Stability of synchronization in coupled time-delay systems using Krasovskii-Lyapunov theory.
Senthilkumar, D V; Kurths, J; Lakshmanan, M
2009-06-01
Stability of synchronization in unidirectionally coupled time-delay systems is studied using the Krasovskii-Lyapunov theory. We have shown that the same general stability condition is valid for different cases, even for the general situation (but with a constraint) where all the coefficients of the error equation corresponding to the synchronization manifold are time dependent. These analytical results are also confirmed by the numerical simulation of paradigmatic examples.
Lyapunov stability in an evolutionary game theory model of the labour market
Directory of Open Access Journals (Sweden)
Ricardo Azevedo Araujo
2014-01-01
Full Text Available In this paper the existence and stability of equilibriums in an evolutionary game theory model of the labour market is studied by using the Lyapunov method. The model displays multiple equilibriums and it is shown that the Nash equilibriums of the static game are evolutionary stable equilibrium in the game theory evolutionary set up. A complete characterization of the dynamics of an evolutionary model of the labour market is provided.
A Lyapunov Stability Theory-Based Control Strategy for Three-Level Shunt Active Power Filter
Directory of Open Access Journals (Sweden)
Yijia Cao
2017-01-01
Full Text Available The three-phase three-wire neutral-point-clamped shunt active power filter (NPC-SAPF, which most adopts classical closed-loop feedback control methods such as proportional-integral (PI, proportional-resonant (PR and repetitive control, can only output 1st–25th harmonic currents with 10–20 kHz switching frequency. The reason for this is that the controller design must make a compromise between system stability and harmonic current compensation ability under the condition of less than 20 kHz switching frequency. To broaden the bandwidth of the compensation current, a Lyapunov stability theory-based control strategy is presented in this paper for NPC-SAPF. The proposed control law is obtained by constructing the switching function on the basis of the mathematical model and the Lyapunov candidate function, which can avoid introducing closed-loop feedback control and keep the system globally asymptotically stable. By means of the proposed method, the NPC-SAPF has compensation ability for the 1st–50th harmonic currents, the total harmonic distortion (THD and each harmonic content of grid currents satisfy the requirements of IEEE Standard 519-2014. In order to verify the superiority of the proposed control strategy, stability conditions of the proposed strategy and the representative PR controllers are compared. The simulation results in MATLAB/Simulink (MathWorks, Natick, MA, USA and the experimental results obtained on a 6.6 kVA NPC-SAPF laboratory prototype validate the proposed control strategy.
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.
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...
DEFF Research Database (Denmark)
Eriksson, Robert
2014-01-01
as possible, and is based on Lyapunov theory considering the nonlinear behavior. The time optimal controller is of a bang-bang type and uses wide-area measurements as feedback signals. Nonlinear simulations are run in the Nordic32 test system implemented in PowerFactory/DIgSILENT with an interface to Matlab...
Greenwood, Nigel J C; Gunton, Jenny E
2014-07-01
This study demonstrated the novel application of a "machine-intelligent" mathematical structure, combining differential game theory and Lyapunov-based control theory, to the artificial pancreas to handle dynamic uncertainties. Realistic type 1 diabetes (T1D) models from the literature were combined into a composite system. Using a mixture of "black box" simulations and actual data from diabetic medical histories, realistic sets of diabetic time series were constructed for blood glucose (BG), interstitial fluid glucose, infused insulin, meal estimates, and sometimes plasma insulin assays. The problem of underdetermined parameters was side stepped by applying a variant of a genetic algorithm to partial information, whereby multiple candidate-personalized models were constructed and then rigorously tested using further data. These formed a "dynamic envelope" of trajectories in state space, where each trajectory was generated by a hypothesis on the hidden T1D system dynamics. This dynamic envelope was then culled to a reduced form to cover observed dynamic behavior. A machine-intelligent autonomous algorithm then implemented game theory to construct real-time insulin infusion strategies, based on the flow of these trajectories through state space and their interactions with hypoglycemic or near-hyperglycemic states. This technique was tested on 2 simulated participants over a total of fifty-five 24-hour days, with no hypoglycemic or hyperglycemic events, despite significant uncertainties from using actual diabetic meal histories with 10-minute warnings. In the main case studies, BG was steered within the desired target set for 99.8% of a 16-hour daily assessment period. Tests confirmed algorithm robustness for ±25% carbohydrate error. For over 99% of the overall 55-day simulation period, either formal controller stability was achieved to the desired target or else the trajectory was within the desired target. These results suggest that this is a stable, high
Analysis of stability problems via matrix Lyapunov functions
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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.
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.
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...
Stability of time-delay systems via Lyapunov functions
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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.
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.
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 analysis: from dynamical systems theory to applications
Cencini, Massimo; Ginelli, Francesco
2013-06-01
The study of deterministic laws of evolution has characterized the development of science since Newton's times. Chaos, namely the manifestation of irregular and unpredictable dynamics (not random but look random [1]), entered the debate on determinism at the end of the 19th century with the discovery of sensitivity to initial conditions, meaning that small infinitesimal differences in the initial state might lead to dramatic differences at later times. Poincaré [2, 3] was the first to realize that solutions of the three-body problem are generically highly sensitive to initial conditions. At about the same time, this property was recognized in geodesic flows with negative curvature by Hadamard [4]. One of the first experimental observations of chaos, as understood much later, was when irregular noise was heard by Van der Pol in 1927 [5] while studying a periodically forced nonlinear oscillator. Nevertheless, it was only with the advent of digital computing that chaos started to attract the interest of the wider scientific community. After the pioneering investigation of ergodicity in a chain of nonlinear oscillators by Fermi, Pasta and Ulam in 1955 [6], it was in the early 1960s that the numerical studies of Lorenz [7] and Hénon and Heiles [8] revealed that irregular and unpredictable motions are a generic feature of low-dimensional nonlinear deterministic systems. The existence and onset of chaos was then rigorously analyzed in several systems. While an exhaustive list of such mathematical proofs is beyond the scope of this preface, one should mention the contributions of Kolmogorov [9, 10], Chirikov [11], Smale [12], Ruelle and Takens [13], Li and Yorke [14] and Feigenbaum [15]. The characteristic Lyapunov exponents introduced by Oseledets in 1968 [16] are the fundamental quantities for measuring the sensitivity to initial conditions. Oseledets' work generalized the concept of Lyapunov stability to irregular trajectories building upon earlier studies of Birkhoff
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.
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.
Stabilization of Parametric Roll Resonance with Active U-Tanks via Lyapunov Control Design
DEFF Research Database (Denmark)
Holden, Christian; Galeazzi, Roberto; Fossen, Thor Inge;
2009-01-01
design of an active u-tank stabilizer is carried out using Lyapunov theory. A nonlinear backstepping controller is developed to provide global exponential stability of roll. An extension of commonly used u-tank models is presented to account for large roll angles, and the control design is tested via...
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.
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)
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.
Institute of Scientific and Technical Information of China (English)
吴东平
2012-01-01
基于Lyapunov运动稳定性理论,经过推导可知,一个单自由度的某一个受迫振动的特解的运动稳定性问题等价于这个单自由度系统自由振动的稳定问题.对于复杂非线性系统的动力稳定性问题,直接应用Lyapunov理论进行系统的动力稳定性判定比较困难,考虑大跨度拱型结构的变形特征,提出一种简洁、实用且适合数值计算的动力稳定性判别方法——位移时程变化法.运用该方法计算结构在承受一般动荷载类型和不同计算条件下的动力稳定性,验证此方法的实用性及正确性.%Based on the stability theory of Lyapunov motion,it can be derived that the motion stability problem of a particular solution to a forced vibration of a single degree of freedom system,is equal to the stability problem of the free vibration system of single degree of freedom.For a dynamic stability problem of complex nonlinear system,it is difficult to judge the dynamic stability of a system by the theory of Lyapunov motion directly.Considering the deformation feature of a large-span arch structure,a simple and practical theory named deformation history theory which is suitable for digital computation to judge the dynamic stability of an arch structure is put forward and used to judge the dynamic stability of an arch structure under common dynamic loading and different conditions.It is shown that the theory is applicable and reasonable.
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.
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.
Lyapunov stability and its application to systems of ordinary differential equations
Kennedy, E. W.
1979-01-01
An outline and a brief introduction to some of the concepts and implications of Lyapunov stability theory are presented. Various aspects of the theory are illustrated by the inclusion of eight examples, including the Cartesian coordinate equations of the two-body problem, linear and nonlinear (Van der Pol's equation) oscillatory systems, and the linearized Kustaanheimo-Stiefel element equations for the unperturbed two-body problem.
GA and Lyapunov theory-based hybrid adaptive fuzzy controller for non-linear systems
Roy, Ananya; Das Sharma, Kaushik
2015-02-01
In this present article, a new hybrid methodology for designing stable adaptive fuzzy logic controllers (AFLCs) for a class of non-linear system is proposed. The proposed design strategy exploits the features of genetic algorithm (GA)-based stochastic evolutionary global search technique and Lyapunov theory-based local adaptation scheme. The objective is to develop a methodology for designing AFLCs with optimised free parameters and guaranteed closed-loop stability. Simultaneously, the proposed method introduces automation in the design process. The stand-alone Lyapunov theory-based design, GA-based design and proposed hybrid GA-Lyapunov design methodologies are implemented for two benchmark non-linear plants in simulation case studies with different reference signals and one experimental case study. The results demonstrate that the hybrid design methodology outperforms the other control strategies on the whole.
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.
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.
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Blackwell, C. C.
1987-01-01
A relevant facet of the application of Lyapunov gradient-generated robust control to unstable linear autonomous plants is explored. It is demonstrated that if the plant, the output, and the nominal stabilizing control satisfy certain conditions, then the robust component alone stabilizes the nominal plant. An example characterized by two zero eigenvalues and two negative real value poles is presented. These results assure that the robust component will fulfill the role of nominal stabilization successfully so long as the possible magnitude of the robust component can overcome the contribution of the instability to positiveness of the Lyapunov rate.
Blackwell, C. C.
1987-01-01
A relevant facet of the application of Lyapunov gradient-generated robust control to unstable linear autonomous plants is explored. It is demonstrated that if the plant, the output, and the nominal stabilizing control satisfy certain conditions, then the robust component alone stabilizes the nominal plant. An example characterized by two zero eigenvalues and two negative real value poles is presented. These results assure that the robust component will fulfill the role of nominal stabilization successfully so long as the possible magnitude of the robust component can overcome the contribution of the instability to positiveness of the Lyapunov rate.
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’.
Energy Technology Data Exchange (ETDEWEB)
Blanchini, F. [Universita di Udine (Italy); Carabelli, S. [Politecnico di Torino (Italy)
1994-12-31
We apply a technique recently proposed in literature for the robust stabilization of linear systems with time-varying uncertain parameters to a magnetic levitation system. This technique allows the construction of a polyhedral Lyapunov function and a linear variable-structure stabilizing controller.
A Lyapunov-Krasovskii methodology for asymptotic stability of discrete time delay systems
Directory of Open Access Journals (Sweden)
Stojanović Sreten B.
2007-01-01
Full Text Available This paper presents a Lyapunov-Krasovskii methodology for asymptotic stability of discrete time delay systems. Based on the methods, delay-independent stability condition is derived. A numerical example has been working out to show the applicability of results derived.
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.
Robust Stabilization for Uncertain Control Systems Using Piecewise Quadratic Lyapunov Functions
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
The sufficient condition based on piecewise quadratic simultaneous Lyapunov functions for robust stabilizationof uncertain control systems via a constant linear state feedback control law is obtained. The objective is to use a robuststability criterion that is less conservative than the usual quadratic stability criterion. Numerical example is given, show-ing the advanteges of the proposed method.
Stability of stationary barotropic modons by Lyapunov's direct method
Sakuma, H.; Ghil, M.
1990-01-01
A new Liapunov stability condition is formulated for the shallow-water equations, using a gage-variable formalism. This sufficient condition is derived for the class of perturbations that conserve the total mass. It is weaker than existing stability criteria, i.e., it applies to a wider class of flows. Formal stability to infinitesimally small perturbations of arbitrary shape is obtained for two classes of large-scale geophysical flows: pseudo-eastward flow with constant shear, and localized coherent structures of modon type.
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.
H2-Stabilization of the Isothermal Euler Equations:a Lyapunov Function Approach
Institute of Scientific and Technical Information of China (English)
Martin GUGAT; Günter LEUGERING; Simona TAMASOIU; Ke WANG
2012-01-01
The authors consider the problem of boundary feedback stabilization of the 1D Euler gas dynamics locally around stationary states and prove the exponential stability with respect to the H2-norm.To this end,an explicit Lyapunov function as a weighted and squared H2-norm of a small perturbation of the stationary solution is constructed.The authors show that by a suitable choice of the boundary feedback conditions,the H2-exponential stability of the stationary solution follows.Due to this fact,the system is stabilized over an infinite time interval.Furthermore,exponential estimates for the C1-norm are derived.
Chu, Chia-Chi; Tsai, Hung-Chi; Chang, Wei-Neng
A Lyapunov-based recurrent neural networks unified power flow controller (UPFC) is developed for improving transient stability of power systems. First, a simple UPFC dynamical model, composed of a controllable shunt susceptance on the shunt side and an ideal complex transformer on the series side, is utilized to analyze UPFC dynamical characteristics. Secondly, we study the control configuration of the UPFC with two major blocks: the primary control, and the supplementary control. The primary control is implemented by standard PI techniques when the power system is operated in a normal condition. The supplementary control will be effective only when the power system is subjected by large disturbances. We propose a new Lyapunov-based UPFC controller of the classical single-machine-infinite-bus system for damping enhancement. In order to consider more complicated detailed generator models, we also propose a Lyapunov-based adaptive recurrent neural network controller to deal with such model uncertainties. This controller can be treated as neural network approximations of Lyapunov control actions. In addition, this controller also provides online learning ability to adjust the corresponding weights with the back propagation algorithm built in the hidden layer. The proposed control scheme has been tested on two simple power systems. Simulation results demonstrate that the proposed control strategy is very effective for suppressing power swing even under severe system conditions.
On the asymptotic stability of linear discrete time-delay systems: The Lyapunov approach
Directory of Open Access Journals (Sweden)
Stojanović Sreten B.
2006-01-01
Full Text Available New conditions for the stability of discrete delay systems of the form x (k+1 = Arjx (k + Aix (k-h are presented in the paper. These new delay-independent conditions were derived using an approach based on the second Lyapunov's method. These results are less conservative than some in the existing literature. A numerical example was worked out to show the applicability of the derived results.
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...
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.
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.
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.
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.
Theory and application of stability for stochastic reaction diffusion systems
Institute of Scientific and Technical Information of China (English)
LUO Qi; DENG FeiQi; MAO XueRong; BAO JunDong; ZHANG YuTian
2008-01-01
So far, the Lyapunov direct method is still the moat effective technique in the study of stability for ordinary differential equations and stochastic differential equations. Due to the shortage of the corresponding Ito formula, this useful method has not been popularized in stochastic partial differential equations. The aim of this work is to try to extend the Lyapunov direct method to the Ito stochastic reaction diffusion systems and to establish the corresponding Lyapunov stability theory, including stability in probablity, asymptotic stability in probablity, end exponential stability in mean square. As the application of the obtained theorems, this paper addresses the stability of the Hopfield neural network and points out that the main results ob-tained by Holden Helge and Liao Xiaoxin et al. can be all regarded as the corollaries of the theorems presented in this paper.
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.
Advances in stability theory at the end of the 20th century
Martynyuk, AA
2003-01-01
This volume presents surveys and research papers on various aspects of modern stability theory, including discussions on modern applications of the theory, all contributed by experts in the field. The volume consists of four sections that explore the following directions in the development of stability theory: progress in stability theory by first approximation; contemporary developments in Lyapunov''s idea of the direct method; the stability of solutions to periodic differential systems; and selected applications. Advances in Stability Theory at the End of the 20th Century will interest postgraduates and researchers in engineering fields as well as those in mathematics.
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.
Energy Technology Data Exchange (ETDEWEB)
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.
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.
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.
Design and Lyapunov Stability Analysis of a Fuzzy Logic Controller for Autonomous Road Following
Directory of Open Access Journals (Sweden)
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.
Kumar, Rajesh; Srivastava, Smriti; Gupta, J R P
2017-03-01
In this paper adaptive control of nonlinear dynamical systems using diagonal recurrent neural network (DRNN) is proposed. The structure of DRNN is a modification of fully connected recurrent neural network (FCRNN). Presence of self-recurrent neurons in the hidden layer of DRNN gives it an ability to capture the dynamic behaviour of the nonlinear plant under consideration (to be controlled). To ensure stability, update rules are developed using lyapunov stability criterion. These rules are then used for adjusting the various parameters of DRNN. The responses of plants obtained with DRNN are compared with those obtained when multi-layer feed forward neural network (MLFFNN) is used as a controller. Also, in example 4, FCRNN is also investigated and compared with DRNN and MLFFNN. Robustness of the proposed control scheme is also tested against parameter variations and disturbance signals. Four simulation examples including one-link robotic manipulator and inverted pendulum are considered on which the proposed controller is applied. The results so obtained show the superiority of DRNN over MLFFNN as a controller. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
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.
Directory of Open Access Journals (Sweden)
Prechanon Kumkratug
2011-01-01
Full Text Available Problem statement: The disturbance in power system is unavoidable situation. It causes in power system oscillation. Approach: This study applied the Static Var Compensator (SVC to damp power system oscillation. The stability criterion of the Lyapunov is applied to derive the control strategy of SVC. The simulation results are tested on a Single Machine Infinite bus. The proposed method is equipped in sample system with disturbance. The generator rotor angle curve of the system without and with a SVC is plotted and compared for various cases. Results: It was found that the system without a SVC has high variation whereas that of the system with a SVC has much smaller variation. Conclusion: From the simulation results, the SVC can damp power system oscillaton.
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
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
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...
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
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.
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.
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.
Azizi, Sajad
2017-05-01
The robust stability of a class of feedback linearizable minimum-phase nonlinear system, having parametric uncertainties, is investigated in this study. The system in new coordinates is represented to an equivalent formulation after the attempt of feedback linearization. Due to the parametric uncertainties the approximately linearized system entails a norm bounded input nonlinearity such that the equilibrium point condition in error dynamics can not be satisfied. Accordingly, to guarantee the regional asymptotic stability a control synthesis problem is proposed by means of sufficient Linear Matrix Inequalities (LMIs) together with an amended nonlinear control term, derived from the Lyapunov redesign method, which tackles zero steady-state error condition. The numerical examples of a general aviation aircraft's longitudinal dynamics and inverted pendulum are simulated to show the proficiency of the proposed control technique. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
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...
