of such algorithms and also, at about the same time, of those necessary for the computation of fractal dimensions and entropies by Grassberger and Procaccia [28], made possible the study of chaotic behavior in physically relevant models. Lyapunov analysis, applied to experimental systems [29], was also made possible by a combination of these numerical methods with ideas from nonlinear time series analysis [30]. As a result, it is nowadays widely recognized that Lyapunov exponents are a central tool of chaos theory, crucial for characterizing a number of interesting physical properties including dynamical entropies and fractal dimensions [31]. Their pivotal role in modern dynamical systems theory has been established by a fruitful exchange between a rigorous (and beautiful) mathematical theory and the algorithmic approaches essential for understanding many physical phenomena. From the 1990s to the present, with the concomitant progress in both theoretical understanding and computer capabilities, there has been a progressive shift of interest from low dimensional towards high dimensional systems. This shift towards dynamics characterized by many degrees of freedom, possibly spatially organized and/or with several characteristic temporal scales, has been accompanied by the need for
Duc, Luu Hoang; Chávez, Joseph Páez; Son, Doan Thai; Siegmund, Stefan
2016-01-01
In biochemical networks transient dynamics plays a fundamental role, since the activation of signalling pathways is determined by thresholds encountered during the transition from an initial state (e.g. an initial concentration of a certain protein) to a steady-state. These thresholds can be defined in terms of the inflection points of the stimulus-response curves associated to the activation processes in the biochemical network. In the present work, we present a rigorous discussion as to the suitability of finite-time Lyapunov exponents and metabolic control coefficients for the detection of inflection points of stimulus-response curves with sigmoidal shape.
Directory of Open Access Journals (Sweden)
Pratap R. Patnaik
2008-04-01
Full Text Available Large-scale fed-batch fermentations are often subject to noise carried by the feed streams. This noise corrupts the process data and may destabilize the fermentation. So it is important to retrieve clear signals from noisy data. This is done by noise filters. The performances of some commonly used filters have been studied for poly-β-hydroxybutyrate production by Ralstonia eutropha. In simulated experiments, Gaussian noise was added to the flow rates of the carbon and nitrogen substrates. The filters were compared by means of the Lyapunov exponents of the outputs and their closeness to the noise-free performance. Negative exponents indicate a stable fermentation. An auto-associative neural filter performed the best, followed by a combination of a cusum filter and an extended Kalman filter. Butterworth filters were inferior and inadequate.
Reducible linear quasi-periodic systems with positive Lyapunov exponent and varying rotation number
Broer, HW; Simo, C
2000-01-01
A linear system in two dimensions is studied. The coefficients are 2 pi -periodic in three angles, 0(j), = 1, 2, 3, and these angles are linear with respect to time, with incommensurable frequencies. The system has positive Lyapunov coefficients and the rotation number changes in a continuous way wh
The maximal Lyapunov exponent of a co-dimension two-bifurcation system excited by a bounded noise
Institute of Scientific and Technical Information of China (English)
Sheng-Hong Li; Xian-Bin Liu
2012-01-01
In the present paper,the maximal Lyapunov exponent is investigated for a co-dimension two bifurcation system that is on a three-dimensional central manifold and subjected to parametric excitation by a bounded noise.By using a perturbation method,the expressions of the invariant measure of a one-dimensional phase diffusion process are obtained for three cases,in which different forms of the matrix B,that is included in the noise excitation term,are assumed and then,as a result,all the three kinds of singular boundaries for one-dimensional phase diffusion process are analyzed.Via Monte-Carlo simulation,we find that the analytical expressions of the invariant measures meet well the numerical ones.And furthermore,the P-bifurcation behaviors are investigated for the one-dimensional phase diffusion process.Finally,for the three cases of singular boundaries for one-dimensional phase diffusion process,analytical expressions of the maximal Lyapunov exponent are presented for the stochastic bifurcation system.
Nair, Sandeep P.; Shiau, Deng-Shan; Principe, Jose C.; Iasemidis, Leonidas D.; Pardalos, Panos M.; Norman, Wendy M.; Carney, Paul R.; Sackellares, J. Chris
2009-01-01
Analysis of intracranial electroencephalographic (iEEG) recordings in patients with temporal lobe epilepsy (TLE) has revealed characteristic dynamical features that distinguish the interictal, ictal, and postictal states and inter-state transitions. Experimental investigations into the mechanisms underlying these observations require the use of an animal model. A rat TLE model was used to test for differences in iEEG dynamics between well-defined states and to test specific hypotheses: 1) the short-term maximum Lyapunov exponent (STLmax), a measure of signal order, is lowest and closest in value among cortical sites during the ictal state, and highest and most divergent during the postictal state; 2) STLmax values estimated from the stimulated hippocampus are the lowest among all cortical sites; and 3) the transition from the interictal to ictal state is associated with a convergence in STLmax values among cortical sites. iEEGs were recorded from bilateral frontal cortices and hippocampi. STLmax and T-index (a measure of convergence/divergence of STLmax between recorded brain areas) were compared among the four different periods. Statistical tests (ANOVA and multiple comparisons) revealed that ictal STLmax was lower (p < 0.05) than other periods, STLmax values corresponding to the stimulated hippocampus were lower than those estimated from other cortical regions, and T-index values were highest during the postictal period and lowest during the ictal period. Also, the T-index values corresponding to the preictal period were lower than those during the interictal period (p < 0.05). These results indicate that a rat TLE model demonstrates several important dynamical signal characteristics similar to those found in human TLE and support future use of the model to study epileptic state transitions. PMID:19100262
Hurst Exponent Analysis of Financial Time Series
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
Statistical properties of stock market time series and the implication of their Hurst exponents are discussed. Hurst exponents of DJ1A (Dow Jones Industrial Average) components are tested using re-scaled range analysis. In addition to the original stock return series, the linear prediction errors of the daily returns are also tested. Numerical results show that the Hurst exponent analysis can provide some information about the statistical properties of the financial time series.
Beijeren, H. van; Zon, R. van; Dorfman, J.R.
2000-01-01
The kinetic theory of gases provides methods for calculating Lyapunov exponents and other quantities, such as Kolmogorov-Sinai entropies, that characterize the chaotic behavior of hard-ball gases. Here we illustrate the use of these methods for calculating the Kolmogorov-Sinai entropy, and the
Ankışhan, Haydar; Yılmaz, Derya
2013-01-01
Snoring, which may be decisive for many diseases, is an important indicator especially for sleep disorders. In recent years, many studies have been performed on the snore related sounds (SRSs) due to producing useful results for detection of sleep apnea/hypopnea syndrome (SAHS). The first important step of these studies is the detection of snore from SRSs by using different time and frequency domain features. The SRSs have a complex nature that is originated from several physiological and physical conditions. The nonlinear characteristics of SRSs can be examined with chaos theory methods which are widely used to evaluate the biomedical signals and systems, recently. The aim of this study is to classify the SRSs as snore/breathing/silence by using the largest Lyapunov exponent (LLE) and entropy with multiclass support vector machines (SVMs) and adaptive network fuzzy inference system (ANFIS). Two different experiments were performed for different training and test data sets. Experimental results show that the multiclass SVMs can produce the better classification results than ANFIS with used nonlinear quantities. Additionally, these nonlinear features are carrying meaningful information for classifying SRSs and are able to be used for diagnosis of sleep disorders such as SAHS. PMID:24194786
Lueptow, Richard M.; Schlick, Conor P.; Umbanhowar, Paul B.; Ottino, Julio M.
2013-11-01
We investigate chaotic advection and diffusion in competitive autocatalytic reactions. To study this subject, we use a computationally efficient method for solving advection-reaction-diffusion equations for periodic flows using a mapping method with operator splitting. In competitive autocatalytic reactions, there are two species, B and C, which both react autocatalytically with species A (A +B -->2B and A +C -->2C). If there is initially a small amount of spatially localized B and C and a large amount of A, all three species will be advected by the velocity field, diffuse, and react until A is completely consumed and only B and C remain. We find that the small scale interactions associated with the chaotic velocity field, specifically the local finite-time Lyapunov exponents (FTLEs), can accurately predict the final average concentrations of B and C after the reaction is complete. The species, B or C, that starts in the region with the larger FTLE has, with high probability, the larger average concentration at the end of the reaction. If species B and C start in regions having similar FTLEs, their average concentrations at the end of the reaction will also be similar. Funded by NSF Grant CMMI-1000469.
Shayegh, F; Sadri, S; Amirfattahi, R; Ansari-Asl, K
2014-01-01
In order to predict epileptic seizures many precursory features, extracted from the EEG signals, have been introduced. Before checking out the performance of features in detection of pre-seizure state, it is required to see whether these features are accurately extracted. Evaluation of feature estimation methods has been less considered, mainly due to the lack of a ground truth for the real EEG signals' features. In this paper, some simulated long-term depth-EEG signals, with known state spaces, are generated via a realistic neural mass model with physiological parameters. Thanks to the known ground truth of these synthetic signals, they are suitable for evaluating different algorithms used to extract the features. It is shown that conventional methods of estimating correlation dimension, the largest Lyapunov exponent, and phase coherence have non-negligible errors. Then, a parameter identification-based method is introduced for estimating the features, which leads to better estimation results for synthetic signals. It is shown that the neural mass model is able to reproduce real depth-EEG signals accurately; thus, assuming this model underlying real depth-EEG signals, can improve the accuracy of features' estimation.
The Calculation of Lyapunov Exponent of Water Molecules Vibration System%基于水分子振动体系的Lyapunov指数的计算
Institute of Scientific and Technical Information of China (English)
刘松红; 庞成群
2012-01-01
采用wolf重构法改进了水分子振动体系最大Lyapunov指数的计算，通过对水分子振动体系的最大Lyapunov指数的计算，得到了计算水分子振动体系的最大Lyapunov指数合适的初始长度、延迟时间以及总的演化时间。%By adopting wolf reconstruction method and improving the method, we gained the expression of the maximum Lyapunov exponent of water molecules vibration system o From the results of calculating the maximum Lyapunov exponent of water molecules vibration system, we received the appropriate initial length, duration and the total evolution time.
Directory of Open Access Journals (Sweden)
Vaidyanathan Sundarapandian
2014-12-01
Full Text Available In this research work, a twelve-term novel 5-D hyperchaotic Lorenz system with three quadratic nonlinearities has been derived by adding a feedback control to a ten-term 4-D hyperchaotic Lorenz system (Jia, 2007 with three quadratic nonlinearities. The 4-D hyperchaotic Lorenz system (Jia, 2007 has the Lyapunov exponents L1 = 0.3684,L2 = 0.2174,L3 = 0 and L4 =−12.9513, and the Kaplan-Yorke dimension of this 4-D system is found as DKY =3.0452. The 5-D novel hyperchaotic Lorenz system proposed in this work has the Lyapunov exponents L1 = 0.4195,L2 = 0.2430,L3 = 0.0145,L4 = 0 and L5 = −13.0405, and the Kaplan-Yorke dimension of this 5-D system is found as DKY =4.0159. Thus, the novel 5-D hyperchaotic Lorenz system has a maximal Lyapunov exponent (MLE, which is greater than the maximal Lyapunov exponent (MLE of the 4-D hyperchaotic Lorenz system. The 5-D novel hyperchaotic Lorenz system has a unique equilibrium point at the origin, which is a saddle-point and hence unstable. Next, an adaptive controller is designed to stabilize the novel 5-D hyperchaotic Lorenz system with unknown system parameters. Moreover, an adaptive controller is designed to achieve global hyperchaos synchronization of the identical novel 5-D hyperchaotic Lorenz systems with unknown system parameters. Finally, an electronic circuit realization of the novel 5-D hyperchaotic Lorenz system using SPICE is described in detail to confirm the feasibility of the theoretical model.
Institute of Scientific and Technical Information of China (English)
Liu Wei-Dong(刘卫东); K.F.Ren; S.Meunier-Guttin-Cluzel; G.Gouesbet
2003-01-01
A method for the global vector-field reconstruction of nonlinear dynamical systems from a time series is studied in this paper. It employs a complete set of polynomials and singular value decomposition (SVD) to estimate a standard function which is central to the algorithm. Lyapunov exponents and dimension, calculated from the differential equations of a standard system, are used for the validation of the reconstruction. The algorithm is proven to be practical by applying it to a Rossler system.
Wang, Aixing; Fang, Chao; Liu, Yibao
2017-01-07
In this article the dynamic features of the highly excited vibrational states of the hypochlorous acid (HOCl) non-integrable system are studied using the dynamic potential and Lyapunov exponent approaches. On the condition that the 3:1 resonance between the H-O stretching and H-O-Cl bending modes accompany the 2:1 Fermi resonance between the O-Cl stretching and H-O-Cl bending modes, it is found that the dynamic potentials of the highly excited vibrational states vary regularly with different Polyad numbers (P numbers). As the P number increases, the dynamic potentials of the H-O stretching mode remain the same, but those of the H-O-Cl bending mode gradually become complex. In order to investigate the chaotic and stable features of the highly excited vibrational states of the HOCl non-integrable system, the Lyapunov exponents of different energy levels lying in the dynamic potentials of the H-O-Cl bending mode (P = 4 and 5) are calculated. It is shown that the Lyapunov exponents of the energy levels staying in the junction of Morse potential and inverse Morse potential are relative large, which indicates the degrees of chaos for these energy levels is relatively high, but the stabilities of the corresponding states are good. These results could be interpreted as the intramolecular vibrational relaxation (IVR) acting strongly via the HOCl bending motion and causing energy transfers among different modes. Based on the previous studies, these conclusions seem to be generally valid to some extent for non-integrable triatomic molecules.
Energy Technology Data Exchange (ETDEWEB)
Angelin Jeba, K.; Latha, M. M., E-mail: lathaisaac@yahoo.com [Department of Physics, Women' s Christian College, Nagercoil 629 001 (India); Jain, Sudhir R. [Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400085 (India)
2015-11-15
The nonlinear dynamics of intra- and inter-spine interaction models of alpha-helical proteins is investigated by proposing a Hamiltonian using the first quantized operators. Hamilton's equations of motion are derived, and the dynamics is studied by constructing the trajectories and phase space plots in both cases. The phase space plots display a chaotic behaviour in the dynamics, which opens questions about the relationship between the chaos and exciton-exciton and exciton-phonon interactions. This is verified by plotting the Lyapunov characteristic exponent curves.
Angelin Jeba, K; Latha, M M; Jain, Sudhir R
2015-11-01
The nonlinear dynamics of intra- and inter-spine interaction models of alpha-helical proteins is investigated by proposing a Hamiltonian using the first quantized operators. Hamilton's equations of motion are derived, and the dynamics is studied by constructing the trajectories and phase space plots in both cases. The phase space plots display a chaotic behaviour in the dynamics, which opens questions about the relationship between the chaos and exciton-exciton and exciton-phonon interactions. This is verified by plotting the Lyapunov characteristic exponent curves.
Institute of Scientific and Technical Information of China (English)
金俐; 王琪; 陆启韶
2001-01-01
A numerical algorithm for calculating Lyapunov exponents of Hamiltonian multibody systems with topological tree configuration is studied. The algorithms for Lyapunov exponents of Hamiltonian multibody systems using the canonical equations of the system and symplectic algorithm for ordinary differential equations are presented, which are used to study the stability of the Hamiltonian multibody systems. An example is given to analyze the stability of a typical Hamiltonian multibody system, including periodic solution and chaos.%研究了树形多体Hamilton系统Lyapunov指数的数值方法.利用多体Hamilton系统的正则方程和辛算法, 给出了多体Hamilton系统Lyapunov指数的计算方法,该算法具有较好的计算精度和通用性.利用该算法可对系统的运动稳定性进行分析.最后用算例说明了该算法的有效性.
Control design and comprehensive stability analysis of acrobots based on Lyapunov functions
Institute of Scientific and Technical Information of China (English)
LAI Xu-zhi; WU Yun-xin; SHE Jin-hua; WU Min
2005-01-01
A design method for controllers and a comprehensive stability analysis for an acrobat based on Lyapunov functions are presented. Three control laws based on three Lyapunov functions are designed to increase the energy so as to move the acrobot into the unstable inverted equilibrium position, and solve the problem of posture and energy. The concept of a non-smooth Lyapunov function is employed to analyze the stability of the whole system. The validity of this strategy is demonstrated by simulations.
Analysis of stability problems via matrix Lyapunov functions
Directory of Open Access Journals (Sweden)
Anatoly A. Martynyuk
1990-01-01
Full Text Available The stability of nonlinear systems is analyzed by the direct Lyapunov's method in terms of Lyapunov matrix functions. The given paper surveys the main theorems on stability, asymptotic stability and nonstability. They are applied to systems of nonlinear equations, singularly-perturbed systems and hybrid systems. The results are demonstrated by an example of a two-component system.
Nonlinear analysis of anesthesia dynamics by Fractal Scaling Exponent.
Gifani, P; Rabiee, H R; Hashemi, M R; Taslimi, P; Ghanbari, M
2006-01-01
The depth of anesthesia estimation has been one of the most research interests in the field of EEG signal processing in recent decades. In this paper we present a new methodology to quantify the depth of anesthesia by quantifying the dynamic fluctuation of the EEG signal. Extraction of useful information about the nonlinear dynamic of the brain during anesthesia has been proposed with the optimum Fractal Scaling Exponent. This optimum solution is based on the best box sizes in the Detrended Fluctuation Analysis (DFA) algorithm which have meaningful changes at different depth of anesthesia. The Fractal Scaling Exponent (FSE) Index as a new criterion has been proposed. The experimental results confirm that our new Index can clearly discriminate between aware to moderate and deep anesthesia levels. Moreover, it significantly reduces the computational complexity and results in a faster reaction to the transients in patients' consciousness levels in relations with the other algorithms.
Analysis of Lyapunov Method for Control of Quantum States
Wang, Xiaoting; Schirmer, Sonia
2009-01-01
The natural trajectory tracking problem is studied for generic quantum states represented by density operators. A control design based on the Hilbert-Schmidt distance as a Lyapunov function is considered. The control dynamics is redefined on an extended space where the LaSalle invariance principle can be correctly applied even for non-stationary target states. LaSalle's invariance principle is used to derive a general characterization of the invariant set, which is shown to always contain the...
Critical exponents in the transition to chaos in one-dimensional discrete systems
Indian Academy of Sciences (India)
G Ambika; N V Sujatha
2002-07-01
We report the numerically evaluated critical exponents associated with the scaling of generalized fractal dimensions during the transition from order to chaos. The analysis is carried out in detail in the context of unimodal and bimodal maps representing typical one-dimensional discrete dynamical systems. The behavior of Lyapunov exponents (LE) in the cross over region is also studied for a complete characterization.
A statistical approach to estimate the LYAPUNOV spectrum in disc brake squeal
Oberst, S.; Lai, J. C. S.
2015-01-01
The estimation of squeal propensity of a brake system from the prediction of unstable vibration modes using the linear complex eigenvalue analysis (CEA) in the frequency domain has its fair share of successes and failures. While the CEA is almost standard practice for the automotive industry, time domain methods and the estimation of LYAPUNOV spectra have not received much attention in brake squeal analyses. One reason is the challenge in estimating the true LYAPUNOV exponents and their discrimination against spurious ones in experimental data. A novel method based on the application of the ECKMANN-RUELLE matrices is proposed here to estimate LYAPUNOV exponents by using noise in a statistical procedure. It is validated with respect to parameter variations and dimension estimates. By counting the number of non-overlapping confidence intervals for LYAPUNOV exponent distributions obtained by moving a window of increasing size over bootstrapped same-length estimates of an observation function, a dispersion measure's width is calculated and fed into a BAYESIAN beta-binomial model. Results obtained using this method for benchmark models of white and pink noise as well as the classical HENON map indicate that true LYAPUNOV exponents can be isolated from spurious ones with high confidence. The method is then applied to accelerometer and microphone data obtained from brake squeal tests. Estimated LYAPUNOV exponents indicate that the pad's out-of-plane vibration behaves quasi-periodically on the brink to chaos while the microphone's squeal signal remains periodic.
Lyapunov vs. geometrical stability analysis of the Kepler and the restricted three body problems
DEFF Research Database (Denmark)
Yahalom, A.; Levitan, J.; Lewkowicz, M.
2011-01-01
to move in a very interesting and intricate but periodic trajectory; however, the standard Lyapunov analysis, as well as methods based on the parametric variation of curvature associated with the Jacobi metric, incorrectly predict chaotic behavior. The geometric approach predicts the correct stable motion...
Partial differential equations with variable exponents variational methods and qualitative analysis
Radulescu, Vicentiu D
2015-01-01
Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis provides researchers and graduate students with a thorough introduction to the theory of nonlinear partial differential equations (PDEs) with a variable exponent, particularly those of elliptic type. The book presents the most important variational methods for elliptic PDEs described by nonhomogeneous differential operators and containing one or more power-type nonlinearities with a variable exponent. The authors give a systematic treatment of the basic mathematical theory and constructive meth
Directory of Open Access Journals (Sweden)
Héctor Armando Durán Peralta
2010-04-01
Full Text Available The stability of reactors having encompassing concentration and temperature parameters, such as continuous flow stirred tank reactors (CSTR, has been widely explored in the literature; however, there are few papers about the stability of tubular reactor having distributed spatial concentration and temperature parameters such as the plow flow tubular reactor (PFTR. This paper analyses the stability of isothermal and non-isothermal PFTR reactors using the Lyapunov functional method. The first order kinetic reaction was selected because one of this paper’s oblectives was to apply Lyapunov functionals to stability analysis of distributed parameter reactors (technique used in electrical engineering systems’ stability analysis. The stability analysis revealed asymptotically stable tempe- rature and concentration profiles for isothermal PFTR, non-isothermal PFTR with kinetic constant independent of temperature and adiabatic non-isothermal PFTR. Analysis revealed an asymptotically stability region for the heat exchange reactor and an uncertain region where it may have oscillations.
Can stability analysis be really simplified? (revisiting Lyapunov, Barbalat, LaSalle and all that)
Barkana, Itzhak
2017-01-01
Even though Lyapunov approach is the most commonly used method for stability analysis, its use has been hindered by the realization that in most applications the so-called Lyapunov derivative is at most negative semidefinite and not negative definite as desired. Many different approaches have been used in an attempt to overcome these difficulties. Until recently, the most widely accepted stability analysis has been based on Barbalat's Lemma which seems to require uniform continuity of practically all signals involved. Recently, stability analysis methods for nonautonomous nonlinear systems have been revisited. Even though new developments based on unknown works of LaSalle attempted to mitigate these continuity conditions, counterexamples are suggested to contradict these results. New analysis shows that these counterexamples, which are making use of well-known mathematical expressions, are actually using them beyond their domain of validity. Therefore, the restrictive condition of uniform continuity required by Barbalat's Lemma and even the milder conditions required by LaSalle's extension of the Invariance Principle to nonautonomous systems can be further mitigated. A new Invariance Principle only required that bounded trajectories cannot pass an infinite distance in finite time. Finally, a new Theorem of Stability, which is formulated as a direct extension and a generalization of Lyapunov's Theorem, not only simplifies the stability analysis of nonlinear systems, but also leads to conclusive results about the system under analysis.
Schubert, Sebastian; Lucarini, Valerio
2016-04-01
One of the most relevant weather regimes in the mid latitudes atmosphere is the persistent deviation from the approximately zonally symmetric jet stream to the emergence of so-called blocking patterns. Such configurations are usually connected to exceptional local stability properties of the flow which come along with an improved local forecast skills during the phenomenon. It is instead extremely hard to predict onset and decay of blockings. Covariant Lyapunov Vectors (CLVs) offer a suitable characterization of the linear stability of a chaotic flow, since they represent the full tangent linear dynamics by a covariant basis which explores linear perturbations at all time scales. Therefore, we will test whether CLVs feature a signature of the blockings. We examine the CLVs for a quasi-geostrophic beta-plane two-layer model in a periodic channel baroclinically driven by a meridional temperature gradient ΔT. An orographic forcing enhances the emergence of localized blocked regimes. We detect the blocking events of the channel flow with a Tibaldi-Molteni scheme adapted to the periodic channel. When blocking occurs, the global growth rates of the fastest growing CLVs are significantly higher. Hence against intuition, globally the circulation is more unstable in blocked phases. Such an increase in the finite time Lyapunov exponents with respect to the long term average is attributed to stronger barotropic and baroclinic conversion in the case of high temperature gradients, while for low values of ΔT, the effect is only due to stronger barotropic instability. For the localization of the CLVs, we compare the meridionally averaged variance of the CLVs during blocked and unblocked phases. We find that on average the variance of the CLVs is clustered around the center of blocking. These results show that the blocked flow affects all time scales and processes described by the CLVs.
Institute of Scientific and Technical Information of China (English)
LIU Yuan-feng; ZHAO Mei
2005-01-01
An algorithm based on the data-adaptive filtering characteristics of singular spectrum analysis (SSA) is proposed to denoise chaotic data. Firstly, the empirical orthogonal functions ( EOFs ) and principal components ( PCs ) of the signal were calculated, reconstruct the signal using the EOFs and PCs, and choose the optimal reconstructing order based on sigular spectrum to obtain the denoised signal. The noise of the signal can influence the calculating precision of maximal Liapunov exponents. The proposed denoising algorithm was applied to the maximal Liapunov exponents calculations of two chaotic system, Henon map and Logistic map. Some numerical results show that this denoising algorithm could improve the calculating precision of maximal Liapunov exponent.
Lyapunov modes in extended systems.
Yang, Hong-Liu; Radons, Günter
2009-08-28
Hydrodynamic Lyapunov modes, which have recently been observed in many extended systems with translational symmetry, such as hard sphere systems, dynamic XY models or Lennard-Jones fluids, are nowadays regarded as fundamental objects connecting nonlinear dynamics and statistical physics. We review here our recent results on Lyapunov modes in extended system. The solution to one of the puzzles, the appearance of good and 'vague' modes, is presented for the model system of coupled map lattices. The structural properties of these modes are related to the phase space geometry, especially the angles between Oseledec subspaces, and to fluctuations of local Lyapunov exponents. In this context, we report also on the possible appearance of branches splitting in the Lyapunov spectra of diatomic systems, similar to acoustic and optical branches for phonons. The final part is devoted to the hyperbolicity of partial differential equations and the effective degrees of freedom of such infinite-dimensional systems.
Lyapunov decay in quantum irreversibility.
García-Mata, Ignacio; Roncaglia, Augusto J; Wisniacki, Diego A
2016-06-13
The Loschmidt echo--also known as fidelity--is a very useful tool to study irreversibility in quantum mechanics due to perturbations or imperfections. Many different regimes, as a function of time and strength of the perturbation, have been identified. For chaotic systems, there is a range of perturbation strengths where the decay of the Loschmidt echo is perturbation independent, and given by the classical Lyapunov exponent. But observation of the Lyapunov decay depends strongly on the type of initial state upon which an average is carried out. This dependence can be removed by averaging the fidelity over the Haar measure, and the Lyapunov regime is recovered, as has been shown for quantum maps. In this work, we introduce an analogous quantity for systems with infinite dimensional Hilbert space, in particular the quantum stadium billiard, and we show clearly the universality of the Lyapunov regime.
Institute of Scientific and Technical Information of China (English)
Zhou Yu; Leung Yee; Yu Zu-Guo
2011-01-01
Multifractal detrended fluctuation analysis (MF-DFA) is a relatively new method of multifractal analysis. It is extended from detrended fluctuation analysis (DFA),which was developed for detecting the long-range correlation and the fractal properties in stationary and non-stationary time series. Although MF-DFA has become a widely used method,some relationships among the exponents established in the original paper seem to be incorrect under the general situation. In this paper,we theoretically and experimentally demonstrate the invalidity of the expression τ(q)＝qh(q)-1 stipulating the relationship between the multifractal exponent τ(q) and the generalized Hurst exponent h(q). As a replacement,a general relationship is established on the basis of the universal multifractal formalism for the stationary series as τ(q)＝qh(q)-qH'-1,where H'is the nonconservation parameter in the universal multifractal formalism. The singular spectra,a and f (a),are also derived according to this new relationship.
Random Matrices and Lyapunov Coefficients Regularity
Gallavotti, Giovanni
2017-02-01
Analyticity and other properties of the largest or smallest Lyapunov exponent of a product of real matrices with a "cone property" are studied as functions of the matrices entries, as long as they vary without destroying the cone property. The result is applied to stability directions, Lyapunov coefficients and Lyapunov exponents of a class of products of random matrices and to dynamical systems. The results are not new and the method is the main point of this work: it is is based on the classical theory of the Mayer series in Statistical Mechanics of rarefied gases.
Detecting Epileptic Seizure from Scalp EEG Using Lyapunov Spectrum
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Truong Quang Dang Khoa
2012-01-01
Full Text Available One of the inherent weaknesses of the EEG signal processing is noises and artifacts. To overcome it, some methods for prediction of epilepsy recently reported in the literature are based on the evaluation of chaotic behavior of intracranial electroencephalographic (EEG recordings. These methods reduced noises, but they were hazardous to patients. In this study, we propose using Lyapunov spectrum to filter noise and detect epilepsy on scalp EEG signals only. We determined that the Lyapunov spectrum can be considered as the most expected method to evaluate chaotic behavior of scalp EEG recordings and to be robust within noises. Obtained results are compared to the independent component analysis (ICA and largest Lyapunov exponent. The results of detecting epilepsy are compared to diagnosis from medical doctors in case of typical general epilepsy.
Detecting epileptic seizure from scalp EEG using Lyapunov spectrum.
Khoa, Truong Quang Dang; Huong, Nguyen Thi Minh; Toi, Vo Van
2012-01-01
One of the inherent weaknesses of the EEG signal processing is noises and artifacts. To overcome it, some methods for prediction of epilepsy recently reported in the literature are based on the evaluation of chaotic behavior of intracranial electroencephalographic (EEG) recordings. These methods reduced noises, but they were hazardous to patients. In this study, we propose using Lyapunov spectrum to filter noise and detect epilepsy on scalp EEG signals only. We determined that the Lyapunov spectrum can be considered as the most expected method to evaluate chaotic behavior of scalp EEG recordings and to be robust within noises. Obtained results are compared to the independent component analysis (ICA) and largest Lyapunov exponent. The results of detecting epilepsy are compared to diagnosis from medical doctors in case of typical general epilepsy.
Zhang, Hongbin; Feng, Gang
2008-10-01
This paper is concerned with stability analysis and H(infinity) decentralized control of discrete-time fuzzy large-scale systems based on piecewise Lyapunov functions. The fuzzy large-scale systems consist of J interconnected discrete-time Takagi-Sugeno (T-S) fuzzy subsystems, and the stability analysis is based on Lyapunov functions that are piecewise quadratic. It is shown that the stability of the discrete-time fuzzy large-scale systems can be established if a piecewise quadratic Lyapunov function can be constructed, and moreover, the function can be obtained by solving a set of linear matrix inequalities (LMIs) that are numerically feasible. The H(infinity) controllers are also designed by solving a set of LMIs based on these powerful piecewise quadratic Lyapunov functions. It is demonstrated via numerical examples that the stability and controller synthesis results based on the piecewise quadratic Lyapunov functions are less conservative than those based on the common quadratic Lyapunov functions.
Zhang, Baoyong; Lam, James; Xu, Shengyuan
2015-07-01
This paper revisits the problem of asymptotic stability analysis for neural networks with distributed delays. The distributed delays are assumed to be constant and prescribed. Since a positive-definite quadratic functional does not necessarily require all the involved symmetric matrices to be positive definite, it is important for constructing relaxed Lyapunov-Krasovskii functionals, which generally lead to less conservative stability criteria. Based on this fact and using two kinds of integral inequalities, a new delay-dependent condition is obtained, which ensures that the distributed delay neural network under consideration is globally asymptotically stable. This stability criterion is then improved by applying the delay partitioning technique. Two numerical examples are provided to demonstrate the advantage of the presented stability criteria.