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.
Yedavalli, R. K.
1992-01-01
The problem of analyzing and designing controllers for linear systems subject to real parameter uncertainty is considered. An elegant, unified theory for robust eigenvalue placement is presented for a class of D-regions defined by algebraic inequalities by extending the nominal matrix root clustering theory of Gutman and Jury (1981) to linear uncertain time systems. The author presents explicit conditions for matrix root clustering for different D-regions and establishes the relationship between the eigenvalue migration range and the parameter range. The bounds are all obtained by one-shot computation in the matrix domain and do not need any frequency sweeping or parameter gridding. The method uses the generalized Lyapunov theory for getting the bounds.
Yedavalli, R. K.
1992-01-01
The problem of analyzing and designing controllers for linear systems subject to real parameter uncertainty is considered. An elegant, unified theory for robust eigenvalue placement is presented for a class of D-regions defined by algebraic inequalities by extending the nominal matrix root clustering theory of Gutman and Jury (1981) to linear uncertain time systems. The author presents explicit conditions for matrix root clustering for different D-regions and establishes the relationship between the eigenvalue migration range and the parameter range. The bounds are all obtained by one-shot computation in the matrix domain and do not need any frequency sweeping or parameter gridding. The method uses the generalized Lyapunov theory for getting the bounds.
Directory of Open Access Journals (Sweden)
Coşkun Yakar
2010-01-01
Full Text Available The qualitative behavior of a perturbed fractional-order differential equation with Caputo's derivative that differs in initial position and initial time with respect to the unperturbed fractional-order differential equation with Caputo's derivative has been investigated. We compare the classical notion of stability to the notion of initial time difference stability for fractional-order differential equations in Caputo's sense. We present a comparison result which again gives the null solution a central role in the comparison fractional-order differential equation when establishing initial time difference stability of the perturbed fractional-order differential equation with respect to the unperturbed fractional-order differential equation.
Institute of Scientific and Technical Information of China (English)
Qiu Fang; Zhang Quan-Xin; Deng Xue-Hui
2012-01-01
This paper investigates the asymptotical stability problem of a neural system with a constant delay.A new delaydependent stability condition is derived by using the novel augmented Lyapunov-Krasovskii function with triple integral terms,and the additional triple integral terms play a key role in the further reduction of conservativeness.Finally,a numerical example is given to demonstrate the effectiveness and lower conservativeness of the proposed method.
Elements of magnetohydrodynamic stability theory
Energy Technology Data Exchange (ETDEWEB)
Spies, G O
1976-11-01
The nonlinear equations of ideal magnetohydrodynamics are discussed along with the following topics: (1) static equilibrium, (2) strict linear theory, (3) stability of a system with one degree of freedom, (4) spectrum and variational principles in magnetohydrodynamics, (5) elementary proof of the modified energy principle, (6) sufficient stability criteria, (7) local stability, and (8) normal modes. (MOW)
Pakmehr, Mehrdad; Fitzgerald, Nathan; Feron, Eric; Shamma, Jeff; Behbahani, Alireza
2012-01-01
This manuscript aims to develop and describe gain scheduling control concept for a gas turbine engine which drives a variable pitch propeller. An architecture for gain-scheduling control is developed that controls the turboshaft engine for large thrust commands in stable fashion with good performance. Fuel ow and propeller pitch angle are the two control inputs of the system. New stability proof has been developed for gain scheduling control of gas turbine engines using global linearization a...
Robinett III, Rush D
2011-01-01
Nonlinear Powerflow Control Design presents an innovative control system design process motivated by renewable energy electric grid integration problems. The concepts developed result from the convergence of three research and development goals: • to create a unifying metric to compare the value of different energy sources – coal-burning power plant, wind turbines, solar photovoltaics, etc. – to be integrated into the electric power grid and to replace the typical metric of costs/profit; • to develop a new nonlinear control tool that applies power flow control, thermodynamics, and complex adaptive systems theory to the energy grid in a consistent way; and • to apply collective robotics theories to the creation of high-performance teams of people and key individuals in order to account for human factors in controlling and selling power into a distributed, decentralized electric power grid. All three of these goals have important concepts in common: exergy flow, limit cycles, and balance between compe...
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
Li, Zhihong; Liu, Lei; Zhu, Quanxin
2016-12-01
This paper studies the mean-square exponential input-to-state stability of delayed Cohen-Grossberg neural networks with Markovian switching. By using the vector Lyapunov function and property of M-matrix, two generalized Halanay inequalities are established. By means of the generalized Halanay inequalities, sufficient conditions are also obtained, which can ensure the exponential input-to-state stability of delayed Cohen-Grossberg neural networks with Markovian switching. Two numerical examples are given to illustrate the efficiency of the derived results. Copyright © 2016 Elsevier Ltd. All rights reserved.
Haddad, Wassim M.; Bernstein, Dennis S.
1991-01-01
Lyapunov function proofs of sufficient conditions for asymptotic stability are given for feedback interconnections of bounded real and positive real transfer functions. Two cases are considered: (1) a proper bounded real (resp., positive real) transfer function with a bounded real (resp., positive real) time-varying memoryless nonlinearity; and (2) two strictly proper bounded real (resp., positive real) transfer functions. A similar treatment is given for the circle and Popov theorems. Application of these results to robust stability with time-varying bounded real, positive real, and sector-bounded uncertainty is discussed.
Haddad, Wassim M.; Bernstein, Dennis S.
1991-01-01
Lyapunov function proofs of sufficient conditions for asymptotic stability are given for feedback interconnections of bounded real and positive real transfer functions. Two cases are considered: (1) a proper bounded real (resp., positive real) transfer function with a bounded real (resp., positive real) time-varying memoryless nonlinearity; and (2) two strictly proper bounded real (resp., positive real) transfer functions. A similar treatment is given for the circle and Popov theorems. Application of these results to robust stability with time-varying bounded real, positive real, and sector-bounded uncertainty is discussed.
Robust root clustering for linear uncertain systems using generalized Lyapunov theory
Yedavalli, R. K.
1993-01-01
Consideration is given to the problem of matrix root clustering in subregions of a complex plane for linear state space models with real parameter uncertainty. The nominal matrix root clustering theory of Gutman & Jury (1981) using the generalized Liapunov equation is extended to the perturbed matrix case, and bounds are derived on the perturbation to maintain root clustering inside a given region. The theory makes it possible to obtain an explicit relationship between the parameters of the root clustering region and the uncertainty range of the parameter space.
Robust root clustering for linear uncertain systems using generalized Lyapunov theory
Yedavalli, R. K.
1993-01-01
Consideration is given to the problem of matrix root clustering in subregions of a complex plane for linear state space models with real parameter uncertainty. The nominal matrix root clustering theory of Gutman & Jury (1981) using the generalized Liapunov equation is extended to the perturbed matrix case, and bounds are derived on the perturbation to maintain root clustering inside a given region. The theory makes it possible to obtain an explicit relationship between the parameters of the root clustering region and the uncertainty range of the parameter space.
Stability theory of differential equations
Bellman, Richard
2008-01-01
Suitable for advanced undergraduates and graduate students, this was the first English-language text to offer detailed coverage of boundedness, stability, and asymptotic behavior of linear and nonlinear differential equations. It remains a classic guide, featuring material from original research papers, including the author's own studies.The linear equation with constant and almost-constant coefficients receives in-depth attention that includes aspects of matrix theory. No previous acquaintance with the theory is necessary, since author Richard Bellman derives the results in matrix theory from
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.
An introduction to stability theory
Pillay, Anand
2008-01-01
This introductory treatment covers the basic concepts and machinery of stability theory. Lemmas, corollaries, proofs, and notes assist readers in working through and understanding the material and applications. Full of examples, theorems, propositions, and problems, it is suitable for graduate students in logic and mathematics, professional mathematicians, and computer scientists. Chapter 1 introduces the notions of definable type, heir, and coheir. A discussion of stability and order follows, along with definitions of forking that follow the approach of Lascar and Poizat, plus a consideration
Institute of Scientific and Technical Information of China (English)
岑丽辉; 席裕庚
2009-01-01
A Lyapunov function is constructed based on the weighted sum of entropies for the case of two open channels in cascade, which is described by the Saint-Venant equations. A class of boundary feedback controllers is presented to guaran-tee the local closed-loop asymptotic stability in a neighborhood of the equilibrium point by means of the Lyapunov design ap-proach.
The Stability Analysis for an Extended Car Following Model Based on Control Theory
Ge, Hong-Xia; Meng, Xiang-Pei; Zhu, Ke-Qiang; Cheng, Rong-Jun
2014-08-01
A new method is proposed to study the stability of the car-following model considering traffic interruption probability. The stability condition for the extended car-following model is obtained by using the Lyapunov function and the condition for no traffic jam is also given based on the control theory. Numerical simulations are conducted to demonstrate and verify the analytical results. Moreover, numerical simulations show that the traffic interruption probability has an influence on driving behavior and confirm the effectiveness of the method on the stability of traffic flow.
Stability theory for dynamic equations on time scales
Martynyuk, Anatoly A
2016-01-01
This monograph is a first in the world to present three approaches for stability analysis of solutions of dynamic equations. The first approach is based on the application of dynamic integral inequalities and the fundamental matrix of solutions of linear approximation of dynamic equations. The second is based on the generalization of the direct Lyapunovs method for equations on time scales, using scalar, vector and matrix-valued auxiliary functions. The third approach is the application of auxiliary functions (scalar, vector, or matrix-valued ones) in combination with differential dynamic inequalities. This is an alternative comparison method, developed for time continuous and time discrete systems. In recent decades, automatic control theory in the study of air- and spacecraft dynamics and in other areas of modern applied mathematics has encountered problems in the analysis of the behavior of solutions of time continuous-discrete linear and/or nonlinear equations of perturbed motion. In the book “Men of Ma...
A simple extension of contraction theory to study incremental stability properties
DEFF Research Database (Denmark)
Jouffroy, Jerome
Contraction theory is a recent tool enabling to study the stability of nonlinear systems trajectories with respect to one another, and therefore belongs to the class of incremental stability methods. In this paper, we extend the original definition of contraction theory to incorporate...... in an explicit manner the control input of the considered system. Such an extension, called universal contraction, is quite analogous in spirit to the well-known Input-to-State Stability (ISS). It serves as a simple formulation of incremental ISS, external stability, and detectability in a differential setting....... The hierarchical combination result of contraction theory is restated in this framework, and a differential small-gain theorem is derived from results already available in Lyapunov theory....
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.
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.
Farivar, Faezeh; Shoorehdeli, Mahdi Aliyari
2012-01-01
In this paper, fault tolerant synchronization of chaotic gyroscope systems versus external disturbances via Lyapunov rule-based fuzzy control is investigated. Taking the general nature of faults in the slave system into account, a new synchronization scheme, namely, fault tolerant synchronization, is proposed, by which the synchronization can be achieved no matter whether the faults and disturbances occur or not. By making use of a slave observer and a Lyapunov rule-based fuzzy control, fault tolerant synchronization can be achieved. Two techniques are considered as control methods: classic Lyapunov-based control and Lyapunov rule-based fuzzy control. On the basis of Lyapunov stability theory and fuzzy rules, the nonlinear controller and some generic sufficient conditions for global asymptotic synchronization are obtained. The fuzzy rules are directly constructed subject to a common Lyapunov function such that the error dynamics of two identical chaotic motions of symmetric gyros satisfy stability in the Lyapunov sense. Two proposed methods are compared. The Lyapunov rule-based fuzzy control can compensate for the actuator faults and disturbances occurring in the slave system. Numerical simulation results demonstrate the validity and feasibility of the proposed method for fault tolerant synchronization.
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.
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.
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.
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.
Cui, Ying; Wang, Rui; Huang, Huang; Zhang, Shunqing
2011-01-01
In this tutorial paper, a comprehensive survey is given on several major systematic approaches in dealing with delay-aware control problems, namely the equivalent rate constraint approach, the Lyapunov stability drift approach and the approximate Markov Decision Process (MDP) approach using stochastic learning. These approaches essentially embrace most of the existing literature regarding delay-aware resource control in wireless systems. They have their relative pros and cons in terms of performance, complexity and implementation issues. For each of the approaches, the problem setup, the general solution and the design methodology are discussed. Applications of these approaches to delay-aware resource allocation are illustrated with examples in single-hop wireless networks. Furthermore, recent results regarding delay-aware multi-hop routing designs in general multi-hop networks are elaborated. Finally, the delay performance of the various approaches are compared through simulations using an example of the upl...
Robust Stability Analysis of Nonlinear Switched Systems with Filippov Solutions
DEFF Research Database (Denmark)
Ahmadi, Mohamadreza; Mojallali, Hamed; Wisniewski, Rafal
2012-01-01
. Based on the theory of differential inclusions, a Lyapunov stability theorem is brought forward. These results are also extended to autonomous switched systems subject to polytopic uncertainty. Furthermore, the proposed stability theorems are reformulated using the sum of squares decomposition method...... which provides sufficient means to construct the corresponding Lyapunov functions via available semi-definite programming techniques....
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.
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....
Directory of Open Access Journals (Sweden)
Seung Kwan Song
2016-10-01
Full Text Available We present two control strategies for an oscillating water column-wave energy converter (OWC-WEC in the time domain. We consider a fixed OWC-WEC on the open sea with an impulse turbine module. This system mainly consists of a chamber, turbine and electric generator. For the time domain analysis, all of the conversion stages considering mutualities among them should be analyzed based on the Newtonian mechanics. According to the analysis of Newtonian mechanics, the hydrodynamics of wave energy absorption in the chamber and the turbine aerodynamic performance are directly coupled and share the internal air pressure term via the incompressible air assumption. The turbine aerodynamics and the dynamics of the electric generator are connected by torque load through the rotor shaft, which depends on an electric terminal load that acts as a control input. The proposed control strategies are an instant maximum turbine efficiency tracking control and a constant angular velocity of the turbine rotor control methods. Both are derived by Lyapunov stability analysis. Numerical simulations are carried out under irregular waves with various heights and periods in the time domain, and the results with the controllers are analyzed. We then compare these results with simulations carried out in the absence of the control strategy in order to prove the performance of the controllers.
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.
Institute of Scientific and Technical Information of China (English)
李安; 宋新宇; 王志祥
2011-01-01
该文研究了非线性微分方程关于初始时刻偏差的实用稳定性,利用扰动Lyapunov函数得到了几个非线性动力系统关于初始时刻偏差的实用稳定性准则,所得结论丰富了非线性微分方程关于初始时刻偏差的实用稳定性理论.%In this paper, the practical stability of nonlinear differential equations with solutions starting off with different initial times is investigated. Several practical stability criteria of nonlinear dynamical systems relative to initial time difference are presented by perturbing Lyapunov functions. The results enrich the theory on practical stability of nonlinear differential equations relative to initial time difference.
Drop stability in wind: theory
Lee, Sungyon
2015-11-01
Water drops may remain pinned on a solid substrate against external forcing due to contact angle hysteresis. Schmucker and White investigated this phenomenon experimentally in a high Reynolds number regime, by measuring the critical wind velocity at which partially wetting water drops depin inside a wind tunnel. Due to the unsteady turbulent boundary layer, droplets are observed to undergo vortex-shedding induced oscillations. By contrast, the overall elongation of the drop prior to depinning occurs on a much slower timescale with self-similar droplet shapes at the onset. Based on these observations, a simple, quasi-static model of depinning droplet is developed by implementing the phenomenological description of the boundary layer. The resultant model successfully captures the critical onset of droplet motion and is the first of on-going studies that connect the classical boundary layer theory with droplet dynamics.
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.
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...
Theory of Arched Structures Strength, Stability, Vibration
Karnovsky, Igor A
2012-01-01
Theory of Arched Structures: Strength, Stability, Vibration presents detailed procedures for analytical analysis of the strength, stability, and vibration of arched structures of different types, using exact analytical methods of classical structural analysis. The material discussed is divided into four parts. Part I covers stress and strain with a particular emphasis on analysis; Part II discusses stability and gives an in-depth analysis of elastic stability of arches and the role that matrix methods play in the stability of the arches; Part III presents a comprehensive tutorial on dynamics and free vibration of arches, and forced vibration of arches; and Part IV offers a section on special topics which contains a unique discussion of plastic analysis of arches and the optimal design of arches.
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.
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.
On formalism and stability of switched systems
DEFF Research Database (Denmark)
Leth, John-Josef; Wisniewski, Rafal
2012-01-01
In this paper, we formulate a uniform mathematical framework for studying switched systems with piecewise linear partitioned state space and state dependent switching. Based on known results from the theory of differential inclusions, we devise a Lyapunov stability theorem suitable for this class...... of switched systems. With this, we prove a Lyapunov stability theorem for piecewise linear switched systems by means of a concrete class of Lyapunov functions. Contrary to existing results on the subject, the stability theorems in this paper include Filippov (or relaxed) solutions and allow infinite switching...
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.
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.
Conditions of asymptotic stability for linear homogeneous switched systems
Ivanov, Gennady; Alferov, Gennady; Sharlay, Artem; Efimova, Polina
2017-07-01
In this article the authors prove the theorems giving the necessary and sufficient conditions for stability of robotic and mechatronic systems motion in terms of Lyapunov functions theory with the use of set-theoretic approach.
Stability theory of Knudsen plasma diodes
Energy Technology Data Exchange (ETDEWEB)
Kuznetsov, V. I., E-mail: victor.kuznetsov@mail.ioffe.ru; Ender, A. Ya. [Ioffe Institute, Russian Academy of Sciences (Russian Federation)
2015-11-15
A stability theory is developed for a plasma diode in which an electron beam supplied from the emitter propagates without collisions in the self-consistent electric field against the immobile ion background. An integral equation for the amplitude of the perturbed field is deduced using the Q,G method for the regime without electron reflection from a potential barrier. Analytic solutions to this equation are obtained for a number of important particular cases, and the plasma dispersion properties are examined.
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...
Handbook of functional equations stability theory
2014-01-01
This handbook consists of seventeen chapters written by eminent scientists from the international mathematical community, who present important research works in the field of mathematical analysis and related subjects, particularly in the Ulam stability theory of functional equations. The book provides an insight into a large domain of research with emphasis to the discussion of several theories, methods and problems in approximation theory, analytic inequalities, functional analysis, computational algebra and applications. The notion of stability of functional equations has its origins with S. M. Ulam, who posed the fundamental problem for approximate homomorphisms in 1940 and with D. H. Hyers, Th. M. Rassias, who provided the first significant solutions for additive and linear mappings in 1941 and 1978, respectively. During the last decade the notion of stability of functional equations has evolved into a very active domain of mathematical research with...
Moduli stabilization in heterotic M-theory
Correia, Filipe Paccetti
2007-01-01
We reconsider the ingredients of moduli stabilization in heterotic M-theory. On this line we close a gap in the literature deriving the Kaehler potential dependence on vector bundle moduli and charged matter. Crucial in this derivation is our superspace formulation of 5d heterotic M-theory taking into account the Bianchi identities modified by brane terms. Likewise, we obtain the Fayet-Iliopolous terms due to brane localised anomalous U(1)'s. After assembling perturbative and non-perturbative contributions to the superpotential, we study supersymmetric (adS) vacua. It is found that the susy condition decouples the bundle moduli from the geometric moduli. We show that M-theory supersymmetric vacua without five-branes can be found, albeit not at phenomenologically interesting values of the geometric moduli. This result is fairly independent of the choice of vector bundle at the observable brane.
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.
Theory and modelling of nanocarbon phase stability.
Energy Technology Data Exchange (ETDEWEB)
Barnard, A. S.
2006-01-01
The transformation of nanodiamonds into carbon-onions (and vice versa) has been observed experimentally and has been modeled computationally at various levels of sophistication. Also, several analytical theories have been derived to describe the size, temperature and pressure dependence of this phase transition. However, in most cases a pure carbon-onion or nanodiamond is not the final product. More often than not an intermediary is formed, known as a bucky-diamond, with a diamond-like core encased in an onion-like shell. This has prompted a number of studies investigating the relative stability of nanodiamonds, bucky-diamonds, carbon-onions and fullerenes, in various size regimes. Presented here is a review outlining results of numerous theoretical studies examining the phase diagrams and phase stability of carbon nanoparticles, to clarify the complicated relationship between fullerenic and diamond structures at the nanoscale.
Liapunov Functions and Stability in Control Theory
Bacciotti, Andrea
2005-01-01
This book presents a modern and self-contained treatment of the Liapunov method for stability analysis, in the framework of mathematical nonlinear control theory. A Particular focus is on the problem of the existence of Liapunov functions (converse Liapunov theorems) and their regularity, whose interest is especially motivated by applications to automatic control. Many recent results in this area have been collected and presented in a systematic way. Some of them are given in extended, unified versions and with new, simpler proofs. In the 2nd edition of this successful book several new section
A New Approach to the Method of Lyapunov Functionals and Its Applications
Directory of Open Access Journals (Sweden)
Yunguo Jin
2013-01-01
Full Text Available We show some results which can replace the graph theory used to construct global Lyapunov functions in some coupled systems of differential equations. We present an example of an epidemic model with stage structure and latency spreading in a heterogeneous host population and obtain a more general threshold for the extinction and persistence of a disease. Using some results obtained by mathematical induction and suitable Lyapunov functionals, we prove the global stability of the endemic equilibrium. For some coupled systems of differential equations, by a similar approach to the discussion of the epidemic model, the conditions of threshold property or global stability can be established without the assumption that the relative matrix is irreducible.
On Designing Lyapunov-Krasovskii Based AQM for Routers Supporting TCP Flows
Labit, Yann; Gouaisbaut, Frédéric; 10.1109/CDC.2007.4434673
2009-01-01
For the last few years, we assist to a growing interest of designing AQM (Active Queue Management) using control theory. In this paper, we focus on the synthesis of an AQM based on the Lyapunov theory for time delay systems. With the help of a recently developed Lyapunov-Krasovskii functional and using a state space representation of a linearized fluid model of TCP, two robust AQMs stabilizing the TCP model are constructed. Notice that our results are constructive and the synthesis problem is reduced to a convex optimization scheme expressed in terms of linear matrix inequalities (LMIs). Finally, an example extracted from the literature and simulations via {\\it NS simulator} support our study.
Stability of Fractional Order Switching Systems
HosseinNia, S Hassan; Vinagre, Blas M
2012-01-01
This paper addresses the stabilization issue for fractional order switching systems. Common Lyapunov method is generalized for fractional order systems and frequency domain stability equivalent to this method is proposed to prove the quadratic stability. Some examples are given to show the applicability and effectiveness of the proposed theory.
Practical stability of nonlinear systems
Lakshmikantham, Vangipuram; Martynyuk, Anatolii Andreevich
1990-01-01
This is the first book that deals with practical stability and its development. It presents a systematic study of the theory of practical stability in terms of two different measures and arbitrary sets and demonstrates the manifestations of general Lyapunov's method by showing how this effective technique can be adapted to investigate various apparently diverse nonlinear problems including control systems and multivalued differential equations.
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.
Stability of melt crystal growth under microgravity conditions
Tatarchenko, V. A.
The conception of dynamic stability of melt crystal growth has been developed. The method based on the Lyapunov stability theory has been used to the study stability of crystallization by capillary shaping techniques including Czokhralsky, Stepanov, Kiropoulos, Verneuil and floating zone methods. Preliminary results of the stability analysis of crystallization by floating zone technique under microgravity conditions are presented here.
Stability analysis of nonlinear systems
Lakshmikantham, Vangipuram; Martynyuk, Anatoly A
2015-01-01
The book investigates stability theory in terms of two different measure, exhibiting the advantage of employing families of Lyapunov functions and treats the theory of a variety of inequalities, clearly bringing out the underlying theme. It also demonstrates manifestations of the general Lyapunov method, showing how this technique can be adapted to various apparently diverse nonlinear problems. Furthermore it discusses the application of theoretical results to several different models chosen from real world phenomena, furnishing data that is particularly relevant for practitioners. Stability Analysis of Nonlinear Systems is an invaluable single-sourse reference for industrial and applied mathematicians, statisticians, engineers, researchers in the applied sciences, and graduate students studying differential equations.
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.
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.
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.
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.
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.
Stability of Linear Stochastic Differential Equations with Respect to Fractional Brownian Motion
Institute of Scientific and Technical Information of China (English)
SHU Hui-sheng; CHEN Chun-li; WEI Guo-liang
2009-01-01
This paper is concerned with the stochastically stability for the m -dimensional linear stochastic differential equations with respect to fractional Brownian motion (FBM) with Hurst parameter H∈ (1/2, 1). On the basis of the pioneering work of Duncan and Hu, a Ito's formula is given.An improved derivative operator to Lyapunov functions is constructed, and the sufficient conditions for the stochastically stability of linear stochastic differential equations driven by FBM are established. These extend the stochastic Lyapunov stability theories.
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 ...
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.
f(R) Theories: stability of stars
Energy Technology Data Exchange (ETDEWEB)
Domont, S.; O' Dwyer, M.; Joras, S.E. [Universidade Federal do Rio de Janeiro (UFRJ), RJ (Brazil). Inst. de Fisica
2011-07-01
Full text: Assuming homogeneity in very large scales, the universe is undergoing a period of accelerated expansion. According to General Relativity (GR), this is caused by the so-called Dark Energy, which accounts for about 70% of the universe. To fill such a huge gap is one of the main reasons to study alternatives to GR. Nevertheless, the very existence of static spherically symmetric solutions puts strong constraints on a particular class of modified gravity theories - the so-called f(R). Even when one is able to find a numeric solution that interpolates the metric from the center of a star up to a de Sitter universe (at spatial infinity), the necessary fine tuning may be simply too strong to allow the existence of such objects in the real world. We will show studies on the stability of such solutions for homogeneous stars. Namely, we will argue that the border between 'good' and 'bad' initial conditions has a fractal dimension, which means that one cannot be sure that it has or not been crossed. Besides, any infinitesimal displacement caused by a inherit fluctuation may drive the system, i.e, the star, to instability. (author)
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.
Xinghua Liu; Hongsheng Xi
2013-01-01
The exponential stability of neutral Markovian jump systems with interval mode-dependent time-varying delays, nonlinear perturbations, and partially known transition rates is investigated. A novel augmented stochastic Lyapunov functional is constructed, which employs the improved bounding technique and contains triple-integral terms to reduce conservativeness; then the delay-range-dependent and rate-dependent exponential stability criteria are developed by Lyapunov stability theory, reciproca...
Stabilization of a Class of Stochastic Systems with Time Delays
Directory of Open Access Journals (Sweden)
Jian Wang
2014-01-01
Full Text Available The problem of exponential stability is investigated for a class of stochastic time-delay systems. By using the decomposition technique and Lyapunov stability theory, two improved exponential stability criteria are derived. Finally, a numerical example is given to illustrate the effectiveness and the benefit of the proposed method.
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.
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.
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.
A Stability Theory in Nonlinear Programming
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
We propose a new method for finding the local optimal points ofthe constrained nonlinear programming by Ordinary Differential Equations (ODE), and prove asymptotic stability of the singular points of partial variables in this paper. The condition of overall uniform, asymptotic stability is also given.
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.
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 ...
Vacuum stability of asymptotically safe gauge-Yukawa theories
DEFF Research Database (Denmark)
Litim, Daniel F.; Mojaza, Matin; Sannino, Francesco
2016-01-01
We study the phase diagram and the stability of the ground state for certain four-dimensional gauge-Yukawa theories whose high-energy behaviour is controlled by an interacting fixed point. We also provide analytical and numerical results for running couplings, their crossover scales, the separatrix......, and the Coleman-Weinberg effective potential. Classical and quantum stability of the vacuum is established....
Vacuum stability of asymptotically safe gauge-Yukawa theories
Litim, Daniel F; Sannino, Francesco
2016-01-01
We study the phase diagram and the stability of the ground state for certain four-dimensional gauge-Yukawa theories whose high-energy behaviour is controlled by an interacting fixed point. We also provide analytical and numerical results for running couplings, their crossover scales, the separatrix, and the Coleman-Weinberg effective potential. Classical and quantum stability of the vacuum is established.
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 and Reconstruction Operations Doctrine and Theory
2014-05-22
Bosnia—Herzegovina, Kosovo , East Timor, and of course Iraq and Afghanistan. A notable exception is Guiding Principles for Stabilization and...Galula advises commanders not to waste resources protecting terrain that would be considered critical in a conventional fight but rather to deploy
Stability in higher-derivative matter fields theories
Tretyakov, Petr V.
2016-09-01
We discuss possible instabilities in higher-derivative matter field theories. These theories have two free parameters β _1 and β _4. By using a dynamical system approach we explicitly demonstrate that for the stability of Minkowski space in an expanding universe we need the condition β _4-1/3β _4, which is needed to avoid a tachyon-like instability.
Stability in higher-derivative matter fields theories
Tretyakov, Petr V
2016-01-01
We discuss possible instabilities in higher-derivative matter fields theories. These theories has two free parameters $\\beta_1$ and $\\beta_4$. By using dynamical system approach we explicitly demonstrate that for stability of Minkowski space in expanding Universe it is need condition $\\beta_4-\\frac{1}{3}\\beta_4$ which is need to avoid tachyon-like instability.
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.
Quantum Stability of Chameleon Field Theories
Upadhye, Amol; Khoury, Justin
2012-01-01
Chameleon scalar fields are dark energy candidates which suppress fifth forces in high density regions of the universe by becoming massive. We consider chameleon models as effective field theories and estimate quantum corrections to their potentials. Requiring that quantum corrections be small, so as to allow reliable predictions of fifth forces, leads to an upper bound $m 0.0042$\\,eV. An improvement of less than a factor of two in the range of fifth force experiments could test all classical chameleon field theories whose quantum corrections are well-controlled and couple to matter with nearly gravitational strength regardless of the specific form of the chameleon potential.
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...
New results in subspace-stabilization control theory
Directory of Open Access Journals (Sweden)
C. D. Johnson
2000-01-01
Full Text Available Subspace-stabilization is a generalization of the classical idea of stabilizing motions of a dynamical system to an equilibrium state. The concept of subspace-stabilization and a theory for designing subspace-stabilizing control laws was introduced in a previously published paper. In the present paper, two new alternative methods for designing control laws that achieve subspace-stabilization are presented. These two alternative design methods are based on: (i a novel application of existing Linear Quadratic Regulator optimal-control theory, and (ii an algebraic method in which the control-law is expressed as a linear feedback of certain “canonical variables.” In some cases, these new design methods may be more effective than existing ones. The results are illustrated by worked examples.
Directory of Open Access Journals (Sweden)
Hao Chen
2015-01-01
Full Text Available This paper concerns the problem of the globally exponential stability of neural networks with discrete and distributed delays. A novel criterion for the globally exponential stability of neural networks is derived by employing the Lyapunov stability theory, homomorphic mapping theory, and matrix theory. The proposed result improves the previously reported global stability results. Finally, two illustrative numerical examples are given to show the effectiveness of our results.
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.
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.
Energy Technology Data Exchange (ETDEWEB)
Kwon, O.M., E-mail: madwind@chungbuk.ac.k [School of Electrical Engineering, Chungbuk National University, Cheongju (Korea, Republic of); Lee, S.M., E-mail: moony@daegu.ac.k [School of Electronics Engineering, Daegu University, Kyongsan (Korea, Republic of); Park, Ju H., E-mail: jessie@ynu.ac.k [Department of Electrical Engineering, Yeungnam University, Kyongsan (Korea, Republic of)
2010-02-22
This Letter investigates the problem of delay-dependent exponential stability analysis for uncertain stochastic neural networks with time-varying delay. Based on the Lyapunov stability theory, improved delay-dependent exponential stability criteria for the networks are established in terms of linear matrix inequalities (LMIs).
Delay-Dependent Exponential Stability Criterion for BAM Neural Networks with Time-Varying Delays
Institute of Scientific and Technical Information of China (English)
Wei-Wei Su; Yi-Ming Chen
2008-01-01
By employing the Lyapunov stability theory and linear matrix inequality (LMI) technique, delay dependent stability criterion is derived to ensure the exponential stability of bi-directional associative memory (BAM) neural networks with time-varying delays. The proposed condition can be checked easily by LMI control toolbox in Matlab. A numerical example is given to demonstrate the effectiveness of our results.
ASYMPTOTIC STABILITY OF A SINGULAR SYSTEM WITH DISTRIBUTED DELAYS
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
Based on the stability theory of functional differential equations, this paper studies the asymptotic stability of a singular system with distributed delays by constructing suitable Lyapunov functionals and applying the linear matrix inequalities. A numerical example is given to show the effectiveness of the main 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...
The theory of stabilization of sawtooth oscillations in TFTR supershots
Energy Technology Data Exchange (ETDEWEB)
Zakharov, L.; Levinton, F. [Princeton Univ., NJ (United States). Plasma Physics Lab.; Migliuolo, S.; Rogers, B. [Massachusetts Inst. of Tech., Cambridge, MA (United States)
1993-11-01
A theoretical concept of onset and stabilization of sawtooth oscillations in tokamaks is formulated based on the analysis of supershots in the Tokamak Fusion Test Reactor (TFTR). While the linear theory, which includes the ideal m = 1 mode, contradicts the experimental data, the criterion of {omega}*-stabilization of the collisionless m = 1 reconnection mode determines an operational space for the sawtooth-free phase in TFTR.
Hegemonic Stability Theories of the International Monetary System
Eichengreen, Barry
1987-01-01
Specialists in international relations have argued that international regimes operate smoothly and exhibit stability only when dominated by a single, exceptionally powerful national economy. In particular, this "theory of hegemonic stability" has been applied to the international monetary system. The maintenance of the Bretton Woods System for a quarter of a century up to 1972 is ascribed to the singular power of the United States in the postwar world, while the persistence of the classical g...
Graphic theory on interval stability of networked control systems
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
A new method on the interval stability of networked control systems (NCSs) with random delay and data packet dropout is studied. Combining interval systems and NCSs, a graphic condition on judging interval stability is presented in terms of the weighted diagraph theory in graph theory. Furthermore, utilizing the graph-theoretic algorithm, the delay-depended controller gains are obtained. Aiming at the same delay and data packed dropout, several controller gains are obtained, simultaneously. The example and simulation illustrate the effectiveness of the proposed method.
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.
Lyapunov Stability of Complementarity and Extended Systems
Camlibel, M. Kanat; Pang, Jong-Shi; Shen, Jinglai
2006-01-01
A linear complementarity system (LCS) is a piecewise linear dynamical system consisting of a linear time-invariant ordinary differential equation (ODE) parameterized by an algebraic variable that is required to be a solution to a finite-dimensional linear complementarity problem (LCP), whose
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.
Stability in higher-derivative matter fields theories
Energy Technology Data Exchange (ETDEWEB)
Tretyakov, Petr V. [Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, Dubna (Russian Federation); Kazan Federal University, Department of General Relativity and Gravitation, Institute of Physics, Kazan (Russian Federation)
2016-09-15
We discuss possible instabilities in higher-derivative matter field theories. These theories have two free parameters β{sub 1} and β{sub 4}. By using a dynamical system approach we explicitly demonstrate that for the stability of Minkowski space in an expanding universe we need the condition β{sub 4} < 0. By using the quantum field theory approach we also find an additional restriction for the parameters, β{sub 1} > -(1)/(3)β{sub 4}, which is needed to avoid a tachyon-like instability. (orig.)
A tutorial on incremental stability analysis using contraction theory
DEFF Research Database (Denmark)
Jouffroy, Jerome; Fossen, Thor I.
2010-01-01
This paper introduces a methodology for dierential nonlinear stability analysis using contraction theory (Lohmiller and Slotine, 1998). The methodology includes four distinct steps: the descriptions of two systems to be compared (the plant and the observer in the case of observer convergence anal...
Vacuum stability of asymptotically safe gauge-Yukawa theories
DEFF Research Database (Denmark)
Litim, Daniel F.; Mojaza, Matin; Sannino, Francesco
2016-01-01
We study the phase diagram and the stability of the ground state for certain four-dimensional gauge-Yukawa theories whose high-energy behaviour is controlled by an interacting fixed point. We also provide analytical and numerical results for running couplings, their crossover scales, the separatrix...
M-theory model-building and proton stability
Energy Technology Data Exchange (ETDEWEB)
Ellis, J. [CERN, Geneva (Switzerland). Theory Div.; Faraggi, A.E. [Florida Univ., Gainesville, FL (United States). Inst. for Fundamental Theory; Nanopoulos, D.V. [Texas A and M Univ., College Station, TX (United States)]|[Houston Advanced Research Center, The Woodlands, TX (United States). Astroparticle Physics Group]|[Academy of Athens (Greece). Div. of Natural Sciences
1997-09-01
The authors study the problem of baryon stability in M theory, starting from realistic four-dimensional string models constructed using the free-fermion formulation of the weakly-coupled heterotic string. Suitable variants of these models manifest an enhanced custodial gauge symmetry that forbids to all orders the appearance of dangerous dimension-five baryon-decay operators. The authors exhibit the underlying geometric (bosonic) interpretation of these models, which have a Z{sub 2} x Z{sub 2} orbifold structure similar, but not identical, to the class of Calabi-Yau threefold compactifications of M and F theory investigated by Voisin and Borcea. A related generalization of their work may provide a solution to the problem of proton stability in M theory.