2009-08-01
setting in Duc & Siegmund [28]: Definition A.10 (Dynamic partition of IR2). Consider the extended phase space, IR2 × I, associated with the flow...Fluid Dynamics, Cambridge University Press, Cam- bridge, 1967. [8] A. Berger, D. T. Son, and S. Siegmund , Nonautonomous finite-time dynamics, Discrete...28] L. H. Duc and S. Siegmund , Hyperbolicity and invariant manifolds for planar nonau- tonomous systems on finite time intervals, Int. J. Bif. Chaos
Schubert, Sebastian; Lucarini, Valerio
2016-04-01
The classical approach for studying atmospheric variability is based on defining a background state and studying the linear stability of the small fluctuations around such a state. Weakly non-linear theories can be constructed using higher order expansions terms. While these methods have undoubtedly great value for elucidating the relevant physical processes, they are unable to follow the dynamics of a turbulent atmosphere. We provide a first example of extension of the classical stability analysis to a non-linearly evolving quasi-geostrophic flow. The so-called covariant Lyapunov vectors (CLVs) provide a covariant basis describing the directions of exponential expansion and decay of perturbations to the non-linear trajectory of the flow. We use such a formalism to re-examine the basic barotropic and baroclinic processes of the atmosphere with a quasi-geostrophic beta-plane two-layer model in a periodic channel driven by a forced meridional temperature gradient ΔT . We explore three settings of ΔT , representative of relatively weak turbulence, well-developed turbulence, and intermediate conditions. We construct the Lorenz energy cycle for each CLV describing the energy exchanges with the background state. A positive baroclinic conversion rate is a necessary but not sufficient condition of instability. Barotropic instability is present only for few very unstable CLVs for large values of ΔT. Slowly growing and decaying hydrodynamic Lyapunov modes closely mirror the properties of the background flow. Following classical necessary conditions for barotropic/baroclinic instability, we find a clear relationship between the properties of the eddy fluxes of a CLV and its instability. CLVs with positive baroclinic conversion seem to form a set of modes for constructing a reduced model of the atmosphere dynamics.
The Lyapunov dimension and its estimation via the Leonov method
Energy Technology Data Exchange (ETDEWEB)
Kuznetsov, N.V., E-mail: nkuznetsov239@gmail.com
2016-06-03
Highlights: • Survey on effective analytical approach for Lyapunov dimension estimation, proposed by Leonov, is presented. • Invariance of Lyapunov dimension under diffeomorphisms and its connection with Leonov method are demonstrated. • For discrete-time dynamical systems an analog of Leonov method is suggested. - Abstract: Along with widely used numerical methods for estimating and computing the Lyapunov dimension there is an effective analytical approach, proposed by G.A. Leonov in 1991. The Leonov method is based on the direct Lyapunov method with special Lyapunov-like functions. The advantage of the method is that it allows one to estimate the Lyapunov dimension of invariant sets without localization of the set in the phase space and, in many cases, to get effectively an exact Lyapunov dimension formula. In this work the invariance of the Lyapunov dimension with respect to diffeomorphisms and its connection with the Leonov method are discussed. For discrete-time dynamical systems an analog of Leonov method is suggested. In a simple but rigorous way, here it is presented the connection between the Leonov method and the key related works: Kaplan and Yorke (the concept of the Lyapunov dimension, 1979), Douady and Oesterlé (upper bounds of the Hausdorff dimension via the Lyapunov dimension of maps, 1980), Constantin, Eden, Foiaş, and Temam (upper bounds of the Hausdorff dimension via the Lyapunov exponents and Lyapunov dimension of dynamical systems, 1985–90), and the numerical calculation of the Lyapunov exponents and dimension.
On the computation of quantum characteristic exponents
Vilela-Mendes, R; Coutinho, Ricardo
1998-01-01
A quantum characteristic exponent may be defined, with the same operational meaning as the classical Lyapunov exponent when the latter is expressed as a functional of densities. Existence conditions and supporting measure properties are discussed as well as the problems encountered in the numerical computation of the quantum exponents. Although an example of true quantum chaos may be exhibited, the taming effect of quantum mechanics on chaos is quite apparent in the computation of the quantum exponents. However, even when the exponents vanish, the functionals used for their definition may still provide a characterization of distinct complexity classes for quantum behavior.
Design and Lyapunov Stability Analysis of a Fuzzy Logic Controller for Autonomous Road Following
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Yi Fu
2010-01-01
Full Text Available Autonomous road following is one of the major goals in intelligent vehicle applications. The development of an autonomous road following embedded system for intelligent vehicles is the focus of this paper. A fuzzy logic controller (FLC is designed for vision-based autonomous road following. The stability analysis of this control system is addressed. Lyapunov's direct method is utilized to formulate a class of control laws that guarantee the convergence of the steering error. Certain requirements for the control laws are presented for designers to choose a suitable rule base for the fuzzy controller in order to make the system stable. Stability of the proposed fuzzy controller is guaranteed theoretically and also demonstrated by simulation studies and experiments. Simulations using the model of the four degree of freedom nonholonomic robotic vehicle are conducted to investigate the performance of the fuzzy controller. The proposed fuzzy controller can achieve the desired steering angle and make the robotic vehicle follow the road successfully. Experiments show that the developed intelligent vehicle is able to follow a mocked road autonomously.
Canada, Antonio
2011-01-01
Several different problems make the study of the so called Lyapunov type inequalities of great interest, both in pure and applied mathematics. Although the original historical motivation was the study of the stability properties of the Hill equation (which applies to many problems in physics and engineering), other questions that arise in systems at resonance, crystallography, isoperimetric problems, Rayleigh type quotients, etc. lead to the study of $L_p$ Lyapunov inequalities ($1\\leq p\\leq \\infty$) for differential equations. In this work we review some recent results on these kinds of questions which can be formulated as optimal control problems. In the case of Ordinary Differential Equations, we consider periodic and antiperiodic boundary conditions at higher eigenvalues and by using a more accurate version of the Sturm separation theory, an explicit optimal result is obtained. Then, we establish Lyapunov inequalities for systems of equations. To this respect, a key point is the characterization of the be...
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Mokaedi V. Lekgari
2014-01-01
Full Text Available We investigate random-time state-dependent Foster-Lyapunov analysis on subgeometric rate ergodicity of continuous-time Markov chains (CTMCs. We are mainly concerned with making use of the available results on deterministic state-dependent drift conditions for CTMCs and on random-time state-dependent drift conditions for discrete-time Markov chains and transferring them to CTMCs.
Lyapunov instabilities of Lennard-Jones fluids.
Yang, Hong-liu; Radons, Günter
2005-03-01
Recent work on many-particle systems reveals the existence of regular collective perturbations corresponding to the smallest positive Lyapunov exponents (LEs), called hydrodynamic Lyapunov modes. Until now, however, these modes have been found only for hard-core systems. Here we report results on Lyapunov spectra and Lyapunov vectors (LVs) for Lennard-Jones fluids. By considering the Fourier transform of the coordinate fluctuation density u((alpha)) (x,t) , it is found that the LVs with lambda approximately equal to 0 are highly dominated by a few components with low wave numbers. These numerical results provide strong evidence that hydrodynamic Lyapunov modes do exist in soft-potential systems, although the collective Lyapunov modes are more vague than in hard-core systems. In studying the density and temperature dependence of these modes, it is found that, when the value of the Lyapunov exponent lambda((alpha)) is plotted as function of the dominant wave number k(max) of the corresponding LV, all data from simulations with different densities and temperatures collapse onto a single curve. This shows that the dispersion relation lambda((alpha)) vs k(max) for hydrodynamical Lyapunov modes appears to be universal for the low-density cases studied here. Despite the wavelike character of the LVs, no steplike structure exists in the Lyapunov spectrum of the systems studied here, in contrast to the hard-core case. Further numerical simulations show that the finite-time LEs fluctuate strongly. We have also investigated localization features of LVs and propose a length scale to characterize the Hamiltonian spatiotemporal chaotic states.
Comparison between covariant and orthogonal Lyapunov vectors.
Yang, Hong-liu; Radons, Günter
2010-10-01
Two sets of vectors, covariant Lyapunov vectors (CLVs) and orthogonal Lyapunov vectors (OLVs), are currently used to characterize the linear stability of chaotic systems. A comparison is made to show their similarity and difference, especially with respect to the influence on hydrodynamic Lyapunov modes (HLMs). Our numerical simulations show that in both Hamiltonian and dissipative systems HLMs formerly detected via OLVs survive if CLVs are used instead. Moreover, the previous classification of two universality classes works for CLVs as well, i.e., the dispersion relation is linear for Hamiltonian systems and quadratic for dissipative systems, respectively. The significance of HLMs changes in different ways for Hamiltonian and dissipative systems with the replacement of OLVs with CLVs. For general dissipative systems with nonhyperbolic dynamics the long-wavelength structure in Lyapunov vectors corresponding to near-zero Lyapunov exponents is strongly reduced if CLVs are used instead, whereas for highly hyperbolic dissipative systems the significance of HLMs is nearly identical for CLVs and OLVs. In contrast the HLM significance of Hamiltonian systems is always comparable for CLVs and OLVs irrespective of hyperbolicity. We also find that in Hamiltonian systems different symmetry relations between conjugate pairs are observed for CLVs and OLVs. Especially, CLVs in a conjugate pair are statistically indistinguishable in consequence of the microreversibility of Hamiltonian systems. Transformation properties of Lyapunov exponents, CLVs, and hyperbolicity under changes of coordinate are discussed in appendices.
Baire classes of Lyapunov invariants
Bykov, V. V.
2017-05-01
It is shown that no relations exist (apart from inherent ones) between Baire classes of Lyapunov transformation invariants in the compact- open and uniform topologies on the space of linear differential systems. It is established that if a functional on the space of linear differential systems with the compact-open topology is the repeated limit of a multisequence of continuous functionals, then these can be chosen to be determined by the values of system coefficients on a finite interval of the half-line (one for each functional). It is proved that the Lyapunov exponents cannot be represented as the limit of a sequence of (not necessarily continuous) functionals such that each of these depends only on the restriction of the system to a finite interval of the half-line. Bibliography: 28 titles.
Stability analysis for impulsive fractional hybrid systems via variational Lyapunov method
Yang, Ying; He, Yong; Wang, Yong; Wu, Min
2017-04-01
This paper investigates the stability properties for a class of impulsive Caputo fractional-order hybrid systems with impulse effects at fixed moments. By utilizing the variational Lyapunov method, a fractional variational comparison principle is established. Some stability and instability criteria in terms of two measures are obtained. These results generalize the known ones, extending the corresponding theory of impulsive fractional differential systems. An example is given to demonstrate their effectiveness.
Covariant Lyapunov vectors for rigid disk systems.
Bosetti, Hadrien; Posch, Harald A
2010-10-05
We carry out extensive computer simulations to study the Lyapunov instability of a two-dimensional hard-disk system in a rectangular box with periodic boundary conditions. The system is large enough to allow the formation of Lyapunov modes parallel to the x-axis of the box. The Oseledec splitting into covariant subspaces of the tangent space is considered by computing the full set of covariant perturbation vectors co-moving with the flow in tangent space. These vectors are shown to be transversal, but generally not orthogonal to each other. Only the angle between covariant vectors associated with immediate adjacent Lyapunov exponents in the Lyapunov spectrum may become small, but the probability of this angle to vanish approaches zero. The stable and unstable manifolds are transverse to each other and the system is hyperbolic.
Regeneration cycle and the covariant Lyapunov vectors in a minimal wall turbulence.
Inubushi, Masanobu; Takehiro, Shin-ichi; Yamada, Michio
2015-08-01
Considering a wall turbulence as a chaotic dynamical system, we study regeneration cycles in a minimal wall turbulence from the viewpoint of orbital instability by employing the covariant Lyapunov analysis developed by [F. Ginelli et al. Phys. Rev. Lett. 99, 130601 (2007)]. We divide the regeneration cycle into two phases and characterize them with the local Lyapunov exponents and the covariant Lyapunov vectors of the Navier-Stokes turbulence. In particular, we show numerically that phase (i) is dominated by instabilities related to the sinuous mode and the streamwise vorticity, and there is no instability in phase (ii). Furthermore, we discuss a mechanism of the regeneration cycle, making use of an energy budget analysis.
A theoretical analysis to current exponent variation regularity and electromigration-induced failure
Wang, Yuexing; Yao, Yao
2017-02-01
The electric current exponent, typically with j-n form, is a key parameter to predict electromigration-induced failure lifetime. It is experimentally observed that the current exponent depends on different damage mechanisms. In the current research, the physical mechanisms including void initiation, void growth, and joule heating effect are all taken into account to investigate the current exponent variation regularity. Furthermore, a physically based model to predict the mean time to failure is developed and the traditional Black's equation is improved with clear physical meaning. It is found that the solution to the resulting void initiation and growth equation yields a current exponent of 2 and 1, respectively. On the other hand, joule heating plays an important role in failure time prediction and will induce the current exponent n > 2 based on the traditional semi-empirical model. The predictions are in agreement with the experimental results.
Directory of Open Access Journals (Sweden)
M. Widi Triyatno
2015-03-01
Full Text Available Disturbances in the operation of the power system may cause disturbance in voltage stability. Therefore, dynamic voltage stability analysis before and after disturbance needs to be performed. This paper proposes dynamic voltage stability prediction using maximum Lyapunov exponent with Lampung’s electrical system as case study. Voltage stability simulation is performed with various types of disturbances that occur at line between of Baturaja substation and Bukit Kemuning substation. Time-series data of voltage measurement of simulation results at GI Baturaja is applied for voltage stability prediction analysis using maximum Lyapunov exponent. With the same number of data samples and the same time for circuit breakers to interrupt disturbances, the simulation results using maximum Lyapunov exponent show that the voltage can be stabilized at 1.7 seconds after the occurrence of the three-phase disturbance, at 1.2 seconds after the occurrence of the phase-to-ground disturbance, at 0,9 second after the occurrence of the disturbance between phase, at 1.2 seconds after the occurrence of the loss of line disturbance and 1.4 seconds after the occurrence of the loss of load disturbance. The amount of data samples used in analysis affect the time for the voltage reaches stability.
Analysis of a bio-dynamic model via Lyapunov principle and small-world network for tuberculosis.
Chung, H-Y; Chung, C-Y; Ou, S-C
2012-10-01
The study will apply Lyapunov principle to construct a dynamic model for tuberculosis (TB). The Lyapunov principle is commonly used to examine and determine the stability of a dynamic system. To simulate the transmissions of vector-borne diseases and discuss the related health policies effects on vector-borne diseases, the authors combine the multi-agent-based system, social network and compartmental model to develop an epidemic simulation model. In the identity level, the authors use the multi-agent-based system and the mirror identity concept to describe identities with social network features such as daily visits, long-distance movement, high degree of clustering, low degree of separation and local clustering. The research will analyse the complex dynamic mathematic model of TB epidemic and determine its stability property by using the popular Matlab/Simulink software and relative software packages. Facing the current TB epidemic situation, the development of TB and its developing trend through constructing a dynamic bio-mathematical system model of TB is investigated. After simulating the development of epidemic situation with the solution of the SMIR epidemic model, the authors will come up with a good scheme to control epidemic situation to analyse the parameter values of a model that influence epidemic situation evolved. The authors will try to find the quarantining parameters that are the most important factors to control epidemic situation. The SMIR epidemic model and the results via numerical analysis may offer effective prevention with reference to controlling epidemic situation of TB.
Large-Signal Lyapunov-Based Stability Analysis of DC/AC Inverters and Inverter-Based Microgrids
Kabalan, Mahmoud
Microgrid stability studies have been largely based on small-signal linearization techniques. However, the validity and magnitude of the linearization domain is limited to small perturbations. Thus, there is a need to examine microgrids with large-signal nonlinear techniques to fully understand and examine their stability. Large-signal stability analysis can be accomplished by Lyapunov-based mathematical methods. These Lyapunov methods estimate the domain of asymptotic stability of the studied system. A survey of Lyapunov-based large-signal stability studies showed that few large-signal studies have been completed on either individual systems (dc/ac inverters, dc/dc rectifiers, etc.) or microgrids. The research presented in this thesis addresses the large-signal stability of droop-controlled dc/ac inverters and inverter-based microgrids. Dc/ac power electronic inverters allow microgrids to be technically feasible. Thus, as a prelude to examining the stability of microgrids, the research presented in Chapter 3 analyzes the stability of inverters. First, the 13 th order large-signal nonlinear model of a droop-controlled dc/ac inverter connected to an infinite bus is presented. The singular perturbation method is used to decompose the nonlinear model into 11th, 9th, 7th, 5th, 3rd and 1st order models. Each model ignores certain control or structural components of the full order model. The aim of the study is to understand the accuracy and validity of the reduced order models in replicating the performance of the full order nonlinear model. The performance of each model is studied in three different areas: time domain simulations, Lyapunov's indirect method and domain of attraction estimation. The work aims to present the best model to use in each of the three domains of study. Results show that certain reduced order models are capable of accurately reproducing the performance of the full order model while others can be used to gain insights into those three areas of
Schubert, Sebastian
2015-01-01
One of the most relevant weather regimes in the mid latitudes atmosphere is the persistent deviation from the approximately zonally symmetric jet stream to the emergence of so-called blocking patterns. Such configurations are usually connected to exceptional local stability properties of the flow which come along with an improved local forecast skills during the phenomenon. It is instead extremely hard to predict onset and decay of blockings. Covariant Lyapunov Vectors (CLVs) offer a suitable characterization of the linear stability of a chaotic flow, since they represent the full tangent linear dynamics by a covariant basis which explores linear perturbations at all time scales. Therefore, we will test whether CLVs feature a signature of the blockings. We examine the CLVs for a quasi-geostrophic beta-plane two-layer model in a periodic channel baroclinically driven by a meridional temperature gradient $\\Delta T$. An orographic forcing enhances the emergence of localized blocked regimes. We detect the blockin...
A Lyapunov method for stability analysis of piecewise-affine systems over non-invariant domains
Rubagotti, Matteo; Zaccarian, Luca; Bemporad, Alberto
2016-05-01
This paper analyses stability of discrete-time piecewise-affine systems, defined on possibly non-invariant domains, taking into account the possible presence of multiple dynamics in each of the polytopic regions of the system. An algorithm based on linear programming is proposed, in order to prove exponential stability of the origin and to find a positively invariant estimate of its region of attraction. The results are based on the definition of a piecewise-affine Lyapunov function, which is in general discontinuous on the boundaries of the regions. The proposed method is proven to lead to feasible solutions in a broader range of cases as compared to a previously proposed approach. Two numerical examples are shown, among which a case where the proposed method is applied to a closed-loop system, to which model predictive control was applied without a-priori guarantee of stability.
A finite-time exponent for random Ehrenfest gas
Energy Technology Data Exchange (ETDEWEB)
Moudgalya, Sanjay; Chandra, Sarthak [Indian Institute of Technology, Kanpur 208016 (India); Jain, Sudhir R., E-mail: srjain@barc.gov.in [Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400085 (India)
2015-10-15
We consider the motion of a system of free particles moving on a plane with regular hard polygonal scatterers arranged in a random manner. Calling this the Ehrenfest gas, which is known to have a zero Lyapunov exponent, we propose a finite-time exponent to characterize its dynamics. As the number of sides of the polygon goes to infinity, when polygon tends to a circle, we recover the usual Lyapunov exponent for the Lorentz gas from the exponent proposed here. To obtain this result, we generalize the reflection law of a beam of rays incident on a polygonal scatterer in a way that the formula for the circular scatterer is recovered in the limit of infinite number of vertices. Thus, chaos emerges from pseudochaos in an appropriate limit. - Highlights: • We present a finite-time exponent for particles moving in a plane containing polygonal scatterers. • The exponent found recovers the Lyapunov exponent in the limit of the polygon becoming a circle. • Our findings unify pseudointegrable and chaotic scattering via a generalized collision rule. • Stretch and fold:shuffle and cut :: Lyapunov:finite-time exponent :: fluid:granular mixing.
Stability analysis of titanium alloy milling by multiscale entropy and Hurst exponent
Rusinek, Rafał; Borowiec, Marek
2015-10-01
This paper discusses the problem of stability in a milling process for titanium super-alloy Ti6242. The phenomenon of chatter vibration is analysed by the multiscale entropy method and Hurst exponent. Although this problem is often considered based on stability lobe diagrams, theoretical findings do not always agree with experimental results. First, a stability lobe diagram is created based on parameters determined by impact testing. Next, cutting forces are measured in an experiment where the axial cutting depth is gradually increased. Finally, the obtained experimental signals are investigated with respect to stability using the multiscale entropy method and Hurst exponent.
Ma, Junhai; Ren, Wenbo; Zhan, Xueli
2017-04-01
Based on the study of scholars at home and abroad, this paper improves the three-dimensional IS-LM model in macroeconomics, analyzes the equilibrium point of the system and stability conditions, focuses on the parameters and complex dynamic characteristics when Hopf bifurcation occurs in the three-dimensional IS-LM macroeconomics system. In order to analyze the stability of limit cycles when Hopf bifurcation occurs, this paper further introduces the first Lyapunov coefficient to judge the limit cycles, i.e. from a practical view of the business cycle. Numerical simulation results show that within the range of most of the parameters, the limit cycle of 3D IS-LM macroeconomics is stable, that is, the business cycle is stable; with the increase of the parameters, limit cycles becomes unstable, and the value range of the parameters in this situation is small. The research results of this paper have good guide significance for the analysis of macroeconomics system.
A Lyapunov-Razumikhin approach for stability analysis of logistics networks with time-delays
Dashkovskiy, Sergey; Karimi, Hamid Reza; Kosmykov, Michael
2012-05-01
Logistics network represents a complex system where different elements that are logistic locations interact with each other. This interaction contains delays caused by time needed for delivery of the material. Complexity of the system, time-delays and perturbations in a customer demand may cause unstable behaviour of the network. This leads to the loss of the customers and high inventory costs. Thus the investigation of the network on stability is desired during its design. In this article we consider local input-to-state stability of such logistics networks. Their behaviour is described by a functional differential equation with a constant time-delay. We are looking for verifiable conditions that guarantee stability of the network under consideration. Lyapunov-Razumikhin functions and the local small gain condition are utilised to obtain such conditions. Our stability conditions for the logistics network are based on the information about the interconnection properties between logistic locations and their production rates. Finally, numerical results are provided to demonstrate the proposed approach.
An analysis of the financial crisis in the KOSPI market using Hurst exponents
Yim, Kyubin; Oh, Gabjin; Kim, Seunghwan
2014-09-01
Recently, the study of the financial crisis has progressed to include the concept of the complex system, thereby improving the understanding of this extreme event from a neoclassical economic perspective. To determine which variables are related to the financial event caused by the 2008 US subprime crisis using temporal correlations, we investigate the diverse variables that may explain the financial system. These variables include return, volatility, trading volume and inter-trade duration data sets within the TAQ data for 27 highly capitalized individual companies listed on the KOSPI stock market. During 2008 and 2009, the Hurst exponent for the return time series over the whole period was less than 0.5, and the Hurst exponents for other variables, such as the volatility, trading volume and inter-trade duration, were greater than 0.5. Additionally, we analyze the relationships between the variation of temporal correlation and market instability based on these Hurst exponents and the degree of multifractality. We find that for the data related to trading volume, the Hurst exponents do not allow us to detect changes in market status, such as changes from normal to abnormal status, whereas other variables, including the return, volatility and weekly inter-trade duration, indicate a significant change in market status after the Lehman Brothers' bankruptcy. In addition, the multifractality and the measurement defined by subtracting the Hurst exponent of the return time series from that of the volatility time series decrease sharply after the US subprime event and recover approximately 50 days after the Lehman Brothers' collapse. Our findings suggest that the temporal features of financial quantities in the TAQ data set and the market complexity perform very well at diagnosing financial market stability.
Kolyada, Sergiy; Rybak, Oleksandr
2013-01-01
We introduce and study the Lyapunov numbers -- quantitative measures of the sensitivity of a dynamical system $(X,f)$ given by a compact metric space $X$ and a continuous map $f:X \\to X$. In particular, we prove that for a minimal topologically weakly mixing system all Lyapunov numbers are the same.
Relative Lyapunov Center Bifurcations
DEFF Research Database (Denmark)
Wulff, Claudia; Schilder, Frank
2014-01-01
Relative equilibria (REs) and relative periodic orbits (RPOs) are ubiquitous in symmetric Hamiltonian systems and occur, for example, in celestial mechanics, molecular dynamics, and rigid body motion. REs are equilibria, and RPOs are periodic orbits of the symmetry reduced system. Relative Lyapunov...... center bifurcations are bifurcations of RPOs from REs corresponding to Lyapunov center bifurcations of the symmetry reduced dynamics. In this paper we first prove a relative Lyapunov center theorem by combining recent results on the persistence of RPOs in Hamiltonian systems with a symmetric Lyapunov...... center theorem of Montaldi, Roberts, and Stewart. We then develop numerical methods for the detection of relative Lyapunov center bifurcations along branches of RPOs and for their computation. We apply our methods to Lagrangian REs of the N-body problem....
Nonlinear Dynamic Analysis of MPEG-4 Video Traffic
Institute of Scientific and Technical Information of China (English)
GE Fei; CAO Yang; WANG Yuan-ni
2005-01-01
The main research motive is to analysis and to verify the inherent nonlinear character of MPEG-4 video. The power spectral density estimation of the video trafiic describes its 1/fβ and periodic characteristics. The principal components analysis of the reconstructed space dimension shows only several principal components can be the representation of all dimensions. The correlation dimension analysis proves its fractal characteristic. To accurately compute the largest Lyapunov exponent, the video traffic is divided into many parts. So the largest Lyapunov exponent spectrum is separately calculated using the small data sets method. The largest Lyapunov exponent spectrum shows there exists abundant nonlinear chaos in MPEG-4 video traffic. The conclusion can be made that MPEG-4 video traffic have complex nonlinear behavior and can be characterized by its power spectral density, principal components, correlation dimension and the largest Lyapunov exponent besides its common statistics.
Lyapunov spectra of Coulombic and gravitational periodic systems
Kumar, Pankaj
2016-01-01
We compute Lyapunov spectra for Coulombic and gravitational versions of the one-dimensional systems of parallel sheets with periodic boundary conditions. Exact time evolution of tangent-space vectors are derived and are utilized toward computing Lypaunov characteristic exponents using an event-driven algorithm. The results indicate that the energy dependence of the largest Lyapunov exponent emulates that of Kolmogorov-entropy density for each system at different degrees of freedom. Our approach forms an effective and approximation-free tool toward studying the dynamical properties exhibited by the Coulombic and gravitational systems and finds applications in investigating indications of thermodynamic transitions in large versions of the spatially periodic systems.
Experimental measurement and elaborate analysis of strain hardening exponent in tensile deformation
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
This paper gives a set of formulae for measuring strain hardeningexponent n in different typical deforming routes by using experimental parameters p (forming load), v (velocity of cross-head) and l (gauge length of specimen). With them the uniform method for measuring n (strain hardening exponent at constant strain rate), nv (strain hardening exponent under constant velocity) and np (strain hardening exponent under constant load) is established when , v or p is constant distinctively. Furthermore, the deviation among n values via different typical deformation route is analyzed. The results indicate that there exists structural sensitivity under superplastic and plastic deformation. In addition, the experimental results also prove that the values of n, nv and np obtained with different sets of constant , v or p curves are different too, even if the formulae are the same. Thus a more profound understanding of the relation between the experimental results and the mathematic expressions of n, nv and np is reached and the parameter n is more subtly analyzed by experiment.
Hydrodynamic Lyapunov modes and strong stochasticity threshold in Fermi-Pasta-Ulam models.
Yang, Hong-Liu; Radons, Günter
2006-06-01
The existence of a strong stochasticity threshold (SST) has been detected in many Hamiltonian lattice systems, including the Fermi-Pasta-Ulam (FPU) model, which is characterized by a crossover of the system dynamics from weak to strong chaos with increasing energy density epsilon. Correspondingly, the relaxation time to energy equipartition and the largest Lyapunov exponent exhibit different scaling behavior in the regimes below and beyond the threshold value. In this paper, we attempt to go one step further in this direction to explore further changes in the energy density dependence of other Lyapunov exponents and of hydrodynamic Lyapunov modes (HLMs). In particular, we find that for the FPU-beta and FPU-alpha(beta) models the scalings of the energy density dependence of all Lyapunov exponents experience a similar change at the SST as that of the largest Lyapunov exponent. In addition, the threshold values of the crossover of all Lyapunov exponents are nearly identical. These facts lend support to the point of view that the crossover in the system dynamics at the SST manifests a global change in the geometric structure of phase space. They also partially answer the question of why the simple assumption that the ambient manifold representing the system dynamics is quasi-isotropic works quite well in the analytical calculation of the largest Lyapunov exponent. Furthermore, the FPU-beta model is used as an example to show that HLMs exist in Hamiltonian lattice models with continuous symmetries. Some measures are defined to indicate the significance of HLMs. Numerical simulations demonstrate that there is a smooth transition in the energy density dependence of these variables corresponding to the crossover in Lyapunov exponents at the SST. In particular, our numerical results indicate that strong chaos is essential for the appearance of HLMs and those modes become more significant with increasing degree of chaoticity.
Biased random walks, lyapunov functions, and stochastic analysis of best fit bin packing
Energy Technology Data Exchange (ETDEWEB)
Kenyon, C. [CNRS, Lyon (France); Rabani, Y. [Technion, Haifa (Israel); Sinclair, A. [Univ. of California, Berkeley, CA (United States)
1996-12-31
We study the average case performance of the Best Fit algorithm for on-line bin packing under the distribution U(j,k), in which the item sizes are uniformly distributed in the discrete range (1/k, 2/k,..., j/k). Our main result is that, in the case j = k - 2, the expected waste for an infinite stream of items remains bounded. This settles an open problem posed recently by Coffman et al. It is also the first result which involves a detailed analysis of the infinite multi-dimensional Markov chain underlying the algorithm.
On the recurrence and Lyapunov time scales of the motion near the chaos border
Shevchenko, Ivan I
2016-01-01
Conditions for the emergence of a statistical relationship between $T_r$, the chaotic transport (recurrence) time, and $T_L$, the local Lyapunov time (the inverse of the numerically measured largest Lyapunov characteristic exponent), are considered for the motion inside the chaotic layer around the separatrix of a nonlinear resonance. When numerical values of the Lyapunov exponents are measured on a time interval not greater than $T_r$, the relationship is shown to resemble the quadratic one. This tentatively explains numerical results presented in the literature.
Bosetti, Hadrien; Posch, Harald A; Dellago, Christoph; Hoover, William G
2010-10-01
Recently, a new algorithm for the computation of covariant Lyapunov vectors and of corresponding local Lyapunov exponents has become available. Here we study the properties of these still unfamiliar quantities for a simple model representing a harmonic oscillator coupled to a thermal gradient with a two-stage thermostat, which leaves the system ergodic and fully time reversible. We explicitly demonstrate how time-reversal invariance affects the perturbation vectors in tangent space and the associated local Lyapunov exponents. We also find that the local covariant exponents vary discontinuously along directions transverse to the phase flow.
Mazinan, A H
2016-03-01
The research addresses a Lyapunov-based constrained control strategy to deal with the autonomous space system in the presence of large disturbances. The aforementioned autonomous space system under control is first represented through a dynamics model and subsequently the proposed control strategy is fully investigated with a focus on the three-axis detumbling and the corresponding pointing mode control approaches. The three-axis detumbling mode control approach is designed to deal with the unwanted angular rates of the system to be zero, while the saturations of the actuators are taken into consideration. Moreover, the three-axis pointing mode control approach is designed in the similar state to deal with the rotational angles of the system to be desirable. The contribution of the research is mathematically made to propose a control law in connection with a new candidate of Lyapunov function to deal with the rotational angles and the related angular rates of the present autonomous space system with respect to state-of-the-art. A series of experiments are carried out to consider the efficiency of the proposed control strategy, as long as a number of benchmarks are realized in the same condition to verify and guarantee the strategy performance in both modes of control approaches.