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)
Robust Backstepping Control Based on a Lyapunov Redesign for Skid-Steered Wheeled Mobile Robots
Directory of Open Access Journals (Sweden)
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.
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.
ROBUST STABILITY ANALYSIS FOR RAILWAY VEHICLE SYSTEMS
Institute of Scientific and Technical Information of China (English)
Wang Yong; Zeng Jing; Cao Dengqing
2003-01-01
The lateral stability for railway vehicle dynamic system with uncertain parameters and nonlinear uncertain force vector is studied by using the Lyapunov stability theory. A robust stability condition for the considered system is derived, and the obtained stability bounds are not necessarily symmetric with respect to the origin in the parameter space. The lateral stability analysis for a railway bogie model is analyzed by using the proposed approach. The symmetric and asymmetric results are both given and the influence of the adjustable parameter ( on the stability bounds is also discussed. With the help of the proposed method, the robust stability analysis can provide a reference for the design of the railway vehicle systems.
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.
Stability Analysis and Design for Nonlinear Singular Systems
Yang, Chunyu; Zhou, Linna
2013-01-01
Singular systems which are also referred to as descriptor systems, semi-state systems, differential- algebraic systems or generalized state-space systems have attracted much attention because of their extensive applications in the Leontief dynamic model, electrical and mechanical models, etc. This monograph presented up-to-date research developments and references on stability analysis and design of nonlinear singular systems. It investigated the problems of practical stability, strongly absolute stability, input-state stability and observer design for nonlinear singular systems and the problems of absolute stability and multi-objective control for nonlinear singularly perturbed systems by using Lyapunov stability theory, comparison principle, S-procedure and linear matrix inequality (LMI), etc. Practical stability, being quite different from stability in the sense of Lyapunov, is a significant performance specification from an engineering point of view. The basic concepts and results on practical stability f...
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.
On the Stability of Phantom K-essence Theories
Abramo, L R; Pinto-Neto, Nelson
2006-01-01
We show that phantom dark energy, if it is described by a K-essence theory, has three fundamental problems: first, its hamiltonian is unbounded from below. Second, classical stability precludes the equation of state from crossing the ``Lambda-barrier'', $w_\\Lambda=-1$. Finally, both the equation of state and the sound speed are unbounded -- the first, from below, the second, from above -- if the kinetic term is not bounded by dynamics.
Research on the stability of control systems described by fractional-order transfer functions
Institute of Scientific and Technical Information of China (English)
Zeng Qingshan; Zhu Xinjian; Cao Guangyi
2005-01-01
The stability of control systems described by fractional-order transfer function form is mainly investigated. The stability analysis of integer-order linear systems was extended to the fractional-order control systems. The stability definition of fractional-order linear control systems is presented in terms of the Lyapunov's stability theory. Using the theorems of the Mittag-Leffler function in two parameters directly derives the stability conclusion. The illustrative examples are also given by simulation results.
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.
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.
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...
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.
Invariance and stability for bounded uncertain systems.
Peng, T. K. C.
1972-01-01
The positive limit sets of the solutions of a contingent differential equation are shown to possess an invariance property. In this connection the 'invariance principle' in the theory of Lyapunov stability is extended to systems with unknown, bounded, time-varying parameters, and thus to a large and important class of nonautonomous systems. Asymptotic stability criteria are obtained and applied to guaranteed cost control problems.
First principles theory of disordered alloys and alloy phase stability
Energy Technology Data Exchange (ETDEWEB)
Stocks, G.M.; Nicholson, D.M.C.; Shelton, W.A. [and others
1993-06-05
These lecture notes review the LDA-KKR-CPA method for treating the electronic structure and energetics of random alloys and the MF-CF and GPM theories of ordering and phase stability built on the LDA- KKR-CPA description of the disordered phase. Section 2 lays out the basic LDA-KKR-CPA theory of random alloys and some applications. Section 3 reviews the progress made in understanding specific ordering phenomena in binary solid solutions base on the MF-CF and GPM theories of ordering and phase stability. Examples are Fermi surface nesting, band filling, off diagonal randomness, charge transfer, size difference or local strain fluctuations, magnetic effects; in each case, an attempt is made to link the ordering and the underlying electronic structure of the disordered phase. Section 4 reviews calculations of electronic structure of {beta}-phase Ni{sub c}Al{sub 1-c} alloys using a version of the LDA-KKR-CPA codes generalized to complex lattices.
Stability analysis of impulsive functional systems of fractional order
Stamova, Ivanka; Stamov, Gani
2014-03-01
In this paper, a class of impulsive fractional functional differential systems is investigated. Sufficient conditions for stability of the zero solution are proved, extending the corresponding theory of impulsive functional differential equations. The investigations are carried out by using the comparison principle, coupled with the Lyapunov function method. We apply our results to an impulsive single species model of Lotka-Volterra type.
Stability Analysis of Uncertain Discrete Time-Delay Control Systems
Institute of Scientific and Technical Information of China (English)
Long Xuming; Duan Ping
2006-01-01
Based on Lyapunov stability theory, a less conservative sufficient conditions for the stabilities of uncertain discrete delay-independent and delay-dependent control systems are obtained by using the linear matrix inequality (LMI) approach. Judgement of the stability of time-delay systems is transformed to judgement of the feasible solution of an LMI, and hence is solved by use of MATLAB. Numerical simulations verify the validity of the proposed method.
Aspects of Moduli Stabilization in Type IIB String Theory
Directory of Open Access Journals (Sweden)
Shaaban Khalil
2016-01-01
Full Text Available We review moduli stabilization in type IIB string theory compactification with fluxes. We focus on KKLT and Large Volume Scenario (LVS. We show that the predicted soft SUSY breaking terms in KKLT model are not phenomenological viable. In LVS, the following result for scalar mass, gaugino mass, and trilinear term is obtained: m0=m1/2=-A0=m3/2, which may account for Higgs mass limit if m3/2~O(1.5 TeV. However, in this case, the relic abundance of the lightest neutralino cannot be consistent with the measured limits. We also study the cosmological consequences of moduli stabilization in both models. In particular, the associated inflation models such as racetrack inflation and Kähler inflation are analyzed. Finally, the problem of moduli destabilization and the effect of string moduli backreaction on the inflation models are discussed.
Finite element exterior calculus: from Hodge theory to numerical stability
Arnold, Douglas N; Winther, Ragnar
2009-01-01
This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are key tools for the continuous problem. After a brief introduction to finite element methods, the discretization methods we consider, we develop an abstract Hilbert space framework for analyzing stability and convergence. In this framework, the differential complex is represented by a complex of Hilbert spaces and stability is obtained by transferring Hodge theoretic structures from the continuous level to the discrete. We show stable discretization discretization is achieved if the finite element spaces satisfy two hypotheses: they form a subcomplex and there exists a bounded cochain projection from the full complex to the subcomplex. Next, we consider the mos...
Aspects of moduli stabilization in type IIB string theory
Khalil, Shaaban; Nassar, Ali
2015-01-01
We review moduli stabilization in type IIB string theory compactification with fluxes. We focus on the KKLT and Large Volume Scenario (LVS). We show that the predicted soft SUSY breaking terms in KKLT model are not phenomenological viable. In LVS, the following result for scalar mass, gaugino mass, and trilinear term is obtained: $m_0 =m_{1/2}= - A_0=m_{3/2}$, which may account for Higgs mass limit if $m_{3/2} \\sim {\\cal O}(1.5)$ TeV. However, in this case the relic abundance of the lightest neutralino can not be consistent with the measured limits. We also study the cosmological consequences of moduli stabilization in both models. In particular, the associated inflation models such as racetrack inflation and K\\"ahler inflation are analyzed. Finally the problem of moduli destabilization and the effect of string moduli backreaction on the inflation models are discussed.
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.
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.
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.
Bounds on scalar masses in theories of moduli stabilization
Acharya, Bobby Samir; Kane, Gordon; Kuflik, Eric
2014-04-01
In recent years it has been realized that pre-BBN decays of moduli can be a significant source of dark matter production, giving a "nonthermal WIMP miracle" and substantially reduced fine-tuning in cosmological axion physics. We study moduli masses and sharpen the claim that moduli dominated the pre-BBN universe. We conjecture that in any string theory with stabilized moduli there will be at least one modulus field whose mass is of order (or less than) the gravitino mass. Cosmology then generically requires the gravitino mass not be less than about 30 TeV and the cosmological history of the universe is nonthermal prior to BBN. Stable LSP's produced in these decays can account for the observed dark matter if they are "wino-like." We briefly consider implications for the LHC, rare decays, and dark matter direct detection and point out that these results could prove challenging for models attempting to realize gauge mediation in string theory.
Vector Lyapunov Functions for Stochastic Interconnected Systems
Boussalis, D.
1985-01-01
Theoretical paper presents set of sufficient conditions for asymptotic and exponential stability with probability 1 for class of stochastic interconnected systems. Theory applicable to complicated, large-scale mechanical or electrical systems, and, for several design problems, it reduces computational difficulty by relating stability criteria to fundamental structural features of system.
Vector Lyapunov Functions for Stochastic Interconnected Systems
Boussalis, D.
1985-01-01
Theoretical paper presents set of sufficient conditions for asymptotic and exponential stability with probability 1 for class of stochastic interconnected systems. Theory applicable to complicated, large-scale mechanical or electrical systems, and, for several design problems, it reduces computational difficulty by relating stability criteria to fundamental structural features of system.
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...
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.
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.
Institute of Scientific and Technical Information of China (English)
Su Weiwei; Chen Yiming
2008-01-01
Delay-dependent robust stability of cellular neural networks with time-varying discrete and distributed time-varying delays is considered. Based on Lyapunov stability theory and the linear matrix inequality (LMIs) technique, delay-dependent stability criteria are derived in terms of LMIs avoiding bounding certain cross terms, which often leads to conservatism. The effectiveness of the proposed stability criteria and the improvement over the existing results are illustrated in the numerical examples.
Plasma stability theory including the resistive wall effects
Pustovitov, V. D.
2015-12-01
> Plasma stabilization due to a nearby conducting wall can provide access to better performance in some scenarios in tokamaks. This was proved by experiments with an essential gain in and demonstrated as a long-lasting effect at sufficiently fast plasma rotation in the DIII-D tokamak (see, for example, Strait et al., Nucl. Fusion, vol. 43, 2003, pp. 430-440). The rotational stabilization is the central topic of this review, though eventually the mode rotation gains significance. The analysis is based on the first-principle equations describing the energy balance with dissipation in the resistive wall. The method emphasizes derivation of the dispersion relations for the modes which are faster than the conventional resistive wall modes, but slower than the ideal magnetohydrodynamics modes. Both the standard thin wall and ideal-wall approximations are not valid in this range. Here, these are replaced by an approach incorporating the skin effect in the wall. This new element in the stability theory makes the energy sink a nonlinear function of the complex growth rate. An important consequence is that a mode rotating above a critical level can provide a damping effect sufficient for instability suppression. Estimates are given and applications are discussed.
Institute of Scientific and Technical Information of China (English)
Huaicheng YAN; Xinhan HUANG; Min WANG
2006-01-01
This paper deals with the problem of delay-dependent stability and stabilization for networked control systems(NCSs)with multiple time-delays. In view of multi-input and multi-output(MIMO) NCSs with many independent sensors and actuators, a continuous time model with distributed time-delays is proposed. Utilizing the Lyapunov stability theory combined with linear matrix inequalities(LMIs) techniques, some new delay-dependent stability criteria for NCSs in terms of generalized Lyapunov matrix equation and LMIs are derived. Stabilizing controller via state feedback is formulated by solving a set of LMIs. Compared with the reported methods, the proposed methods give a less conservative delay bound and more general results. Numerical example and simulation show that the methods are less conservative and more effective.
Lyapunov exponent for aging process in induction motor
Bayram, Duygu; Ünnü, Sezen Yıdırım; Şeker, Serhat
2012-09-01
focused on the controlling the mechanical parameters of the electrical machines. Brushless DC motor (BLDCM) and the other general purpose permanent magnet (PM) motors are the most widely examined motors [1, 8, 9]. But the researches, about Lyapunov Exponent, subjected to the induction motors are mostly focused on the control theory of the motors. Flux estimation of rotor, external load disturbances and speed tracking and vector control position system are the main research areas for induction motors [10, 11, 12-14]. For all the data sets which can be collected from an induction motor, vibration data have the key role for understanding the mechanical behaviours like aging, bearing damage and stator insulation damage [15-18]. In this paper aging of an induction motor is investigated by using the vibration signals. The signals consist of new and aged motor data. These data are examined by their 2 dimensional phase portraits and the geometric interpretation is applied for detecting the Lyapunov Exponents. These values are compared in order to define the character and state estimation of the aging processes.
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.
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...
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.
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.
Jahanpanah, Jafar
2015-01-01
The vibrational motion equations of both homo and hetero-nuclei diatomic molecules are here derived for the first time. A diatomic molecule is first considered as a one dimensional quantum mechanics oscillator. The second and third-order Hamiltonian operators are then formed by substituting the number operator for the quantum number in the corresponding vibrational energy eigenvalues. The expectation values of relative position and linear momentum operators of two oscillating atoms are calculated by solving Heisenbergs equations of motion. Subsequently, the expectation values of potential and kinetics energy operators are evaluated in all different vibrational levels of Morse potential. On the other hand, the stability theory of optical oscillators (lasers) is exploited to determine the stability conditions of an oscillating diatomic molecule.It is peculiarly turned out that the diatomic molecules are exactly dissociated at the energy level in which their equations of motion become unstable. We also determine...
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.
Institute of Scientific and Technical Information of China (English)
S. Lakshmanan; P. Balasubramaniarn
2011-01-01
This paper studies the problem of linear matrix inequality(LMI)approach to robust stability analysis for stochastic neural networks with a time-varying delay. By developing a delay decomposition approach, the information of the delayed plant states can be taken into full consideration. Based on the new Lyapunov-Krasovskii functional, some inequality techniques and stochastic stability theory, new delay-dependent stability criteria are obtained in terms of LMIs. The proposed results prove the less conservatism, which are realized by choosing new Lyapunov matrices in the decomposed integral intervals. Finally, numerical examples are provided to demonstrate the less conservatism and effectiveness of the proposed LMI method.
Razumikhin's method in the qualitative theory of processes with delay
Directory of Open Access Journals (Sweden)
Anatoly D. Myshkis
1995-01-01
Full Text Available B.S. Razumikhin's concept in the qualitative theory of systems delay is clarified and discussed. Various ways of improvements of stability conditions are considered. The author shows that the guiding role of Lyapunov functions and demonstrates Razumikhin's method as a practical case of continuous version of the mathematical induction. Several examples demonstrate the obtained results.
Stability and response bounds of non-conservative linear systems
DEFF Research Database (Denmark)
Pommer, Christian
2003-01-01
For a linear system of second order differential equations the stability is studied by Lyapunov's direct method. The Lyapunov matrix equation is solved and a sufficient condition for stability is expressed by the system matrices. For a system which satisfies the condition for stability the Lyapunov...
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.
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.
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.
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.
Wirtinger-Type Inequality and the Stability Analysis of Delayed Lur'e System
Directory of Open Access Journals (Sweden)
Zixin Liu
2013-01-01
Full Text Available This paper proposes a new delay-depended stability criterion for a class of delayed Lur'e systems with sector and slope restricted nonlinear perturbation. The proposed method employs an improved Wirtinger-type inequality for constructing a new Lyapunov functional with triple integral items. By using the convex expression of the nonlinear perturbation function, the original nonlinear Lur'e system is transformed into a linear uncertain system. Based on the Lyapunov stable theory, some novel delay-depended stability criteria for the researched system are established in terms of linear matrix inequality technique. Three numerical examples are presented to illustrate the validity of the main results.
Global asymptotic stability for Hopfield-type neural networks with diffusion effects
Institute of Scientific and Technical Information of China (English)
YAN Xiang-ping; LI Wan-tong
2007-01-01
The existence, uniqueness and global asymptotic stability for the equilibrium of Hopfield-type neural networks with diffusion effects are studied. When the activation functions are monotonously nondecreasing, differentiable, and the interconnected matrix is related to the Lyapunov diagonal stable matrix, the sufficient conditions guaranteeing the existence of the equilibrium of the system are obtained by applying the topological degree theory. By means of constructing the suitable average Lyapunov functions, the global asymptotic stability of the equilibrium of the system is also investigated. It is shown that the equilibrium (if it exists) is globally asymptotically stable and this implies that the equilibrium of the system is unique.
Zhao, Xueyan; Deng, Feiqi
2016-07-01
In this paper, a particular property of Lyapunov functions for functional differential equations (FDEs) is developed, that is the direct dependence of the signs of the derivatives of the Lyapunov functions on the initial data. This property implies that the derivatives of the Lyapunov functions for FDEs cannot be guaranteed to be negative definite generally, and then makes the FDEs differ from the ordinary differential equations constitutionally. With this property, we give some enlightenments for the research methods for establishing stability theorems or criteria for FDEs, which may help us to form a common view about the choice of the investigation methods on the stability of FDEs. The conclusion is stated in both the deterministic and stochastic versions. Two illustrative examples are given to show and verify our conclusion through the paper.
DEFF Research Database (Denmark)
Ribard, Nicolas; Wisniewski, Rafael; Sloth, Christoffer
2016-01-01
In the paper, we strive to develop an algorithm that simultaneously computes a polynomial control and a polynomial Lyapunov function. This ensures asymptotic stability of the designed feedback system. The above problem is translated to a certificate of positivity. To this end, we use the represen......In the paper, we strive to develop an algorithm that simultaneously computes a polynomial control and a polynomial Lyapunov function. This ensures asymptotic stability of the designed feedback system. The above problem is translated to a certificate of positivity. To this end, we use...... the representation of the given control system in Bernstein basis. Subsequently, the control synthesis problem is reduced to finite number of evaluations of a polynomial on vertices of cubes in the space of parameters representing admissible controls and Lyapunov functions....
A Criterion for Stability of Synchronization and Application to Coupled Chua's Systems
Institute of Scientific and Technical Information of China (English)
WANG Hai-Xia; LU Qi-Shao; WANG Qing-Yun
2009-01-01
We investigate synchronization in an array network of nearest-neighbor coupled chaotic oscillators. By using of the Lyapunov stability theory and matrix theory, a criterion for stability of complete synchronization is deduced. Meanwhile, an estimate of the critical coupling strength is obtained to ensure achieving chaos synchronization. As an example application, a model of coupled Chua's circuits with linearly bidirectional coupling is studied to verify the validity of the criterion.