Lyapunov vectors and assimilation in the unstable subspace: theory and applications
Palatella, Luigi; Carrassi, Alberto; Trevisan, Anna
2013-06-01
Based on a limited number of noisy observations, estimation algorithms provide a complete description of the state of a system at current time. Estimation algorithms that go under the name of assimilation in the unstable subspace (AUS) exploit the nonlinear stability properties of the forecasting model in their formulation. Errors that grow due to sensitivity to initial conditions are efficiently removed by confining the analysis solution in the unstable and neutral subspace of the system, the subspace spanned by Lyapunov vectors with positive and zero exponents, while the observational noise does not disturb the system along the stable directions. The formulation of the AUS approach in the context of four-dimensional variational assimilation (4DVar-AUS) and the extended Kalman filter (EKF-AUS) and its application to chaotic models is reviewed. In both instances, the AUS algorithms are at least as efficient but simpler to implement and computationally less demanding than their original counterparts. As predicted by the theory when error dynamics is linear, the optimal subspace dimension for 4DVar-AUS is given by the number of positive and null Lyapunov exponents, while the EKF-AUS algorithm, using the same unstable and neutral subspace, recovers the solution of the full EKF algorithm, but dealing with error covariance matrices of a much smaller dimension and significantly reducing the computational burden. Examples of the application to a simplified model of the atmospheric circulation and to the optimal velocity model for traffic dynamics are given. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to applications’.
Computing Lyapunov spectra with continuous Gram-Schmidt orthonormalization
Christiansen, F; Christiansen, Freddy; Rugh, Hans Henrik
1996-01-01
We present a straightforward and reliable continuous method for computing the full or a partial Lyapunov spectrum associated with a dynamical system specified by a set of differential equations. We do this by introducing a stability parameter beta>0 and augmenting the dynamical system with an orthonormal k-dimensional frame and a Lyapunov vector such that the frame is continuously Gram-Schmidt orthonormalized and at most linear growth of the dynamical variables is involved. We prove that the method is strongly stable when beta > -lambda_k where lambda_k is the k'th Lyapunov exponent in descending order and we show through examples how the method is implemented. It extends many previous results.
Yoon, J.; von Hoyningen-Huene, W.; Kokhanovsky, A. A.; Vountas, M.; Burrows, J. P.
2012-06-01
Regular aerosol observations based on well-calibrated instruments have led to a better understanding of the aerosol radiative budget on Earth. In recent years, these instruments have played an important role in the determination of the increase of anthropogenic aerosols by means of long-term studies. Only few investigations regarding long-term trends of aerosol optical characteristics (e.g. aerosol optical thickness (AOT) and Ångström exponent (ÅE)) have been derived from ground-based observations. This paper aims to derive and discuss linear trends of AOT (440, 675, 870, and 1020 nm) and ÅE (440-870 nm) using AErosol RObotic NETwork (AERONET) level 2.0 spectral observations. Additionally, temporal trends of coarse- and fine-mode dominant AOTs (CdAOT and FdAOT) have been estimated by applying an aerosol classification based on accurate ÅE and Ångström exponent difference (ÅED). In order to take into account the fact that cloud disturbance is having a significant influence on the trend analysis of aerosols, we introduce a weighted least squares regression depending on two weights: (1) monthly standard deviation (σt) and (2) number of observations per month (nt). Temporal increase of FdAOTs (440 nm) prevails over newly industrializing countries in East Asia (weighted trends; +6.23% yr-1 at Beijing) and active agricultural burning regions in South Africa (+1.89% yr-1 at Mongu). On the other hand, insignificant or negative trends for FdAOTs are detected over Western Europe (+0.25% yr-1 at Avignon and -2.29% yr-1 at Ispra) and North America (-0.52% yr-1 for GSFC and -0.01% yr-1 at MD_Science_Center). Over desert regions, both increase and decrease of CdAOTs (+3.37% yr-1 at Solar_Village and -1.18% yr-1 at Ouagadougou) are observed depending on meteorological conditions.
Characteristic exponents of complex networks
Nicosia, Vincenzo; Latora, Vito
2013-01-01
We propose a method to characterize and classify complex networks based on the time series generated by random walks and different node properties. The analysis of the fluctuations of the time series reveals the presence of long-range correlations, and allows to define, for each network, a set of characteristic exponents that capture its essential structural properties. By considering a large data set of real-world networks, we show that the characteristic exponents can be used to classify complex networks according to their function, and are able to discriminate social from biological and technological systems.
Estimating the Lyapunov spectrum of time delay feedback systems from scalar time series.
Hegger, R
1999-08-01
On the basis of a recently developed method for modeling time delay systems, we propose a procedure to estimate the spectrum of Lyapunov exponents from a scalar time series. It turns out that the spectrum is approximated very well and allows for good estimates of the Lyapunov dimension even if the sampling rate of the time series is so low that the infinite dimensional tangent space is spanned quite sparsely.
Lyapunov, Floquet, and singular vectors for baroclinic waves
Directory of Open Access Journals (Sweden)
R. M. Samelson
2001-01-01
Full Text Available The dynamics of the growth of linear disturbances to a chaotic basic state is analyzed in an asymptotic model of weakly nonlinear, baroclinic wave-mean interaction. In this model, an ordinary differential equation for the wave amplitude is coupled to a partial differential equation for the zonal flow correction. The leading Lyapunov vector is nearly parallel to the leading Floquet vector f1 of the lowest-order unstable periodic orbit over most of the attractor. Departures of the Lyapunov vector from this orientation are primarily rotations of the vector in an approximate tangent plane to the large-scale attractor structure. Exponential growth and decay rates of the Lyapunov vector during individual Poincaré section returns are an order of magnitude larger than the Lyapunov exponent l ≈ 0.016. Relatively large deviations of the Lyapunov vector from parallel to f1 are generally associated with relatively large transient decays. The transient growth and decay of the Lyapunov vector is well described by the transient growth and decay of the leading Floquet vectors of the set of unstable periodic orbits associated with the attractor. Each of these vectors is also nearly parallel to f1. The dynamical splitting of the complete sets of Floquet vectors for the higher-order cycles follows the previous results on the lowest-order cycle, with the vectors divided into wave-dynamical and decaying zonal flow modes. Singular vectors and singular values also generally follow this split. The primary difference between the leading Lyapunov and singular vectors is the contribution of decaying, inviscidly-damped wave-dynamical structures to the singular vectors.
Jumping property of Lyapunov values
Institute of Scientific and Technical Information of China (English)
毛锐; 王铎
1996-01-01
A sufficient condition for fcth Lyapunov value to be zero for planar polynomial vector fields is given, which extends the result of "jumping property’ of Lyapunov values obtained by Wang Duo to more general cases. A concrete example that the origin cannot be weak focus of order 1, 2, 4, 5, 8 is presented.
Broucke, R.
1982-01-01
It is pointed out that the Lyapunov Characteristic Numbers constitute a new tool for determining stability of trajectories of dynamical systems, or, even more generally, of solutions of systems of ordinary differential equations. In contrast with the characteristic exponents, which apply only to periodic solutions, the Lyapunov Characteristic Numbers apply to arbitrary nonperiodic solutions as well. A description is presented of the numerical experiments which have been made in order to investigate the practical value of the Lyapunov Characteristic Number and the Kolmogorov Entropy for the purpose of estimating the stability of trajectories and/or numerical integration methods in celestial mechanics. It is found that the Lyapunov Characteristic Numbers are extremely useful for the classification of the solutions of nonintegrable dynamical systems, especially in order to distinguish between quasi-periodic and chaotic solutions. However, the Lyapunov Characteristics Numbers do not appear to be useful for the purpose of evaluating numerical integration methods.
Broucke, R.
1982-01-01
It is pointed out that the Lyapunov Characteristic Numbers constitute a new tool for determining stability of trajectories of dynamical systems, or, even more generally, of solutions of systems of ordinary differential equations. In contrast with the characteristic exponents, which apply only to periodic solutions, the Lyapunov Characteristic Numbers apply to arbitrary nonperiodic solutions as well. A description is presented of the numerical experiments which have been made in order to investigate the practical value of the Lyapunov Characteristic Number and the Kolmogorov Entropy for the purpose of estimating the stability of trajectories and/or numerical integration methods in celestial mechanics. It is found that the Lyapunov Characteristic Numbers are extremely useful for the classification of the solutions of nonintegrable dynamical systems, especially in order to distinguish between quasi-periodic and chaotic solutions. However, the Lyapunov Characteristics Numbers do not appear to be useful for the purpose of evaluating numerical integration methods.
Yang, Hong-Liu; Radons, Günter
2008-01-01
Crossover from weak to strong chaos in high-dimensional Hamiltonian systems at the strong stochasticity threshold (SST) was anticipated to indicate a global transition in the geometric structure of phase space. Our recent study of Fermi-Pasta-Ulam models showed that corresponding to this transition the energy density dependence of all Lyapunov exponents is identical apart from a scaling factor. The current investigation of the dynamic XY model discovers an alternative scenario for the energy dependence of the system dynamics at SSTs. Though similar in tendency, the Lyapunov exponents now show individually different energy dependencies except in the near-harmonic regime. Such a finding restricts the use of indices such as the largest Lyapunov exponent and the Ricci curvatures to characterize the global transition in the dynamics of high-dimensional Hamiltonian systems. These observations are consistent with our conjecture that the quasi-isotropy assumption works well only when parametric resonances are the dominant sources of dynamical instabilities. Moreover, numerical simulations demonstrate the existence of hydrodynamical Lyapunov modes (HLMs) in the dynamic XY model and show that corresponding to the crossover in the Lyapunov exponents there is also a smooth transition in the energy density dependence of significance measures of HLMs. In particular, our numerical results confirm that strong chaos is essential for the appearance of HLMs.
Struzik, Zbigniew R.; van Wijngaarden, Willem J.
We introduce a special purpose cumulative indicator, capturing in real time the cumulative deviation from the reference level of the exponent h (local roughness, Hölder exponent) of the fetal heartbeat during labour. We verify that the indicator applied to the variability component of the heartbeat coincides with the fetal outcome as determined by blood samples. The variability component is obtained from running real time decomposition of fetal heartbeat into independent components using an adaptation of an oversampled Haar wavelet transform. The particular filters used and resolutions applied are motivated by obstetricial insight/practice. The methodology described has the potential for real-time monitoring of the fetus during labour and for the prediction of the fetal outcome, allerting the attending staff in the case of (threatening) hypoxia.
Controller design for TS models using delayed nonquadratic Lyapunov functions.
Lendek, Zsofia; Guerra, Thierry-Marie; Lauber, Jimmy
2015-03-01
In the last few years, nonquadratic Lyapunov functions have been more and more frequently used in the analysis and controller design for Takagi-Sugeno fuzzy models. In this paper, we developed relaxed conditions for controller design using nonquadratic Lyapunov functions and delayed controllers and give a general framework for the use of such Lyapunov functions. The two controller design methods developed in this framework outperform and generalize current state-of-the-art methods. The proposed methods are extended to robust and H∞ control and α -sample variation.
Hristov, Jordan
2010-01-01
The heat-balance integral method of Goodman has been thoroughly analyzed in the case of a parabolic profile with unspecified exponent depending on the boundary condition imposed. That the classical Good man's boundary conditions defining the time-dependent coefficients of the prescribed temperature profile do not work efficiently at the front of the thermal layers if the specific parabolic profile at issue is employed. Additional constraints based on physical assumption enhance the heat-balance integral method and form a robust algorithm defining the parabola exponent . The method has been compared by results provided by the Veinik's method that is by far different from the Good man's idea but also assume forma tion of thermal layer penetrating the heat body. The method has been demonstrated through detailed solutions of 4 1-D heat-conduction problems in Cartesian co-ordinates including a spherical problem (through change of vari ables) and over-specified boundary condition at the face of the thermal layer.
Schaefer, Alexander; Brach, Jennifer S; Perera, Subashan; Sejdić, Ervin
2014-01-30
The time evolution and complex interactions of many nonlinear systems, such as in the human body, result in fractal types of parameter outcomes that exhibit self similarity over long time scales by a power law in the frequency spectrum S(f)=1/f(β). The scaling exponent β is thus often interpreted as a "biomarker" of relative health and decline. This paper presents a thorough comparative numerical analysis of fractal characterization techniques with specific consideration given to experimentally measured gait stride interval time series. The ideal fractal signals generated in the numerical analysis are constrained under varying lengths and biases indicative of a range of physiologically conceivable fractal signals. This analysis is to complement previous investigations of fractal characteristics in healthy and pathological gait stride interval time series, with which this study is compared. The results of our analysis showed that the averaged wavelet coefficient method consistently yielded the most accurate results. Class dependent methods proved to be unsuitable for physiological time series. Detrended fluctuation analysis as most prevailing method in the literature exhibited large estimation variances. The comparative numerical analysis and experimental applications provide a thorough basis for determining an appropriate and robust method for measuring and comparing a physiologically meaningful biomarker, the spectral index β. In consideration of the constraints of application, we note the significant drawbacks of detrended fluctuation analysis and conclude that the averaged wavelet coefficient method can provide reasonable consistency and accuracy for characterizing these fractal time series. Copyright © 2013 Elsevier B.V. All rights reserved.
Stability, Resonance and Lyapunov Inequalities for Periodic Conservative Systems
Canada, Antonio
2010-01-01
This paper is devoted to the study of Lyapunov type inequalities for periodic conservative systems. The main results are derived from a previous analysis which relates the best Lyapunov constants to some especial (constrained or unconstrained) minimization problems. We provide some new results on the existence and uniqueness of solutions of nonlinear resonant and periodic systems. Finally, we present some new conditions which guarantee the stable boundedness of linear periodic conservative systems.
A Spectral Lyapunov Function for Exponentially Stable LTV Systems
Zhu, J. Jim; Liu, Yong; Hang, Rui
2010-01-01
This paper presents the formulation of a Lyapunov function for an exponentially stable linear timevarying (LTV) system using a well-defined PD-spectrum and the associated PD-eigenvectors. It provides a bridge between the first and second methods of Lyapunov for stability assessment, and will find significant applications in the analysis and control law design for LTV systems and linearizable nonlinear time-varying systems.
Chaotic Griffiths Phase with Anomalous Lyapunov Spectra in Coupled Map Networks.
Shinoda, Kenji; Kaneko, Kunihiko
2016-12-16
Dynamics of coupled chaotic oscillators on a network are studied using coupled maps. Within a broad range of parameter values representing the coupling strength or the degree of elements, the system repeats formation and split of coherent clusters. The distribution of the cluster size follows a power law with the exponent α, which changes with the parameter values. The number of positive Lyapunov exponents and their spectra are scaled anomalously with the power of the system size with the exponent β, which also changes with the parameters. The scaling relation α∼2(β+1) is uncovered, which is universal independent of parameters and among random networks.
Chaotic Griffiths Phase with Anomalous Lyapunov Spectra in Coupled Map Networks
Shinoda, Kenji; Kaneko, Kunihiko
2016-12-01
Dynamics of coupled chaotic oscillators on a network are studied using coupled maps. Within a broad range of parameter values representing the coupling strength or the degree of elements, the system repeats formation and split of coherent clusters. The distribution of the cluster size follows a power law with the exponent α , which changes with the parameter values. The number of positive Lyapunov exponents and their spectra are scaled anomalously with the power of the system size with the exponent β , which also changes with the parameters. The scaling relation α ˜2 (β +1 ) is uncovered, which is universal independent of parameters and among random networks.
Lyapunov spectra and conjugate-pairing rule for confined atomic fluids
DEFF Research Database (Denmark)
Bernadi, Stefano; Todd, B.D.; Hansen, Jesper Schmidt
2010-01-01
In this work we present nonequilibrium molecular dynamics simulation results for the Lyapunov spectra of atomic fluids confined in narrow channels of the order of a few atomic diameters. We show the effect that realistic walls have on the Lyapunov spectra. All the degrees of freedom of the confined...... the spectrum reflects the presence of two different dynamics in the system: one for the unthermostatted fluid atoms and the other one for the thermostatted and tethered wall atoms. In particular the Lyapunov spectrum of the whole system does not satisfy the conjugate-pairing rule. Two regions are instead...... distinguishable, one with negative pairs' sum and one with a sum close to zero. To locate the different contributions to the spectrum of the system, we computed "approximate" Lyapunov exponents belonging to the phase space generated by the thermostatted area and the unthermostatted area alone. To achieve this, we...
DEFF Research Database (Denmark)
Skjoldan, P.F.; Hansen, Morten Hartvig
2009-01-01
Structures with isotropic bladed rotors can be modally analyzed by eigenvalue analysis of time-invariant Coleman transformed equations of motion related to the inertial frame or by Floquet analysis of the periodic equations of motion. The Coleman transformation is here shown to be a special case ...
Persistence probabilities \\& exponents
Aurzada, Frank
2012-01-01
This article deals with the asymptotic behaviour as $t\\to +\\infty$ of the survival function $P[T > t],$ where $T$ is the first passage time above a non negative level of a random process starting from zero. In many cases of physical significance, the behaviour is of the type $P[T > t]=t^{-\\theta + o(1)}$ for a known or unknown positive parameter $\\theta$ which is called a persistence exponent. The problem is well understood for random walks or L\\'evy processes but becomes more difficult for integrals of such processes, which are more related to physics. We survey recent results and open problems in this field.
The isentropic exponent in plasmas
K.T.A.L. Burm,; W. J. Goedheer,; D.C. Schram,
1999-01-01
The isentropic exponent for gases is a physical quantity that can ease significantly the hydrodynamic modeling effort. In gas dynamics the isentropic exponent depends only on the number of degrees of freedom of the considered gas. The isentropic exponent for a plasma is lower due to an extra degree
Schaefer, Alexander; Brach, Jennifer S.; Perera, Subashan; Sejdić, Ervin
2013-01-01
Background The time evolution and complex interactions of many nonlinear systems, such as in the human body, result in fractal types of parameter outcomes that exhibit self similarity over long time scales by a power law in the frequency spectrum S(f) = 1/fβ. The scaling exponent β is thus often interpreted as a “biomarker” of relative health and decline. New Method This paper presents a thorough comparative numerical analysis of fractal characterization techniques with specific consideration given to experimentally measured gait stride interval time series. The ideal fractal signals generated in the numerical analysis are constrained under varying lengths and biases indicative of a range of physiologically conceivable fractal signals. This analysis is to complement previous investigations of fractal characteristics in healthy and pathological gait stride interval time series, with which this study is compared. Results The results of our analysis showed that the averaged wavelet coefficient method consistently yielded the most accurate results. Comparison with Existing Methods: Class dependent methods proved to be unsuitable for physiological time series. Detrended fluctuation analysis as most prevailing method in the literature exhibited large estimation variances. Conclusions The comparative numerical analysis and experimental applications provide a thorough basis for determining an appropriate and robust method for measuring and comparing a physiologically meaningful biomarker, the spectral index β. In consideration of the constraints of application, we note the significant drawbacks of detrended fluctuation analysis and conclude that the averaged wavelet coefficient method can provide reasonable consistency and accuracy for characterizing these fractal time series. PMID:24200509
Directory of Open Access Journals (Sweden)
Jesus Manuel Munoz-Pacheco
2013-01-01
Full Text Available An algorithm to compute the Lyapunov exponents of piecewise linear function-based multidirectional multiscroll chaotic oscillators is reported. Based on the m regions in the piecewise linear functions, the suggested algorithm determines the individual expansion rate of Lyapunov exponents from m-piecewise linear variational equations and their associated m-Jacobian matrices whose entries remain constant during all computation cycles. Additionally, by considering OpAmp-based chaotic oscillators, we study the impact of two analog design procedures on the magnitude of Lyapunov exponents. We focus on analyzing variations of both frequency bandwidth and voltage/current dynamic range of the chaotic signals at electronic system level. As a function of the design parameters, a renormalization factor is proposed to estimate correctly the Lyapunov spectrum. Numerical simulation results in a double-scroll type chaotic oscillator and complex chaotic oscillators generating multidirectional multiscroll chaotic attractors on phase space confirm the usefulness of the reported algorithm.
Allometric Exponent and Randomness
Yi, Su Do; Minnhagen, Petter; 10.1088/1367-2630/15/4/043001
2013-01-01
An allometric height-mass exponent $\\gamma$ gives an approximative power-law relation $ \\propto H^\\gamma$ between the average mass $$ and the height $H$, for a sample of individuals. The individuals in the present study are humans but could be any biological organism. The sampling can be for a specific age of the individuals or for an age-interval. The body-mass index (BMI) is often used for practical purposes when characterizing humans and it is based on the allometric exponent $\\gamma=2$. It is here shown that the actual value of $\\gamma$ is to large extent determined by the degree of correlation between mass and height within the sample studied: no correlation between mass and height means $\\gamma=0$, whereas if there was a precise relation between mass and height such that all individuals had the same shape and density then $\\gamma=3$. The connection is demonstrated by showing that the value of $\\gamma$ can be obtained directly from three numbers characterizing the spreads of the relevant random Gaussian ...
Institute of Scientific and Technical Information of China (English)
Feng Guo-Lin; Gong Zhi-Qiang; Zhi Rong; Zhang Da-Quan
2008-01-01
Precipitation sequence is a typical nonlinear and chaotic observational series, and studies on precipitation forecasts are restricted to the use of traditional linear statistical methods, especially when analysing the regional characteristics of precipitation. In the context of 20 stations' daily precipitation series (from 1956 to 2000) in South China (SC) and North China (NC), we divide each precipitation series into many self-stationary segments by using the heuristic segmentation algorithm (briefly BG algorithm). For each station's precipitation series, we calculate the exponent of power-law tail (EPT) of the cumulative probability distribution of segments with a length larger than l for precipitation and temperature series. Our results show that the power-law decay of the cumulative probability distribution of stationary segments might be a common attribution for precipitation and other nonstationary time series; the EPT somewhat indicates the precipitation duration and its spatial distribution that might be different from area to area. The EPT in NC is larger than in SC; Meanwhile, EPT might be another effective way to study the abrupt changes in nonlinear and nonstationary time series.
Institute of Scientific and Technical Information of China (English)
何岱海; 徐健学; 陈永红; 谭宁
2000-01-01
本文研究条件Lyapunov指数与τ-条件Lyapunov指数的定义、求解技术及其应用.两种指数从不同角度对系统本质特性进行刻划.条件Lyapunov指数在混沌同步中有重要应用,近来它还被用来进行相空间重构问题的研究.时间τ-条件Lyapunov指数是一类利用状态变量的离散采样作驱动信号的脉冲方式同步的重要定量指标.本文提出一种简便的求解技术,在Wolf求解Lyapunov指数谱程序的基础上,稍加改动即可使其适用于Lyapunov指数、条件Lyapunov指数和时间τ-条件Lyapunov指数的计算,对其正确性进行了验证.研究发现对时间τ-条件Lyapunov指数的计算,可以准确估计脉冲方式同步的最大间隔τ和最优区间,对于实际工作具有重要意义.
Multifractal Analysis of Local Entropies for Gibbs Measures
Takens, Floris; Verbitski, Evgeni
1998-01-01
Recently a complete multifractal analysis of local dimensions, entropies and Lyapunov exponents of conformal expanding maps and surface Axion A diffeomorphisms for Gibbs measures was performed. The main goal of this was primarily the analysis of the local (pointwise) dimensions. This is an extremely
Yoon, J.; von Hoyningen-Huene, W.; Kokhanovsky, A. A.; Vountas, M.; Burrows, J. P.
2011-08-01
Regular aerosol observations based on well-calibrated instruments have led to a better understanding of the aerosol radiative budget on Earth. In recent years, these instruments have played an important role in the determination of the increase of anthropogenic aerosols by means of long-term studies. Only few investigations regarding long-term trends of aerosol optical characteristics (e.g. Aerosol Optical Thickness (AOT) and Ångström Exponent (ÅE)) have been derived from ground-based observations. This paper aims to derive and discuss linear trends of AOT (440, 675, 870, and 1020 nm) and ÅE (440-870 nm) using AErosol RObotic NETwork (AERONET) spectral observations. Additionally, temporal trends of Coarse- and Fine-mode dominant AOTs (CAOT and FAOT) have been estimated by applying an aerosol classification based on accurate ÅE and Ångström Exponent Difference (ÅED). In order to take into account the fact that cloud disturbance is having a significant influence on the trend analysis of aerosols, we introduce a weighted least squares regression depending on two weights: (1) monthly standard deviation and (2) Number of Observations (NO) per month. Temporal increase of FAOTs prevails over regions dominated by emerging economy or slash-burn agriculture in East Asia and South Africa. On the other hand, insignificant or negative trends for FAOTs are detected over Western Europe and North America. Over desert regions, both increase and decrease of CAOTs are observed depending on meteorological conditions.
Fuwape, Ibiyinka A.; Ogunjo, Samuel T.
2016-12-01
Radio refractivity index is used to quantify the effect of atmospheric parameters in communication systems. Scaling and dynamical complexities of radio refractivity across different climatic zones of Nigeria have been studied. Scaling property of the radio refractivity across Nigeria was estimated from the Hurst Exponent obtained using two different scaling methods namely: The Rescaled Range (R/S) and the detrended fluctuation analysis(DFA). The delay vector variance (DVV), Largest Lyapunov Exponent (λ1) and Correlation Dimension (D2) methods were used to investigate nonlinearity and the results confirm the presence of deterministic nonlinear profile in the radio refractivity time series. The recurrence quantification analysis (RQA) was used to quantify the degree of chaoticity in the radio refractivity across the different climatic zones. RQA was found to be a good measure for identifying unique fingerprint and signature of chaotic time series data. Microwave radio refractivity was found to be persistent and chaotic in all the study locations. The dynamics of radio refractivity increases in complexity and chaoticity from the Coastal region towards the Sahelian climate. The design, development and deployment of robust and reliable microwave communication link in the region will be greatly affected by the chaotic nature of radio refractivity in the region.
Hashimoto, Y.; Shimizu, C.; Kishi, S.; Chao, Y. E.; Wan-Chung, L.
2016-12-01
Changes in stress state with seismic cycles are significant to understand the magnitude and nature of earthquakes. 1999 Chi-Chi earthquake occurred along the Chelung-pu fault, Taiwan. Structural data was obtained from core samples from Taiwan Chelung-pu fault Drilling Project (TCDP). Paleo-stress analysis revealed that the paleo-stress changes between horizontal compression and horizontal extension, which possibly corresponds to stress change before and after earthquake (Hashimoto et al., 2015). The microfault inversion analysis provided an opportunity to classify the micro-faults corresponding to the stress state before and after earthquake. In this study, we have analyzed roughness of micro-faults using power spectrum density and Hurst exponent to understand the roughness change with seismic cycles. Micro-faults were classified into two groups related to stress state before and after earthquake using misfit angle. Misfit angle is the angle between calculated slip direction and observed slip direction for each estimated stress state. Microfaults were sampled from TCDP core. 12 of each samples for horizontal compression and horizontal extension stress state were analyzed. 3D surface data of the slip surface of microfaults were obtained by 3D macro scope (Keyence, VR-3200). Three lines in a surface were analyzed to get power spectrum density-wave number relationships. Hurst exponent is a parameter related to a slope of a log-linear decreasing line in power spectrum density-wave number relationship. Power spectrum density before earthquake is smaller than that after earthquake. Hurst exponent is constant around 0.73-0.75 both in the stress states before and after earthquake. The differences between the directions parallel or vertical to the slip direction were not observed. These results suggest that amplitude decreases with slip at the time of horizontal compression (stress state before earthquake) at to keep the slope in power spectrum density-wave number
Taniguchi, Tooru; Morriss, Gary P
2005-01-01
The time-dependent mode structure of the Lyapunov vectors associated with the stepwise structure of the Lyapunov spectra and its relation to the momentum autocorrelation function are discussed in quasi-one-dimensional many-hard-disk systems. We obtain the complete mode structures (Lyapunov modes) for all components of the Lyapunov vectors, including the longitudinal and transverse components of both the spatial and momentum parts, and their phase relations. These mode structures are suggested by the form of the Lyapunov vectors for the zero-Lyapunov exponents. The spatial node structures of these modes are explained by the reflection properties of the hard walls used in the models. Our main result is that the largest time-oscillating period of the Lyapunov modes is twice as long as the time-oscillating period of the longitudinal momentum autocorrelation function. This relation is satisfied irrespective of the number of particles and the boundary conditions. A simple explanation for this relation is given based on the form of the time-dependent Lyapunov mode.
Construction of the Lyapunov Spectrum in a Chaotic System Displaying Phase Synchronization
Energy Technology Data Exchange (ETDEWEB)
Carlo, Leonardo De, E-mail: neoleodeo@gmail.com [Gran Sasso Science Institute (GSSI) (Italy); Gentile, Guido, E-mail: gentile@mat.uniroma3.it; Giuliani, Alessandro, E-mail: giuliani@mat.uniroma3.it [Università degli Studi Roma Tre, Dipartimento di Matematica e Fisica (Italy)
2016-06-15
We consider a three-dimensional chaotic system consisting of the suspension of Arnold’s cat map coupled with a clock via a weak dissipative interaction. We show that the coupled system displays a synchronization phenomenon, in the sense that the relative phase between the suspension flow and the clock locks to a special value, thus making the motion fall onto a lower dimensional attractor. More specifically, we construct the attractive invariant manifold, of dimension smaller than three, using a convergent perturbative expansion. Moreover, we compute via convergent series the Lyapunov exponents, including notably the central one. The result generalizes a previous construction of the attractive invariant manifold in a similar but simpler model. The main novelty of the current construction relies in the computation of the Lyapunov spectrum, which consists of non-trivial analytic exponents. Some conjectures about a possible smoothening transition of the attractor as the coupling is increased are also discussed.
Lyapunov inequalities for Partial Differential Equations at radial higher eigenvalues
Canada, Antonio
2011-01-01
This paper is devoted to the study of $L_{p}$ Lyapunov-type inequalities ($ \\ 1 \\leq p \\leq +\\infty$) for linear partial differential equations at radial higher eigenvalues. More precisely, we treat the case of Neumann boundary conditions on balls in $\\real^{N}$. It is proved that the relation between the quantities $p$ and $N/2$ plays a crucial role to obtain nontrivial and optimal Lyapunov inequalities. By using appropriate minimizing sequences and a detailed analysis about the number and distribution of zeros of radial nontrivial solutions, we show significant qualitative differences according to the studied case is subcritical, supercritical or critical.
Guastello, Stephen J; Nathan, Dominic E; Johnson, Michelle J
2009-01-01
The principles of attractors and Lyapunov exponents were used to develop a reaching-to-grasp model for use in a robotic therapy system for stroke patients. Previously known models for these movements, the fifth order minimum jerk and the seventh order polynomial, do not account for the change in grasp aperture of the hand. The Lyapunov model was tested with reaching-to-grasp movements performed by five neurologically intact subjects and produced an average R-square = .97 over 15 replications for 41 different task events, reflecting a notable advantage over the fifth order (average R-square = .58) and seventh order (average R-square = .67) models. A similar level of success was obtained for the Lyapunov model that was specific to grasp aperture. The results indicated that intentional movements can be accurately characterized as attractor trajectories, and as functions of position along two Cartesian coordinates rather than as functions of time. The Lyapunov exponent model requires fewer parameters and provides an efficient platform for real-time implementation.
On the Computation of Lyapunov Functions for Interconnected Systems
DEFF Research Database (Denmark)
Sloth, Christoffer
2016-01-01
This paper addresses the computation of additively separable Lyapunov functions for interconnected systems. The presented results can be applied to reduce the complexity of the computations associated with stability analysis of large scale systems. We provide a necessary and sufficient condition...
Lyapunov spectra and conjugate-pairing rule for confined atomic fluids.
Bernardi, Stefano; Todd, B D; Hansen, J S; Searles, Debra J; Frascoli, Federico
2010-06-28
In this work we present nonequilibrium molecular dynamics simulation results for the Lyapunov spectra of atomic fluids confined in narrow channels of the order of a few atomic diameters. We show the effect that realistic walls have on the Lyapunov spectra. All the degrees of freedom of the confined system have been considered. Two different types of flow have been simulated: planar Couette flow and planar Poiseuille flow. Several studies exist on the former for homogeneous flows, so a direct comparison with previous results is performed. An important outcome of this work is the demonstration of how the spectrum reflects the presence of two different dynamics in the system: one for the unthermostatted fluid atoms and the other one for the thermostatted and tethered wall atoms. In particular the Lyapunov spectrum of the whole system does not satisfy the conjugate-pairing rule. Two regions are instead distinguishable, one with negative pairs' sum and one with a sum close to zero. To locate the different contributions to the spectrum of the system, we computed "approximate" Lyapunov exponents belonging to the phase space generated by the thermostatted area and the unthermostatted area alone. To achieve this, we evolved Lyapunov vectors projected into a reduced dimensional phase space. We finally observe that the phase-space compression due to the thermostat remains confined into the wall region and does not significantly affect the purely Newtonian fluid region.
Are Bred Vectors The Same As Lyapunov Vectors?
Kalnay, E.; Corazza, M.; Cai, M.
in the local dimen- sion starts to occur at about 6 BVs, and is essentially complete when the number of vectors is about 10-15 (Corazza et al, 2001a). This should be contrasted with the re- sults of Snyder and Joly (1998) and Palmer et al (1998) who showed that hundreds of Lyapunov vectors with positive Lyapunov exponents are needed to represent the attractor of the system in quasi-geostrophic models. 4) Since only a few bred vectors are needed, and background errors project strongly in the subspace of bred vectors, Corazza et al (2001b) were able to develop cost-efficient methods to improve the 3D-Var data assimilation by adding to the background error covariance terms proportional to the outer product of the bred vectors, thus represent- ing the "errors of the day". This approach led to a reduction of analysis error variance of about 40% at very low cost. 5) The fact that BVs have finite amplitude provides a natural way to filter out instabil- ities present in the system that have fast growth, but saturate nonlinearly at such small amplitudes that they are irrelevant for ensemble perturbations. As shown by Lorenz (1996) Lyapunov vectors (and singular vectors) of models including these physical phenomena would be dominated by the fast but small amplitude instabilities, unless they are explicitly excluded from the linearized models. Bred vectors, on the other 2 hand, through the choice of an appropriate size for the perturbation, provide a natural filter based on nonlinear saturation of fast but irrelevant instabilities. 6) Every bred vector is qualitatively similar to the *leading* LV. LVs beyond the leading LV are obtained by orthogonalization after each time step with respect to the previous LVs subspace. The orthogonalization requires the introduction of a norm. With an enstrophy norm, the successive LVs have larger and larger horizontal scales, and a choice of a stream function norm would lead to successively smaller scales in the LVs. Beyond the first few LVs
Lyapunov Function Synthesis - Algorithm and Software
DEFF Research Database (Denmark)
Leth, Tobias; Wisniewski, Rafal; Sloth, Christoffer
2016-01-01
In this paper we introduce an algorithm for the synthesis of polynomial Lyapunov functions for polynomial vector fields. The Lyapunov function is a continuous piecewisepolynomial defined on simplices, which compose a collection of simplices. The algorithm is elaborated and crucial features...
Rank-one LMIs and Lyapunov's inequality
Henrion, D.; Meinsma, Gjerrit
2001-01-01
We describe a new proof of the well-known Lyapunov's matrix inequality about the location of the eigenvalues of a matrix in some region of the complex plane. The proof makes use of standard facts from quadratic and semi-definite programming. Links are established between the Lyapunov matrix,
Rank-one LMIs and Lyapunov's inequality
Henrion, D.; Meinsma, G.
2001-01-01
We describe a new proof of the well-known Lyapunov's matrix inequality about the location of the eigenvalues of a matrix in some region of the complex plane. The proof makes use of standard facts from quadratic and semi-definite programming. Links are established between the Lyapunov matrix, rank-on
Lyapunov Function Synthesis - Algorithm and Software
DEFF Research Database (Denmark)
Leth, Tobias; Sloth, Christoffer; Wisniewski, Rafal
2016-01-01
In this paper we introduce an algorithm for the synthesis of polynomial Lyapunov functions for polynomial vector fields. The Lyapunov function is a continuous piecewisepolynomial defined on simplices, which compose a collection of simplices. The algorithm is elaborated and crucial features...
Predictability of chaotic dynamics a finite-time Lyapunov exponents approach
Vallejo, Juan C
2017-01-01
This book is primarily concerned with the computational aspects of predictability of dynamical systems – in particular those where observation, modeling and computation are strongly interdependent. Unlike with physical systems under control in laboratories, for instance in celestial mechanics, one is confronted with the observation and modeling of systems without the possibility of altering the key parameters of the objects studied. Therefore, the numerical simulations offer an essential tool for analyzing these systems. With the widespread use of computer simulations to solve complex dynamical systems, the reliability of the numerical calculations is of ever-increasing interest and importance. This reliability is directly related to the regularity and instability properties of the modeled flow. In this interdisciplinary scenario, the underlying physics provide the simulated models, nonlinear dynamics provides their chaoticity and instability properties, and the computer sciences provide the actual numerica...
Robust lyapunov controller for uncertain systems
Laleg-Kirati, Taous-Meriem
2017-02-23
Various examples of systems and methods are provided for Lyapunov control for uncertain systems. In one example, a system includes a process plant and a robust Lyapunov controller configured to control an input of the process plant. The robust Lyapunov controller includes an inner closed loop Lyapunov controller and an outer closed loop error stabilizer. In another example, a method includes monitoring a system output of a process plant; generating an estimated system control input based upon a defined output reference; generating a system control input using the estimated system control input and a compensation term; and adjusting the process plant based upon the system control input to force the system output to track the defined output reference. An inner closed loop Lyapunov controller can generate the estimated system control input and an outer closed loop error stabilizer can generate the system control input.
Infinitesimal Lyapunov functions and singular-hyperbolicity
Araujo, Vitor
2012-01-01
We present an extension of the notion of infinitesimal Lyapunov function to singular flows on three-dimensional manifolds, and show how this technique provides a characterization of partially hyperbolic structures for invariant sets for such flows, and also of singular-hyperbolicity. In the absence of singularities, we can also rephrase uniform hyperbolicity with the language of infinitesimal Lyapunov functions. These conditions are expressed using the vector field X and its space derivative DX together with an infinitesimal Lyapunov function only and are reduced to checking that a certain symmetric operator is positive definite on the trapping region: we show how to express partial hyperbolicity using only the interplay between the infinitesimal generator X of the flow X_t, its derivative DX and the infinitesimal Lyapunov function.
Coordinate-invariant incremental Lyapunov functions
Zamani, Majid
2011-01-01
The notion of incremental stability was proposed by several researchers as a strong property of dynamical and control systems. In this type of stability, the focus is on the convergence of trajectories with respect to themselves, rather than with respect to an equilibrium point or a particular trajectory. Similarly to stability, Lyapunov functions play an important role in the study of incremental stability. In this paper, we propose coordinate-invariant notions of incremental Lyapunov function and provide the description of incremental stability in terms of existence of the proposed Lyapunov functions. Moreover, we develop a backstepping design approach providing a recursive way of constructing controllers as well as incremental Lyapunov functions. The effectiveness of our method is illustrated by synthesizing a controller rendering a single-machine infinite-bus electrical power system incrementally stable.
Localization properties of covariant Lyapunov vectors for quasi-one-dimensional hard disks.
Morriss, G P
2012-05-01
The Lyapunov exponent spectrum and covariant Lyapunov vectors are studied for a quasi-one-dimensional system of hard disks as a function of density and system size. We characterize the system using the angle distributions between covariant vectors and the localization properties of both Gram-Schmidt and covariant vectors. At low density there is a kinetic regime that has simple scaling properties for the Lyapunov exponents and the average localization for part of the spectrum. This regime shows strong localization in a proportion of the first Gram-Schmidt and covariant vectors and this can be understood as highly localized configurations dominating the vector. The distribution of angles between neighboring covariant vectors has characteristic shapes depending upon the difference in vector number, which vary over the continuous region of the spectrum. At dense gas- or liquid-like densities the behavior of the covariant vectors are quite different. The possibility of tangencies between different components of the unstable manifold and between the stable and unstable manifolds is explored but it appears that exact tangencies do not occur for a generic chaotic trajectory.
Analisis Kestabilan Model Matematika Penyakit Leukimia dengan Fungsi Lyapunov
2015-01-01
This study aims to analyze the stability of the equilibrium point of the mathematical model of leukemia before and after undergoing chemotherapy. Analysis of the stability of the model is done by analyzing the model by using a Lyapunov function. By using MATLAB program will be described stability of the model before chemotherapy and after chemotherapy. The results showed that the equilibrium point of stem cell compartment model is asymptotically stable for certain parameter values. This is be...
Stabilization of the Ball on the Beam System by Means of the Inverse Lyapunov Approach
Directory of Open Access Journals (Sweden)
Carlos Aguilar-Ibañez
2012-01-01
Full Text Available A novel inverse Lyapunov approach in conjunction with the energy shaping technique is applied to derive a stabilizing controller for the ball on the beam system. The proposed strategy consists of shaping a candidate Lyapunov function as if it were an inverse stability problem. To this purpose, we fix a suitable dissipation function of the unknown energy function, with the property that the selected dissipation divides the corresponding time derivative of the candidate Lyapunov function. Afterwards, the stabilizing controller is directly obtained from the already shaped Lyapunov function. The stability analysis of the closed-loop system is carried out by using the invariance theorem of LaSalle. Simulation results to test the effectiveness of the obtained controller are presented.
MIMO Lyapunov Theory-Based RBF Neural Classifier for Traffic Sign Recognition
Directory of Open Access Journals (Sweden)
King Hann Lim
2012-01-01
Full Text Available Lyapunov theory-based radial basis function neural network (RBFNN is developed for traffic sign recognition in this paper to perform multiple inputs multiple outputs (MIMO classification. Multidimensional input is inserted into RBF nodes and these nodes are linked with multiple weights. An iterative weight adaptation scheme is hence designed with regards to the Lyapunov stability theory to obtain a set of optimum weights. In the design, the Lyapunov function has to be well selected to construct an energy space with a single global minimum. Weight gain is formed later to obey the Lyapunov stability theory. Detail analysis and discussion on the proposed classifier’s properties are included in the paper. The performance comparisons between the proposed classifier and some existing conventional techniques are evaluated using traffic sign patterns. Simulation results reveal that our proposed system achieved better performance with lower number of training iterations.
Stability of dynamical systems on the role of monotonic and non-monotonic Lyapunov functions
Michel, Anthony N; Liu, Derong
2015-01-01
The second edition of this textbook provides a single source for the analysis of system models represented by continuous-time and discrete-time, finite-dimensional and infinite-dimensional, and continuous and discontinuous dynamical systems. For these system models, it presents results which comprise the classical Lyapunov stability theory involving monotonic Lyapunov functions, as well as corresponding contemporary stability results involving non-monotonicLyapunov functions.Specific examples from several diverse areas are given to demonstrate the applicability of the developed theory to many important classes of systems, including digital control systems, nonlinear regulator systems, pulse-width-modulated feedback control systems, and artificial neural networks. The authors cover the following four general topics: - Representation and modeling of dynamical systems of the types described above - Presentation of Lyapunov and Lagrange stability theory for dynamical sy...
A survey of quantum Lyapunov control methods.
Cong, Shuang; Meng, Fangfang
2013-01-01
The condition of a quantum Lyapunov-based control which can be well used in a closed quantum system is that the method can make the system convergent but not just stable. In the convergence study of the quantum Lyapunov control, two situations are classified: nondegenerate cases and degenerate cases. For these two situations, respectively, in this paper the target state is divided into four categories: the eigenstate, the mixed state which commutes with the internal Hamiltonian, the superposition state, and the mixed state which does not commute with the internal Hamiltonian. For these four categories, the quantum Lyapunov control methods for the closed quantum systems are summarized and analyzed. Particularly, the convergence of the control system to the different target states is reviewed, and how to make the convergence conditions be satisfied is summarized and analyzed.
Critical exponents from cluster coefficients
Rotman, Z.; Eisenberg, E.
2009-09-01
For a large class of repulsive interaction models, the Mayer cluster integrals can be transformed into a tridiagonal real symmetric matrix Rmn , whose elements converge to two constants. This allows for an effective extrapolation of the equation of state for these models. Due to a nearby (nonphysical) singularity on the negative real z axis, standard methods (e.g., Padé approximants based on the cluster integrals expansion) fail to capture the behavior of these models near the ordering transition, and, in particular, do not detect the critical point. A recent work [E. Eisenberg and A. Baram, Proc. Natl. Acad. Sci. U.S.A. 104, 5755 (2007)] has shown that the critical exponents σ and σ' , characterizing the singularity of the density as a function of the activity, can be exactly calculated if the decay of the R matrix elements to their asymptotic constant follows a 1/n2 law. Here we employ renormalization group (RG) arguments to extend this result and analyze cases for which the asymptotic approach of the R matrix elements toward their limiting value is of a more general form. The relevant asymptotic correction terms (in RG sense) are identified, and we then present a corrected exact formula for the critical exponents. We identify the limits of usage of the formula and demonstrate one physical model, which is beyond its range of validity. The formula is validated numerically and then applied to analyze a number of concrete physical models.
Lyapunov functions for fractional order systems
Aguila-Camacho, Norelys; Duarte-Mermoud, Manuel A.; Gallegos, Javier A.
2014-09-01
A new lemma for the Caputo fractional derivatives, when 0<α<1, is proposed in this paper. This result has proved to be useful in order to apply the fractional-order extension of Lyapunov direct method, to demonstrate the stability of many fractional order systems, which can be nonlinear and time varying.
Inertia theorems for operator Lyapunov inequalities
Sasane, AJ; Curtain, RF
2001-01-01
We study operator Lyapunov inequalities and equations for which the infinitesimal generator is not necessarily stable, but it satisfies the spectrum decomposition assumption and it has at most finitely many unstable eigenvalues. Moreover, the input or output operators are not necessarily bounded, bu
Lyapunov Function Synthesis - Infeasibility and Farkas' Lemma
DEFF Research Database (Denmark)
Leth, Tobias; Wisniewski, Rafal; Sloth, Christoffer
2017-01-01
In this paper we prove the convergence of an algorithm synthesising continuous piecewise-polynomial Lyapunov functions for polynomial vector elds dened on simplices. We subsequently modify the algorithm to sub-divide locally by utilizing information from infeasible linear problems. We prove...
Inertia theorems for operator Lyapunov inequalities
Sasane, AJ; Curtain, RF
2001-01-01
We study operator Lyapunov inequalities and equations for which the infinitesimal generator is not necessarily stable, but it satisfies the spectrum decomposition assumption and it has at most finitely many unstable eigenvalues. Moreover, the input or output operators are not necessarily bounded,
Controllability of semilinear matrix Lyapunov systems
Directory of Open Access Journals (Sweden)
Bhaskar Dubey
2013-02-01
Full Text Available In this article, we establish some sufficient conditions for the complete controllability of semilinear matrix Lyapunov systems involving Lipschitzian and non-Lipschitzian nonlinearities. In case of non-Lipschitzian nonlinearities, we assume that nonlinearities are of monotone type.
Analysis and control for a new chaotic system via piecewise linear feedback
Energy Technology Data Exchange (ETDEWEB)
Zhang Jianxiong [Institute of Systems Engineering, Tianjin University, Tianjin 300072 (China)], E-mail: jxzhang@tju.edu.cn; Tang Wansheng [Institute of Systems Engineering, Tianjin University, Tianjin 300072 (China)
2009-11-30
This paper presents a new three-dimensional chaotic system containing two system parameters and a nonlinear term in the form of arc-hyperbolic sine function. The complicated dynamics are studied by virtue of theoretical analysis, numerical simulation and Lyapunov exponents spectrum. The system proposed is converted to an uncertain piecewise linear system. Then, based on piecewise quadratic Lyapunov function technique, the global control of the new chaotic system with {alpha}-stability constraint via piecewise linear state feedback is studied, where the optimal controller maximizing the decay rate {alpha} can be obtained by solving an optimization problem under bilinear matrix inequalities (BMIs) constraints.
On stability of discontinuous systems via vector Lyapunov functions
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
This paper deals with the stability of systems with discontinuous righthand side (with solutions in Filippov's sense) via locally Lipschitz continuous and regular vector Lyapunov functions. A new type of "set-valued derivative" of vector Lyapunov functions is introduced, some generalized comparison principles on dis(c)ontinuous systems are shown. Furthermore, Lyapunov stability theory is developed for a class of discontinuous systems based on locally Lipschitz continuous and regular vector Lyapunov functions.
Nonlinear Dynamic Analysis of the Whole Vehicle on Bumpy Road
Institute of Scientific and Technical Information of China (English)
王威; 李瑰贤; 宋玉玲
2010-01-01
Through the research into the characteristics of 7-DoF high dimensional nonlinear dynamics of a vehicle on bumpy road, the periodic movement and chaotic behavior of the vehicle were found.The methods of nonlinear frequency response analysis, global bifurcation, frequency chart and Poincaré maps were used simultaneously to derive strange super chaotic attractor.According to Lyapunov exponents calculated by Gram-Schmidt method, the unstable region was compartmentalized and the super chaotic characteristic of ...
BAL: A library for the brute-force analysis of dynamical systems
Linaro, Daniele; Storace, Marco
2016-04-01
This paper describes the functionality and usage of BAL, a C/C++ library with a Python front-end for the brute-force analysis of continuous-time dynamical systems described by ordinary differential equations (ODEs). BAL provides an easy-to-use wrapper for the efficient numerical integration of ODEs and, by detecting intersections of the trajectory with appropriate Poincaré sections, allows to classify the asymptotic trajectory of a dynamical system for bifurcation analysis. Some examples of application are discussed, concerning two-dimensional bifurcation diagrams, Lyapunov exponents and finite-time Lyapunov exponents, basins of attraction, simulation of switching ODE systems, and integration with AUTO, a software package for continuation analysis.
Control of acrobot based on Lyapunov function
Institute of Scientific and Technical Information of China (English)
赖旭芝; 吴敏; 佘锦华
2004-01-01
Fuzzy control based on Lyapunov function was employed to control the posture and the energy of an acrobot to make the transition from upswing control to balance control smoothly and stably. First, a control law based on Lyapunov function was used to control the angle and the angular velocity of the second link towards zero when the energy of the acrobot reaches the potential energy at the unstable straight-up equilibrium position in the upswing process. The controller based on Lyapunov function makes the second link straighten nature relatively to the first link. At the same time, a fuzzy controller was designed to regulate the parameters of the upper control law to keep the change of the energy of the acrobot to a minimum, so that the switching from upswing to balance can be properly carried out and the acrobot can enter the balance quickly. The results of simulation show that the switching from upswing to balance can be completed smoothly, and the control effect of the acrobot is improved greatly.
Diophantine exponents for mildly restricted approximation
DEFF Research Database (Denmark)
Bugeaud, Yann; Kristensen, Simon
We are studying the Diophantine exponent defined for integers and a vector by letting , where is the scalar product and denotes the distance to the nearest integer and is the generalised cone consisting of all vectors with the height attained among the first coordinates. We show that the exponent...
Developing Students' Understanding of Exponents and Logarithms.
Weber, Keith
In this paper, we describe instruction designed to teach students about exponents and logarithms and report a pilot study to test the effectiveness of this instruction. Based on the theoretical work of Dubinsky and Sfard, we postulate a set of mental constructions that a student could make to understand the concepts of exponents and logarithms. We…
Bol loops of odd prime exponent
Foguel, Tuval
2009-01-01
Although any finite Bol loop of odd prime exponent is solvable, we show there exist such Bol loops with trivial center. We also construct finitely generated, infinite, simple Bruck loops of odd prime exponent for sufficiently large primes. This shows that the Burnside problem for Bruck loops has a negative answer.
Diophantine exponents for mildly restricted approximation
DEFF Research Database (Denmark)
Bugeaud, Yann; Kristensen, Simon
We are studying the Diophantine exponent defined for integers and a vector by letting , where is the scalar product and denotes the distance to the nearest integer and is the generalised cone consisting of all vectors with the height attained among the first coordinates. We show that the exponent...
Time-varying Hurst exponent for US stock markets
Alvarez-Ramirez, Jose; Alvarez, Jesus; Rodriguez, Eduardo; Fernandez-Anaya, Guillermo
2008-10-01
In this work, the dynamical behavior of the US stock markets is characterized on the basis of the temporal variations of the Hurst exponent estimated with detrended fluctuation analysis (DFA) over moving windows for the historical Dow Jones (1928-2007) and the S&P-500 (1950-2007) daily indices. According to the results drawn: (i) the Hurst exponent displays an erratic dynamics with some episodes alternating low and high persistent behavior, (ii) the major breakthrough of the long-term trend of the scaling behavior occurred in 1972, at the end of the Bretton Woods system, when the Hurst exponent shifted form a positive to a negative long-term trend. Other effects, such as the 1987 crisis and the emergence of anti-correlated behavior in the recent two years, are also discussed.
Critical exponents of a three dimensional O(4) spin model
Kanaya, K; Kanaya, K; Kaya, S
1995-01-01
By Monte Carlo simulation we study the critical exponents governing the transition of the three-dimensional classical O(4) Heisenberg model, which is considered to be in the same universality class as the finite-temperature QCD with massless two flavors. We use the single cluster algorithm and the histogram reweighting technique to obtain observables at the critical temperature. After estimating an accurate value of the inverse critical temperature \\Kc=0.9360(1) we make non-perturbative estimates for various critical exponents by finite-size scaling analysis. They are in excellent agreement with those obtained with the 4-\\epsilon expansion method with errors reduced to about halves of them.
The Scaling Exponent Distinguishes the Injured Sick Hearts Against Normal Healthy Hearts
Yazawa, Toru; Tanaka, Katsunori
2009-05-01
We analyzed heartbeat-intervals with our own program of detrended fluctuation analysis (DFA) to quantify the irregularity of the heartbeat. The present analysis revealed that normal healthy subjects have the scaling exponent of 1.0, and ischemic heart disease pushes the scaling exponent up to 1.2-1.5. We conclude that the scaling exponent, calculated by the DFA, reflects a risk for the "failing" heart. The scaling exponents could determine whether the subjects are under sick or in healthy conditions on the basis of cardiac physiology.
A Converse Sum of Squares Lyapunov Result with a Degree Bound
Peet, Matthew M
2012-01-01
Sum of Squares programming has been used extensively over the past decade for the stability analysis of nonlinear systems but several questions remain unanswered. In this paper, we show that exponential stability of a polynomial vector field on a bounded set implies the existence of a Lyapunov function which is a sum-of-squares of polynomials. In particular, the main result states that if a system is exponentially stable on a bounded nonempty set, then there exists an SOS Lyapunov function which is exponentially decreasing on that bounded set. The proof is constructive and uses the Picard iteration. A bound on the degree of this converse Lyapunov function is also given. This result implies that semidefinite programming can be used to answer the question of stability of a polynomial vector field with a bound on complexity.
Hydrodynamische Lyapunov-Moden in mehrkomponentigen Lennard-Jones-Flüssigkeiten
Drobniewski, Christian
2011-01-01
Die Charakterisierung hochdimensionaler Systeme mit Lyapunov-Instabilität wird durch das Lyapunov-Spektrum und die zugehörigen Lyapunov-Vektoren ermöglicht. Für eine Vielzahl von derartigen Systemen (Coupled-Map-Lattices, Hartkugel-Systeme, Systeme mit ausgedehnten Potentialen ...) konnte durch die Untersuchung der Lyapunov-Vektoren die Existenz von hydrodynamischen Lyapunov-Moden nachgewiesen werden. Diese kollektiven Anregungen zeigen sich in Lyapunov-Vektoren, deren Lyapunov-Exponenten de...
DYNAMICAL ANALYSIS OF A 3-D CHAOTIC SYSTEM WITH ONLY TWO QUADRATIC NONLINEARITIES
Institute of Scientific and Technical Information of China (English)
Zeraoulia ELHADJ
2008-01-01
The paper reports the dynamical study of a three-dimensional quadratic autonomous chaotic system with only two quadratic nonlinearities, which is a special case of the so-called conjugate Lü system. Basic properties of this system are analyzed by means of Lyapunov exponent spectrum and bifurcation diagram. The analysis shows that the system has complex dynamics with some interesting characteristics in which there are several periodic regions, but each of them has quite different periodic orbits.
Time-Delay Systems Lyapunov Functionals and Matrices
Kharitonov, Vladimir L
2013-01-01
Stability is one of the most studied issues in the theory of time-delay systems, but the corresponding chapters of published volumes on time-delay systems do not include a comprehensive study of a counterpart of classical Lyapunov theory for linear delay free systems. The principal goal of the book is to fill this gap, and to provide readers with a systematic and exhaustive treatment of the basic concepts of the Lyapunov-Krasovskii approach to the stability analysis of linear time-delay systems. The book is organized into two parts. The first part is dedicated to the case of retarded type time-delay systems; it consists of four chapters, which respectively deal with results concerning the existence and uniqueness of the solutions of an initial value problem, the class of linear systems with one delay, the case of systems with several delays, and the case of systems with distributed delays. The second part of the book studies the case of neutral type time-delay systems, containing three chapters that e...
Circular orbit spacecraft control at the L4 point using Lyapunov functions
Agrawal, Rachana
2015-01-01
The objective of this work is to demonstrate the utility of Lyapunov functions in control synthesis for the purpose of maintaining and stabilizing a spacecraft in a circular orbit around the L4 point in the circular restricted three body problem (CRTBP). Incorporating the requirements of a fixed radius orbit and a desired angular momentum, a Lyapunov function is constructed and the requisite analysis is performed to obtain a controller. Asymptotic stability is proved in a defined region around the L4 point using LaSalle's principle.
Lyapunov function and the basin of attraction for a single-joint muscle-skeletal model.
Giesl, Peter; Wagner, Heiko
2007-04-01
This paper provides an explicit Lyapunov function for a general single-joint muscle-skeletal model. Using this Lyapunov function one can determine analytically large subsets of the basin of attraction of an asymptotically stable equilibrium. Besides providing an analytical tool for the analysis of such a system we consider an elbow model and show that the theoretical predictions are in agreement with experimental results. Moreover, we can thus distinguish between regions where the self-stabilizing properties of the muscle-skeletal system guarantee stability and regions where nerval control and reflexes are necessary.
Universal scalings of universal scaling exponents
Energy Technology Data Exchange (ETDEWEB)
Llave, Rafael de la [Department of Mathematics, University of Texas, Austin, TX 78712 (United States); Olvera, Arturo [IIMAS-UNAM, FENOMEC, Apdo. Postal 20-726, Mexico DF 01000 (Mexico); Petrov, Nikola P [Department of Mathematics, University of Oklahoma, Norman, OK 73019 (United States)
2007-06-08
In the last decades, renormalization group (RG) ideas have been applied to describe universal properties of different routes to chaos (quasi-periodic, period doubling or tripling, Siegel disc boundaries, etc). Each of the RG theories leads to universal scaling exponents which are related to the action of certain RG operators. The goal of this announcement is to show that there is a principle that organizes many of these scaling exponents. We give numerical evidence that the exponents of different routes to chaos satisfy approximately some arithmetic relations. These relations are determined by combinatorial properties of the route and become exact in an appropriate limit. (fast track communication)
Nonuniform exponential dichotomies and Lyapunov functions
Barreira, Luis; Dragičević, Davor; Valls, Claudia
2017-05-01
For the nonautonomous dynamics defined by a sequence of bounded linear operators acting on an arbitrary Hilbert space, we obtain a characterization of the notion of a nonuniform exponential dichotomy in terms of quadratic Lyapunov sequences. We emphasize that, in sharp contrast with previous results, we consider the general case of possibly noninvertible linear operators, thus requiring only the invertibility along the unstable direction. As an application, we give a simple proof of the robustness of a nonuniform exponential dichotomy under sufficiently small linear perturbations.
Lyapunov Functions and Solutions of the Lyapunov Matrix Equation for Marginally Stable Systems
DEFF Research Database (Denmark)
Kliem, Wolfhard; Pommer, Christian
2000-01-01
of the Lyapunov matrix equation and characterize the set of matrices $(B, C)$ which guarantees marginal stability. The theory is applied to gyroscopic systems, to indefinite damped systems, and to circulatory systems, showing how to choose certain parameter matrices to get sufficient conditions for marginal...
Nonlinear analysis of chaotic flow in a 3D closed-loop pulsating heat pipe
Pouryoussefi, S M
2016-01-01
Numerical simulation has been conducted for the chaotic flow in a 3D closed-loop pulsating heat pipe (PHP). Heat flux and constant temperature boundary conditions were applied for evaporator and condenser sections, respectively. Water and ethanol were used as working fluids. Volume of Fluid (VOF) method has been employed for two-phase flow simulation. Spectral analysis of temperature time series was carried out using Power Spectrum Density (PSD) method. Existence of dominant peak in PSD diagram indicated periodic or quasi-periodic behavior in temperature oscillations at particular frequencies. Correlation dimension values for ethanol as working fluid was found to be higher than that for water under the same operating conditions. Similar range of Lyapunov exponent values for the PHP with water and ethanol as working fluids indicated strong dependency of Lyapunov exponent to the structure and dimensions of the PHP. An O-ring structure pattern was obtained for reconstructed 3D attractor at periodic or quasi-peri...
Dynamic exponents for potts model cluster algorithms
Coddington, Paul D.; Baillie, Clive F.
We have studied the Swendsen-Wang and Wolff cluster update algorithms for the Ising model in 2, 3 and 4 dimensions. The data indicate simple relations between the specific heat and the Wolff autocorrelations, and between the magnetization and the Swendsen-Wang autocorrelations. This implies that the dynamic critical exponents are related to the static exponents of the Ising model. We also investigate the possibility of similar relationships for the Q-state Potts model.
Universality of Tail Exponents of Price Changes?
Huang, Luwen; Farmer, Doyne
2007-03-01
We study the tail exponents of the distribution of logarithmic price changes in financial markets, and investigate the conjecture that they are universal with an exponent near three. Using data from the London Stock Exchange, we construct the empirical distributions of price returns on several different time scales and study their variation as a function of parameters such as trading volume and tick size (the minimal unit of price variation).
The Hurst exponent in energy futures prices
Serletis, Apostolos; Rosenberg, Aryeh Adam
2007-07-01
This paper extends the work in Elder and Serletis [Long memory in energy futures prices, Rev. Financial Econ., forthcoming, 2007] and Serletis et al. [Detrended fluctuation analysis of the US stock market, Int. J. Bifurcation Chaos, forthcoming, 2007] by re-examining the empirical evidence for random walk type behavior in energy futures prices. In doing so, it uses daily data on energy futures traded on the New York Mercantile Exchange, over the period from July 2, 1990 to November 1, 2006, and a statistical physics approach-the ‘detrending moving average’ technique-providing a reliable framework for testing the information efficiency in financial markets as shown by Alessio et al. [Second-order moving average and scaling of stochastic time series, Eur. Phys. J. B 27 (2002) 197-200] and Carbone et al. [Time-dependent hurst exponent in financial time series. Physica A 344 (2004) 267-271; Analysis of clusters formed by the moving average of a long-range correlated time series. Phys. Rev. E 69 (2004) 026105]. The results show that energy futures returns display long memory and that the particular form of long memory is anti-persistence.
Stability of quantized time-delay nonlinear systems : A Lyapunov-Krasowskii-functional approach
Persis, Claudio De; Mazenc, Frédéric
2009-01-01
Lyapunov-Krasowskii functionals are used to design quantized control laws for nonlinear continuous-time systems in the presence of time-invariant constant delays in the input. The quantized control law is implemented via hysteresis to avoid chattering. Under appropriate conditions, our analysis appl
Toropov, A. V.; Toropova, L.V.