Heisenberg Picture Approach to the Stability of Quantum Markov Systems
Pan, Yu; Amini, Hadis; Miao, Zibo; Gough, John; Ugrinovskii, Valery; James, Matthew R.
2014-01-01
Quantum Markovian systems, modeled as unitary dilations in the quantum stochastic calculus of Hudson and Parthasarathy, have become standard in current quantum technological applications. This paper investigates the stability theory of such systems. Lyapunov-type conditions in the Heisenberg picture are derived in order to stabilize the evolution of system operators as well as the underlying dynamics of the quantum states. In particular, using the quantum Markov semigroup associated with this...
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.
Exponential stability of cellular neural networks with multiple time delays and impulsive effects
Institute of Scientific and Technical Information of China (English)
Li Dong; Wang Hui; Yang Dan; Zhang Xiao-Hong; Wang Shi-Long
2008-01-01
In this work,the stability issues of the equilibrium points of the cellular neural networks with multiple time delays and impulsive effects are investigated.Based on the stability theory of Lyapunov-Krasovskii,the method of linear matrix inequality (LMI) and parametrized first-order model transformation,several novel conditions guaranteeing the delaydependent and the delay-independent exponential stabilities are obtained.A numerical example is given to illustrate the effectiveness of our results.
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.
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.
Random matrix theory for portfolio optimization: a stability approach
Sharifi, S.; Crane, M.; Shamaie, A.; Ruskin, H.
2004-04-01
We apply random matrix theory (RMT) to an empirically measured financial correlation matrix, C, and show that this matrix contains a large amount of noise. In order to determine the sensitivity of the spectral properties of a random matrix to noise, we simulate a set of data and add different volumes of random noise. Having ascertained that the eigenspectrum is independent of the standard deviation of added noise, we use RMT to determine the noise percentage in a correlation matrix based on real data from S&P500. Eigenvalue and eigenvector analyses are applied and the experimental results for each of them are presented to identify qualitatively and quantitatively different spectral properties of the empirical correlation matrix to a random counterpart. Finally, we attempt to separate the noisy part from the non-noisy part of C. We apply an existing technique to cleaning C and then discuss its associated problems. We propose a technique of filtering C that has many advantages, from the stability point of view, over the existing method of cleaning.
Stability Theory for Interfacial Patterns in Magnetic Pulse Welding
Nassiri, Ali; Chini, Gregory; Kinsey, Brad; UNH Team
2013-11-01
Magnetic Pulse Welding (MPW) is a solid state, high strain-rate joining process in which a weld of dissimilar or similar materials can be created via high-speed oblique impact of two workpieces. Experiments routinely show the emergence of a distinctive wavy pattern, with a well defined amplitude and wavelength of approximately 20 and 70 micrometers, respectively, at the interface between the two welded materials. Although the origin of the wavy pattern has been the subject of much investigation, a unique fundamental physical theory for this phenomenon is as yet not widely accepted. Some researchers have proposed that the interfacial waves are formed in a process akin to Kelvin-Helmholtz instability, with relative shear movement of the flyer and base plates providing the energy source. Here, we employ a linear stability analysis to investigate whether the wavy pattern could be the signature of a shear-driven high strain-rate instability of an elastic-plastic solid material. Preliminary results confirm that an instability giving rise to a wavy interfacial pattern is possible.
Bounds on Scalar Masses in Theories of Moduli Stabilization
Acharya, Bobby Samir; Kuflik, Eric
2014-01-01
In recent years it has been realised that pre-BBN decays of moduli can be a significant source of dark matter production, giving a `non-thermal WIMP miracle' and substantially reduced fine-tuning in cosmological axion physics. We study moduli masses and sharpen the claim that moduli dominated the pre-BBN Universe. We conjecture that in any string theory with stabilized moduli there will be at least one modulus field whose mass is of order (or less than) the gravitino mass and we prove this for a large class of models based on Calabi-Yau extra dimensions. Cosmology then generically requires the gravitino mass not be less than about 30 TeV and the cosmological history of the Universe is non-thermal prior to BBN. Stable LSP's produced in these decays can account for the observed dark matter if they are `wino-like,' which is consistent with the PAMELA data for positrons and antiprotons. With WIMP dark matter, there is an upper limit on the gravitino mass of order 250 TeV. We briefly consider implications for the ...
Lyapunov based nonlinear control of electrical and mechanical systems
Behal, Aman
This Ph.D. dissertation describes the design and implementation of various control strategies centered around the following applications: (i) an improved indirect field oriented controller for the induction motor, (ii) partial state feedback control of an induction motor with saturation effects, (iii) tracking control of an underactuated surface vessel, and (iv) an attitude tracking controller for an underactuated spacecraft. The theory found in each of these sections is demonstrated through simulation or experimental results. An introduction to each of these four primary chapters can be found in chapter one. In the second chapter, the previously published tracking control of [16] 1 is presented in the indirect field oriented control (IFOC) notation to achieve exponential rotor velocity/rotor flux tracking. Specifically, it is illustrated how the proposed IFOC controller can be rewritten in the manner of [16] to allow for a direct Lyapunov stability proof. Experimental results (implemented with the IFOC algorithm) are provided to corroborate the efficacy of the algorithm. In the third chapter, a singularity-free, rotor position tracking controller is presented for the full order, nonlinear dynamic model of the induction motor that includes the effects of magnetic saturation. Specifically, by utilizing the pi-equivalent saturation model, an observer/controller strategy is designed that achieves semi-global exponential rotor position tracking and only requires stator current, rotor velocity, and rotor position measurements. Simulation and experimental results are included to demonstrate the efficacy of the proposed algorithm. In the fourth chapter, a continuous, time-varying tracking controller is designed that globally exponentially forces the position/orientation tracking error of an under-actuated surface vessel to a neighborhood about zero that can be made arbitrarily small (i.e., global uniformly ultimately boundedness (GUUB)). The result is facilitated by
Mizuta, Keisuke; Tokita, Takashi; Ito, Yatsuji; Aoki, Mitsuhiro; Kuze, Bunya
2009-12-01
largest Lyapunov exponents has a different significance from instability of standing posture indicated by a conventional analysis. We propose that the largest Lyapunov exponents may be an useful subsidiary measure to evaluate postural stability and its change due to vestibular dysfunction.
A proposal for studying off-shell stability of vacuum geometries in string theory
Suneeta, V
2008-01-01
We briefly review studies of off-shell stability of vacuum geometries in semiclassical gravity. We propose a study of off-shell stability of vacua in string theory by a distinct, though somewhat related approach -- by studying their stability under suitable world-sheet sigma model renormalization group (RG) flows. Stability under RG flow is a mathematically well-posed and tractable problem in many cases, as we illustrate through examples. The advantage is that we can make definite predictions about late time behaviour and endpoints of off-shell processes in string theory. This is a contribution to the proceedings of Theory Canada 4, CRM Montreal.
Theory and methods of global stability analysis for high arch dam
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
The global stability of high arch dam is one of the key problems in the safety study of arch dams,but no feasible method with theoretical basis is available.In this paper,based on the stability theory of mechanical system,it is demonstrated that the global failure of high arch dams belongs to a physical instability starting from local strength failure,which is the extreme point instability according to the characteristics of load-displacement curve obtained from the failure process of dam-foundation system. So the global failure of dam-foundation system should be studied with the stability theory of mechanical system.It is also pointed out that the current stability analysis methods used in engineering are consistent with the stability theory,but not established according to the mechanical system stability theory directly.A perfect method can be obtained through the study of physical disturbance equations.
Study Of The Theory Of Optical Stabilizing Image
Zhijian, Wang; Jianping, Zheng
1989-01-01
In this paper, all varieties of the optical stabilizing image methods have been summarized into an optical stabilization pattern, and a mathematical model of the optical stabilizing image are proposed. Some representative systems are analyzed by means of this model in orde to show how to use this model.
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.
Exponential Stability of Stochastic Nonlinear Dynamical Price System with Delay
Directory of Open Access Journals (Sweden)
Wenli Zhu
2013-01-01
Full Text Available Based on Lyapunov stability theory, Itô formula, stochastic analysis, and matrix theory, we study the exponential stability of the stochastic nonlinear dynamical price system. Using Taylor's theorem, the stochastic nonlinear system with delay is reduced to an n-dimensional semilinear stochastic differential equation with delay. Some sufficient conditions of exponential stability and corollaries for such price system are established by virtue of Lyapunov function. The time delay upper limit is solved by using our theoretical results when the system is exponentially stable. Our theoretical results show that if the classical price Rayleigh equation is exponentially stable, so is its perturbed system with delay provided that both the time delay and the intensity of perturbations are small enough. Two examples are presented to illustrate our results.
Ghabraei, Soheil; Moradi, Hamed; Vossoughi, Gholamreza
2016-06-01
Large amplitude oscillation of the power transmission lines, which is also known as galloping phenomenon, has hazardous consequences such as short circuiting and failure of transmission line. In this article, to suppress the undesirable vibrations of the transmission lines, first the governing equations of transmission line are derived via mode summation technique. Then, due to the occurrence of large amplitude vibrations, nonlinear quadratic and cubic terms are included in the derived linear equations. To suppress the vibrations, arbitrary number of the piezoelectric actuators is assumed to exert the actuation forces. Afterwards, a Lyapunov based approach is proposed for the robust adaptive suppression of the undesirable vibrations in the finite time. To compensate the supposed parametric uncertainties with unknown bands, proper adaption laws are introduced. To avoid the vibration devastating consequences as quickly as possible, appropriate control laws are designed. The vibration suppression in the finite time with supposed adaption and control laws is mathematically proved via Lyapunov finite time stability theory. Finally, to illustrate and validate the efficiency and robustness of the proposed finite time control scheme, a parametric case study with three piezoelectric actuators is performed. It is observed that the proposed active control strategy is more efficient and robust than the passive control methods.
Stabilisation of a class of 2-DOF underactuated mechanical systems via direct Lyapunov approach
Turker, Turker; Gorgun, Haluk; Cansever, Galip
2013-06-01
This paper represents an alternative stabilisation procedure for a class of two degree-of-freedom underactuated mechanical systems based on a set of transformations and a Lyapunov function. After simplifying dynamic equations of the system via partial feedback linearisation and coordinate changes, the stability of the system is provided with Lyapunov's direct method. Proposed control scheme is used on two different examples and asymptotic convergence for each system is proven by means of La Salle's invariance principle. The designed controller is successfully illustrated through numerical simulations for each example.
A Stability Criterion for Time-Delay Tension Leg Platform Systems Subjected to External Force
Institute of Scientific and Technical Information of China (English)
Chen-Yuan CHEN; Chien-wen SHEN; Cheng-Wu CHEN; Kevin Fong-Rey LIU; Ming-Jen CHENG
2009-01-01
Stability analysis plays a central role in nonlinear system theory and engineering application.Over the past few yeats,the stability analysis of fuzzy systems has been proposed and there are many successful applications in practical engineering.Therefore,in this paper firstly proposed is the stability analysis on oceanic structure by fuzzy models.In the present study,Takagi-Sugeno (T-S) fuzzy model is proposed for a time delay tension leg platform (TLP) system subjected to an external wave force.In terms of stability analysis,linear matrix inequality (LMI) conditions are derived via Lyapunov theory to guarantee the stability of the TLP system.
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.
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.
Fuzzy Logic Controller Stability Analysis Using a Satisfiability Modulo Theories Approach
Arnett, Timothy; Cook, Brandon; Clark, Matthew A.; Rattan, Kuldip
2017-01-01
While many widely accepted methods and techniques exist for validation and verification of traditional controllers, at this time no solutions have been accepted for Fuzzy Logic Controllers (FLCs). Due to the highly nonlinear nature of such systems, and the fact that developing a valid FLC does not require a mathematical model of the system, it is quite difficult to use conventional techniques to prove controller stability. Since safety-critical systems must be tested and verified to work as expected for all possible circumstances, the fact that FLC controllers cannot be tested to achieve such requirements poses limitations on the applications for such technology. Therefore, alternative methods for verification and validation of FLCs needs to be explored. In this study, a novel approach using formal verification methods to ensure the stability of a FLC is proposed. Main research challenges include specification of requirements for a complex system, conversion of a traditional FLC to a piecewise polynomial representation, and using a formal verification tool in a nonlinear solution space. Using the proposed architecture, the Fuzzy Logic Controller was found to always generate negative feedback, but inconclusive for Lyapunov stability.
Lyapunov-based boundary feedback control in multi-reach canals
Institute of Scientific and Technical Information of China (English)
CEN LiHui; XI YuGeng
2009-01-01
This paper presents a Lyapunov-based approach to design the boundary feedback control for an open-channel network composed of a cascade of multi-reach canals, each described by a pair of Saint-Venant equations. The weighted sum of entropies of the multi-reaches is adopted to construct the Lyapunov function. The time derivative of the Lyapunov function is expressed by the water depth variations at the gate boundaries, based on which a class of boundary feedback controllers is presented to guarantee the local asymptotic closed-loop stability. The advantage of this approach is that only the water level depths at the gate boundaries are measured as the feedback.
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.
Institute of Scientific and Technical Information of China (English)
Cuimei ZHANG; Wencheng CHEN; Yu YANG
2006-01-01
In this paper, we study the existence and global asymptotic stability of positive periodic solutions of a delayed periodic predator-prey system with Holling Ⅱ type functional response. By use of the continuation theorem of coincidence degree theory and the method of Lyapunov function, some sufficient conditions are obtained.
Lyapunov-Based Feedback Preparation of GHZ Entanglement of N-Qubit Systems.
Liu, Yanan; Kuang, Sen; Cong, Shuang
2016-07-09
The Greenberger-Horne-Zeilinger (GHZ) entangled states are a typical class of entangled states in multiparticle systems and play an important role in the applications of quantum communication and quantum computation. For a general quantum system of N qubits, degenerate measurement operators are often met, which cause the convergence obstacle in the state preparation or stabilization problem. This paper first generalizes the traditional quantum state continuous reduction theory to the case of a degenerate measurement operator and chooses a measurement operator for an arbitrarily given target GHZ entangled state, then presents a state stabilization control strategy based on the Lyapunov method and achieves the feedback preparation of the target GHZ state. In our stabilization strategy, we separate the target GHZ state and all the other GHZ states that often form the equilibrium points of the closed-loop system by dividing the state space into several different regions; and formally design a switching control law between the regions, which contains the control Hamiltonians to be constructed. By analyzing the stability of the closed-loop system in the different regions, we propose a systematic method for constructing the control Hamiltonians and solve the convergence problem caused by the degenerate measurement operator. The global stability of the whole closed-loop stochastic system is strictly proved. Also, we perform some simulation experiments on a three-qubit system and prepare a three-qubit GHZ entangled state. At the same time, the simulation results show the effectiveness of the switching control law and the construction method for the control Hamiltonians proposed in this paper.
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’.
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.
Direct adaptive control for nonlinear uncertain system based on control Lyapunov function method
Institute of Scientific and Technical Information of China (English)
Chen Yimei; Han Zhengzhi; Tang Houjun
2006-01-01
The problem of adaptive stabilization of a class of multi-input nonlinear systems with unknown parameters both in the state vector-field and the input vector-field has been considered. By employing the control Lyapunov function method, a direct adaptive controller is designed to complete the global adaptive stability of the uncertain system. At the same time, the controller is also verified to possess the optimality. Example and simulations are provided to illustrate the effectiveness of the proposed method.
A Low Order Theory of Arctic Sea Ice Stability
Moon, W
2011-01-01
We analyze the stability of a low-order coupled sea ice and climate model and extract the essential physics governing the time scales of response as a function of greenhouse gas forcing. Under present climate conditions the stability is controlled by longwave radiation driven heat conduction. However, as greenhouse gas forcing increases and the ice cover decays, the destabilizing influence of ice-albedo feedback acts on equal footing with longwave stabilization. Both are seasonally out of phase and as the system warms towards a seasonal ice state these effects, which underlie the bifurcations between climate states, combine to extend the intrinsic relaxation time scale from ~ 2 yr to 5 yr.
Stability of the Einstein static universe in modified theories of gravity
Boehmer, Christian G.; Hollenstein, Lukas; Lobo, Francisco S. N.; Seahra, Sanjeev S.
2010-01-01
We present a brief overview of the stability analysis of the Einstein static universe in various modified theories of gravity, like f(R) gravity, Gauss-Bonnet or f(G) gravity, and Horava-Lifshitz gravity.
Stability of the Einstein static universe in modified theories of gravity
Boehmer, Christian G.; Hollenstein, Lukas; Lobo, Francisco S. N.; Seahra, Sanjeev S.
2010-01-01
We present a brief overview of the stability analysis of the Einstein static universe in various modified theories of gravity, like f(R) gravity, Gauss-Bonnet or f(G) gravity, and Horava-Lifshitz gravity.
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.
EXISTENCE AND UNIQUENESS AND STABILITY OF SOLUTIONS FOR STOCHASTIC IMPULSIVE SYSTEMS
Institute of Scientific and Technical Information of China (English)
Bin LIU; Xinzhi LIU; Xiaoxin LIAO
2007-01-01
This paper studies the existence,uniqueness,and stability of solutions for stochastic impul sive systems.By employing Lyapunov-like functions,some sufficient conditions of the global existence,uniqueness,and stability of solutions for stochastic impulsive systems are established.Furthermore,the results are specialized to the case of linear stochastic impulsive systems.Finally,some examples are given to illustrate the applications of our theory.
Stability analysis and design of fuzzy control system with bounded uncertain delays
Institute of Scientific and Technical Information of China (English)
Jianguo GUO; Juntao LI; Fengqi ZHOU; Jun ZHOU
2005-01-01
Fuzzy control problems for systems with bounded uncertain delays were studied.Based on Lyapunov stability theory and matrix theory,a nonlinear state feedback fuzzy controller was designed by linear matrix inequalities (LMI) approach,and the global exponential stability of the closed-loop system was strictly proved.For a fuzzy control system with bounded uncertain delays,under the global exponential stability condition which is reduced to p linear matrix inequalities,the controller guarantees stability performances of state variables.Finally,the simulation shows the validity of the method in this paper.
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.
NONLINEAR THEORY OF DYNAMIC STABILITY FOR LAMINATED COMPOSITE CYLINDRICAL SHELLS
Institute of Scientific and Technical Information of China (English)
周承倜; 王列东
2001-01-01
Hamilton Principle was uaed to derive the general governing equations of nonlinear dynamic stability for laminated cylindrical shells in which, factors of nonlinear large deflection, transverse shear and longitudinal inertia force were concluded. Equations were solved by variational method. Analysis reveals that under the action of dynamic load,laminated cylindrical shells will fall into a state of parametric resonance and enter into the dynamic unstable region that causes dynamic instability of shells. Laminated shells of three typical composites were computed: i.e. T300/5 208 graphite epoxy E-glass epoxy, and ARALL shells. Results show that all factors will induce important influence for dynamic stability of laminated shells. So, in research of dynamic stability for laminated shells, to consider these factors is important.