2014-01-01
Тhe problem of synthesis of nonlinear speed controller asynchronized switched motor is considered. To find the optimal control law by, the method of Bellman - Lyapunov by concept of "immersion" is used. Modeling and comparative analysis of the system with the standard PI - controller, as well as the synthesized regulators are made
Vannitsem, Stéphane; Lucarini, Valerio
2016-06-01
We study a simplified coupled atmosphere-ocean model using the formalism of covariant Lyapunov vectors (CLVs), which link physically-based directions of perturbations to growth/decay rates. The model is obtained via a severe truncation of quasi-geostrophic equations for the two fluids, and includes a simple yet physically meaningful representation of their dynamical/thermodynamical coupling. The model has 36 degrees of freedom, and the parameters are chosen so that a chaotic behaviour is observed. There are two positive Lyapunov exponents (LEs), sixteen negative LEs, and eighteen near-zero LEs. The presence of many near-zero LEs results from the vast time-scale separation between the characteristic time scales of the two fluids, and leads to nontrivial error growth properties in the tangent space spanned by the corresponding CLVs, which are geometrically very degenerate. Such CLVs correspond to two different classes of ocean/atmosphere coupled modes. The tangent space spanned by the CLVs corresponding to the positive and negative LEs has, instead, a non-pathological behaviour, and one can construct robust large deviations laws for the finite time LEs, thus providing a universal model for assessing predictability on long to ultra-long scales along such directions. Interestingly, the tangent space of the unstable manifold has substantial projection on both atmospheric and oceanic components. The results show the difficulties in using hyperbolicity as a conceptual framework for multiscale chaotic dynamical systems, whereas the framework of partial hyperbolicity seems better suited, possibly indicating an alternative definition for the chaotic hypothesis. They also suggest the need for an accurate analysis of error dynamics on different time scales and domains and for a careful set-up of assimilation schemes when looking at coupled atmosphere-ocean models.
Lyapunov指数与混沌同步的计算研究%Computation of Lyapunov exponents and chaos synchronization
Institute of Scientific and Technical Information of China (English)
谌龙; 王德石
2003-01-01
条件Lyapunov指数是混沌系统同步的重要指标.文中以已知方程的Lyapunov指数谱计算方法为基础,通过数值计算考察了Lyapunov指数随矢量长度、演化时间、置换次数的变化规律,为在现有各种算法中选择参数提供参考.同时,用其计算了混沌同步系统的条件Lyapunov指数,并研究了混沌同步系统的稳定性.
A calculation method of Lyapunov exponent and its realization%一种Lyapunov指数算法及其实现
Institute of Scientific and Technical Information of China (English)
冯明库; 丘水生; 晋建秀
2007-01-01
提出了一种利用周期轨道不同权重计算Lyapunov指数的算法.对混沌序列的周期轨道进行统计,并计算不同的周期轨道的Lyapunov指数,依据周期轨道的权重加权求和得到整个混沌吸引子的平均Lyapunov指数.深入讨论了初始值等对平均Lyapunov指数的影响.该算法不用舍去开始迭代点,适用于复杂混沌系统.
Research on Lyapunov Exponents Algorithm and its Application%Lyapunov指数计算研究及应用
Institute of Scientific and Technical Information of China (English)
廖德玮; 朱伟强
2008-01-01
Lyapunov指数是判定系统是否处于混沌状态的简捷方法之一,但计算Lyapunov指数的诸多方法普遍存在精度不高、受噪声影响大且计算量大等问题而使应用受到限制.借助计算机代数系统Maple建立基于Wolf算法的Lyapunov指数的机械化算法,可以方便地计算Lyapunov指数,从而可以迅速判定系统的混沌性.
Stability of time-delay systems via Lyapunov functions
Directory of Open Access Journals (Sweden)
Carlos F. Alastruey
2002-01-01
Full Text Available In this paper, a Lyapunov function candidate is introduced for multivariable systems with inner delays, without assuming a priori stability for the nondelayed subsystem. By using this Lyapunov function, a controller is deduced. Such a controller utilizes an input–output description of the original system, a circumstance that facilitates practical applications of the proposed approach.
Construction of Lyapunov functions by the localization method
Krishchenko, A. P.; Kanatnikov, A. N.
2017-07-01
In this paper, we examine the problem of construction of Lyapunov functions for asymptotically stable equilibrium points. We exploit conditions of asymptotic stability in terms of compact invariant sets and positively invariant sets. Our results are methods of verification of these conditions and construction of Lyapunov functions by the localization method of compact invariant sets. These results are illustrated by an example.
Scaling Exponents in Financial Markets
Kim, Kyungsik; Kim, Cheol-Hyun; Kim, Soo Yong
2007-03-01
We study the dynamical behavior of four exchange rates in foreign exchange markets. A detrended fluctuation analysis (DFA) is applied to detect the long-range correlation embedded in the non-stationary time series. It is for our case found that there exists a persistent long-range correlation in volatilities, which implies the deviation from the efficient market hypothesis. Particularly, the crossover is shown to exist in the scaling behaviors of the volatilities.
Preparing entangled states by Lyapunov control
Shi, Z. C.; Wang, L. C.; Yi, X. X.
2016-09-01
By Lyapunov control, we present a protocol to prepare entangled states such as Bell states in the context of cavity QED system. The advantage of our method is of threefold. Firstly, we can only control the phase of classical fields to complete the preparation process. Secondly, the evolution time is sharply shortened when compared to adiabatic control. Thirdly, the final state is steady after removing control fields. The influence of decoherence caused by the atomic spontaneous emission and the cavity decay is discussed. The numerical results show that the control scheme is immune to decoherence, especially for the atomic spontaneous emission from |2rangle to |1rangle . This can be understood as the state staying in an invariant subspace. Finally, we generalize this method in preparation of W state.
Preparation of topological modes by Lyapunov control.
Shi, Z C; Zhao, X L; Yi, X X
2015-09-08
By Lyapunov control, we present a proposal to drive quasi-particles into a topological mode in quantum systems described by a quadratic Hamiltonian. The merit of this control is the individual manipulations on the boundary sites. We take the Kitaev's chain as an illustration for Fermi systems and show that an arbitrary excitation mode can be steered into the Majorana zero mode by manipulating the chemical potential of the boundary sites. For Bose systems, taking the noninteracting Su-Schrieffer-Heeger (SSH) model as an example, we illustrate how to drive the system into the edge mode. The sensitivity of the fidelity to perturbations and uncertainties in the control fields and initial modes is also examined. The experimental feasibility of the proposal and the possibility to replace the continuous control field with square wave pulses is finally discussed.
Preparing entangled states by Lyapunov control
Shi, Z. C.; Wang, L. C.; Yi, X. X.
2016-12-01
By Lyapunov control, we present a protocol to prepare entangled states such as Bell states in the context of cavity QED system. The advantage of our method is of threefold. Firstly, we can only control the phase of classical fields to complete the preparation process. Secondly, the evolution time is sharply shortened when compared to adiabatic control. Thirdly, the final state is steady after removing control fields. The influence of decoherence caused by the atomic spontaneous emission and the cavity decay is discussed. The numerical results show that the control scheme is immune to decoherence, especially for the atomic spontaneous emission from |2rangle to |1rangle . This can be understood as the state staying in an invariant subspace. Finally, we generalize this method in preparation of W state.
Numerical solution of large Lyapunov equations
Saad, Youcef
1989-01-01
A few methods are proposed for solving large Lyapunov equations that arise in control problems. The common case where the right hand side is a small rank matrix is considered. For the single input case, i.e., when the equation considered is of the form AX + XA(sup T) + bb(sup T) = 0, where b is a column vector, the existence of approximate solutions of the form X = VGV(sup T) where V is N x m and G is m x m, with m small is established. The first class of methods proposed is based on the use of numerical quadrature formulas, such as Gauss-Laguerre formulas, applied to the controllability Grammian. The second is based on a projection process of Galerkin type. Numerical experiments are presented to test the effectiveness of these methods for large problems.
Fuzzy Lyapunov Reinforcement Learning for Non Linear Systems.
Kumar, Abhishek; Sharma, Rajneesh
2017-03-01
We propose a fuzzy reinforcement learning (RL) based controller that generates a stable control action by lyapunov constraining fuzzy linguistic rules. In particular, we attempt at lyapunov constraining the consequent part of fuzzy rules in a fuzzy RL setup. Ours is a first attempt at designing a linguistic RL controller with lyapunov constrained fuzzy consequents to progressively learn a stable optimal policy. The proposed controller does not need system model or desired response and can effectively handle disturbances in continuous state-action space problems. Proposed controller has been employed on the benchmark Inverted Pendulum (IP) and Rotational/Translational Proof-Mass Actuator (RTAC) control problems (with and without disturbances). Simulation results and comparison against a) baseline fuzzy Q learning, b) Lyapunov theory based Actor-Critic, and c) Lyapunov theory based Markov game controller, elucidate stability and viability of the proposed control scheme.
Experimentally realizable control fields in quantum Lyapunov control
Yi, X X; Wu, Chunfeng; Feng, X L; Oh, C H
2011-01-01
As a hybrid of techniques from open-loop and feedback control, Lyapunov control has the advantage that it is free from the measurement-induced decoherence but it includes the system's instantaneous message in the control loop. Often, the Lyapunov control is confronted with time delay in the control fields and difficulty in practical implementations of the control. In this paper, we study the effect of time-delay on the Lyapunov control, and explore the possibility of replacing the control field with a pulse train or a bang-bang signal. The efficiency of the Lyapunov control is also presented through examining the convergence time of the controlled system. These results suggest that the Lyapunov control is robust gainst time delay, easy to realize and effective for high-dimensional quantum systems.
Lyapunov functionals and stability of stochastic functional differential equations
Shaikhet, Leonid
2013-01-01
Stability conditions for functional differential equations can be obtained using Lyapunov functionals. Lyapunov Functionals and Stability of Stochastic Functional Differential Equations describes the general method of construction of Lyapunov functionals to investigate the stability of differential equations with delays. This work continues and complements the author’s previous book Lyapunov Functionals and Stability of Stochastic Difference Equations, where this method is described for discrete- and continuous-time difference equations. The text begins with a description of the peculiarities of deterministic and stochastic functional differential equations. There follow basic definitions for stability theory of stochastic hereditary systems, and a formal procedure of Lyapunov functionals construction is presented. Stability investigation is conducted for stochastic linear and nonlinear differential equations with constant and distributed delays. The proposed method is used for stability investigation of di...
Determination of the Turbulent Decay Exponent
Perot, J.; Zusi, Chris
2011-11-01
All theories concerning the decay of isotropic turbulence agree that the turbulent kinetic energy has a power law dependence on time. However, there is significant disagreement about what the value of the exponent should be for this power law. The primary theories, proposed by researchers such as Batchelor, Townsend, and Kolmogorov, have the decay exponent at values of 1, 6/5, 10/7, 3/2, 2, and 5/2. The debate over the decay exponent has remained unresolved for many decades because the decay exponent is an extremely sensitive quantity. Experiments have decay times which are too short to be able to accurately differentiate between the various theoretical possibilities, and all prior numerical simulations of decaying turbulence impose the decay rate a priori via the choice of initial conditions. In this work, direct numerical simulation is used to achieve very long decay times, and the initial turbulence is generated by the Navier-Stokes equations and is not imposed. The initial turbulence is created by the stirring action of the flow past 768 small randomly placed cubes. Stirring occurs at 1/30th of the simulation domain size so that the low wavenumber and large scale behavior of the turbulent spectrum which dictates the decay rate is generated by the fluid and is not imposed. It is shown that in all 16 simulations the decay exponent closely matches the theoretical predictions of Saffman at both high and low Reynolds numbers. Perot, AIP Advances 1, 022104 (2011).
Wu, Jing; Wang, Yu; Zhang, Weiwei; Nie, Zhenhua; Lin, Rong; Ma, Hongwei
2017-01-01
This study proposes a novel small defect detection approach for steel pipes using the Lyapunov dimension (D) of the Duffing chaotic system based on ultrasonic guided waves. In this paper, inspection model is constructed by inputting the measured guided wave signal into the Duffing equation as the external turbulent driving force term and then Dis calculated. The properties of the Duffing system's noise immunity are first demonstrated theoretically based on the Lyapunov exponents. By comparing Dof the Duffing inspection system between the conditions of the inputted pure noise and the guided wave signal, the amplitude of the periodic force (F), the important parameter of the Duffing inspection system, could be determined. The values of other parameters of the Duffing inspection system are subsequently determined according to the numerical investigation. Furthermore, a time-moving window function is constructed to scan along the measured signal to locate the defect. And the small defect echo signal polluted by the noise is illustrated to prove the availability of the proposed method. Both numerical and experimental results show that the proposed approach can be used to improve the sensitivity of small defect detection and locate the small defect in pipes.
Interaction of Lyapunov vectors in the formulation of the nonlinear extension of the Kalman filter.
Palatella, Luigi; Trevisan, Anna
2015-04-01
When applied to strongly nonlinear chaotic dynamics the extended Kalman filter (EKF) is prone to divergence due to the difficulty of correctly forecasting the forecast error probability density function. In operational forecasting applications ensemble Kalman filters circumvent this problem with empirical procedures such as covariance inflation. This paper presents an extension of the EKF that includes nonlinear terms in the evolution of the forecast error estimate. This is achieved starting from a particular square-root implementation of the EKF with assimilation confined in the unstable subspace (EKF-AUS), that is, the span of the Lyapunov vectors with non-negative exponents. When the error evolution is nonlinear, the space where it is confined is no more restricted to the unstable and neutral subspace causing filter divergence. The algorithm presented here, denominated EKF-AUS-NL, includes the nonlinear terms in the error dynamics: These result from the nonlinear interaction among the leading Lyapunov vectors and account for all directions where the error growth may take place. Numerical results show that with the nonlinear terms included, filter divergence can be avoided. We test the algorithm on the Lorenz96 model, showing very promising results.
The random phase property and the Lyapunov spectrum for disordered multi-channel systems
Roemer, Rudolf A
2009-01-01
A random phase property establishing a link between quasi-one-dimensional random Schroedinger operators and full random matrix theory is advocated. Briefly summarized it states that the random transfer matrices placed into a normal system of coordinates act on the isotropic frames and lead to a Markov process with a unique invariant measure which is of geometric nature. On the elliptic part of the transfer matrices, this measure is invariant under the full hermitian symplectic group of the universality class under study. While the random phase property can up to now only be proved in special models or in a restricted sense, we provide strong numerical evidence that it holds in the Anderson model of localization. A main outcome of the random phase property is a perturbative calculation of the Lyapunov exponents which shows that the Lyapunov spectrum is equidistant and that the localization lengths for large systems in the unitary, orthogonal and symplectic ensemble differ by a factor 2 each. In an Anderson-And...
A Self-Check System for Mental Health Care based on Nonlinear and Chaos Analysis
Oyama-Higa, Mayumi; Miao, Tiejun; Cheng, Huaichang; Tang, Yuan Guang
2007-11-01
We applied nonlinear and chaos analysis to fingertip pulse wave data. The largest Lyapunov exponent, a measure of the "divergence" of the trajectory of the attractor in phase space, was found to be a useful index of mental health in humans, particularly for the early detection of dementia and depressive psychosis, and for monitoring mental changes in healthy persons. Most of the methods used for assessing mental health are subjective. A few of existing objective methods, such as those using EEG and ECG, for example, are not simple to use and expansive. Therefore, we developed an easy-to-use economical device, a PC mouse with an integrated sensor for measuring the pulse waves, and its required software, to make the measurements. After about 1 min of measurement, the Lyapunov exponent is calculated and displayed as a graph on the PC. An advantage of this system is that the measurements can be made very easily, and hence mental health can be assessed during operating a PC using the pulse wave mouse. Moreover, the measured data can be saved according to the time and date, so diurnal changes and changes over longer time periods can be monitored as a time series and history. At the time the pulse waves are measured, we ask the subject about his or her physical health and mood, and use their responses, along with the Lyapunov exponents, as factors causing variation in the divergence. The changes in the Lyapunov exponent are displayed on the PC as constellation graphs, which we developed to facilitate simpler self-diagnosis and problem resolution.
Non-trivial exponents in coarsening phenomena
Derrida, B.
1997-02-01
One of the simplest examples of stochastic automata is the Glauber dynamics of ferromagnetic spin models such as Ising or Potts models. At zero temperature, if the initial condition is random, one observes a pattern of growing domains with a characteristic size which increases with time like t {1}/{2}. In this self-similar regime, the fraction of spins which never flip up to time t decreases like t- θ where the exponent θ is non-trivial and depends both on the number q of states of the Potts model and on the dimension of space. This exponent can be calculated exactly in one dimension. Similar non-trivial exponents are also present in even simpler models of coarsening, where the dynamical rule is deterministic.
Critical exponent of the fractional Langevin equation.
Burov, S; Barkai, E
2008-02-22
We investigate the dynamical phase diagram of the fractional Langevin equation and show that critical exponents mark dynamical transitions in the behavior of the system. For a free and harmonically bound particle the critical exponent alpha(c)=0.402+/-0.002 marks a transition to a nonmonotonic underdamped phase. The critical exponent alpha(R)=0.441... marks a transition to a resonance phase, when an external oscillating field drives the system. Physically, we explain these behaviors using a cage effect, where the medium induces an elastic type of friction. Phase diagrams describing the underdamped, the overdamped and critical frequencies of the fractional oscillator, recently used to model single protein experiments, show behaviors vastly different from normal.
Error Exponents of Optimum Decoding for the Interference Channel
Etkin, Raul; Ordentlich, Erik
2008-01-01
Exponential error bounds for the finite-alphabet interference channel (IFC) with two transmitter-receiver pairs, are investigated under the random coding regime. Our focus is on optimum decoding, as opposed to heuristic decoding rules that have been used in previous works, like joint typicality decoding, decoding based on interference cancellation, and decoding that considers the interference as additional noise. Indeed, the fact that the actual interfering signal is a codeword and not an i.i.d. noise process complicates the application of conventional techniques to the performance analysis of the optimum decoder. Using analytical tools rooted in statistical physics, we derive a single letter expression for error exponents achievable under optimum decoding and demonstrate strict improvement over error exponents obtainable using suboptimal decoding rules, but which are amenable to more conventional analysis.
Critical exponents from large mass expansion
Yamada, Hirofumi
2014-01-01
We perform estimation of critical exponents via large mass expansion under crucial help of delta-expansion. We address to the three dimensional Ising model at high temperature and estimate omega, the correction-to-scaling exponent, nu, eta and gamma in unbiased and self-contained manner. The results read at the highest 25th order expansion omega=0.8002, nu=0.6295, eta=0.0369 and gamma=1.2357. Estimation biased by omega=0.84(4) is also performed and proved to be in agreement with the summary of recent literatures.
Covariant Lyapunov vectors of chaotic Rayleigh-Bénard convection.
Xu, M; Paul, M R
2016-06-01
We explore numerically the high-dimensional spatiotemporal chaos of Rayleigh-Bénard convection using covariant Lyapunov vectors. We integrate the three-dimensional and time-dependent Boussinesq equations for a convection layer in a shallow square box geometry with an aspect ratio of 16 for very long times and for a range of Rayleigh numbers. We simultaneously integrate many copies of the tangent space equations in order to compute the covariant Lyapunov vectors. The dynamics explored has fractal dimensions of 20≲D_{λ}≲50, and we compute on the order of 150 covariant Lyapunov vectors. We use the covariant Lyapunov vectors to quantify the degree of hyperbolicity of the dynamics and the degree of Oseledets splitting and to explore the temporal and spatial dynamics of the Lyapunov vectors. Our results indicate that the chaotic dynamics of Rayleigh-Bénard convection is nonhyperbolic for all of the Rayleigh numbers we have explored. Our results yield that the entire spectrum of covariant Lyapunov vectors that we have computed are tangled as indicated by near tangencies with neighboring vectors. A closer look at the spatiotemporal features of the Lyapunov vectors suggests contributions from structures at two different length scales with differing amounts of localization.
Large-deviation joint statistics of the finite-time Lyapunov spectrum in isotropic turbulence
Energy Technology Data Exchange (ETDEWEB)
Johnson, Perry L., E-mail: pjohns86@jhu.edu; Meneveau, Charles [Department of Mechanical Engineering and Center for Environmental and Applied Fluid Mechanics, The Johns Hopkins University, 3400 N. Charles Street, Baltimore, Maryland 21218 (United States)
2015-08-15
One of the hallmarks of turbulent flows is the chaotic behavior of fluid particle paths with exponentially growing separation among them while their distance does not exceed the viscous range. The maximal (positive) Lyapunov exponent represents the average strength of the exponential growth rate, while fluctuations in the rate of growth are characterized by the finite-time Lyapunov exponents (FTLEs). In the last decade or so, the notion of Lagrangian coherent structures (which are often computed using FTLEs) has gained attention as a tool for visualizing coherent trajectory patterns in a flow and distinguishing regions of the flow with different mixing properties. A quantitative statistical characterization of FTLEs can be accomplished using the statistical theory of large deviations, based on the so-called Cramér function. To obtain the Cramér function from data, we use both the method based on measuring moments and measuring histograms and introduce a finite-size correction to the histogram-based method. We generalize the existing univariate formalism to the joint distributions of the two FTLEs needed to fully specify the Lyapunov spectrum in 3D flows. The joint Cramér function of turbulence is measured from two direct numerical simulation datasets of isotropic turbulence. Results are compared with joint statistics of FTLEs computed using only the symmetric part of the velocity gradient tensor, as well as with joint statistics of instantaneous strain-rate eigenvalues. When using only the strain contribution of the velocity gradient, the maximal FTLE nearly doubles in magnitude, highlighting the role of rotation in de-correlating the fluid deformations along particle paths. We also extend the large-deviation theory to study the statistics of the ratio of FTLEs. The most likely ratio of the FTLEs λ{sub 1} : λ{sub 2} : λ{sub 3} is shown to be about 4:1:−5, compared to about 8:3:−11 when using only the strain-rate tensor for calculating fluid volume
Bergman kernel and complex singularity exponent
Institute of Scientific and Technical Information of China (English)
LEE; HanJin
2009-01-01
We give a precise estimate of the Bergman kernel for the model domain defined by Ω F={(z,w) ∈ C n+1:Im w |F (z)| 2 > 0},where F=(f 1,...,f m) is a holomorphic map from C n to C m,in terms of the complex singularity exponent of F.
Bergman kernel and complex singularity exponent
Institute of Scientific and Technical Information of China (English)
CHEN BoYong; LEE HanJin
2009-01-01
We give a precise estimate of the Bergman kernel for the model domain defined by Ω_F = {(z,w) ∈ C~(n+1) : Imw - |F(z)|~2 > 0},where F = (f_1,... ,f_m) is a holomorphic map from C~n to C~m,in terms of the complex singularity exponent of F.
Diophantine exponents for mildly restricted approximation
DEFF Research Database (Denmark)
Bugeaud, Yann; Kristensen, S.
2009-01-01
We are studying the Diophantine exponent μ n,l defined for integers 1≤l
Circular orbits, Lyapunov stability and Manev-type forces
Blaga, Cristina
2016-01-01
In this article we study the stability in the sense of Lyapunov of the circular orbits in the generalized Manev two bodies problem. First, we explore the existence of the circular orbits and determine their radius. Then, using the first integrals of motion we build a positive definite function, known as a Lyapunov function. It's existence proves that the circular orbit is stable in the sense of Lyapunov. In the end, we consider several real systems of two bodies and compare the characteristics of the circular orbits in Newtonian and modified Manev gravitational field, arguing about our possibilities to observe the differences between the motion in these two fields.
Automatically Discovering Relaxed Lyapunov Functions for Polynomial Dynamical Systems
Liu, Jiang; Zhao, Hengjun
2011-01-01
The notion of Lyapunov function plays a key role in design and verification of dynamical systems, as well as hybrid and cyber-physical systems. In this paper, to analyze the asymptotic stability of a dynamical system, we generalize standard Lyapunov functions to relaxed Lyapunov functions (RLFs), by considering higher order Lie derivatives of certain functions along the system's vector field. Furthermore, we present a complete method to automatically discovering polynomial RLFs for polynomial dynamical systems (PDSs). Our method is complete in the sense that it is able to discover all polynomial RLFs by enumerating all polynomial templates for any PDS.
Stabilization of nonlinear systems based on robust control Lyapunov function
Institute of Scientific and Technical Information of China (English)
CAI Xiu-shan; HAN Zheng-zhi; LU Gan-yun
2007-01-01
This paper deals with the robust stabilization problem for a class of nonlinear systems with structural uncertainty. Based on robust control Lyapunov function, a sufficient and necessary condition for a function to be a robust control Lyapunov function is given. From this condition, simply sufficient condition for the robust stabilization (robust practical stabilization) is deduced. Moreover, if the equilibrium of the closed-loop system is unique, the existence of such a robust control Lyapunov function will also imply robustly globally asymptotical stabilization. Then a continuous state feedback law can be constructed explicitly. The simulation shows the effectiveness of the method.
Estimation of Hurst Exponent for the Financial Time Series
Kumar, J.; Manchanda, P.
2009-07-01
Till recently statistical methods and Fourier analysis were employed to study fluctuations in stock markets in general and Indian stock market in particular. However current trend is to apply the concepts of wavelet methodology and Hurst exponent, see for example the work of Manchanda, J. Kumar and Siddiqi, Journal of the Frankline Institute 144 (2007), 613-636 and paper of Cajueiro and B. M. Tabak. Cajueiro and Tabak, Physica A, 2003, have checked the efficiency of emerging markets by computing Hurst component over a time window of 4 years of data. Our goal in the present paper is to understand the dynamics of the Indian stock market. We look for the persistency in the stock market through Hurst exponent and fractal dimension of time series data of BSE 100 and NIFTY 50.
Critical dynamics and global persistence exponent on Taiwan financial market
Chen, I C; Li, P C; Chen, H J; Tseng, Hsen-Che; Li, Ping-Cheng; Chen, Hung-Jung
2006-01-01
We investigated the critical dynamics on the daily Taiwan stock exchange index (TSE) from 1971 to 2005, and the 5-min intraday data from 1996 to 2005. A global persistence exponent $\\theta_{p}$ was defined for non-equilibrium critical phenomena \\cite{Janssen,Majumdar}, and describing dynamic behavior in an economic index \\cite{Zheng}. In recent numerical analysis studies of literatures, it is illustrated that the persistence probability has a universal scaling form $P(t) \\sim t^{-\\theta_{p}}$ \\cite{Zheng1}. In this work, we analyzed persistence properties of universal scaling behavior on Taiwan financial market, and also calculated the global persistence exponent $\\theta_{p}$. We found our analytical results in good agreement with the same universality.
Scaling exponents for fracture surfaces in opal glass
Energy Technology Data Exchange (ETDEWEB)
Chavez-Guerrero, L., E-mail: guerreroleo@hotmail.com [Facultad de Ingenieria Mecanica y Electrica. Cd. Universitaria s/n, C.P. 66450, Universidad Autonoma de Nuevo Leon, Nuevo Leon (Mexico); Center of Innovation, Research and Development on Engineering and Technology, Universidad Autonoma de Nuevo Leon Monterrey, C.P. 66600, Apodaca, Nuevo Leon (Mexico); Garza, F.J., E-mail: fjgarza@gama.fime.uanl.mx [Facultad de Ciencias Quimicas, Cd. Universitaria s/n, C.P. 66450, Universidad Autonoma de Nuevo Leon, Nuevo Leon (Mexico); Hinojosa, M., E-mail: hinojosa@gama.fime.uanl.mx [Facultad de Ingenieria Mecanica y Electrica. Cd. Universitaria s/n, C.P. 66450, Universidad Autonoma de Nuevo Leon, Nuevo Leon (Mexico); Center of Innovation, Research and Development on Engineering and Technology, Universidad Autonoma de Nuevo Leon Monterrey, C.P. 66600, Apodaca, Nuevo Leon (Mexico)
2010-09-25
We have investigated the scaling properties of fracture surfaces in opal glass. Specimens with two different opacifying particle sizes (1 {mu}m and 0.4 {mu}m) were broken by three-point bending test and the resulting fracture surfaces were analyzed using Atomic Force Microscopy. The analysis of the self-affine behavior was performed using the Variable Bandwidth and Height-Height Correlation Methods, and both the roughness exponent, {zeta}, and the correlation length, {xi}, were determined. It was found that the roughness exponent obtained in both samples is {zeta} {approx} 0.8; whereas the correlation length in both fractures is of the order of the particle size, demonstrating the dependence of this self-affine parameter on the microstructure of opal glass.
Beaudette, Shawn M; Howarth, Samuel J; Graham, Ryan B; Brown, Stephen H M
2016-10-01
Several different state-space reconstruction methods have been employed to assess the local dynamic stability (LDS) of a 3D kinematic system. One common method is to use a Euclidean norm (N) transformation of three orthogonal x, y, and z time-series' followed by the calculation of the maximum finite-time Lyapunov exponent (λmax) from the resultant N waveform (using a time-delayed state space reconstruction technique). By essentially acting as a weighted average, N has been suggested to account for simultaneous expansion and contraction along separate degrees of freedom within a 3D system (e.g. the coupling of dynamic movements between orthogonal planes). However, when estimating LDS using N, non-linear transformations inherent within the calculation of N should be accounted for. Results demonstrate that the use of N on 3D time-series data with arbitrary magnitudes of relative bias and zero-crossings cause the introduction of error in estimates of λmax obtained through N. To develop a standard for the analysis of 3D dynamic kinematic waveforms, we suggest that all dimensions of a 3D signal be independently shifted to avoid the incidence of zero-crossings prior to the calculation of N and subsequent estimation of LDS through the use of λmax.
A Lyapunov approach to strong stability of semigroups
Paunonen, L.T.; Zwart, Heiko J.
2013-01-01
In this paper we present Lyapunov based proofs for the well-known Arendt–Batty–Lyubich–Vu Theorem for strongly continuous and discrete semigroups. We also study the spectral properties of the limit isometric groups used in the proofs.
Lyapunov functionals and stability of stochastic difference equations
Shaikhet, Leonid
2011-01-01
This book offers a general method of Lyapunov functional construction which lets researchers analyze the degree to which the stability properties of differential equations are preserved in their difference analogues. Includes examples from physical systems.
Lyapunov Computational Method for Two-Dimensional Boussinesq Equation
Mabrouk, Anouar Ben
2010-01-01
A numerical method is developed leading to Lyapunov operators to approximate the solution of two-dimensional Boussinesq equation. It consists of an order reduction method and a finite difference discretization. It is proved to be uniquely solvable and analyzed for local truncation error for consistency. The stability is checked by using Lyapunov criterion and the convergence is studied. Some numerical implementations are provided at the end of the paper to validate the theoretical results.
Stabilization of discrete nonlinear systems based on control Lyapunov functions
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
The stabilization of discrete nonlinear systems is studied.Based on control Lyapunov functions,asufficient and necessary condition for a quadratic function to be a control Lyapunov function is given.From this condition,a continuous state feedback law is constructed explicitly.It can globally asymptotically stabilize the equilibrium of the closed-loop system.A simulation example shows the effectiveness of the proposed method.
Lyapunov control of quantum systems with impulsive control fields.
Yang, Wei; Sun, Jitao
2013-01-01
We investigate the Lyapunov control of finite-dimensional quantum systems with impulsive control fields, where the studied quantum systems are governed by the Schrödinger equation. By three different Lyapunov functions and the invariant principle of impulsive systems, we study the convergence of quantum systems with impulsive control fields and propose new results for the mentioned quantum systems in the form of sufficient conditions. Two numerical simulations are presented to illustrate the effectiveness of the proposed control method.
A unified perspective on robot control - The energy Lyapunov function approach
Wen, John T.
1990-01-01
A unified framework for the stability analysis of robot tracking control is presented. By using an energy-motivated Lyapunov function candidate, the closed-loop stability is shown for a large family of control laws sharing a common structure of proportional and derivative feedback and a model-based feedforward. The feedforward can be zero, partial or complete linearized dynamics, partial or complete nonlinear dynamics, or linearized or nonlinear dynamics with parameter adaptation. As result, the dichotomous approaches to the robot control problem based on the open-loop linearization and nonlinear Lyapunov analysis are both included in this treatment. Furthermore, quantitative estimates of the trade-offs between different schemes in terms of the tracking performance, steady state error, domain of convergence, realtime computation load and required a prior model information are derived.
The evolution of the exponent of Zipf's law in language ontogeny.