Supersymmetry Breaking due to Moduli Stabilization in String Theory
Linde, Andrei; Olive, Keith A
2011-01-01
We consider the phenomenological consequences of fixing compactification moduli. In the simplest KKLT constructions, stabilization of internal dimensions is rather soft: weak scale masses for moduli are generated, and are of order m_\\sigma ~ m_{3/2}. As a consequence one obtains a pattern of soft supersymmetry breaking masses found in gravity and/or anomaly mediated supersymmetry breaking (AMSB) models. These models may lead to destabilization of internal dimensions in the early universe, unless the Hubble constant during inflation is very small. Fortunately, strong stabilization of compactified dimensions can be achieved by a proper choice of the superpotential (e.g in the KL model with a racetrack superpotential). This allows for a solution of the cosmological moduli problem and for a successful implementation of inflation in supergravity. We show that strong moduli stabilization leads a very distinct pattern of soft supersymmetry breaking masses. In general, we find that soft scalar masses remain of order ...
The stability concept of evolutionary game theory a dynamic approach
1992-01-01
These Notes grew from my research in evolutionary biology, specifically on the theory of evolutionarily stable strategies (ESS theory), over the past ten years. Personally, evolutionary game theory has given me the opportunity to transfer my enthusiasm for abstract mathematics to more practical pursuits. I was fortunate to have entered this field in its infancy when many biologists recognized its potential but were not prepared to grant it general acceptance. This is no longer the case. ESS theory is now a rapidly expanding (in both applied and theoretical directions) force that no evolutionary biologist can afford to ignore. Perhaps, to continue the life-cycle metaphor, ESS theory is now in its late adolescence and displays much of the optimism and exuberance of this exciting age. There are dangers in writing a text about a theory at this stage of development. A comprehensive treatment would involve too many loose ends for the reader to appreciate the central message. On the other hand, the current central m...
Generalized invariance principles and the theory of stability.
Lasalle, J. P.
1971-01-01
Description of some recent extensions of the invariance principle to more generalized dynamical systems where the state space is not locally compact and the flow is unique only in the forward direction of time. A sufficient condition for asymptotic stability of an invariant set is obtained which does not require that the Liapunov function be positive-definite. A recently developed generalized invariance principle is described which is applicable to functional differential equations, partial differential equations, and, in particular, to certain stability problems arising in thermoelasticity, viscoelasticity, and distributed nonlinear networks.
Stability condition of FAST TCP in high speed network Oil the basis of control theory
Institute of Scientific and Technical Information of China (English)
Zhao Fuzhe; Zhou Jianzhong; Luo Zhimeng; Xiao Yang
2008-01-01
Considering the instability of data transferred existing in high speed network.a near method is proposed for improving the stability using control theory.Under this method,the mathematical model of such a network is established.Stability condition is derived from the mathematical model.Several sivaulation experiments are performed.The results show that the method can increase the stability of data transferred in terms of the congestion window,queue size,and sending rate of the source.
Institute of Scientific and Technical Information of China (English)
Feng Yi-Fu; Zhang Qing-Ling; Feng De-Zhi
2012-01-01
The global stability problem of Takagi-Sugeno (T S) fuzzy Hopfield neural networks (FHNNs) with time delays is investigated.Novel LMI-based stability criteria are obtained by using Lyapunov functional theory to guarantee the asymptotic stability of the FHNNs with less conservatism.Firstly,using both Finsler's lemma and an improved homogeneous matrix polynomial technique,and applying an affine parameter-dependent Lyapunov-Krasovskii functional,we obtain the convergent LMI-based stability criteria.Algebraic properties of the fuzzy membership functions in the unit simplex are considered in the process of stability analysis via the homogeneous matrix polynomials technique.Secondly,to further reduce the conservatism,a new right-hand-side slack variables introducing technique is also proposed in terms of LMIs,which is suitable to the homogeneous matrix polynomials setting.Finally,two illustrative examples are given to show the efficiency of the proposed approaches.
A Robust Stability and Control Theory for Hybrid Dynamical Systems
2006-09-30
IEEE Transactions on Automatic Control , to...Dual Linear Differential Inclusions", IEEE Transactions on Automatic Control , Vol. 51, Issue 4, April 2006, pp. 661-666. D. Liberzon and J. Hespanha...34Stabilization of nonlinear systems with limited information feedback", IEEE Transactions on Automatic Control , vol. 50, no. 6, pp. 910-915,
A stability dependent theory for air-sea gas exchange
Erickson, David J.
1993-05-01
The influence of thermal stability at the air-sea interface on computed values of the transfer velocities of trace gases is examined. The novel "whitecap" model for air-sea gas exchange of Monahan and Spillane (1984), extended here to include thermal stability effects, is linked with an atmospheric general circulation model to compute global transfer velocity patterns of a climate reactive gas, CO2. The important terms in the model equations such as the whitecap coverage, friction velocity, neutral and local drag coefficients and the stability parameter ψm(Z/L) are discussed and analyzed. The atmospheric surface level air temperature, relative humidity, wind speed and sea surface temperature, obtained from the National Center for Atmospheric Research Community Climate Model 1 (CCM1) are used to drive algorithms describing the air-sea transfer velocity of trace gases. The transfer velocity for CO2 (kCO2) is then computed for each 2.8° × 2.8° latitudinal-longitudinal area every 24 hours for 5 years of the seasonal-hydro runs of the CCM1. The new model results are compared to previously proposed formulations using the identical CCM1 forcing terms. Air-sea thermal stability effects on the transfer velocity for CO2 are most important at mid-high wind speeds. Where cold air from continental interiors is transported over relatively warm oceanic waters, the transfer velocities are enhanced over neutral stability values. The depression of computed kCO2 values when warm air resides over cold water is especially important, due to asymmetry in the stability dependence of the drag coefficient. The stability influence is 20% to 50% of kCO2 for modest air-sea temperature differences and up to 100% for extreme cases of stability or instability. The stability dependent "whitecap" model, using the transfer velocity coefficients for whitecap and nonwhitecap areas suggested by Monahan and Spillane (1984), produces CO2 transfer velocities that range from 13 to 50 cm h-1 for a
Design of a Helicopter Stability and Control Augmentation System Using Optimal Control Theory.
technique is described for the design of multivariable feedback controllers based upon results in optimal control theory . For a specified performance...helicopter flight envelope. The results show that optimal control theory can be used to design a helicopter stability and control augmentation system
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.
Analysis on stability of strategic alliance: A game theory perspective
Institute of Scientific and Technical Information of China (English)
CHEN Fei-qiong; FAN Liang-cong
2006-01-01
Strategic alliance has suffered much instabilities since its first implementation. Scholars have carried out many embedded, precise and comprehensive researches from both theory and empiricism. Here we try to find certain stable solutions by employing game theory, in an attempt to construct theoretical bases for strategic alliance, which people called "one of the most important organizational innovation in the end of the 20th century" (Shi, 2001), to exploit its advantages in the process of globalization. Finally, this article puts forward some advices for its success.
Gyrokinetic stability theory of electron-positron plasmas
Helander, Per
2016-01-01
The linear gyrokinetic stability properties of magnetically confined electron-positron plasmas are investigated in the parameter regime most likely to be relevant for the first laboratory experiments involving such plasmas, where the density is small enough that collisions can be ignored and the Debye length substantially exceeds the gyroradius. Although the plasma beta is very small, electromagnetic effects are retained, but magnetic compressibility can be neglected. The work of a previous publication (Helander, 2014) is thus extended to include electromagnetic instabilities, which are of importance in closed-field-line configurations, where such instabilities can occur at arbitrarily low pressure. It is found that gyrokinetic instabilities are completely absent if the magnetic field is homogeneous: any instability must involve magnetic curvature or shear. Furthermore, in dipole magnetic fields, the stability threshold for interchange modes with wavelengths exceeding the Debye radius coincides with that in i...
Stability of Spatial Structure of Urban Agglomeration in China Based on Central Place Theory
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
This paper brings forward the concept of stability of the spatial structure of urban agglomeration (UA) based on Central Place Theory by introducing centrality index and fractal theory. Before assessment, K=4 is selected as parameter to calculate centrality index and fractal dimension (K represents the quantitive relationship between city and the counties in Central Place Theory), and then found the number of nodes, the type of spatial structure, the spatial allocation of nodes with different hierarchy affecting the stability of spatial structure. According to spatial contact direction and the level of stability, UAs in China are classified into five types. Finally, it is posed as a further question that how to use hierarchical relation K=6 and K=7 in central place system to coordinate with the assessment of stability of spatial structure is brought forward.
Graph theory and stability analysis of protein complex interaction networks.
Huang, Chien-Hung; Chen, Teng-Hung; Ng, Ka-Lok
2016-04-01
Protein complexes play an essential role in many biological processes. Complexes can interact with other complexes to form protein complex interaction network (PCIN) that involves in important cellular processes. There are relatively few studies on examining the interaction topology among protein complexes; and little is known about the stability of PCIN under perturbations. We employed graph theoretical approach to reveal hidden properties and features of four species PCINs. Two main issues are addressed, (i) the global and local network topological properties, and (ii) the stability of the networks under 12 types of perturbations. According to the topological parameter classification, we identified some critical protein complexes and validated that the topological analysis approach could provide meaningful biological interpretations of the protein complex systems. Through the Kolmogorov-Smimov test, we showed that local topological parameters are good indicators to characterise the structure of PCINs. We further demonstrated the effectiveness of the current approach by performing the scalability and data normalization tests. To measure the robustness of PCINs, we proposed to consider eight topological-based perturbations, which are specifically applicable in scenarios of targeted, sustained attacks. We found that the degree-based, betweenness-based and brokering-coefficient-based perturbations have the largest effect on network stability.
Stability of glycol nanofluids -- the consensus between theory and measurement
Palabiyik, Ibrahim; Musina, Zenfira; Ding, Yulong
2012-01-01
Formulation of stable nanofluids containing ZnO, Al2O3 and TiO2 nanoparticles in propylene glycol (PG), ethylene glycol (EG) and 50wt% mixtures of PG and EG in water (WPG, WEG) were investigated, with and without the presence of surfactants. Nanofluid samples of particle concentrations 1-9wt% were prepared by dispersive method. Surfactant presence was in the range of 0-1wt%/wt% of nanoparticles. Visual observation, particle size measurement and zeta potential analysis were performed to evaluate the dispersion stability. In overall the PG-based samples were found to be the most stable suspensions. The effect of base fluid on particle size and the effect of day light on nanofluid stability were also examined as a function of time. TiO2-PG samples showed a colour change when exposed to sunlight. Sunlight also caused the PG based TiO2 and Al2O3 nanofluid to increase their particle sizes by up to 45% in the course of 3 days. As for stability, the sedimentation velocity was observed to be a key parameter. Finally b...
Extension to nonlinear stability theory of the circular Couette flow
Yau, Pun Wong; Wang, Shixiao; Rusak, Zvi
2016-11-01
A nonlinear stability analysis of the viscous circular Couette flow to axisymmetric perturbations under axial periodic boundary conditions is developed. The analysis is based on investigating the properties of a reduced Arnol'd energy-Casimir function Ard of Wang (2009). We show that all the inviscid flow effects as well as all the viscous-dependent terms related to the flow boundaries vanish. The evolution of ΔArd depends solely on the viscous effects of the perturbation's dynamics inside the flow domain. The requirement for the temporal decay of ΔArd leads to novel sufficient conditions for the nonlinear stability of the circular Couette flow in response to axisymmetric perturbations. Comparisons with historical studies show that our results shed light on the experimental measurements of Wendt (1933) and significantly extend the classical nonlinear stability results of Serrin (1959) and Joseph & Hung (1971). When the flow is nonlinearly stable and evolves axisymmetrically for all time, then it always decays asymptotically in time to the circular Couette flow determined uniquely by the setup of the rotating cylinders. This study provides new physical insights into a classical flow problem that was studied for decades.
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...
Narayan, K S
2002-01-01
In this thesis, we discuss two topics—marginal stability in gauge theories and little string theories at the Hagedorn temperature. The spectrum of stable supersymmetric charged particle states can change discontinuously as we change the vacuum on the Coulomb branch of four dimensional gauge theories with extended supersymmetry. This discontinuous change manifests itself via the decay of some of these states which become marginally unstable across certain submanifolds in the Coulomb branch. We describe how this decay process can be studied through semiclassical field configurations, purely within the low energy effective action on the Coulomb branch, even at strong coupling. We then describe how these semiclassical field configurations naturally give rise to and generalize the string web description of these supersyrnmetric states found in D-brane constructions for some gauge theories. After a brief study of string web interactions in theories with sixteen supercharges, we move on to study the supers...
Stability with respect to initial time difference for generalized delay differential equations
Directory of Open Access Journals (Sweden)
Ravi Agarwal
2015-02-01
Full Text Available Stability with initial data difference for nonlinear delay differential equations is introduced. This type of stability generalizes the known concept of stability in the literature. It gives us the opportunity to compare the behavior of two nonzero solutions when both initial values and initial intervals are different. Several sufficient conditions for stability and for asymptotic stability with initial time difference are obtained. Lyapunov functions as well as comparison results for scalar ordinary differential equations are employed. Several examples are given to illustrate the theory.
PARTIAL STABILIZATION OF A CLASS OF CONTINUOUS NONLINEAR CONTROL SYSTEMS WITH SEPARATED VARIABLES
Institute of Scientific and Technical Information of China (English)
Jigui JIAN; Xiaoxin LIAO
2005-01-01
In this paper, the partial stabilization problem for a class of nonlinear continuous control systems with separated variables is investigated. Several stabilizing controllers are constructed based on the partial stability theory of Lyapunov and the property of M-matrix, and some of these stabilizing controllers are only related to partial state variables. The controllers constructed here are shown to guarantee partial asymptotic stability of the closed-loop systems and these sufficient conditions may give some instructions to actual engineering application. A example is also given to illustrate the design method.
A Less Conservative Stability Criterion for Delayed Stochastic Genetic Regulatory Networks
Directory of Open Access Journals (Sweden)
Tingting Yu
2014-01-01
Full Text Available This paper concerns the problem of stability analysis for delayed stochastic genetic regulatory networks. By introducing an appropriate Lyapunov-Krasovskii functional and employing delay-range partition approach, a new stability criterion is given to ensure the mean square stability of genetic regulatory networks with time-varying delays and stochastic disturbances. The stability criterion is given in the form of linear matrix inequalities, which can be easily tested by the LMI Toolbox of MATLAB. Moreover, it is theoretically shown that the obtained stability criterion is less conservative than the one in W. Zhang et al., 2012. Finally, a numerical example is presented to illustrate our theory.
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 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.
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.
Theory, Investigation and Stability of Cathode Electrocatalytic Activity
Energy Technology Data Exchange (ETDEWEB)
Ding, Dong; Liu, Mingfei; Lai, Samson; Blinn, Kevin; Liu, Meilin
2012-09-30
The main objective of this project is to systematically characterize the surface composition, morphology, and electro-catalytic properties of catalysts coated on LSCF, aiming to establish the scientific basis for rational design of high-performance cathodes by combining a porous backbone (such as LSCF) with a thin catalyst coating. The understanding gained will help us to optimize the composition and morphology of the catalyst layer and microstructure of the LSCF backbone for better performance. More specifically, the technical objectives include: (1) to characterize the surface composition, morphology, and electro-catalytic properties of catalysts coated on LSCF; (2) to characterize the microscopic details and stability of the LSCF-catalyst (e.g., LSM) interfaces; (3) to establish the scientific basis for rational design of high-performance cathodes by combining a porous backbone (such as LSCF) with a thin catalyst coating; and (4) to demonstrate that the performance and stability of porous LSCF cathodes can be enhanced by the application of a thin-film coating of LSM through a solution infiltration process in small homemade button cells and in commercially available cells of larger dimension. We have successfully developed dense, conformal LSM films with desired structure, composition, morphology, and thickness on the LSCF surfaces by two different infiltration processes: a non-aqueous and a water-based sol-gel process. It is demonstrated that the activity and stability of LSCF cathodes can be improved by the introduction of a thin-film LSM coating through an infiltration process. Surface and interface of the LSM-coated LSCF cathode were systematically characterized using advanced microscopy and spectroscopy techniques. TEM observation suggests that a layer of La and Sr oxide was formed on LSCF surfaces after annealing. With LSM infiltration, in contrast, we no longer observe such La/Sr oxide layer on the LSM-coated LSCF samples after annealing under similar
On fuzzy sampled-data control of chaotic systems via a time-dependent Lyapunov functional approach.
Wang, Zi-Peng; Wu, Huai-Ning
2015-04-01
In this paper, a novel approach to fuzzy sampled-data control of chaotic systems is presented by using a time-dependent Lyapunov functional. The advantage of the new method is that the Lyapunov functional is continuous at sampling times but not necessarily positive definite inside the sampling intervals. Compared with the existing works, the constructed Lyapunov functional makes full use of the information on the piecewise constant input and the actual sampling pattern. In terms of a new parameterized linear matrix inequality (LMI) technique, a less conservative stabilization condition is derived to guarantee the exponential stability for the closed-loop fuzzy sampled-data system. By solving a set of LMIs, the fuzzy sampled-data controller can be easily obtained. Finally, the chaotic Lorenz system and Rössler's system are employed to illustrate the feasibility and effectiveness of the proposed method.
Stability by fixed point theory for functional differential equations
Burton, T A
2006-01-01
This book is the first general introduction to stability of ordinary and functional differential equations by means of fixed point techniques. It contains an extensive collection of new and classical examples worked in detail and presented in an elementary manner. Most of this text relies on three principles: a complete metric space, the contraction mapping principle, and an elementary variation of parameters formula. The material is highly accessible to upper-level undergraduate students in the mathematical sciences, as well as working biologists, chemists, economists, engineers, mathematicia
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 Interval Stability of an Electricity Market Model
Directory of Open Access Journals (Sweden)
Weijuan Wang
2014-01-01
Full Text Available Combined with the electric power market dynamic model put forward by Alvarado, an interval model of electricity markets is established and investigated in this paper pertaining to the range of demand elasticity with suppliers and consumers. The stability of an electricity market framework with demand elasticity interval is analyzed. The conclusions characterizing the interval model provided are derived by constructing a suitable Lyapunov function and using the theory of interval dynamical system in differential equations and matrix inequality theory and so forth. Applying the corollary obtained can judge the system stability by available data about demand elasticity. The obtained results are validated and illustrated by a case example.