Directory of Open Access Journals (Sweden)
Jaume Baixeries
Full Text Available It is well-known that word frequencies arrange themselves according to Zipf's law. However, little is known about the dependency of the parameters of the law and the complexity of a communication system. Many models of the evolution of language assume that the exponent of the law remains constant as the complexity of a communication systems increases. Using longitudinal studies of child language, we analysed the word rank distribution for the speech of children and adults participating in conversations. The adults typically included family members (e.g., parents or the investigators conducting the research. Our analysis of the evolution of Zipf's law yields two main unexpected results. First, in children the exponent of the law tends to decrease over time while this tendency is weaker in adults, thus suggesting this is not a mere mirror effect of adult speech. Second, although the exponent of the law is more stable in adults, their exponents fall below 1 which is the typical value of the exponent assumed in both children and adults. Our analysis also shows a tendency of the mean length of utterances (MLU, a simple estimate of syntactic complexity, to increase as the exponent decreases. The parallel evolution of the exponent and a simple indicator of syntactic complexity (MLU supports the hypothesis that the exponent of Zipf's law and linguistic complexity are inter-related. The assumption that Zipf's law for word ranks is a power-law with a constant exponent of one in both adults and children needs to be revised.
A variational approach to Lyapunov type inequalities from ODEs to PDEs
Cañada, Antonio
2015-01-01
This book highlights the current state of Lyapunov-type inequalities through a detailed analysis. Aimed toward researchers and students working in differential equations and those interested in the applications of stability theory and resonant systems, the book begins with an overview Lyapunov’s original results and moves forward to include prevalent results obtained in the past ten years. Detailed proofs and an emphasis on basic ideas are provided for different boundary conditions for ordinary differential equations, including Neumann, Dirichlet, periodic, and antiperiodic conditions. Novel results of higher eigenvalues, systems of equations, partial differential equations as well as variational approaches are presented. To this respect, a new and unified variational point of view is introduced for the treatment of such problems and a systematic discussion of different types of boundary conditions is featured. Various problems make the study of Lyapunov-type inequalities of interest to those in pure and ...
Universality of modulation length and time exponents.
Chakrabarty, Saurish; Dobrosavljević, Vladimir; Seidel, Alexander; Nussinov, Zohar
2012-10-01
We study systems with a crossover parameter λ, such as the temperature T, which has a threshold value λ(*) across which the correlation function changes from exhibiting fixed wavelength (or time period) modulations to continuously varying modulation lengths (or times). We introduce a hitherto unknown exponent ν(L) characterizing the universal nature of this crossover and compute its value in general instances. This exponent, similar to standard correlation length exponents, is obtained from motion of the poles of the momentum (or frequency) space correlation functions in the complex k-plane (or ω-plane) as the parameter λ is varied. Near the crossover (i.e., for λ→λ(*)), the characteristic modulation wave vector K(R) in the variable modulation length "phase" is related to that in the fixed modulation length "phase" q via |K(R)-q|[proportionality]|T-T(*)|(νL). We find, in general, that ν(L)=1/2. In some special instances, ν(L) may attain other rational values. We extend this result to general problems in which the eigenvalue of an operator or a pole characterizing general response functions may attain a constant real (or imaginary) part beyond a particular threshold value λ(*). We discuss extensions of this result to multiple other arenas. These include the axial next-nearest-neighbor Ising (ANNNI) model. By extending our considerations, we comment on relations pertaining not only to the modulation lengths (or times), but also to the standard correlation lengths (or times). We introduce the notion of a Josephson time scale. We comment on the presence of aperiodic "chaotic" modulations in "soft-spin" and other systems. These relate to glass-type features. We discuss applications to Fermi systems, with particular application to metal to band insulator transitions, change of Fermi surface topology, divergent effective masses, Dirac systems, and topological insulators. Both regular periodic and glassy (and spatially chaotic behavior) may be found in
Critical exponents of the classical Heisenberg ferromagnet
Holm, C; Holm, Christian; Janke, Wolfhard
1997-01-01
In a recent letter, R.G. Brown and M. Ciftan (Phys. Rev. Lett. 76, 1352, 1996) reported high precision Monte Carlo (MC) estimates of the static critical exponents of the classical 3D Heisenberg model, which stand in sharp contrast to values obtained by four independent approaches, namely by other recent high statistics MC simulations, high-temperature series analyses, field theoretical methods, and experimental studies. In reply to the above cited work we submitted this paper as a comment to Phys. Rev. Lett.
On Bruck Loops of 2-power Exponent
Baumeister, Barbara; Stroth, Gernot
2009-01-01
We classify "nice" loop envelopes to Bruck loops of 2-power exponent under the assumption that every nonabelian simple section of $G$ is either passive or isomorphic to $\\PSL_2(q)$, $q-1 \\ge 4$ a 2-power. The hypothesis is verified in a separate paper. This paper is a continuation of the work by Aschbacher, Kinyon and Phillips on finite Bruck loops [AKP]. In [BS3] we applied these results and get a neat description of the structure of the finite Bruck loops.
On monochromatic arm exponents for 2D critical percolation
Beffara, Vincent
2009-01-01
We investigate the so-called monochromatic arm exponents for critical percolation in two dimensions. These exponents, describing the probability of observing j disjoint macroscopic paths, are shown to exist and to form a different family from the (now well-understood) polychromatic exponents.
Lipschitz Properties in Variable Exponent Problems via Relative Rearrangement
Institute of Scientific and Technical Information of China (English)
Jean-Michel RAKOTOSON
2010-01-01
The author first studies the Lipschitz properties of the monotone and relative rearrangement mappings in variable exponent Lebesgue spaces completing the result given in[9].This paper is ended by establishing the Lipschitz properties for quasilinear problems with variable exponent when the right-hand side is in some dual spaces of a suitable Sobolev space associated to variable exponent.
Seismic Activity Seen Through Evolution of the Hurst Exponent Model in 3D
Patiño Ortiz, J.; Carreño Aguilera, R.; Balankin, A. S.; Patiño Ortiz, M.; Tovar Rodriguez, J. C.; Acevedo Mosqueda, M. A.; Martinez Cruz, M. A.; Yu, Wen
2016-10-01
The dynamics seismic activity occurred in the Cocos Plate—Mexico is analyzed through the evolution of Hurst exponent and 3D fractal dimension, under the mathematical fractal structure based on seismic activity time series, taking into account the magnitude (M) as the main parameter to be estimated. The seismic activity time series and, annual intervals are considered first for finding the Hurst exponent of each year since 1988 (the year in which the database is consistent) until 2012, and then the following years are accumulated describing the cumulative Hurst exponent. The seismic activity description is based on Cocos Plate data information; during a period conform from 1 January 1988 to 31 December 2012. Analyses were performed following methods, mainly considering that the Hurst exponent analysis provides the ability to find the seismicity behavior time-space, described by parameters obtained under the fractal dimension and complex systems.
THE SECOND EXPONENT SET OF PRIMITIVE DIGRAPHS
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
Let D = (V,E) be a primitive digraph. The exponent of D, denoted by γ(D), is the least integer k such that for any u, v ∈ V there is a directed walk of length k from u to v. The local exponent of D at a vertex u ∈ V, denoted by expD (u), is the least integer k such that there is a directed walk of length k from u to v for each v ∈ V. Let V = {1,2,... ,n}. Following [1], the vertices of V are ordered so that exPD (1) ≤expD (2)≤…≤expD (n) ＝γ(D). Let En(i) :＝ {expD (i) | D ∈ PDn}, where PDn is the set of all primitive digraphs of order n. It is known that En(n) = {γ(D) | D ∈ PDn} has been completely settled by [7]. In 1998, En(1)was characterized by [5]. In this paper, the authors describe En(2) for all n≥2.
Loops with exponent three in all isotopes
Kinyon, Michael
2011-01-01
It was shown by van Rees \\cite{vR} that a latin square of order $n$ cannot have more than $n^2(n-1)/18$ latin subsquares of order 3. He conjectured that this bound is only achieved if $n$ is a power of 3. We show that it can only be achieved if $n\\equiv3\\bmod6$. We also state several conditions that are equivalent to achieving the van Rees bound. One of these is that the Cayley table of a loop achieves the van Rees bound if and only if every loop isotope has exponent 3. We call such loops \\emph{van Rees loops} and show that they form an equationally defined variety. We also show that (1) In a van Rees loop, any subloop of index 3 is normal, (2) There are exactly 6 nonassociative van Rees loops of order 27 with a non-trivial nucleus, (3) There is a Steiner quasigroup associated with every van Rees loop and (4) Every Bol loop of exponent 3 is a van Rees loop.
Some comments on scaling exponents of turbulence
Baudet, C.; Ciliberto, S.; Nhan Tien, Phan
1993-03-01
Several authors have reported that in turbulence the scaling exponent of the first order velocity structure function increases when the Reynolds number Re decreases. From this result some important consequences on the transition to turbulence could be obtained. However we report experiemntal evidence that this result is coming only from an improper definition of the inertial range. Our data clearly show that the scaling exponents remain constant as a function of Re which is consistent with the selfsimilarity of spectra. Quelques auteurs ont prétendu observer dans les écoulements turbulents un accroissement de l'exposant de la loi de puissance pour la fonction de structure du premier ordre de la vitesse lorsque le nombre de Reynolds décroît. D'importantes déductions relatives à la transition vers la turbulence pourraient être tirées de ce résultat. Cependant, il ressort des résultats expérimentaux que nous avons obtenus récemment que ce résultat ne provient que d'une définition incorrecte du domaine inertiel. Nos données expérimentales prouvent clairement que les exposants des fonctions de structrue de la vitesse ne varient pas avec le nombre de Reynolds, ce qui est cohérent avec le caractère auto-similaire des fluctuations de vitesse en turbulence.
On Controllability and Observability of Fuzzy Dynamical Matrix Lyapunov Systems
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M. S. N. Murty
2008-04-01
Full Text Available We provide a way to combine matrix Lyapunov systems with fuzzy rules to form a new fuzzy system called fuzzy dynamical matrix Lyapunov system, which can be regarded as a new approach to intelligent control. First, we study the controllability property of the fuzzy dynamical matrix Lyapunov system and provide a sufficient condition for its controllability with the use of fuzzy rule base. The significance of our result is that given a deterministic system and a fuzzy state with rule base, we can determine the rule base for the control. Further, we discuss the concept of observability and give a sufficient condition for the system to be observable. The advantage of our result is that we can determine the rule base for the initial value without solving the system.
FFT Bifurcation Analysis of Routes to Chaos via Quasiperiodic Solutions
Directory of Open Access Journals (Sweden)
L. Borkowski
2015-01-01
Full Text Available The dynamics of a ring of seven unidirectionally coupled nonlinear Duffing oscillators is studied. We show that the FFT analysis presented in form of a bifurcation graph, that is, frequency distribution versus a control parameter, can provide a valuable and helpful complement to the corresponding typical bifurcation diagram and the course of Lyapunov exponents, especially in context of detailed identification of the observed attractors. As an example, bifurcation analysis of routes to chaos via 2-frequency and 3-frequency quasiperiodicity is demonstrated.
Nonlinear analysis and prediction of time series in multiphase reactors
Liu, Mingyan
2014-01-01
This book reports on important nonlinear aspects or deterministic chaos issues in the systems of multi-phase reactors. The reactors treated in the book include gas-liquid bubble columns, gas-liquid-solid fluidized beds and gas-liquid-solid magnetized fluidized beds. The authors take pressure fluctuations in the bubble columns as time series for nonlinear analysis, modeling and forecasting. They present qualitative and quantitative non-linear analysis tools which include attractor phase plane plot, correlation dimension, Kolmogorov entropy and largest Lyapunov exponent calculations and local non-linear short-term prediction.
Nonlinear Direct Robust Adaptive Control Using Lyapunov Method
Directory of Open Access Journals (Sweden)
Chunbo Xiu
2013-07-01
Full Text Available The problem of robust adaptive stabilization of a class of multi-input nonlinear systems with arbitrary unknown parameters and unknown structure of bounded variation have been considered. By employing the direct adaptive and control Lyapunov function method, a robust adaptive controller is designed to complete the globally adaptive stability of the system states. By employing our result, a kind of nonlinear system is analyzed, the concrete form of the control law is given and the meaningful quadratic control Lyapunov function for the system is constructed. Simulation of parallel manipulator is provided to illustrate the effectiveness of the proposed method.
The Lyapunov stabilization of satellite equations of motion using integrals
Nacozy, P. E.
1973-01-01
A method is introduced that weakens the Lyapunov or in track instability of satellite equations of motion. The method utilizes a linearized energy integral of satellite motion as a constraint on solutions obtained by numerical integration. The procedure prevents local numerical error from altering the frequency associated with the fast angular variable and thereby reduces the Lyapunov instability and the global numerical error. Applications of the method to satellite motion show accuracy improvements of two to three orders of magnitude in position and velocity after 50 revolutions. A modification of the method is presented that allows the use of slowly varying integrals of motion.
An iterative decoupling solution method for large scale Lyapunov equations
Athay, T. M.; Sandell, N. R., Jr.
1976-01-01
A great deal of attention has been given to the numerical solution of the Lyapunov equation. A useful classification of the variety of solution techniques are the groupings of direct, transformation, and iterative methods. The paper summarizes those methods that are at least partly favorable numerically, giving special attention to two criteria: exploitation of a general sparse system matrix structure and efficiency in resolving the governing linear matrix equation for different matrices. An iterative decoupling solution method is proposed as a promising approach for solving large-scale Lyapunov equation when the system matrix exhibits a general sparse structure. A Fortran computer program that realizes the iterative decoupling algorithm is also discussed.
The Lyapunov stabilization of satellite equations of motion using integrals
Nacozy, P. E.
1973-01-01
A method is introduced that weakens the Lyapunov or in track instability of satellite equations of motion. The method utilizes a linearized energy integral of satellite motion as a constraint on solutions obtained by numerical integration. The procedure prevents local numerical error from altering the frequency associated with the fast angular variable and thereby reduces the Lyapunov instability and the global numerical error. Applications of the method to satellite motion show accuracy improvements of two to three orders of magnitude in position and velocity after 50 revolutions. A modification of the method is presented that allows the use of slowly varying integrals of motion.
An iterative decoupling solution method for large scale Lyapunov equations
Athay, T. M.; Sandell, N. R., Jr.
1976-01-01
A great deal of attention has been given to the numerical solution of the Lyapunov equation. A useful classification of the variety of solution techniques are the groupings of direct, transformation, and iterative methods. The paper summarizes those methods that are at least partly favorable numerically, giving special attention to two criteria: exploitation of a general sparse system matrix structure and efficiency in resolving the governing linear matrix equation for different matrices. An iterative decoupling solution method is proposed as a promising approach for solving large-scale Lyapunov equation when the system matrix exhibits a general sparse structure. A Fortran computer program that realizes the iterative decoupling algorithm is also discussed.
Critical exponents for diluted resistor networks.
Stenull, O; Janssen, H K; Oerding, K
1999-05-01
An approach by Stephen [Phys. Rev. B 17, 4444 (1978)] is used to investigate the critical properties of randomly diluted resistor networks near the percolation threshold by means of renormalized field theory. We reformulate an existing field theory by Harris and Lubensky [Phys. Rev. B 35, 6964 (1987)]. By a decomposition of the principal Feynman diagrams, we obtain diagrams which again can be interpreted as resistor networks. This interpretation provides for an alternative way of evaluating the Feynman diagrams for random resistor networks. We calculate the resistance crossover exponent phi up to second order in epsilon=6-d, where d is the spatial dimension. Our result phi=1+epsilon/42+4epsilon(2)/3087 verifies a previous calculation by Lubensky and Wang, which itself was based on the Potts-model formulation of the random resistor network.
Ising exponents from the functional renormalisation group
Litim, Daniel F
2010-01-01
We study the 3d Ising universality class using the functional renormalisation group. With the help of background fields and a derivative expansion up to fourth order we compute the leading index, the subleading symmetric and anti-symmetric corrections to scaling, the anomalous dimension, the scaling solution, and the eigenperturbations at criticality. We also study the cross-correlations of scaling exponents, and their dependence on dimensionality. We find a very good numerical convergence of the derivative expansion, also in comparison with earlier findings. Evaluating the data from all functional renormalisation group studies to date, we estimate the systematic error which is found to be small and in good agreement with findings from Monte Carlo simulations, \\epsilon-expansion techniques, and resummed perturbation theory.
A conjecture on the norm of Lyapunov mapping
Institute of Scientific and Technical Information of China (English)
Daizhan CHENG; Yahong ZHU; Hongsheng QI
2009-01-01
A conjecture that the norm of Lyapunov mapping LA equals to its restriction to the symmetric set,S,i.e.,‖LA‖ = ‖LA |s‖ was proposed in [1].In this paper,a method for numerical testing is provided first.Then,some recent progress on this conjecture is presented.
Construction of Lyapunov Function for Dissipative Gyroscopic System
Institute of Scientific and Technical Information of China (English)
XU Wei; YUAN Bo; AO Ping
2011-01-01
@@ We introduce a force decomposition to construct a potential function in deterministic dynamics described by ordinary differential equations in the context of dissipative gyroscopic systems.Such a potential function serves as the corresponding Lyapunov function for the dynamics,hence it gives both quantitative and qualitative descriptions for stability of motion.As an example we apply our force decomposition to a four-dimensional dissipative gyroscopic system.We explicitly obtain the potential function for all parameter regimes in the linear limit,including those regimes where the Lyapunov function was previously believed not to exist.%We introduce a force decomposition to construct a potential function in deterministic dynamics described by ordinary differential equations in the context of dissipative gyroscopic systems. Such a potential function serves as the corresponding Lyapunov function for the dynamics, hence it gives both quantitative and qualitative descriptions for stability of motion. As an example we apply our force decomposition to a four-dimensional dissipative gyroscopic system. We explicitly obtain the potential function for all parameter regimes in the linear limit, including those regimes where the Lyapunov function was previously believed not to exist.
Quadratic Lyapunov Function and Exponential Dichotomy on Time Scales
Institute of Scientific and Technical Information of China (English)
ZHANG JI; LIU ZHEN-XIN
2011-01-01
In this paper, we study the relationship between exponential dichotomy and quadratic Lyapunov function for the linear equation x△ ＝ A(t)x on time scales.Moreover, for the nonlinear perturbed equation x△ ＝ A(t)x + f(t,x) we give the instability of the zero solution when f is sufficiently small.
Alignment of Lyapunov Vectors: A Quantitative Criterion to Predict Catastrophes?
Beims, Marcus W; Gallas, Jason A C
2016-11-15
We argue that the alignment of Lyapunov vectors provides a quantitative criterion to predict catastrophes, i.e. the imminence of large-amplitude events in chaotic time-series of observables generated by sets of ordinary differential equations. Explicit predictions are reported for a Rössler oscillator and for a semiconductor laser with optoelectronic feedback.
Control Lyapunov Stabilization of Nonlinear Systems with Structural Uncertainty
Institute of Scientific and Technical Information of China (English)
CAI Xiu-shan; HAN Zheng-zhi; TANG Hou-jun
2005-01-01
This paper deals with global stabilization problem for the nonlinear systems with structural uncertainty.Based on control Lyapunov function, a sufficient and necessary condition for the globally and asymptotically stabilizing the equailibrium of the closed system is given. Moreovery, an almost smooth state feedback control law is constructed. The simulation shows the effectiveness of the method.
Alignment of Lyapunov Vectors: A Quantitative Criterion to Predict Catastrophes?
Beims, Marcus W.; Gallas, Jason A. C.
2016-11-01
We argue that the alignment of Lyapunov vectors provides a quantitative criterion to predict catastrophes, i.e. the imminence of large-amplitude events in chaotic time-series of observables generated by sets of ordinary differential equations. Explicit predictions are reported for a Rössler oscillator and for a semiconductor laser with optoelectronic feedback.
STABILIZATION OF NONLINEAR TIME-VARYING SYSTEMS: A CONTROL LYAPUNOV FUNCTION APPROACH
Institute of Scientific and Technical Information of China (English)
Zhongping JIANG; Yuandan LIN; Yuan WANG
2009-01-01
This paper presents a control Lyapunov function approach to the global stabilization problem for general nonlinear and time-varying systems. Explicit stabilizing feedback control laws are proposed based on the method of control Lyapunov functions and Sontag's universal formula.
Lyapunov matrices approach to the parametric optimization of time-delay systems
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Duda Józef
2015-09-01
Full Text Available In the paper a Lyapunov matrices approach to the parametric optimization problem of time-delay systems with a P-controller is presented. The value of integral quadratic performance index of quality is equal to the value of Lyapunov functional for the initial function of the time-delay system. The Lyapunov functional is determined by means of the Lyapunov matrix
Lyapunov Matrices Approach to the Parametric Optimization of a System with Two Delays
Directory of Open Access Journals (Sweden)
Duda Jozef
2016-09-01
Full Text Available In the paper a Lyapunov matrices approach to the parametric optimization problem of time-delay systems with two commensurate delays and a P-controller is presented. The value of integral quadratic performance index of quality is equal to the value of the Lyapunov functional for the initial function of time-delay system. The Lyapunov functional is determined by means of the Lyapunov matrix.
Analysis and Study of Modern Fault Diagnosis Methods of Mechanical Equipments
Institute of Scientific and Technical Information of China (English)
HOU Rong-tao
2008-01-01
Fast Fourier Transform(FFT)fiequeney spectrum analysis,signal decomposing and reconstruction by wavelet analysis,fractal theory and chaos theory are hot research topics for fault diagnosis and prediction of complex machinery so far.In this paper,the characteristics of the FFT method.wavelet method,fractal method,and largest Lyapunov exponent method are studied and analyzed in detail.The advantages and shortcomings of these methods are pointed out respectively.Some unsolved problems are presented here.
Directory of Open Access Journals (Sweden)
S. Vaidyanathan
2013-09-01
Full Text Available This research work describes the modelling of two novel 3-D chaotic systems, the first with a hyperbolic sinusoidal nonlinearity and two quadratic nonlinearities (denoted as system (A and the second with a hyperbolic cosinusoidal nonlinearity and two quadratic nonlinearities (denoted as system (B. In this work, a detailed qualitative analysis of the novel chaotic systems (A and (B has been presented, and the Lyapunov exponents and Kaplan-Yorke dimension of these chaotic systems have been obtained. It is found that the maximal Lyapunov exponent (MLE for the novel chaotic systems (A and (B has a large value, viz. for the system (A and for the system (B. Thus, both the novel chaotic systems (A and (B display strong chaotic behaviour. This research work also discusses the problem of finding adaptive controllers for the global chaos synchronization of identical chaotic systems (A, identical chaotic systems (B and nonidentical chaotic systems (A and (B with unknown system parameters. The adaptive controllers for achieving global chaos synchronization of the novel chaotic systems (A and (B have been derived using adaptive control theory and Lyapunov stability theory. MATLAB simulations have been shown to illustrate the novel chaotic systems (A and (B, and also the adaptive synchronization results derived for the novel chaotic systems (A and (B.
Hurst exponents for short time series
Qi, Jingchao; Yang, Huijie
2011-12-01
A concept called balanced estimator of diffusion entropy is proposed to detect quantitatively scalings in short time series. The effectiveness is verified by detecting successfully scaling properties for a large number of artificial fractional Brownian motions. Calculations show that this method can give reliable scalings for short time series with length ˜102. It is also used to detect scalings in the Shanghai Stock Index, five stock catalogs, and a total of 134 stocks collected from the Shanghai Stock Exchange Market. The scaling exponent for each catalog is significantly larger compared with that for the stocks included in the catalog. Selecting a window with size 650, the evolution of scaling for the Shanghai Stock Index is obtained by the window's sliding along the series. Global patterns in the evolutionary process are captured from the smoothed evolutionary curve. By comparing the patterns with the important event list in the history of the considered stock market, the evolution of scaling is matched with the stock index series. We can find that the important events fit very well with global transitions of the scaling behaviors.
Efficiency analysis of control algorithms in spatially distributed systems with chaotic behavior
Directory of Open Access Journals (Sweden)
Korus Łukasz
2014-12-01
Full Text Available The paper presents results of examination of control algorithms for the purpose of controlling chaos in spatially distributed systems like the coupled map lattice (CML. The mathematical definition of the CML, stability analysis as well as some basic results of numerical simulation exposing complex, spatiotemporal and chaotic behavior of the CML were already presented in another paper. The main purpose of this article is to compare the efficiency of controlling chaos by simple classical algorithms in spatially distributed systems like CMLs. This comparison is made based on qualitative and quantitative evaluation methods proposed in the previous paper such as the indirect Lyapunov method, Lyapunov exponents and the net direction phase indicator. As a summary of this paper, some conclusions which can be useful for creating a more efficient algorithm of controlling chaos in spatially distributed systems are made.
Lyapunov inequalities for the periodic boundary value problem at higher eigenvalues
Canada, Antonio
2009-01-01
This paper is devoted to provide some new results on Lyapunov type inequalities for the periodic boundary value problem at higher eigenvalues. Our main result is derived from a detailed analysis on the number and distribution of zeros of nontrivial solutions and their first derivatives, together with the study of some special minimization problems, where the Lagrange multiplier Theorem plays a fundamental role. This allows to obtain the optimal constants. Our applications include the Hill's equation where we give some new conditions on its stability properties and also the study of periodic and nonlinear problems at resonance where we show some new conditions which allow to prove the existence and uniqueness of solutions.
The direct Lyapunov method for the stabilisation of the Furuta pendulum
Aguilar-Ibañez, Carlos; Suárez-Castañón, Miguel S.; Gutiérres-Frias, Oscar O.
2010-11-01
A nonlinear controller for the stabilisation of the Furuta pendulum is presented. The control strategy is based on a partial feedback linearisation. In a first stage only the actuated coordinate of the Furuta pendulum is linearised. Then, the stabilising feedback controller is obtained by applying the Lyapunov direct method. That is, using this method we prove local asymptotic stability and demonstrate that the closed-loop system has a large region of attraction. The stability analysis is carried out by means of LaSalle's invariance principle. To assess the controller effectiveness, the results of the corresponding numerical simulations are presented.
Complementarity Properties of the Lyapunov Transformation over Symmetric Cones
Institute of Scientific and Technical Information of China (English)
Yuan Min LI; Xing Tao WANG; De Yun WEI
2012-01-01
The well-known Lyapunov's theorem in matrix theory/continuous dynamical systems asserts that a square matrix A is positive stable if and only if there exists a positive definite matrix X such that AX+XA* is positive definite.In this paper,we extend this theorem to the setting of any Euclidean Jordan algebra V.Given any element a ∈ V,we consider the corresponding Lyapunov transformation La and show that the P and S-properties are both equivalent to a being positive. Then we characterize the Ro-property for La and show that La has the R0-property if and only if a is invertible.Finally,we provide La with some characterizatious of the E0-property and the nondegeneracy property.
Quantum synchronization in an optomechanical system based on Lyapunov control.
Li, Wenlin; Li, Chong; Song, Heshan
2016-06-01
We extend the concepts of quantum complete synchronization and phase synchronization, which were proposed in A. Mari et al., Phys. Rev. Lett. 111, 103605 (2013)PRLTAO0031-900710.1103/PhysRevLett.111.103605, to more widespread quantum generalized synchronization. Generalized synchronization can be considered a necessary condition or a more flexible derivative of complete synchronization, and its criterion and synchronization measure are proposed and analyzed in this paper. As examples, we consider two typical generalized synchronizations in a designed optomechanical system. Unlike the effort to construct a special coupling synchronization system, we purposefully design extra control fields based on Lyapunov control theory. We find that the Lyapunov function can adapt to more flexible control objectives, which is more suitable for generalized synchronization control, and the control fields can be achieved simply with a time-variant voltage. Finally, the existence of quantum entanglement in different generalized synchronizations is also discussed.
Bickel, David R.; Verklan, M. Terese; Moon, Jon
1998-11-01
The scaling exponent of the root mean square (rms) displacement quantifies the roughness of fractal or multifractal time series; it is equivalent to other second-order measures of scaling, such as the power-law exponents of the spectral density and autocorrelation function. For self-similar time series, the rms scaling exponent equals the Hurst parameter, which is related to the fractal dimension. A scaling exponent of 0.5 implies that the process is normal diffusion, which is equivalent to an uncorrelated random walk; otherwise, the process can be modeled as anomalous diffusion. Higher exponents indicate that the increments of the signal have positive correlations, while exponents below 0.5 imply that they have negative correlations. Scaling exponent estimates of successive segments of the increments of a signal are used to test the null hypothesis that the signal is normal diffusion, with the alternate hypothesis that the diffusion is anomalous. Dispersional analysis, a simple technique which does not require long signals, is used to estimate the scaling exponent from the slope of the linear regression of the logarithm of the standard deviation of binned data points on the logarithm of the number of points per bin. Computing the standard error of the scaling exponent using successive segments of the signal is superior to previous methods of obtaining the standard error, such as that based on the sum of squared errors used in the regression; the regression error is more of a measure of the deviation from power-law scaling than of the uncertainty of the scaling exponent estimate. Applying this test to preterm neonate heart rate data, it is found that time intervals between heart beats can be modeled as anomalous diffusion with negatively correlated increments. This corresponds to power spectra between 1/f2 and 1/f, whereas healthy adults are usually reported to have 1/f spectra, suggesting that the immaturity of the neonatal nervous system affects the scaling
Kolmogorov complexity, Lovasz local lemma and critical exponents
Rumyantsev, Andrey
2010-01-01
D. Krieger and J. Shallit have proved that every real number greater than 1 is a critical exponent of some sequence. We show how this result can be derived from some general statements about sequences whose subsequences have (almost) maximal Kolmogorov complexity. In this way one can also construct a sequence that has no "approximate" fractional powers with exponent that exceeds a given value.
The Exponent Set of Central Symmetric Primitive Matrices
Institute of Scientific and Technical Information of China (English)
陈佘喜; 胡亚辉
2004-01-01
This paper first establishes a distance inequality of the associated diagraph of a central symmetric primitive matrix, then characters the exponent set of central symmetric primitive matrices, and proves that the exponent set of central symmetric primitive matrices of order n is {1, 2,… ,n-1}. There is no gap in it.
Scaling Exponents for Lattice Quantum Gravity in Four Dimensions
Hamber, Herbert W
2015-01-01
In this work nonperturbative aspects of quantum gravity are investigated using the lattice formulation, and some new results are presented for critical exponents, amplitudes and invariant correlation functions. Values for the universal scaling dimensions are compared with other nonperturbative approaches to gravity in four dimensions, and specifically to the conjectured value for the universal critical exponent $\
Lyapunov functions for a dengue disease transmission model
Energy Technology Data Exchange (ETDEWEB)
Tewa, Jean Jules [Department of Mathematics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde (Cameroon)], E-mail: tewa@univ-metz.fr; Dimi, Jean Luc [Department of Mathematics, Faculty of Science, University Marien Ngouabi, P.O. Box 69, Brazzaville (Congo, The Democratic Republic of the)], E-mail: jldimi@yahoo.fr; Bowong, Samuel [Department of Mathematics and Computer Science, Faculty of Science, University of Douala, P.O. Box 24157, Douala (Cameroon)], E-mail: samuelbowong@yahoo.fr
2009-01-30
In this paper, we study a model for the dynamics of dengue fever when only one type of virus is present. For this model, Lyapunov functions are used to show that when the basic reproduction ratio is less than or equal to one, the disease-free equilibrium is globally asymptotically stable, and when it is greater than one there is an endemic equilibrium which is also globally asymptotically stable.
Using Lyapunov function to design optimal controller for AQM routers
Institute of Scientific and Technical Information of China (English)
ZHANG Peng; YE Cheng-qing; MA Xue-ying; CHEN Yan-hua; LI Xin
2007-01-01
It was shown that active queue management schemes implemented in the routers of communication networks supporting transmission control protocol (TCP) flows can be modelled as a feedback control system. In this paper based on Lyapunov function we developed an optimal controller to improve active queue management (AQM) router's stability and response time,which are often in conflict with each other in system performance. Ns-2 simulations showed that optimal controller outperforms PI controller significantly.
Lyapunov Criteria for Structural Stability of Supply Chain System
Institute of Scientific and Technical Information of China (English)
LU Ying-jin; TANG Xiao-wo; ZHOU Zong-fang
2004-01-01
In this paper, based on Cobb-Douglas production function, the structural stability of the supply chain system are analyzed by employing Lyapunov criteria. That the supply chain system structure,with the variance of the rate of re-production input funding, becomes unstable is proved. Noticeably, the solutions shows that when the optimal combination of input parameter element, the qualitative properties of supply chain system change and the supply chain system becomes unstable.