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.
Stability of 2-body orbits in retarded gravitation theory (RGT)
Raju, C K
2015-01-01
The recently formulated retarded gravitation theory (RGT) explains the non-Newtonian velocities of stars in spiral galaxies, *without any new hypothesis*, and may hence be tested even in the laboratory. However, doubts have been expressed that those higher rotation velocities in RGT may be due to instabilities. We resolve these doubts by solving the full functional differential equations of RGT for a model 2-body planetary system. The solution is stable and closely agrees with the Newtonian solution for this planetary case. Thus, the big difference between RGT and Newtonian gravity for a spiral galaxy is not due to any instability in RGT.
Chen, Jiyang; Li, Chuandong; Huang, Tingwen; Yang, Xujun
2017-02-01
In this paper, the memristor-based fractional-order neural networks (MFNN) with delay and with two types of stabilizing control are described in detail. Based on the Lyapunov direct method, the theories of set-value maps, differential inclusions and comparison principle, some sufficient conditions and assumptions for global stabilization of this neural network model are established. Finally, two numerical examples are presented to demonstrate the effectiveness and practicability of the obtained results.
Institute of Scientific and Technical Information of China (English)
Xia ZHOU; Shou Ming ZHONG
2011-01-01
In this paper the asymptotical stability in p-moment of neutral stochastic differential equations with discrete and distributed time-varying delays is discussed. The authors apply the fixed-point theory rather than the Lyapunov functions. We give a sufficient condition for asymptotical stability in p-moment when the coefficient functions of equations are not required to be fixed values. Since more general form of system is considered, this paper improves Luo Jiaowan's results.
Practical Stability of Mooring System Based on Linear Complementarity Theory%基于线性互补理论的系泊系统实用稳定性
Institute of Scientific and Technical Information of China (English)
熊先巍; 韦灼彬; 项成安
2012-01-01
为了研究船艇系泊系统的实用稳定性,将系泊系统视为混合系统,基于混合系统理论对混合系统中的线性互补系统的实用稳定性进行研究,重点对李亚普诺夫函数法进行了分析,给出了一种判断线性互补系统实用稳定性的方法.利用线性互补模型对一类系泊系统进行建模,并用给出的方法对该系统的实用稳定性进行了分析,证明了这种方法的可行性.%To study the practical stability of mooring system, the mooring system is seen as the hybrid system. Based on the practical stability of linear complementarity system in the hybrid system studied by the hybrid system theory, the Lyapunov function approach is mainly analyzed. A method determining the practical stability of linear complementarity system is presented. The linear complementarity model of a typical mooring system is constructed, and its practical stability is analyzed. The feasibility of this approach is verified.
Stability of singular networked control systems with control constraint
Institute of Scientific and Technical Information of China (English)
Qiu Zhanzhi; Zhang Qingling; Zhao Zhiwu
2007-01-01
Based on bounded network-induced time-delay, the networked control system is modeled as a linear time-variant singular system.Using the Lyapunov theory and the linear matrix inequality approach, the criteria for delay-independent stability and delay-dependent stability of singular networked control systems are derived and transformed to a feasibility problem of linear matrix inequality formulation, which can be solved by the Matlab LMI toolbox, and the feasible solutions provide the maximum allowable delay bound that makes the system stable.A numerical example is provided, which shows that the analysis method is valid and the stability criteria are feasible.
Robust stabilization for a class of nonlinear networked control systems
Institute of Scientific and Technical Information of China (English)
Jinfeng GAO; Hongye SU; Xiaofu JI; Jian CHU
2008-01-01
The problem of robust stabilization for a class of uncertain networked control systems(NCSs)with nonlinearities satisfying a given sector condition is investigated in this paper.By introducing a new model of NCSs with parameter uncertainty,network.induced delay,nonlinearity and data packet dropout in the transmission,a strict linear matrix inequality(LMI)criterion is proposed for robust stabilization of the uncenmn nonlinear NCSs based on the Lyapunov stability theory.The maximum allowable transfer interval(MATI)can be derived by solving the feasibility problem of the corresponding LMI.Some numerical examples are provided to demonstrate the applicability of the proposed algorithm.
Stabilization Strategies of Supply Networks with Stochastic Switched Topology
Directory of Open Access Journals (Sweden)
Shukai Li
2013-01-01
Full Text Available In this paper, a dynamical supply networks model with stochastic switched topology is presented, in which the stochastic switched topology is dependent on a continuous time Markov process. The goal is to design the state-feedback control strategies to stabilize the dynamical supply networks. Based on Lyapunov stability theory, sufficient conditions for the existence of state feedback control strategies are given in terms of matrix inequalities, which ensure the robust stability of the supply networks at the stationary states and a prescribed H∞ disturbance attenuation level with respect to the uncertain demand. A numerical example is given to illustrate the effectiveness of the proposed method.
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.
Stabilization of Ultracold Molecules Using Optimal Control Theory
Koch, C P; Kosloff, R; Koch, Christiane P.; Palao, Jos\\'e P.; Kosloff, Ronnie
2004-01-01
In recent experiments on ultracold matter, molecules have been produced from ultracold atoms by photoassociation, Feshbach resonances, and three-body recombination. The created molecules are translationally cold, but vibrationally highly excited. This will eventually lead them to be lost from the trap due to collisions. We propose shaped laser pulses to transfer these highly excited molecules to their ground vibrational level. Optimal control theory is employed to find the light field that will carry out this task with minimum intensity. We present results for the sodium dimer. The final target can be reached to within 99% if the initial guess field is physically motivated. We find that the optimal fields contain the transition frequencies required by a good Franck-Condon pumping scheme. The analysis is able to identify the ranges of intensity and pulse duration which are able to achieve this task before other competing process take place. Such a scheme could produce stable ultracold molecular samples or even...
Liu, Yanbin; Liu, Mengying; Sun, Peihua
2014-01-01
A typical model of hypersonic vehicle has the complicated dynamics such as the unstable states, the nonminimum phases, and the strong coupling input-output relations. As a result, designing a robust stabilization controller is essential to implement the anticipated tasks. This paper presents a robust stabilization controller based on the guardian maps theory for hypersonic vehicle. First, the guardian maps theories are provided to explain the constraint relations between the open subsets of complex plane and the eigenvalues of the state matrix of closed-loop control system. Then, a general control structure in relation to the guardian maps theories is proposed to achieve the respected design demands. Furthermore, the robust stabilization control law depending on the given general control structure is designed for the longitudinal model of hypersonic vehicle. Finally, a simulation example is provided to verify the effectiveness of the proposed methods.
On the consistency of Reynolds stress turbulence closures with hydrodynamic stability theory
Speziale, Charles G.; Abid, Ridha; Blaisdell, Gregory A.
1995-01-01
The consistency of second-order closure models with results from hydrodynamic stability theory is analyzed for the simplified case of homogeneous turbulence. In a recent study, Speziale, Gatski, and MacGiolla Mhuiris showed that second-order closures are capable of yielding results that are consistent with hydrodynamic stability theory for the case of homogeneous shear flow in a rotating frame. It is demonstrated in this paper that this success is due to the fact that the stability boundaries for rotating homogeneous shear flow are not dependent on the details of the spatial structure of the disturbances. For those instances where they are -- such as in the case of elliptical flows where the instability mechanism is more subtle -- the results are not so favorable. The origins and extent of this modeling problem are examined in detail along with a possible resolution based on rapid distortion theory (RDT) and its implications for turbulence modeling.
Stability of boundary layers with porous suction strips: Experiment and theory
Reynolds, G. A.; Saric, W. S.; Reed, H. L.; Nayfeh, A. H.
1982-01-01
Low turbulence tunnel experiments on the stability and transition of 2 D boundary layers on flat plates with and without suction are described. A number of general suction cases are discussed. Test results showed that the maximum stabilization occurred when the suction was moved toward the Branch I neutral point. An analytical study of the stability of two dimensional, incompressible boundary layer flows over plates with suction through porous strips was performed. The mean flow was calculated using linearized triple deck, closed form solutions. The stability results of the triple deck theory are shown to be in good agreement with those of the interacting boundary layers. An analytical optimization scheme for the suction configuration was developd. Numerical calculations were performed corresponding to the experimental configurations. In each case, the theory correctly predicts the experimental results.
Theory, Computation and Experiment on Criticality and Stability of Vortices Separating from Edges
2016-08-15
AFRL-AFOSR-VA-TR-2016-0313 Theory, computation and experiment on criticality and stability of vortices separating from edges Ashok Gopalarathnam...computation and experiment on criticality and stability of vortices separating from edges 5a. CONTRACT NUMBER - 5b. GRANT NUMBER FA9550-13-1-0179...leading-edge suction on a finite wing reaches a critical value, LEV initiation takes place. The critical value is the same as that for the corresponding
Effects of heavy modes on vacuum stability in supersymmetric theories
Brizi, Leonardo
2010-01-01
We study the effects induced by heavy fields on the masses of light fields in supersymmetric theories, under the assumption that the heavy mass scale is much higher than the supersymmetry breaking scale. We show that the square-masses of light scalar fields can get two different types of significant corrections when a heavy multiplet is integrated out. The first is an indirect level-repulsion effect, which may arise from heavy chiral multiplets and is always negative. The second is a direct coupling contribution, which may arise from heavy vector multiplets and can have any sign. We then apply these results to the sGoldstino mass and study the implications for the vacuum metastability condition. We find that the correction from heavy chiral multiplets is always negative and tends to compromise vacuum metastability, whereas the contribution from heavy vector multiplets is always positive and tends on the contrary to reinforce it. These two effects are controlled respectively by Yukawa couplings and gauge charg...
Stability Analysis of Grasps with a Robotic Multifingered Hand
Institute of Scientific and Technical Information of China (English)
WAN An-hua
2005-01-01
Stability is a significant property for a robot hand grasp to perform complex tasks similar to human hands. The common method to investigate the stability of robotic multi-fingered grasp system is Lyapunov direct method, but usually it is rather difficult to construct a proper Lyapunov function. Avoiding the hard work of constructing a Lyapunov function, we propose the sufficient conditions for stability of the robotic grasp system.
W-Stability of Multistable Nonlinear Discrete-Time Systems
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Zhishuai Ding
2012-01-01
Full Text Available Motivated by the importance and application of discrete dynamical systems, this paper presents a new Lyapunov characterization which is an extension of conventional Lyapunov characterization for multistable discrete-time nonlinear systems. Based on a new type stability notion of W-stability introduced by D. Efimov, the estimates of solution and the Lyapunov stability theorem and converse theorem are proposed for multi-stable discrete-time nonlinear systems.
Opportunistic Channel Scheduling for Ad Hoc Networks with Queue Stability
Dong, Lei; Wang, Yongchao
2015-03-01
In this paper, a distributed opportunistic channel access strategy in ad hoc network is proposed. We consider the multiple sources contend for the transmission opportunity, the winner source decides to transmit or restart contention based on the current channel condition. Owing to real data assumption at all links, the decision still needs to consider the stability of the queues. We formulate the channel opportunistic scheduling as a constrained optimization problem which maximizes the system average throughput with the constraints that the queues of all links are stable. The proposed optimization model is solved by Lyapunov stability in queueing theory. The successive channel access problem is decoupled into single optimal stopping problem at every frame and solved with Lyapunov algorithm. The threshold for every frame is different, and it is derived based on the instantaneous queue information. Finally, computer simulations are conducted to demonstrate the validity of the proposed strategy.
Equilibrium and stability of relativistic stars in extended theories of gravity
Energy Technology Data Exchange (ETDEWEB)
Wojnar, Aneta [Maria Curie-Sklodowska University, Institute of Physics, Lublin (Poland); Univ. di Monte S. Angelo, Napoli (Italy); Universita' di Napoli Federico II, Complesso Universitario di Monte S. Angelo, Dipartimento di Fisica ' ' E. Pancini' ' , Naples (Italy); INFN, Napoli (Italy); Velten, Hermano [Universidade Federal do Espirito Santo (UFES), Vitoria (Brazil)
2016-12-15
We study static, spherically symmetric equilibrium configurations in extended theories of gravity (ETG) following the notation introduced by Capozziello et al. We calculate the differential equations for the stellar structure in such theories in a very generic form i.e., the Tolman-Oppenheimer-Volkoff generalization for any ETG is introduced. Stability analysis is also investigated with special focus on the particular example of scalar-tensor gravity. (orig.)
Equilibrium and stability of relativistic stars in extended theories of gravity
Wojnar, Aneta
2016-01-01
We study static, spherically symmetric equilibrium configurations in extended theories of gravity (ETG) following the notation introduced by Capozziello et {\\it al}. We calculate the differential equations for the stellar structure in such theories in a very generic form i.e., the Tolman-Oppenheimer-Volkoff generalization for any ETG is introduced. Stability analysis is also investigated with special focus on the particular example of scalar-tensor gravity.
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…
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
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.
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 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.
Institute of Scientific and Technical Information of China (English)
M.Syed Ali
2011-01-01
In this paper,the global stability of Takagi-Sugeno(TS)uncertain stochastic fuzzy recurrent neural networks with discrete and distributed time-varying delays(TSUSFRNNs)is considered.A novel LMI-based stability criterion is obtained by using Lyapunov functional theory to guarantee the asymptotic stability of TSUSFRNNs.The proposed stability conditions are demonstrated through numerical examples.Furthermore,the supplementary requirement that the time derivative of time-varying delays must be smaller than one is removed.Comparison results are demonstrated to show that the proposed method is more able to guarantee the widest stability region than the other methods available in the existing literature.
Stationary Stability for Evolutionary Dynamics in Finite Populations
Directory of Open Access Journals (Sweden)
Marc Harper
2016-08-01
Full Text Available We demonstrate a vast expansion of the theory of evolutionary stability to finite populations with mutation, connecting the theory of the stationary distribution of the Moran process with the Lyapunov theory of evolutionary stability. We define the notion of stationary stability for the Moran process with mutation and generalizations, as well as a generalized notion of evolutionary stability that includes mutation called an incentive stable state (ISS candidate. For sufficiently large populations, extrema of the stationary distribution are ISS candidates and we give a family of Lyapunov quantities that are locally minimized at the stationary extrema and at ISS candidates. In various examples, including for the Moran and Wright–Fisher processes, we show that the local maxima of the stationary distribution capture the traditionally-defined evolutionarily stable states. The classical stability theory of the replicator dynamic is recovered in the large population limit. Finally we include descriptions of possible extensions to populations of variable size and populations evolving on graphs.
Direct Lyapunov-based control law design for spacecraft attitude maneuvers
Institute of Scientific and Technical Information of China (English)
HU Likun; ANG Qingchao
2006-01-01
A direct Lyapunov-based control law is presented to perform on-orbit stability for spacecraft attitude maneuvers. Spacecraft attitude kinematic equations and dynamic equations are coupled, nonlinear, multi-input multi-output(MIMO), which baffles controller design. Orbit angular rates are taken into account in kinematic equations and influence of gravity gradient moments and disturbance moments on the spacecraft attitude in dynamic equations is considered to approach the practical environment, which enhance the problem complexity to some extent. Based on attitude tracking errors and angular rates, a Lyapunov function is constructed, through which the stabilizing feedback control law is deduced via Lie derivation of the Lyapunov function. The proposed method can deal with the case that the spacecraft is subjected to mass property variations or centroidal inertia matrix variations due to fuel assumption or flexibility, and disturbance moments, which shows the proposed controller is robust for spacecraft attitude maneuvers. The unlimited controller and the limited controller are taken into account respectively in simulations. Simulation results are demonstrated to validate effectiveness and feasibility of the proposed method.
Salari, J.W.O.; Leermakers, F.A.M.; Klumperman, B.
2011-01-01
The assembly of sterically stabilized colloids at liquid–liquid interfaces is studied with the self-consistent field (SCF) theory using the discretization scheme that was developed by Scheutjens, Fleer, and co-workers. The model is based on a poly(methyl methacrylate) (pMMA) particle with poly(isobu
Gravity in a stabilized brane world model in five-dimensional Brans-Dicke theory
Mikhailov, A S; Smolyakov, M N; Volobuev, I P
2008-01-01
Linearized equations of motion for gravitational and scalar fields are found and solved in a stabilized brane world model in five-dimensional Brans-Dicke theory. The physical degrees of freedom are isolated, the mass spectrum of Kaluza-Klein excitations is found and the coupling constants of these excitations to matter on the negative tension brane are calculated.
Stability of Delayed Cellular Neural Networks Basedon M-matrix Theory%基于 M 矩阵理论的时滞细胞神经网络稳定性分析*
Institute of Scientific and Technical Information of China (English)
江梅; 何汉林; 严路
2015-01-01
This paper deals with the stability of delayed cellular neural networks .By using the M‐matrix theory with its judgment lemma and applying appropriate linear parameter transformation ,the condition of the stability of the system is de‐duced .Compared with the Lyapunov method ,this paper provides a simpler one which reduces the original conservative con‐clusions and improvement the sufficient condition of origin as globally asymptotical stability equilibrium points .The simula‐tion example demonstrates the method is effective .%研究了时滞细胞神经网络的稳定性问题。通过M‐矩阵理论及其判定引理，运用适当的线性参数变换，推导出时滞细胞神经网络的稳定性条件，相比常用的Lyapunov方法，论文为研究多时滞细胞神经网络的稳定性提供了一个更为简单的新方法，降低了原有结论的保守性，进一步推导完善了全局渐近稳定平衡点为原点时的充分条件。仿真实例证明了文章提供的方法有效可行。
Cao, Fangfei; Liu, Jinkun
2017-10-01
Considering full state constraints, this paper designs a boundary controller for a two-link rigid-flexible manipulator via Barrier Lyapunov Function. The dynamic model of the two-link rigid-flexible manipulator is described by coupled ordinary differential equations- partial differential equations (ODEs-PDEs). Based on the original model without neglecting the high-frequency modes, boundary controller is proposed to regulate the joint positions and eliminate the elastic vibration simultaneously. To ensure that the full state constraints which include position, speed and vibration constraints are not transgressed, a Barrier Lyapunov Function is employed in the proposed controller. The asymptotic stability of the closed-loop system is rigorously proved by the LaSalle's Invariance Principle. Simulations are given to verify the effectiveness of the proposed controller with state constraints.
Liu, Xiaoyang; Yu, Wenwu; Cao, Jinde; Chen, Shun
2015-08-01
This paper is concerned with the sampled-data state estimation and H(∞) filtering for a class of Markovian jump systems with the discontinuous Lyapunov approach. The system measurements are sampled and then transmitted to the estimator and filter in order to estimate the state of the jumped system under consideration. The corresponding error dynamics is represented by a system with two types of delays: one is from the system itself, and the other from the sampling period. As the delay due to sampling is discontinuous, a corresponding discontinuous Lyapunov functional is constructed, and sufficient conditions are established so as to guarantee both the asymptotic mean-square stability and the H(∞) performance for the filtering error systems. The explicit expressions of the desired estimator and filter are further provided. Finally, two simulation examples are given to illustrate the design procedures and performances of the proposed method.