Magnetic entropy change and critical exponents in double perovskite Y2NiMnO6
Sharma, G.; Tripathi, T. S.; Saha, J.; Patnaik, S.
2014-11-01
We report the magnetic entropy change (ΔSM) and the critical exponents in the double perovskite manganite Y2NiMnO6 with a ferromagnetic to paramagnetic transition TC~85 K. For a magnetic field change ΔH=80 kOe, a maximum magnetic entropy change ΔSM=-6.57 J/kg K is recorded around TC. The critical exponents β=0.363±0.05 and γ=1.331±0.09 obtained from power law fitting to spontaneous magnetization MS(T) and the inverse initial susceptibility χ0-1(T) satisfy well to values derived for a 3D-Heisenberg ferromagnet. The critical exponent δ=4.761±0.129 is determined from the isothermal magnetization at TC. The scaling exponents corresponding to second order phase transition are consistent with the exponents from Kouvel-Fisher analysis and satisfy Widom's scaling relation δ=1+(γ/β). Additionally, they also satisfy the single scaling equation M(H,ɛ)=ɛβf±(H/ɛ) according to which the magnetization-field-temperature data around TC should collapse into two curves for temperatures below and above TC.
Laminar Flame Velocity and Temperature Exponent of Diluted DME-Air Mixture
Naseer Mohammed, Abdul; Anwar, Muzammil; Juhany, Khalid A.; Mohammad, Akram
2017-03-01
In this paper, the laminar flame velocity and temperature exponent diluted dimethyl ether (DME) air mixtures are reported. Laminar premixed mixture of DME-air with volumetric dilutions of carbon dioxides (CO2) and nitrogen (N2) are considered. Experiments were conducted using a preheated mesoscale high aspect-ratio diverging channel with inlet dimensions of 25 mm × 2 mm. In this method, flame velocities are extracted from planar flames that were stabilized near adiabatic conditions inside the channel. The flame velocities are then plotted against the ratio of mixture temperature and the initial reference temperature. A non-linear power law regression is observed suitable. This regression analysis gives the laminar flame velocity at the initial reference temperature and temperature exponent. Decrease in the laminar flame velocity and increase in temperature exponent is observed for CO2 and N2 diluted mixtures. The addition of CO2 has profound influence when compared to N2 addition on both flame velocity and temperature exponent. Numerical prediction of the similar mixture using a detailed reaction mechanism is obtained. The computational mechanism predicts higher magnitudes for laminar flame velocity and smaller magnitudes of temperature exponent compared to experimental data.
ON THE UPPER GENERALIZED EXPONENTS OF MINISTRONG DIGRAPHS
Institute of Scientific and Technical Information of China (English)
周波
2001-01-01
@@ 1. Introduction A digraph G is called primitive if there exists a positive integer k such that there is a walk of length k from u to v for each ordered pair of not necessarily distinct vertices u and v. The smallest such k is called the exponent of G, denoted by γ(G). Exponents for primitive digraphs have been studied extensively due to their intrinsic importance in graph theory, combinatorics, matrix theory, and their applications in communication problems. As a generalization of exponents, Brualdi and Liu[1] introduced the concept of upper generalized exponents for primitive digraphs. Recently, Shao, Wu and Hwang[2'3] extended this concept of upper generalized exponents from primitive digraphs to general digraphs which are not necessarily primitive.
Scaling Exponent Determined by a Bio-Signal Computation for the Healthy and Diseased Heartbeat
Directory of Open Access Journals (Sweden)
Tomoo Katsuyama
2009-04-01
Full Text Available We analyzed heartbeat-intervals by using our own program of detrended fluctuation analysis (DFA. "Alternans" is an arrhythmia exhibiting alternating amplitude or alternating interval from heartbeat to heartbeat, which was first described in 1872 by Traube. Recently, alternans was finally recognized as the harbinger of a cardiac disease because physicians noticed that an ischemic heart exhibits alternans. To quantify irregularity of the heartbeat including alternans, we used the DFA and revealed that the alternans rhythm lowers the scaling exponent. We conclude that the scaling exponent calculated by the DFA reflects a risk for the "failing" heart. The scaling exponents could determine whether the subjects are under sick or healthy conditions on the basis of cardiac physiology.
High-resolution satellite image segmentation using Hölder exponents
Indian Academy of Sciences (India)
Debasish Chakraborty; Gautam Kumar Sen; Sugata Hazra
2009-10-01
Texture in high-resolution satellite images requires substantial amendment in the conventional segmentation algorithms. A measure is proposed to compute the Hölder exponent (HE) to assess the roughness or smoothness around each pixel of the image. The localized singularity information is incorporated in computing the HE. An optimum window size is evaluated so that HE reacts to localized singularity. A two-step iterative procedure for clustering the transformed HE image is adapted to identify the range of HE, densely occupied in the kernel and to partition Hölder exponents into a cluster that matches with the range. Hölder exponent values (noise or not associated with the other cluster) are clubbed to a nearest possible cluster using the local maximum likelihood analysis.
Shelhamer, Mark; Lowen, Steven B.
2017-01-01
Extraction of fractal exponents via the slope of the power spectrum is common in the analysis of many physiological time series. The fractal structure thus characterized is a manifestation of long-term correlations, for which the temporal order of the sample values is crucial. However, missing data points due to artifacts and dropouts are common in such data sets, which can seriously disrupt the computation of fractal parameters. We evaluated a number of methods for replacing missing data in time series to enable reliable extraction of the fractal exponent and make recommendations as to the preferred replacement method depending on the proportion of missing values and any a priori estimate of the fractal exponent. PMID:28271060
Universality of persistence exponents in two-dimensional Ostwald ripening.
Soriano, Jordi; Braslavsky, Ido; Xu, Di; Krichevsky, Oleg; Stavans, Joel
2009-11-27
We measured persistence exponents theta(phi) of Ostwald ripening in two dimensions, as a function of the area fraction phi occupied by coarsening domains. The values of theta(phi) in two systems, succinonitrile and brine, quenched to their liquid-solid coexistence region, compare well with one another, providing compelling evidence for the universality of the one-parameter family of exponents. For small phi, theta(phi) approximately = 0.39phi, as predicted by a model that assumes no correlations between evolving domains. These constitute the first measurements of persistence exponents in the case of phase transitions with a conserved order parameter.
Bilinear Approximate Model-Based Robust Lyapunov Control for Parabolic Distributed Collectors
Elmetennani, Shahrazed
2016-11-09
This brief addresses the control problem of distributed parabolic solar collectors in order to maintain the field outlet temperature around a desired level. The objective is to design an efficient controller to force the outlet fluid temperature to track a set reference despite the unpredictable varying working conditions. In this brief, a bilinear model-based robust Lyapunov control is proposed to achieve the control objectives with robustness to the environmental changes. The bilinear model is a reduced order approximate representation of the solar collector, which is derived from the hyperbolic distributed equation describing the heat transport dynamics by means of a dynamical Gaussian interpolation. Using the bilinear approximate model, a robust control strategy is designed applying Lyapunov stability theory combined with a phenomenological representation of the system in order to stabilize the tracking error. On the basis of the error analysis, simulation results show good performance of the proposed controller, in terms of tracking accuracy and convergence time, with limited measurement even under unfavorable working conditions. Furthermore, the presented work is of interest for a large category of dynamical systems knowing that the solar collector is representative of physical systems involving transport phenomena constrained by unknown external disturbances.
Passivity/Lyapunov based controller design for trajectory tracking of flexible joint manipulators
Sicard, Pierre; Wen, John T.; Lanari, Leonardo
1992-01-01
A passivity and Lyapunov based approach for the control design for the trajectory tracking problem of flexible joint robots is presented. The basic structure of the proposed controller is the sum of a model-based feedforward and a model-independent feedback. Feedforward selection and solution is analyzed for a general model for flexible joints, and for more specific and practical model structures. Passivity theory is used to design a motor state-based controller in order to input-output stabilize the error system formed by the feedforward. Observability conditions for asymptotic stability are stated and verified. In order to accommodate for modeling uncertainties and to allow for the implementation of a simplified feedforward compensation, the stability of the system is analyzed in presence of approximations in the feedforward by using a Lyapunov based robustness analysis. It is shown that under certain conditions, e.g., the desired trajectory is varying slowly enough, stability is maintained for various approximations of a canonical feedforward.
Intrinsic modulation of ENSO predictability viewed through a local Lyapunov lens
Karamperidou, Christina; Cane, Mark A.; Lall, Upmanu; Wittenberg, Andrew T.
2014-01-01
The presence of rich ENSO variability in the long unforced simulation of GFDL's CM2.1 motivates the use of tools from dynamical systems theory to study variability in ENSO predictability, and its connections to ENSO magnitude, frequency, and physical evolution. Local Lyapunov exponents (LLEs) estimated from the monthly NINO3 SSTa model output are used to characterize periods of increased or decreased predictability. The LLEs describe the growth of infinitesimal perturbations due to internal variability, and are a measure of the immediate predictive uncertainty at any given point in the system phase-space. The LLE-derived predictability estimates are compared with those obtained from the error growth in a set of re-forecast experiments with CM2.1. It is shown that the LLEs underestimate the error growth for short forecast lead times (less than 8 months), while they overestimate it for longer lead times. The departure of LLE-derived error growth rates from the re-forecast rates is a linear function of forecast lead time, and is also sensitive to the length of the time series used for the LLE calculation. The LLE-derived error growth rate is closer to that estimated from the re-forecasts for a lead time of 4 months. In the 2,000-year long simulation, the LLE-derived predictability at the 4-month lead time varies (multi)decadally only by 9-18 %. Active ENSO periods are more predictable than inactive ones, while epochs with regular periodicity and moderate magnitude are classified as the most predictable by the LLEs. Events with a deeper thermocline in the west Pacific up to five years prior to their peak, along with an earlier deepening of the thermocline in the east Pacific in the months preceding the peak, are classified as more predictable. Also, the GCM is found to be less predictable than nature under this measure of predictability.
Asymptotic expansions of Feynman integrals of exponentials with polynomial exponent
Kravtseva, A. K.; Smolyanov, O. G.; Shavgulidze, E. T.
2016-10-01
In the paper, an asymptotic expansion of path integrals of functionals having exponential form with polynomials in the exponent is constructed. The definition of the path integral in the sense of analytic continuation is considered.
Critical exponents of O(N) models in fractional dimensions
Codello, A; D'Odorico, G
2014-01-01
We compute critical exponents of O(N) models in fractal dimensions between two and four, and for continuos values of the number of field components N, in this way completing the RG classification of universality classes for these models. In d=2 the N-dependence of the correlation length critical exponent gives us the last piece of information needed to establish a RG derivation of the Mermin-Wagner theorem. We also report critical exponents for multi-critical universality classes in the cases N>1 and N=0. Finally, in the large-N limit our critical exponents correctly approach those of the spherical model, allowing us to set N~100 as threshold for the quantitative validity of leading order large-N estimates.
A MONTE-CARLO METHOD FOR ESTIMATING THE CORRELATION EXPONENT
MIKOSCH, T; WANG, QA
1995-01-01
We propose a Monte Carlo method for estimating the correlation exponent of a stationary ergodic sequence. The estimator can be considered as a bootstrap version of the classical Hill estimator. A simulation study shows that the method yields reasonable estimates.
A MONTE-CARLO METHOD FOR ESTIMATING THE CORRELATION EXPONENT
MIKOSCH, T; WANG, QA
We propose a Monte Carlo method for estimating the correlation exponent of a stationary ergodic sequence. The estimator can be considered as a bootstrap version of the classical Hill estimator. A simulation study shows that the method yields reasonable estimates.
ACCURATE ESTIMATES OF CHARACTERISTIC EXPONENTS FOR SECOND ORDER DIFFERENTIAL EQUATION
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
In this paper, a second order linear differential equation is considered, and an accurate estimate method of characteristic exponent for it is presented. Finally, we give some examples to verify the feasibility of our result.
Lyapunov Orbits in the Jupiter System Using Electrodynamic Tethers
Bokelmann, Kevin; Russell, Ryan P.; Lantoine, Gregory
2013-01-01
Various researchers have proposed the use of electrodynamic tethers for power generation and capture from interplanetary transfers. The effect of tether forces on periodic orbits in Jupiter-satellite systems are investigated. A perturbation force is added to the restricted three-body problem model and a series of simplifications allows development of a conservative system that retains the Jacobi integral. Expressions are developed to find modified locations of equilibrium positions. Modified families of Lyapunov orbits are generated as functions of tether size and Jacobi integral. Zero velocity curves and stability analyses are used to evaluate the dynamical properties of tether-modified orbits.
Continuation of probability density functions using a generalized Lyapunov approach
Energy Technology Data Exchange (ETDEWEB)
Baars, S., E-mail: s.baars@rug.nl [Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, P.O. Box 407, 9700 AK Groningen (Netherlands); Viebahn, J.P., E-mail: viebahn@cwi.nl [Centrum Wiskunde & Informatica (CWI), P.O. Box 94079, 1090 GB, Amsterdam (Netherlands); Mulder, T.E., E-mail: t.e.mulder@uu.nl [Institute for Marine and Atmospheric research Utrecht, Department of Physics and Astronomy, Utrecht University, Princetonplein 5, 3584 CC Utrecht (Netherlands); Kuehn, C., E-mail: ckuehn@ma.tum.de [Technical University of Munich, Faculty of Mathematics, Boltzmannstr. 3, 85748 Garching bei München (Germany); Wubs, F.W., E-mail: f.w.wubs@rug.nl [Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, P.O. Box 407, 9700 AK Groningen (Netherlands); Dijkstra, H.A., E-mail: h.a.dijkstra@uu.nl [Institute for Marine and Atmospheric research Utrecht, Department of Physics and Astronomy, Utrecht University, Princetonplein 5, 3584 CC Utrecht (Netherlands); School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY (United States)
2017-05-01
Techniques from numerical bifurcation theory are very useful to study transitions between steady fluid flow patterns and the instabilities involved. Here, we provide computational methodology to use parameter continuation in determining probability density functions of systems of stochastic partial differential equations near fixed points, under a small noise approximation. Key innovation is the efficient solution of a generalized Lyapunov equation using an iterative method involving low-rank approximations. We apply and illustrate the capabilities of the method using a problem in physical oceanography, i.e. the occurrence of multiple steady states of the Atlantic Ocean circulation.
Abstraction of Continuous Dynamical Systems Utilizing Lyapunov Functions
DEFF Research Database (Denmark)
Sloth, Christoffer; Wisniewski, Rafal
2010-01-01
This paper considers the development of a method for abstracting continuous dynamical systems by timed automata. The method is based on partitioning the state space of dynamical systems with invariant sets, which form cells representing locations of the timed automata. To enable verification...... of the dynamical system based on the abstraction, conditions for obtaining sound, complete, and refinable abstractions are set up. It is proposed to partition the state space utilizing sub-level sets of Lyapunov functions, since they are positive invariant sets. The existence of sound abstractions for Morse......-Smale systems and complete and refinable abstractions for linear systems are shown....
Suppressing chaos via Lyapunov-Krasovskii's method
Energy Technology Data Exchange (ETDEWEB)
Kuang, J.L. [Faculty of Science and Engineering, City University of Hong Kong, Hong Kong (China)] e-mail: kuangjinlu@hotmail.com; Meehan, P.A. [Department of Mechanical Engineering, University of Queensland, Brisbane, Qld 4072 (Australia)] e-mail: meehan@uq.edu.au; Leung, A.Y.T. [Faculty of Science and Engineering, City University of Hong Kong, Hong Kong (China)] e-mail: bcaleung@cityu.edu.hk
2006-03-01
An algorithm for suppressing the chaotic oscillations in non-linear dynamical systems with singular Jacobian matrices is developed using a linear feedback control law based upon the Lyapunov-Krasovskii (LK) method. It appears that the LK method can serve effectively as a generalised method for the suppression of chaotic oscillations for a wide range of systems. Based on this method, the resulting conditions for undisturbed motions to be locally or globally stable are sufficient and conservative. The generalized Lorenz system and disturbed gyrostat equations are exemplified for the validation of the proposed feedback control rule.
Lyapunov Orbits in the Jupiter System Using Electrodynamic Tethers
Bokelmann, Kevin; Russell, Ryan P.; Lantoine, Gregory
2013-01-01
Various researchers have proposed the use of electrodynamic tethers for power generation and capture from interplanetary transfers. The effect of tether forces on periodic orbits in Jupiter-satellite systems are investigated. A perturbation force is added to the restricted three-body problem model and a series of simplifications allows development of a conservative system that retains the Jacobi integral. Expressions are developed to find modified locations of equilibrium positions. Modified families of Lyapunov orbits are generated as functions of tether size and Jacobi integral. Zero velocity curves and stability analyses are used to evaluate the dynamical properties of tether-modified orbits.
Design of Connectivity Preserving Flocking Using Control Lyapunov Function
Directory of Open Access Journals (Sweden)
Bayu Erfianto
2016-01-01
Full Text Available This paper investigates cooperative flocking control design with connectivity preserving mechanism. During flocking, interagent distance is measured to determine communication topology of the flocks. Then, cooperative flocking motion is built based on cooperative artificial potential field with connectivity preserving mechanism to achieve the common flocking objective. The flocking control input is then obtained by deriving cooperative artificial potential field using control Lyapunov function. As a result, we prove that our flocking protocol establishes group stabilization and the communication topology of multiagent flocking is always connected.
Avrami exponent under transient and heterogeneous nucleation transformation conditions
Sinha, I; Mandal, R. K.
2010-01-01
The Kolmogorov-Johnson-Mehl-Avrami model for isothermal transformation kinetics is universal under specific assumptions. However, the experimental Avrami exponent deviates from the universal value. In this context, we study the effect of transient heterogeneous nucleation on the Avrami exponent for bulk materials and also for transformations leading to nanostructured materials. All transformations are assumed to be polymorphic. A discrete version of the KJMA model is modified for this purpose...
Construction of a Smooth Lyapunov Function for the Robust and Exact Second-Order Differentiator
2016-01-01
Differentiators play an important role in (continuous) feedback control systems. In particular, the robust and exact second-order differentiator has shown some very interesting properties and it has been used successfully in sliding mode control, in spite of the lack of a Lyapunov based procedure to design its gains. As contribution of this paper, we provide a constructive method to determine a differentiable Lyapunov function for such a differentiator. Moreover, the Lyapunov function is used...
Lyapunov functions for a class of nonlinear systems using Caputo derivative
Fernandez-Anaya, G.; Nava-Antonio, G.; Jamous-Galante, J.; Muñoz-Vega, R.; Hernández-Martínez, E. G.
2017-02-01
This paper presents an extension of recent results that allow proving the stability of Caputo nonlinear and time-varying systems, by means of the fractional order Lyapunov direct method, using quadratic Lyapunov functions. This article introduces a new way of building polynomial Lyapunov functions of any positive integer order as a way of determining the stability of a greater variety of systems when the order of the derivative is 0 < α < 1. Some examples are given to validate these results.
2013-06-01
STABILITY BY COMPUTING A SINGLE QUADRATIC LYAPUNOV FUNCTION Mehrdad Pakmehr∗ PhD Candidate Decision and Control Laboratory (DCL) School of Aerospace...linearization and linear matrix inequality (LMI) techniques. Using convex optimization tools, a single quadratic Lyapunov function is computed for multiple...Scheduling Control of Gas Turbine Engines: Stability by Computing a Single Quadratic Lyapunov Function 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c
Covariant hydrodynamic Lyapunov modes and strong stochasticity threshold in Hamiltonian lattices.
Romero-Bastida, M; Pazó, Diego; López, Juan M
2012-02-01
We scrutinize the reliability of covariant and Gram-Schmidt Lyapunov vectors for capturing hydrodynamic Lyapunov modes (HLMs) in one-dimensional Hamiltonian lattices. We show that, in contrast with previous claims, HLMs do exist for any energy density, so that strong chaos is not essential for the appearance of genuine (covariant) HLMs. In contrast, Gram-Schmidt Lyapunov vectors lead to misleading results concerning the existence of HLMs in the case of weak chaos.
Hurst's Exponent Determination for Radial Distribution Functions of In, Sn and In-40 wt％Sn Melt
Institute of Scientific and Technical Information of China (English)
周永志; 李梅; 耿浩然; 杨中喜; 孙春静
2011-01-01
Hurst's exponent of radial distribution functions (RDFs) within the short-range scope of In, Sn and In-40wt%Sn melts are determined by the rescaled range analysis method. Hurst's exponents H are between 0.94 and 0.97, which display long-range dependence. Within short-range scope, the number of particles from a reference particle belongs to fractional Brownian motion. After RDF serials are randomly scrambled, Hurst's exponents all dramatically dropped, which proves long-range dependence. H irregularly varies as the temperature rises, but the change tendency is not consistent with the correlation radius rc.%Hurst's exponent of radial distribution functions(RDFs)within the short-range scope of In,Sn and In-40 wt％Sn melts are determined by the rescaled range analysis method.Hurst's exponents H are between 0.94 and 0.97,which display long-range dependence.Within short-range scope,the number of particles from a reference particle belongs to fractional Brownian motion.After RDF serials are randomly scrambled,Hurst's exponents all dramatically dropped,which proves long-range dependence.H irregularly varies as the temperature rises,but the change tendency is not consistent with the correlation radius rc.
C. Wang (Chenguang); K. Allegaert (Karel); M.Y. Peeters (Mariska); D. Tibboel (Dick); M. Danhof (Meindert); C.A.J. Knibbe (Catherijne)
2014-01-01
textabstractAim For scaling clearance between adults and children, allometric scaling with a fixed exponent of 0.75 is often applied. In this analysis, we performed a systematic study on the allometric exponent for scaling propofol clearance between two subpopulations selected from neonates,
Prasath, V B Surya; Vorotnikov, Dmitry; Pelapur, Rengarajan; Jose, Shani; Seetharaman, Guna; Palaniappan, Kannappan
2015-12-01
Edge preserving regularization using partial differential equation (PDE)-based methods although extensively studied and widely used for image restoration, still have limitations in adapting to local structures. We propose a spatially adaptive multiscale variable exponent-based anisotropic variational PDE method that overcomes current shortcomings, such as over smoothing and staircasing artifacts, while still retaining and enhancing edge structures across scale. Our innovative model automatically balances between Tikhonov and total variation (TV) regularization effects using scene content information by incorporating a spatially varying edge coherence exponent map constructed using the eigenvalues of the filtered structure tensor. The multiscale exponent model we develop leads to a novel restoration method that preserves edges better and provides selective denoising without generating artifacts for both additive and multiplicative noise models. Mathematical analysis of our proposed method in variable exponent space establishes the existence of a minimizer and its properties. The discretization method we use satisfies the maximum-minimum principle which guarantees that artificial edge regions are not created. Extensive experimental results using synthetic, and natural images indicate that the proposed multiscale Tikhonov-TV (MTTV) and dynamical MTTV methods perform better than many contemporary denoising algorithms in terms of several metrics, including signal-to-noise ratio improvement and structure preservation. Promising extensions to handle multiplicative noise models and multichannel imagery are also discussed.
Institute of Scientific and Technical Information of China (English)
丁丹平; 田立新
2000-01-01
自1990年,美国马里兰大学的Ott,Grebogi和Yorke三人首先从理论上提出控制混沌的方法,即OGY方法,混沌控制已成了非线性理论及应用中重要的组成部分.但混沌控制(OGY)方法在数学理论上还有许多工作需要完善,从数学理论上对OGY方法进一步论证和探讨,对混沌控制理论的建立和体系化有很重要的意义.而笔者利用Lyapunov指数讨论了混沌控制(OGY方法)有效的充分条件,获得了具体的表示式.并将此方法用于讨论具体的控制参数的选择及控制参数所须满足的条件.最后对二维Henon映射的轨道稳定化控制的有效性给出了解释.
Lyapunov特性指数谱的研究及其应用%The Study of Lyapunov Characteristic Exponents Spectrum and Its Applications
Institute of Scientific and Technical Information of China (English)
虞文锦; 蔡艳岭; 高英台
2002-01-01
本文对于已有的运动方程的Lyapunov特性指数谱计算公式,从易于数值实现的角度作了简化,并通过Henon映射和Lorena方程对Lyapunov特性指数与倍周期分岔的关系作了分析和比较,最后通过对受特定参数下系统周期轨道控制时的混沌Lorenz系统的Lyapunov特性指数谱的计算和分析,进一步验证了Lyapunov特性指数与倍周期分岔的内在关系.
Research of Judging the Chaotic Characteristics with the Lyapunov Exponents%Lyapunov指数混沌特性判定研究
Institute of Scientific and Technical Information of China (English)
郁俊莉; 王其文
2004-01-01
探讨了Lyapunov指数混沌特性判据原理,分析了时间序列Lyapunov指数的计算过程,通过计算我国上证综合指数收益率时间序列的Lyapunov指数谱系,分析了我国资本市场的混沌特性.
A new theoretical interpretation of Archie's saturation exponent
Glover, Paul W. J.
2017-07-01
This paper describes the extension of the concepts of connectedness and conservation of connectedness that underlie the generalized Archie's law for n phases to the interpretation of the saturation exponent. It is shown that the saturation exponent as defined originally by Archie arises naturally from the generalized Archie's law. In the generalized Archie's law the saturation exponent of any given phase can be thought of as formally the same as the phase (i.e. cementation) exponent, but with respect to a reference subset of phases in a larger n-phase medium. Furthermore, the connectedness of each of the phases occupying a reference subset of an n-phase medium can be related to the connectedness of the subset itself by Gi = GrefSini. This leads naturally to the idea of the term Sini for each phase i being a fractional connectedness, where the fractional connectednesses of any given reference subset sum to unity in the same way that the connectednesses sum to unity for the whole medium. One of the implications of this theory is that the saturation exponent of any phase can be now be interpreted as the rate of change of the fractional connectedness with saturation and connectivity within the reference subset.
Groups of order p^8 and exponent p
Directory of Open Access Journals (Sweden)
Michael Vaughan-Lee
2015-12-01
Full Text Available We prove that for p>7 there are p^4 +2p^3 +20p^2 +147p+(3p+29gcd(p−1,3+5gcd(p−1,4+1246 groups of order p^8 with exponent p. If P is a group of order p^8 and exponent p, and if P has class c>1 then P is a descendant of P/γ c (P. For each group of exponent p with order less than p^8 we calculate the number of descendants of order p^8 with exponent p. In all but one case we are able to obtain a complete and irredundant list of the descendants. But in the case of the three generator class two group of order p^6 and exponent p (p>3 , while we are able to calculate the number of descendants of order p^8, we have not been able to obtain a list of the descendants.
González, Julián J; Pereda, Ernesto
2004-04-01
The short-term cardiovascular control system is reviewed from the analysis of the heart rate, respiration and blood pressure beat-to-beat variability signals. The present state of the art concerning fractal and non-linear techniques as applied to the cardiovascular system and the differences between both approaches are highlighted. We present results obtained in mammals from statistics, such as the fractal exponent, the correlation dimension or the maximal Lyapunov exponent and discuss the convenience of these indexes for characterizing the irregularity present in the signals. Finally, the interdependence between the systems involved in the cardiovascular control is addressed. Recent results obtained from interdependence indexes between the cardio, respiratory and vascular signals are discussed and their convenience in physiological studies and clinical applications are stressed.
Chaotic time series analysis of vision evoked EEG
Zhang, Ningning; Wang, Hong
2010-01-01
To investigate the human brain activities for aesthetic processing, beautiful woman face picture and ugly buffoon face picture were applied. Twelve subjects were assigned the aesthetic processing task while the electroencephalogram (EEG) was recorded. Event-related brain potential (ERP) was required from the 32 scalp electrodes and the ugly buffoon picture produced larger amplitudes for the N1, P2, N2, and late slow wave components. Average ERP from the ugly buffoon picture were larger than that from the beautiful woman picture. The ERP signals shows that the ugly buffoon elite higher emotion waves than the beautiful woman face, because some expression is on the face of the buffoon. Then, chaos time series analysis was carried out to calculate the largest Lyapunov exponent using small data set method and the correlation dimension using G-P algorithm. The results show that the largest Lyapunov exponents of the ERP signals are greater than zero, which indicate that the ERP signals may be chaotic. The correlations dimensions coming from the beautiful woman picture are larger than that from the ugly buffoon picture. The comparison of the correlations dimensions shows that the beautiful face can excite the brain nerve cells. The research in the paper is a persuasive proof to the opinion that cerebrum's work is chaotic under some picture stimuli.
Universal construction of control Lyapunov functions for a class of nonlinear systems
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
A method is developed by which control Lyapunov functions of a class of nonlinear systems can be constructed systematically.Based on the control Lyapunov function,a feedback control is obtained to stabilize the closed-loop system.In addition,this method is applied to stabilize the Benchmark system.A simulation shows the effectiveness of the method.
Average Transient Lifetime and Lyapunov Dimension for Transient Chaos in a High-Dimensional System
Institute of Scientific and Technical Information of China (English)
陈洪; 汤建新; 唐少炎; 向红; 陈新
2001-01-01
The average transient lifetime of a chaotic transient versus the Lyapunov dimension of a chaotic saddle is studied for high-dimensional nonlinear dynamical systems. Typically the average lifetime depends upon not only the system parameter but also the Lyapunov dimension of the chaotic saddle. The numerical example uses the delayed feedback differential equation.
Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility
Korobeinikov, Andrei
2013-01-01
We consider global asymptotic properties for the SIR and SEIR age structured models for infectious diseases where the susceptibility depends on the age. Using the direct Lyapunov method with Volterra type Lyapunov functions, we establish conditions for the global stability of a unique endemic steady state and the infection-free steady state.
Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility.
Melnik, Andrey V; Korobeinikov, Andrei
2013-04-01
We consider global asymptotic properties for the SIR and SEIR age structured models for infectious diseases where the susceptibility depends on the age. Using the direct Lyapunov method with Volterra type Lyapunov functions, we establish conditions for the global stability of a unique endemic steady state and the infection-free steady state.
Directory of Open Access Journals (Sweden)
Weiwei Fang
2014-01-01
Full Text Available The recent advent of satellite swarm technologies has enabled space exploration with a massive number of picoclass, low-power, and low-weight spacecraft. However, developing swarm-based satellite systems, from conceptualization to validation, is a complex multidisciplinary activity. One of the primary challenges is how to achieve energy-efficient data transmission between the satellite swarm and terrestrial terminal stations. Employing Lyapunov optimization techniques, we present an online control algorithm to optimally dispatch traffic load among different satellite-ground links for minimizing overall energy consumption over time. Our algorithm is able to independently and simultaneously make control decisions on traffic dispatching over intersatellite-links and up-down-links so as to offer provable energy and delay guarantees, without requiring any statistical information of traffic arrivals and link condition. Rigorous analysis and extensive simulations have demonstrated the performance and robustness of the proposed new algorithm.
Colburn, B. K.; Boland, J. S., III
1976-01-01
A new nonlinear stability criterion is developed by use of a class of Lyapunov functionals for model-reference adaptive systems (MRAS). Results are compared with traditional results, and a comparative design technique is used to illustrate its function in improving the transient response of an MRAS controller. For a particular system structure and class of input signals, the new stability criterion is shown to include traditional sufficiency stability conditions as a special case. An example is cited to illustrate the use of the nonlinear criterion and its definite advantages in helping improve the adaptive error transient response of a system. Analysis of results is effected by use of a linearization technique on the resulting adaptive equations.
Wen, Guanghui; Yu, Wenwu; Hu, Guoqiang; Cao, Jinde; Yu, Xinghuo
2015-12-01
This paper studies the global pinning synchronization problem for a class of complex networks with switching directed topologies. The common assumption in the existing related literature that each possible network topology contains a directed spanning tree is removed in this paper. Using tools from M -matrix theory and stability analysis of the switched nonlinear systems, a new kind of network topology-dependent multiple Lyapunov functions is proposed for analyzing the synchronization behavior of the whole network. It is theoretically shown that the global pinning synchronization in switched complex networks can be ensured if some nodes are appropriately pinned and the coupling is carefully selected. Interesting issues of how many and which nodes should be pinned for possibly realizing global synchronization are further addressed. Finally, some numerical simulations on coupled neural networks are provided to verify the theoretical results.
Colburn, B. K.; Boland, J. S., III
1976-01-01
A new nonlinear stability criterion is developed by use of a class of Lyapunov functionals for model-reference adaptive systems (MRAS). Results are compared with traditional results, and a comparative design technique is used to illustrate its function in improving the transient response of an MRAS controller. For a particular system structure and class of input signals, the new stability criterion is shown to include traditional sufficiency stability conditions as a special case. An example is cited to illustrate the use of the nonlinear criterion and its definite advantages in helping improve the adaptive error transient response of a system. Analysis of results is effected by use of a linearization technique on the resulting adaptive equations.
Testing Universality in Critical Exponents: the Case of Rainfall
Deluca, Anna; Corral, Alvaro
2015-01-01
One of the key clues to consider rainfall as a self-organized critical phenomenon is the existence of power-law distributions for rain-event sizes. We have studied the problem of universality in the exponents of these distributions by means of a suitable statistic whose distribution is inferred by several variations of a permutational test. In contrast to more common approaches, our procedure does not suffer from the difficulties of multiple testing and does not require the precise knowledge of the uncertainties associated to the power-law exponents. When applied to seven sites monitored by the Atmospheric Radiation Measurement Program the test lead to the rejection of the universality hypothesis, despite the fact that the exponents are rather close to each other.
On the average exponent of elliptic curves modulo $p$
Freiberg, Tristan
2012-01-01
Given an elliptic curve $E$ defined over $\\mathbb{Q}$ and a prime $p$ of good reduction, let $\\tilde{E}(\\mathbb{F}_p)$ denote the group of $\\mathbb{F}_p$-points of the reduction of $E$ modulo $p$, and let $e_p$ denote the exponent of said group. Assuming a certain form of the Generalized Riemann Hypothesis (GRH), we study the average of $e_p$ as $p \\le X$ ranges over primes of good reduction, and find that the average exponent essentially equals $p\\cdot c_{E}$, where the constant $c_{E} > 0$ depends on $E$. For $E$ without complex multiplication (CM), $c_{E}$ can be written as a rational number (depending on $E$) times a universal constant. Without assuming GRH, we can determine the average exponent when $E$ has CM, as well as give an upper bound on the average in the non-CM case.
Adaptive Fuzzy-Lyapunov Controller Using Biologically Inspired Swarm Intelligence
Directory of Open Access Journals (Sweden)
Alejandro Carrasco Elizalde
2008-01-01
Full Text Available The collective behaviour of swarms produces smarter actions than those achieved by a single individual. Colonies of ants, flocks of birds and fish schools are examples of swarms interacting with their environment to achieve a common goal. This cooperative biological intelligence is the inspiration for an adaptive fuzzy controller developed in this paper. Swarm intelligence is used to adjust the parameters of the membership functions used in the adaptive fuzzy controller. The rules of the controller are designed using a computing-with-words approach called Fuzzy-Lyapunov synthesis to improve the stability and robustness of an adaptive fuzzy controller. Computing-with-words provides a powerful tool to manipulate numbers and symbols, like words in a natural language.
A Lyapunov theory based UPFC controller for power flow control
Energy Technology Data Exchange (ETDEWEB)
Zangeneh, Ali; Kazemi, Ahad; Hajatipour, Majid; Jadid, Shahram [Center of Excellence for Power Systems Automation and Operation, Iran University of Science and Technology, Tehran (Iran)
2009-09-15
Unified power flow controller (UPFC) is the most comprehensive multivariable device among the FACTS controllers. Capability of power flow control is the most important responsibility of UPFC. According to high importance of power flow control in transmission lines, the proper controller should be robust against uncertainty and disturbance and also have suitable settling time. For this purpose, a new controller is designed based on the Lyapunov theory and its stability is also evaluated. The Main goal of this paper is to design a controller which enables a power system to track reference signals precisely and to be robust in the presence of uncertainty of system parameters and disturbances. The performance of the proposed controller is simulated on a two bus test system and compared with a conventional PI controller. The simulation results show the power and accuracy of the proposed controller. (author)
Time-delay effects and simplified control fields in quantum Lyapunov control
Energy Technology Data Exchange (ETDEWEB)
Yi, X X; Wu, S L [School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024 (China); Wu, Chunfeng; Feng, X L; Oh, C H, E-mail: yixx@dlut.edu.cn, E-mail: phyohch@nus.edu.sg [Centre for Quantum Technologies and Department of Physics, National University of Singapore, 117543 (Singapore)
2011-10-14
Lyapunov-based quantum control has the advantage that it is free from the measurement-induced decoherence and it includes the instantaneous information of the system in the control. The Lyapunov control is often confronted with time delay in the control fields and difficulty in practical implementations of the control. In this paper, we study the effect of time delay on the Lyapunov control and explore the possibility of replacing the control field with a pulse train or a bang-bang signal. The efficiency of the Lyapunov control is also presented through examining the convergence time of the system. These results suggest that the Lyapunov control is robust against time delay, easy to realize and effective for high-dimensional quantum systems.
Construction of a Smooth Lyapunov Function for the Robust and Exact Second-Order Differentiator
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Tonametl Sanchez
2016-01-01
Full Text Available Differentiators play an important role in (continuous feedback control systems. In particular, the robust and exact second-order differentiator has shown some very interesting properties and it has been used successfully in sliding mode control, in spite of the lack of a Lyapunov based procedure to design its gains. As contribution of this paper, we provide a constructive method to determine a differentiable Lyapunov function for such a differentiator. Moreover, the Lyapunov function is used to provide a procedure to design the differentiator’s parameters. Also, some sets of such parameters are provided. The determination of the positive definiteness of the Lyapunov function and negative definiteness of its derivative is converted to the problem of solving a system of inequalities linear in the parameters of the Lyapunov function candidate and also linear in the gains of the differentiator, but bilinear in both.
Parameter-dependent Lyapunov functional for systems with multiple time delays
Institute of Scientific and Technical Information of China (English)
Min WU; Yong HE
2004-01-01
The separation of the Lyapunov matrices and system matrices plays an important role when one uses parameter-dependent Lyapunov functional handling systems with polytopic type uncertainties.The delay-dependent robust stability problem for systems with polytopic type uncertainties is discussed by using parameter-dependent Lyapunov functional.The derivative term in the derivative of Lyapunov functional is reserved and the free weighting matrices are employed to express the relationship between the terms in the system equation such that the Lyapunov matrices are not involved in any product terms with the system matrices.In addition,the relationships between the terms in the Leibniz Newton formula are also described by some free weighting matrices and some delay-dependent stability conditions are derived.Numerical examples demonstrate that the proposed criteria are more effective than the previous results.
Lyapunov Functions, Stationary Distributions, and Non-equilibrium Potential for Reaction Networks
DEFF Research Database (Denmark)
Anderson, David F; Craciun, Gheorghe; Gopalkrishnan, Manoj;
2015-01-01
We consider the relationship between stationary distributions for stochastic models of reaction systems and Lyapunov functions for their deterministic counterparts. Specifically, we derive the well-known Lyapunov function of reaction network theory as a scaling limit of the non-equilibrium potent......We consider the relationship between stationary distributions for stochastic models of reaction systems and Lyapunov functions for their deterministic counterparts. Specifically, we derive the well-known Lyapunov function of reaction network theory as a scaling limit of the non......-equilibrium potential of the stationary distribution of stochastically modeled complex balanced systems. We extend this result to general birth-death models and demonstrate via example that similar scaling limits can yield Lyapunov functions even for models that are not complex or detailed balanced, and may even have...
Vannitsem, Stephane
2015-01-01
We study a simplified coupled atmosphere-ocean model using the formalism of covariant Lyapunov vectors (CLVs), which link physically-based directions of perturbations to growth/decay rates. The model is obtained via a severe truncation of quasi-geostrophic equations for the two fluids, and includes a simple yet physically meaningful representation of their dynamical/thermodynamical coupling. The model has 36 degrees of freedom, and the parameters are chosen so that a chaotic behaviour is observed. One finds two positive Lyapunov exponents (LEs), sixteen negative LEs, and eighteen near-zero LEs. The presence of many near-zero LEs results from the vast time-scale separation between the characteristic time scales of the two fluids, and leads to nontrivial error growth properties in the tangent space spanned by the corresponding CLVs, which are geometrically very degenerate. Such CLVs correspond to two different classes of ocean/atmosphere coupled modes. The tangent space spanned by the CLVs corresponding to the ...
Gaps and the exponent of convergence of an integer sequence
Grekos, Georges; Sleziak, Martin
2012-01-01
Professor Tibor \\v{S}al\\'at, at one of his seminars at Comenius University, Bratislava, asked to study the influence of gaps of an integer sequence A={a_1
Lattice Based Attack on Common Private Exponent RSA
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Santosh Kumar Ravva
2012-03-01
Full Text Available Lattice reduction is a powerful concept for solving diverse problems involving point lattices. Lattice reduction has been successfully utilizing in Number Theory, Linear algebra and Cryptology. Not only the existence of lattice based cryptosystems of hard in nature, but also has vulnerabilities by lattice reduction techniques. In this paper, we show that Wieners small private exponent attack, when viewed as a heuristic lattice based attack, is extended to attack many instances of RSA when they have the same small private exponent.
Critical Exponents for Fast Diffusion Equations with Nonlinear Boundary Sources
Institute of Scientific and Technical Information of China (English)
WANG LU-SHENG; WANG ZE-JIA
2011-01-01
In this paper, we study the large time behavior of solutions to a class of fast diffusion equations with nonlinear boundary sources on the exterior domain of the unit ball. We are interested in the critical global exponent q0 and the critical Fujita exponent qc for the problem considered, and show that q0 ＝ qc for the multidimensional Non-Newtonian polytropic filtration equation with nonlinear boundary sources, which is quite different from the known results that q0 ＜ qc for the onedimensional case; moreover, the value is different from the slow case.
Second-order relative exponent of isotropic turbulence
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
Theoretical results on the scaling properties of turbulent velocity fields are reported in this letter.Based on the Kolmogorov equation and typical models of the second-order statistical moments (energy spectrum and the second-order structure function),we have studied the relative scaling using the ESS method.It is found that the relative EES scaling exponent S_2 is greater than the real or theoretical inertial range scaling exponentξ_2,which is attributed to an evident bump in the ESS range.
Directory of Open Access Journals (Sweden)
A.B. Demchyshyn
2012-03-01
Full Text Available Differences between critical exponents of this model and the continuous percolation model indicate that the dependence of the modified structure area on the dose and the angle related with the correlation between individual tracks. It results in next effect: angular dependence of the surface area of the branched structure has maximum value at certain «critical» angle of ions incidence. Differences between critical exponents of this model and the continuous percolation model indicate that the dependence of the modified structure area on the dose and the angle related with the correlation between individual tracks. It results in next effect: angular dependence of the surface area of the branched structure has maximum value at certain «critical» angle of ions incidence. Differences between critical exponents of this model and the continuous percolation model indicate that the dependence of the modified structure area on the dose and the angle related with the correlation between individual tracks. It results in next effect: angular dependence of the surface area of the branched structure has maximum value at certain «critical» angle of ions incidence.
Isotropic Brownian motions over complex fields as a solvable model for May-Wigner stability analysis
Ipsen, J. R.; Schomerus, H.
2016-09-01
We consider matrix-valued stochastic processes known as isotropic Brownian motions, and show that these can be solved exactly over complex fields. While these processes appear in a variety of questions in mathematical physics, our main motivation is their relation to a May-Wigner-like stability analysis, for which we obtain a stability phase diagram. The exact results establish the full joint probability distribution of the finite-time Lyapunov exponents, and may be used as a starting point for a more detailed analysis of the stability-instability phase transition. Our derivations rest on an explicit formulation of a Fokker-Planck equation for the Lyapunov exponents. This formulation happens to coincide with an exactly solvable class of models of the Calgero-Sutherland type, originally encountered for a model of phase-coherent transport. The exact solution over complex fields describes a determinantal point process of biorthogonal type similar to recent results for products of random matrices, and is also closely related to Hermitian matrix models with an external source.
Construction of Control Lyapunov Functions for a Class of Nonlinear Systems%一类非线性系统控制Lyapunov函数的构造
Institute of Scientific and Technical Information of China (English)
蔡秀珊; 韩正之; 汪晓东
2006-01-01
The construction of control Lyapunov functions for a class of nonlinear systems is considered. We develop a method by which a control Lyapunov function for the feedback linearizable part can be constructed systematically via Lyapunov equation. Moreover, by a control Lyapunov function of the feedback linearizable part and a Lyapunov function of the zero dynamics, a control Lyapunov function for the overall nonlinear system is established.
Analytical Computation of Critical Exponents in Several Holographic Superconductors
Zeng, Hua-Bi; Jiang, Yu; Zong, Hong-Shi
2010-01-01
It is very interesting that all holographic superconductors like $s$-wave, $p$-wave and $d$-wave holographic superconductors shows the universal mean-field critical exponent $1/2$ at the critical temperature just like Gindzburg-Landau (G-L) theory for second order phase transitions. Now it is believed that the universal critical exponents appear since the dual gravity theory is classic in the large $N$ limit. However, there is an exception called "non-mean-field theory" even in the large $N$ limit: An extension of the $s$-wave model with a cubic term of the charged scalar field provides a different critical exponent $1$. In this paper, we try to use analytical calculation to get the critical exponents for these models to see how these properties of the gravity action decides the appearance of the mean-field or "non-mean-field" behaviors. It will be seen that like the G-L theory, it is the fundamental symmetries rather than the detail parameters of the bulk theory result in the universal properties of the holo...
QUASILINEAR ELLIPTIC PROBLEMS WITH CRITICAL EXPONENT AND HARDY TERM
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
This paper is concerned with a p-Laplacian elliptic problem with critical Sobolev-Hardy exponent and Hardy term. By variational methods and genus theory, we guarantee that this problem has at least one positive solution and admits many solutions with negative energy under sufficient conditions.
Inverted rank distributions: Macroscopic statistics, universality classes, and critical exponents
Eliazar, Iddo; Cohen, Morrel H.
2014-01-01
An inverted rank distribution is an infinite sequence of positive sizes ordered in a monotone increasing fashion. Interlacing together Lorenzian and oligarchic asymptotic analyses, we establish a macroscopic classification of inverted rank distributions into five “socioeconomic” universality classes: communism, socialism, criticality, feudalism, and absolute monarchy. We further establish that: (i) communism and socialism are analogous to a “disordered phase”, feudalism and absolute monarchy are analogous to an “ordered phase”, and criticality is the “phase transition” between order and disorder; (ii) the universality classes are characterized by two critical exponents, one governing the ordered phase, and the other governing the disordered phase; (iii) communism, criticality, and absolute monarchy are characterized by sharp exponent values, and are inherently deterministic; (iv) socialism is characterized by a continuous exponent range, is inherently stochastic, and is universally governed by continuous power-law statistics; (v) feudalism is characterized by a continuous exponent range, is inherently stochastic, and is universally governed by discrete exponential statistics. The results presented in this paper yield a universal macroscopic socioeconophysical perspective of inverted rank distributions.
Predicted and verified evolution of power-law exponent in product market
Hisano, Ryohei; Mizuno, Takayuki
2011-01-01
Power-law distributions constitute a generic empirical statistical regularity found in many complex systems. A recently developed theory finds that the interplay between one of the most universal ingredient, i.e., stochastic proportional growth, and stochastic birth and death processes, leads to generic power law distributions together with a non-universal exponent which depends explicitly on the characteristics of growth, birth and death. In particular, the theory rationalizes Zipf's law and explains deviations from it, for instance for the distribution of firm and of city sizes. Here, we report the first complete test of the theory, based on the empirical analysis from a real world complex phenomenon, namely the dynamics of market shares in the consumer electronics market. We estimate directly from the data the average growth rate of market shares, their standard deviation, the birth rates as well as the "death" hazard rate of products. When plugged in the theory, this predicts the power law exponent of the...
Dependence of exponents on text length versus finite-size scaling for word-frequency distributions
Corral, Álvaro; Font-Clos, Francesc
2017-08-01
Some authors have recently argued that a finite-size scaling law for the text-length dependence of word-frequency distributions cannot be conceptually valid. Here we give solid quantitative evidence for the validity of this scaling law, using both careful statistical tests and analytical arguments based on the generalized central-limit theorem applied to the moments of the distribution (and obtaining a novel derivation of Heaps' law as a by-product). We also find that the picture of word-frequency distributions with power-law exponents that decrease with text length [X. Yan and P. Minnhagen, Physica A 444, 828 (2016), 10.1016/j.physa.2015.10.082] does not stand with rigorous statistical analysis. Instead, we show that the distributions are perfectly described by power-law tails with stable exponents, whose values are close to 2, in agreement with the classical Zipf's law. Some misconceptions about scaling are also clarified.
Liu, Jie; Regenauer-Lieb, Klaus
2011-01-01
Percolation theory provides a tool for linking microstructure and macroscopic material properties. In this paper, percolation theory is applied to the analysis of microtomographic images for the purpose of deriving scaling laws for upscaling of properties. We have tested the acquisition of quantities such as percolation threshold, crossover length, fractal dimension, and critical exponent of correlation length from microtomography. By inflating or deflating the target phase and percolation analysis, we can get a critical model and an estimation of the percolation threshold. The crossover length is determined from the critical model by numerical simulation. The fractal dimension can be obtained either from the critical model or from the relative size distribution of clusters. Local probabilities of percolation are used to extract the critical exponent of the correlation length. For near-isotropic samples such as sandstone and bread, the approach works very well. For strongly anisotropic samples, such as highly deformed rock (mylonite) and a tree branch, the percolation threshold and fractal dimension can be assessed with accuracy. However, the uncertainty of the correlation length makes it difficult to accurately extract its critical exponents. Therefore, this aspect of percolation theory cannot be reliably used for upscaling properties of strongly anisotropic media. Other methods of upscaling have to be used for such media.
Improved estimation of anomalous diffusion exponents in single-particle tracking experiments.
Kepten, Eldad; Bronshtein, Irena; Garini, Yuval
2013-05-01
The mean square displacement is a central tool in the analysis of single-particle tracking experiments, shedding light on various biophysical phenomena. Frequently, parameters are extracted by performing time averages on single-particle trajectories followed by ensemble averaging. This procedure, however, suffers from two systematic errors when applied to particles that perform anomalous diffusion. The first is significant at short-time lags and is induced by measurement errors. The second arises from the natural heterogeneity in biophysical systems. We show how to estimate and correct these two errors and improve the estimation of the anomalous parameters for the whole particle distribution. As a consequence, we manage to characterize ensembles of heterogeneous particles even for rather short and noisy measurements where regular time-averaged mean square displacement analysis fails. We apply this method to both simulations and in vivo measurements of telomere diffusion in 3T3 mouse embryonic fibroblast cells. The motion of telomeres is found to be subdiffusive with an average exponent constant in time. Individual telomere exponents are normally distributed around the average exponent. The proposed methodology has the potential to improve experimental accuracy while maintaining lower experimental costs and complexity.
a Comparison of Three Hurst Exponent Approaches to Predict Nascent Bubbles in S&P500 Stocks
Fernández-Martínez, M.; Sánchez-Granero, M. A.; Muñoz Torrecillas, M. J.; McKelvey, Bill
Since the pioneer contributions due to Vandewalle and Ausloos, the Hurst exponent has been applied by econophysicists as a useful indicator to deal with investment strategies when such a value is above or below 0.5, the Hurst exponent of a Brownian motion. In this paper, we hypothesize that the self-similarity exponent of financial time series provides a reliable indicator for herding behavior (HB) in the following sense: if there is HB, then the higher the price, the more the people will buy. This will generate persistence in the stocks which we shall measure by their self-similarity exponents. Along this work, we shall explore whether there is some connections between the self-similarity exponent of a stock (as a HB indicator) and the stock’s future performance under the assumption that the HB will last for some time. With this aim, three approaches to calculate the self-similarity exponent of a time series are compared in order to determine which performs best to identify the transition from random efficient market behavior to HB and hence, to detect the beginning of a bubble. Generalized Hurst Exponent, Detrended Fluctuation Analysis, and GM2 algorithms have been tested. Traditionally, researchers have focused on identifying the beginning of a crash. We study the beginning of the transition from efficient market behavior to a market bubble, instead. Our empirical results support that the higher (respectively the lower) the self-similarity index, the higher (respectively the lower) the mean of the price change, and hence, the better (respectively the worse) the performance of the corresponding stock. This would imply, as a consequence, that the transition process from random efficient market to HB has started. For experimentation purposes, S&P500 stock Index constituted our main data source.
Dynamical Analysis of the Hindmarsh-Rose Neuron With Time Delays.
Lakshmanan, S; Lim, C P; Nahavandi, S; Prakash, M; Balasubramaniam, P
2016-05-25
This brief is mainly concerned with a series of dynamical analyses of the Hindmarsh-Rose (HR) neuron with state-dependent time delays. The dynamical analyses focus on stability, Hopf bifurcation, as well as chaos and chaos control. Through the stability and bifurcation analysis, we determine that increasing the external current causes the excitable HR neuron to exhibit periodic or chaotic bursting/spiking behaviors and emit subcritical Hopf bifurcation. Furthermore, by choosing a fixed external current and varying the time delay, the stability of the HR neuron is affected. We analyze the chaotic behaviors of the HR neuron under a fixed external current through time series, bifurcation diagram, Lyapunov exponents, and Lyapunov dimension. We also analyze the synchronization of the chaotic time-delayed HR neuron through nonlinear control. Based on an appropriate Lyapunov-Krasovskii functional with triple integral terms, a nonlinear feedback control scheme is designed to achieve synchronization between the uncontrolled and controlled models. The proposed synchronization criteria are derived in terms of linear matrix inequalities to achieve the global asymptotical stability of the considered error model under the designed control scheme. Finally, numerical simulations pertaining to stability, Hopf bifurcation, periodic, chaotic, and synchronized models are provided to demonstrate the effectiveness of the derived theoretical results.
Stability of Cellular Automata Trajectories Revisited: Branching Walks and Lyapunov Profiles
Baetens, Jan M.; Gravner, Janko
2016-10-01
We study non-equilibrium defect accumulation dynamics on a cellular automaton trajectory: a branching walk process in which a defect creates a successor on any neighborhood site whose update it affects. On an infinite lattice, defects accumulate at different exponential rates in different directions, giving rise to the Lyapunov profile. This profile quantifies instability of a cellular automaton evolution and is connected to the theory of large deviations. We rigorously and empirically study Lyapunov profiles generated from random initial states. We also introduce explicit and computationally feasible variational methods to compute the Lyapunov profiles for periodic configurations, thus developing an analog of Floquet theory for cellular automata.
Lyapunov Functions, Stationary Distributions, and Non-equilibrium Potential for Reaction Networks.
Anderson, David F; Craciun, Gheorghe; Gopalkrishnan, Manoj; Wiuf, Carsten
2015-09-01
We consider the relationship between stationary distributions for stochastic models of reaction systems and Lyapunov functions for their deterministic counterparts. Specifically, we derive the well-known Lyapunov function of reaction network theory as a scaling limit of the non-equilibrium potential of the stationary distribution of stochastically modeled complex balanced systems. We extend this result to general birth-death models and demonstrate via example that similar scaling limits can yield Lyapunov functions even for models that are not complex or detailed balanced, and may even have multiple equilibria.
Global stabilization of nonlinear systems based on vector control lyapunov functions
Karafyllis, Iasson
2012-01-01
This paper studies the use of vector Lyapunov functions for the design of globally stabilizing feedback laws for nonlinear systems. Recent results on vector Lyapunov functions are utilized. The main result of the paper shows that the existence of a vector control Lyapunov function is a necessary and sufficient condition for the existence of a smooth globally stabilizing feedback. Applications to nonlinear systems are provided: simple and easily checkable sufficient conditions are proposed to guarantee the existence of a smooth globally stabilizing feedback law. The obtained results are applied to the problem of the stabilization of an equilibrium point of a reaction network taking place in a continuous stirred tank reactor.
Kajiwara, Tsuyoshi; Sasaki, Toru; Takeuchi, Yasuhiro
2015-02-01
We present a constructive method for Lyapunov functions for ordinary differential equation models of infectious diseases in vivo. We consider models derived from the Nowak-Bangham models. We construct Lyapunov functions for complex models using those of simpler models. Especially, we construct Lyapunov functions for models with an immune variable from those for models without an immune variable, a Lyapunov functions of a model with absorption effect from that for a model without absorption effect. We make the construction clear for Lyapunov functions proposed previously, and present new results with our method.
Flexible Lyapunov Functions and Applications to Fast Mechatronic Systems
Lazar, M
2010-01-01
The property that every control system should posses is stability, which translates into safety in real-life applications. A central tool in systems theory for synthesizing control laws that achieve stability are control Lyapunov functions (CLFs). Classically, a CLF enforces that the resulting closed-loop state trajectory is contained within a cone with a fixed, predefined shape, and which is centered at and converges to a desired converging point. However, such a requirement often proves to be overconservative, which is why most of the real-time controllers do not have a stability guarantee. Recently, a novel idea that improves the design of CLFs in terms of flexibility was proposed. The focus of this new approach is on the design of optimization problems that allow certain parameters that define a cone associated with a standard CLF to be decision variables. In this way non-monotonicity of the CLF is explicitly linked with a decision variable that can be optimized on-line. Conservativeness is significantly ...
Flexible Lyapunov Functions and Applications to Fast Mechatronic Systems
Directory of Open Access Journals (Sweden)
Mircea Lazar
2010-03-01
Full Text Available The property that every control system should posses is stability, which translates into safety in real-life applications. A central tool in systems theory for synthesizing control laws that achieve stability are control Lyapunov functions (CLFs. Classically, a CLF enforces that the resulting closed-loop state trajectory is contained within a cone with a fixed, predefined shape, and which is centered at and converges to a desired converging point. However, such a requirement often proves to be overconservative, which is why most of the real-time controllers do not have a stability guarantee. Recently, a novel idea that improves the design of CLFs in terms of flexibility was proposed. The focus of this new approach is on the design of optimization problems that allow certain parameters that define a cone associated with a standard CLF to be decision variables. In this way non-monotonicity of the CLF is explicitly linked with a decision variable that can be optimized on-line. Conservativeness is significantly reduced compared to classical CLFs, which makes flexible CLFs more suitable for stabilization of constrained discrete-time nonlinear systems and real-time control. The purpose of this overview is to highlight the potential of flexible CLFs for real-time control of fast mechatronic systems, with sampling periods below one millisecond, which are widely employed in aerospace and automotive applications.
Fukushima, Toshio
2012-04-01
By extending the exponent of floating point numbers with an additional integer as the power index of a large radix, we compute fully normalized associated Legendre functions (ALF) by recursion without underflow problem. The new method enables us to evaluate ALFs of extremely high degree as 232 = 4,294,967,296, which corresponds to around 1 cm resolution on the Earth's surface. By limiting the application of exponent extension to a few working variables in the recursion, choosing a suitable large power of 2 as the radix, and embedding the contents of the basic arithmetic procedure of floating point numbers with the exponent extension directly in the program computing the recurrence formulas, we achieve the evaluation of ALFs in the double-precision environment at the cost of around 10% increase in computational time per single ALF. This formulation realizes meaningful execution of the spherical harmonic synthesis and/or analysis of arbitrary degree and order.
Determination of the decay exponent in mechanically stirred isotropic turbulence
Directory of Open Access Journals (Sweden)
J. Blair Perot
2011-06-01
Full Text Available Direct numerical simulation is used to investigate the decay exponent of isotropic homogeneous turbulence over a range of Reynolds numbers sufficient to display both high and low Re number decay behavior. The initial turbulence is generated by the stirring action of the flow past many small randomly placed cubes. Stirring occurs at 1/30th of the simulation domain size so that the low-wavenumber and large scale behavior of the turbulent spectrum is generated by the fluid and is not imposed. It is shown that the decay exponent in the resulting turbulence matches the theoretical predictions for a k2 low-wavenumber spectrum at both high and low Reynolds numbers. The transition from high Reynolds number behavior to low Reynolds number behavior occurs relatively abruptly at a turbulent Reynolds number of around 250 ( Re λ≈41.
Determination of critical exponents of inhomogeneous Gd films
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Rosales-Rivera, A., E-mail: arosalesr@unal.edu.co [Laboratorio de Magnetismo y Materiales Avanzados, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Colombia, Sede Manizales, Manizales (Colombia); Salazar, N.A. [Laboratorio de Magnetismo y Materiales Avanzados, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Colombia, Sede Manizales, Manizales (Colombia); Hovorka, O.; Idigoras, O.; Berger, A. [CIC nanoGUNE Consolider, Tolosa Hiribidea 76, E-20018 Donostia-San Sebastian (Spain)
2012-08-15
The role of inhomogeneity on the critical behavior is studied for non-epitaxial Gd films. For this purpose, the film inhomogeneity was varied experimentally by annealing otherwise identical samples at different temperatures T{sub AN}=200, 400, and 500 Degree-Sign C. Vibrating sample magnetometry (VSM) was used for magnetization M vs. T measurements at different external fields H. A method based upon the linear superposition of different sample parts having different Curie temperatures T{sub C} was used to extract the critical exponents and the intrinsic distribution of Curie temperatures. We found that this method allows extracting reliable values of the critical exponents for all annealing temperatures, which enabled us to study the effects of disorder onto the universality class of Gd films.
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
The asymptotic Lyapunov stability of one quasi-integrable Hamiltonian system with time-delayed feedback control is studied by using Lyapunov functions and stochastic averaging method.First,a quasi-integrable Hamiltonian system with time-delayed feedback control subjected to Gaussian white noise excitation is approximated by a quasi-integrable Hamiltonian system without time delay.Then,stochastic averaging method for quasi-integrable Hamiltonian system is used to reduce the dimension of the original system,and after that the Lyapunov function of the averaged It? equation is taken as the optimal linear combination of the corresponding independent first integrals in involution.Finally,the stability of the system is determined by using the largest eigenvalue of the linearized system.Two examples are used to illustrate the proposed procedure and the effects of delayed time on the Lyapunov stability are discussed as well.
Lyapunov-type inequalities for quasilinear elliptic equations with Robin boundary condition.
Dinlemez Kantar, Ülkü; Özden, Tülay
2017-01-01
The aim of this study is to prove Lyapunov-type inequalities for a quasilinear elliptic equation in [Formula: see text]. Also the lower bound for the first positive eigenvalue of the boundary value problem is obtained.
Robust Backstepping Control Based on a Lyapunov Redesign for Skid-Steered Wheeled Mobile Robots
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Eun-Ju Hwang
2013-01-01
Full Text Available This paper represents a robust backstepping tracking control based on a Lyapunov redesign for Skid‐Steered Wheeled Mobile Robots (WMRs. We present kinematic and dynamic models that explicitly relate the perturbations to the skidding in order to improve the tracking performance during real running. A robust controller is synthesized in the backstepping approach and the Lyapunov redesign technique, which forces the error dynamics to stabilize to the reference trajectories. We design an additional feedback control ‐ a Lyapunov redesign ‐ such that the overall control stabilizes the actual system in the presence of uncertainty and perturbation with the knowledge of the Lyapunov function. Simulation results are provided to validate and analyse the performance and stability of the proposed controller.
Dynamical behaviors determined by the Lyapunov function in competitive Lotka-Volterra systems.
Tang, Ying; Yuan, Ruoshi; Ma, Yian
2013-01-01
Dynamical behaviors of the competitive Lotka-Volterra system even for 3 species are not fully understood. In this paper, we study this problem from the perspective of the Lyapunov function. We construct explicitly the Lyapunov function using three examples of the competitive Lotka-Volterra system for the whole state space: (1) the general 2-species case, (2) a 3-species model, and (3) the model of May-Leonard. The basins of attraction for these examples are demonstrated, including cases with bistability and cyclical behavior. The first two examples are the generalized gradient system, where the energy dissipation may not follow the gradient of the Lyapunov function. In addition, under a new type of stochastic interpretation, the Lyapunov function also leads to the Boltzmann-Gibbs distribution on the final steady state when multiplicative noise is added.