Asymptotic theory of neutral stability curve of the Couette flow of vibrationally excited gas
Grigor'ev, Yu N.; Ershov, I. V.
2016-06-01
The asymptotic theory of neutral stability curve of the supersonic plane Couette flow of vibrationally excited gas is constructed. The system of two-temperature viscous gas dynamics equations was used as original mathematical model. Spectral problem for an eighth order linear system of ordinary differential equations was obtained from the system within framework of classical theory of linear stability. Transformations of the spectral problem universal for all shear flows were carried along the classical Dunn — Lin scheme. As a result the problem was reduced to secular algebraic equation with a characteristic division on “inviscid” and “viscous” parts which was solved numerically. The calculated neutral stability curves coincide in limits of 10% with corresponding results of direct numerical solution of original spectral problem.
Color embeddings, charge assignments, and proton stability in unified gauge theories
Energy Technology Data Exchange (ETDEWEB)
Gell-Mann, M.; Ramond, P.; Slansky, R.
1978-10-01
Three problems in hypothetical unified theories of electromagnetic, weak, and strong interactions are discussed here. First, the problem of embedding color in any simple gauge group is solved, and a complete classification of theories where the fermion color is restricted to 1/sup c/, 3/sup c/, and 3/sup c/ of SU/sup c//sub 3/ is given. Generalizations are also indicated. Second, an unbroken U/sub 1/ generated by electric charge is embedded into each of the above theories and the charge assignments analyzed. Third, the general problem of stabilizing the proton by a suitable atomic mass number A is studied for the same set of theories. It is always possible to define A if the gauge group is not too small. In many of these theories there must be fermions with weird values of A: leptons with Anot =0 and quarks with Anot =1/3. Examples are discussed. Some future directions of research are indicated.
Ward, Cindy L P; Wilson, Anne E
2015-09-01
Temporal self-appraisal theory suggests that people can regulate current self-view by recalling former selves in ways that flatter present identity. People critique their subjectively distant (but not recent) former selves, creating the illusion of improvement over time. However, this revisionist strategy might not apply to everyone: People with fixed (entity) beliefs may not benefit from critiquing even distant selves. In three studies, we found that implicit theories of change and stability moderate the effects of subjective distance on the remembered self. In all studies, participants rated past selves portrayed as subjectively close or distant (controlling calendar time). Incremental theorists (but not entity theorists) were more critical of their subjectively distant (but not recent) past attributes. We found the same pattern when measuring existing implicit theories (Studies 1, 2) or manipulating them (Study 3). The present research is the first to integrate temporal self-appraisal theory and the implicit theories literature.
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.
Solution of the Lyapunov matrix equation for a system with a time-dependent stiffness matrix
DEFF Research Database (Denmark)
Pommer, Christian; Kliem, Wolfhard
2004-01-01
The stability of the linearized model of a rotor system with non-symmetric strain and axial loads is investigated. Since we are using a fixed reference system, the differential equations have the advantage to be free of Coriolis and centrifugal forces. A disadvantage is nevertheless the occurrenc...... of time-dependent periodic terms in the stiffness matrix. However, by solving the Lyapunov matrix equation we can formulate several stability conditions for the rotor system. Hereby the positive definiteness of a certain averaged stiffness matrix plays a crucial role....
Total Stability Properties Based on Fixed Point Theory for a Class of Hybrid Dynamic Systems
Directory of Open Access Journals (Sweden)
M. De la Sen
2009-01-01
Full Text Available Robust stability results for nominally linear hybrid systems are obtained from total stability theorems for purely continuous-time and discrete-time systems by using the powerful tool of fixed point theory. The class of hybrid systems dealt consists, in general, of coupled continuous-time and digital systems subject to state perturbations whose nominal (i.e., unperturbed parts are linear and, in general, time-varying. The obtained sufficient conditions on robust stability under a wide class of harmless perturbations are dependent on the values of the parameters defining the over-bounding functions of those perturbations. The weakness of the coupling dynamics in terms of norm among the analog and digital substates of the whole dynamic system guarantees the total stability provided that the corresponding uncoupled nominal subsystems are both exponentially stable. Fixed point stability theory is used for the proofs of stability. A generalization of that result is given for the case that sampling is not uniform. The boundedness of the state-trajectory solution at sampling instants guarantees the global boundedness of the solutions for all time. The existence of a fixed point for the sampled state-trajectory solution at sampling instants guarantees the existence of a fixed point of an extended auxiliary discrete system and the existence of a global asymptotic attractor of the solutions which is either a fixed point or a limit n globally stable asymptotic oscillation.
Directory of Open Access Journals (Sweden)
Victoria Cabrera García
2014-01-01
Full Text Available The explanation of marital satisfaction and stability in trajectories of couple relationships has been the central interest in different studies (Karney, Bradbury. & Johnson, 1999; Sabatelli & Ripoll, 2004; Schoebi, Karney & Bradbury, 2012. However, there are still several questions and unknown aspects surrounding the topic. Within this context, the present reflection seeks to analyze whether the principles of Evolutionary Theory suffice to explain three marital trajectories in terms of satisfaction and stability. With this in mind, we have included other explanations proposed by the Psychosocial Theory that Evolutionary Theory does not refer to in order to better understand mating behavior. Moreover, other factors that could account for satisfied and stable relationships were analyzed. Suggestions for future investigations include the analysis of other marital trajectories that may or may not end in separation or divorce but are not included in this article.
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.
Stabilization effect of multiple drivers' desired velocities in car-following theory
Zhang, Geng; Zhao, Min; Sun, Di-Hua; Liu, Wei-Ning; Li, Hua-Min
2016-01-01
In order to reveal the influence of driver's individual behavior on traffic flow more accurately, a new car-following model is proposed with consideration of multiple drives' desired velocities. The stability criterion of the new model is derived through linear stability theory and the results show that the current driver's desired velocity can stabilize traffic flow but the preceding driver's desired velocity can damage traffic stability. Through nonlinear analysis, the traffic jamming transition characteristics near the critical point can be described by the kink-antikink soliton of the mKdV equation. Numerical simulation confirms the analytical results, which shows that the multiple drivers' desired velocities play an important role in traffic evolution.
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.
Stability Analysis of MEMS Gyroscope Dynamic Systems
M. Naser-Moghadasi; S. A. Olamaei; F. Setoudeh
2013-01-01
In this paper, the existence of a common quadratic Lyapunov function for stability analysis of MEMS Gyroscope dynamic systems has been studied then a new method based on stochastic stability of MEMS Gyroscope system has been proposed.
Institute of Scientific and Technical Information of China (English)
Zhang Yougang; Xu Bugong
2006-01-01
Decentralized robust stabilization problem of discrete-time fuzzy large-scale systems with parametric uncertainties is considered. This uncertain fuzzy large-scale system consists of N interconnected T-S fuzzy subsystems, and the parametric uncertainties are unknown but norm-bounded. Based on Lyapunov stability theory and decentralized control theory of large-scale system, the design schema of decentralized parallel distributed compensation (DPDC) fuzzy controllers to ensure the asymptotic stability of the whole fuzzy large-scale system is proposed. The existence conditions for these controllers take the forms of LMIs. Finally a numerical simulation example is given to show the utility of the method proposed.
Global exponential stability of Cohen-Grossberg neural networks with variable delays
Institute of Scientific and Technical Information of China (English)
ZHANG Li-juan; SHI Bao
2009-01-01
A class of generalized Cohen-Grossberg neural networks(CGNNs) with variable delays are investigated. By introducing a new type of Lyapunov functional and applying the homeomorphism theory and inequality technique, some new conditions are derived ensuring the existence and uniqueness of the equilibrium point and its global exponential stability for CGNNs. These results obtained are independent of delays, develop the existent outcome in the earlier literature and are very easily checked in practice.
Institute of Scientific and Technical Information of China (English)
LIU Hai-feng; WANG Chun-hua; WEI Guo-liang
2008-01-01
The exponential stability problem is investigated fora class of stochastic recurrent neural networks with time delay and Markovian switching.By using It(o)'s differential formula and the Lyapunov stabifity theory,sufficient condition for the solvability of this problem is derived in telm of linear matrix inequalities,which can be easily checked by resorting to available software packages.A numerical example and the simulation are exploited to demonstrate the effectiveness of the proposed results.
Stability and Control of Large-Scale Dynamical Systems A Vector Dissipative Systems Approach
Haddad, Wassim M
2011-01-01
Modern complex large-scale dynamical systems exist in virtually every aspect of science and engineering, and are associated with a wide variety of physical, technological, environmental, and social phenomena, including aerospace, power, communications, and network systems, to name just a few. This book develops a general stability analysis and control design framework for nonlinear large-scale interconnected dynamical systems, and presents the most complete treatment on vector Lyapunov function methods, vector dissipativity theory, and decentralized control architectures. Large-scale dynami
Impulsive stabilization and synchronization of a class of chaotic delay systems.
Li, Chuandong; Liao, Xiaofeng; Yang, Xiaofan; Huang, Tingwen
2005-12-01
The problems of control and synchronization of a class of chaotic systems with time delay via the impulsive control approach are investigated. Based on the Lyapunov-like stability theory for impulsive functional differential equations, several sufficient conditions are derived to guarantee chaos control and synchronization. Furthermore, we address the chaos quasisynchronization in the presence of single-parameter mismatch. Several illustrated examples are also given to show the effectiveness of the proposed methods.
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.
1979-09-01
without determinantal divisors, Linear and Multilinear Algebra 7(1979), 107-109. 4. The use of integral operators in number theory (with C. Ryavec and...Gersgorin revisited, to appear in Letters in Linear Algebra. 15. A surprising determinantal inequality for real matrices (with C.R. Johnson), to appear in...Analysis: An Essay Concerning the Limitations of Some Mathematical Methods in the Social , Political and Biological Sciences, David Berlinski, MIT Press
Mu, Xiaoqun; Alpak, Faruk O; Chapman, Walter G
2016-01-01
Density gradient theory (DGT) allows fast and accurate determination of surface tension and density profile through a phase interface. Several algorithms have been developed to apply this theory in practical calculations. While the conventional algorithm requires a reference substance of the system, a modified "stabilized density gradient theory" (SDGT) algorithm is introduced in our work to solve DGT equations for multiphase pure and mixed systems. This algorithm makes it possible to calculate interfacial properties accurately at any domain size larger than the interface thickness without choosing a reference substance or assuming the functional form of the density profile. As part of DGT inputs, the perturbed chain statistical associating fluid theory (PC-SAFT) equation of state (EoS) was employed for the first time with the SDGT algorithm. PC-SAFT has excellent performance in predicting liquid phase properties as well as phase behaviors. The SDGT algorithm with the PC-SAFT EoS was tested and compared with ...
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.
[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.
Directory of Open Access Journals (Sweden)
Chen Qin
2013-01-01
Full Text Available This paper considers the problems of the robust stability and robust H∞ controller design for time-varying delay switched systems using delta operator approach. Based on the average dwell time approach and delta operator theory, a sufficient condition of the robust exponential stability is presented by choosing an appropriate Lyapunov-Krasovskii functional candidate. Then, a state feedback controller is designed such that the resulting closed-loop system is exponentially stable with a guaranteed H∞ performance. The obtained results are formulated in the form of linear matrix inequalities (LMIs. Finally, a numerical example is provided to explicitly illustrate the feasibility and effectiveness of the proposed method.
Stabilization of the Yang-Mills chaos in non-Abelian Born-Infeld theory
Galtsov, D V
2003-01-01
We investigate dynamics of the homogeneous time-dependent SU(2) Yang-Mills fields governed by the non-Abelian Born-Infeld lagrangian which arises in superstring theory as a result of summation of all orders in the string slope parameter $\\alpha'$. It is shown that generically the Born-Infeld dynamics is less chaotic than that in the ordinary Yang-Mills theory, and at high enough field strength the Yang-Mills chaos is stabilized. More generally, a smothering effect of the string non-locality on behavior of classical fields is conjectured.
Renormalization theory and ultraviolet stability for scalar fields via renormalization group methods
Energy Technology Data Exchange (ETDEWEB)
Gallavotti, G.
1985-04-01
A self-contained analysis is given of the simplest quantum fields from the renormalization group point of view: multiscale decomposition, general renormalization theory, resummations of renormalized series via equations of the Callan-Symanzik type, asymptotic freedom, and proof of ultraviolet stability for sine-Gordon fields in two dimensions and for other super-renormalizable scalar fields. Renormalization in four dimensions (Hepp's theorem and the De Calan--Rivasseau nexclamation bound) is presented and applications are made to the Coulomb gases in two dimensions and to the convergence of the planar graph expansions in four-dimensional field theories (t' Hooft--Rivasseau theorem).
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.
Energy Technology Data Exchange (ETDEWEB)
Tokuda, Shinji [Japan Atomic Energy Research Inst., Naka, Ibaraki (Japan). Naka Fusion Research Establishment; Watanabe, Tomoko
1996-11-01
A theory and a numerical method are presented for the asymptotic matching analysis of resistive magnetohydrodynamic stability in a negative magnetic shear configuration with two rational surfaces. The theory formulates the problem of solving both the Newcomb equations in the ideal MHD region and the inner-layer equations around rational surfaces as boundary value/eigenvalue problems to which the finite element method and the finite difference method can be applied. Hence, the problem of stability analysis is solved by a numerically stable method. The present numerical method has been applied to model equations having analytic solutions in a negative magnetic shear configuration. Comparison of the numerical solutions with the analytical ones verifies the validity of the numerical method proposed. (author)
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.
Defect-Mediated Stability: An Effective Hydrodynamic Theory of Spatio-Temporal Chaos
Chow, Carson C.; Hwa, Terence
1994-01-01
Spatiotemporal chaos (STC) exhibited by the Kuramoto-Sivashinsky (KS) equation is investigated analytically and numerically. An effective stochastic equation belonging to the KPZ universality class is constructed by incorporating the chaotic dynamics of the small KS system in a coarse-graining procedure. The bare parameters of the effective theory are computed approximately. Stability of the system is shown to be mediated by space-time defects that are accompanied by stochasticity. The method...
Müller, H W; Wagner, C; Albers, J; Knorr, K
1996-01-01
We present an analytical stability theory for the onset of the Faraday instability, applying over a wide frequency range between shallow water gravity and deep water capillary waves. For sufficiently thin fluid layers the surface is predicted to occur in harmonic rather than subharmonic resonance with the forcing. An experimental confirmation of this result is given. PACS: 47.20.Ma, 47.20.Gv, 47.15.Cb
On the stability of (M theory) stars against collapse: Role of anisotropic pressures
Kalyana Rama, S.
2015-09-01
Unitarity of evolution in gravitational collapses implies existence of macroscopic stable horizonless objects. With such objects in mind, we study the effects of anisotropy of pressures on the stability of stars. We consider stars in four or higher dimensions and also stars in M theory made up of (intersecting) branes. Taking the stars to be static, spherically symmetric and the equations of state to be linear, we study “singular solutions” and the asymptotic perturbations around them. Oscillatory perturbations are likely to imply instability. We find that nonoscillatory perturbations, which may imply stability, are possible if an appropriate amount of anisotropy is present. This result suggests that it may be possible to have stable horizonless objects in four or any higher dimensions, and that anisotropic pressures may play a crucial role in ensuring their stability.
On the Stability of (M theory) Stars against Collapse : Role of Anisotropic Pressures
Rama, S Kalyana
2015-01-01
Unitarity of evolution in gravitational collapses implies existence of macroscopic stable horizonless objects. With such objects in mind, we study the effects of anisotropy of pressures on the stability of stars. We consider stars in four or higher dimensions and also stars in M theory made up of (intersecting) branes. Taking the stars to be static, spherically symmetric and the equations of state to be linear, we study the asymptotic solutions and perturbations around them. Oscillatory perturbations are known to imply instability. We find that non oscillatory perturbations, which may imply stability, are possible if an appropriate amount of anisotropy is present. This result suggests that it may be possible to have stable horizonless objects in four or any higher dimensions, and that anisotropic pressures may play a crucial role in ensuring their stability.
Asymptotic theory of neutral stability of the Couette flow of a vibrationally excited gas
Grigor'ev, Yu. N.; Ershov, I. V.
2017-01-01
An asymptotic theory of the neutral stability curve for a supersonic plane Couette flow of a vibrationally excited gas is developed. The initial mathematical model consists of equations of two-temperature viscous gas dynamics, which are used to derive a spectral problem for a linear system of eighth-order ordinary differential equations within the framework of the classical linear stability theory. Unified transformations of the system for all shear flows are performed in accordance with the classical Lin scheme. The problem is reduced to an algebraic secular equation with separation into the "inviscid" and "viscous" parts, which is solved numerically. It is shown that the thus-calculated neutral stability curves agree well with the previously obtained results of the direct numerical solution of the original spectral problem. In particular, the critical Reynolds number increases with excitation enhancement, and the neutral stability curve is shifted toward the domain of higher wave numbers. This is also confirmed by means of solving an asymptotic equation for the critical Reynolds number at the Mach number M ≤ 4.
Junhai Ma; Yuehong Guo
2014-01-01
This paper studied system dynamics characteristics of closed-loop supply chain using repeated game theory and complex system theory. It established decentralized decision-making game model and centralized decision-making game model and then established and analyzed the corresponding continuity system. Drew the region local stability of Nash equilibrium and Stackelberg equilibrium, and a series of chaotic system characteristics, have an detail analysis of the Lyapunov index which is under the ...
Stability of Equilibrium Points of Fractional Difference Equations with Stochastic Perturbations
Directory of Open Access Journals (Sweden)
Shaikhet Leonid
2008-01-01
Full Text Available It is supposed that the fractional difference equation , has an equilibrium point and is exposed to additive stochastic perturbations type of that are directly proportional to the deviation of the system state from the equilibrium point . It is shown that known results in the theory of stability of stochastic difference equations that were obtained via V. Kolmanovskii and L. Shaikhet general method of Lyapunov functionals construction can be successfully used for getting of sufficient conditions for stability in probability of equilibrium points of the considered stochastic fractional difference equation. Numerous graphical illustrations of stability regions and trajectories of solutions are plotted.
Stability of the Newton-Like algorithm in optimization flow control
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
The stability of the Newton-like algorithm in optimization flow control is considered in this paper.This algorithm is proved to be globally stable under a general network topology by means of Lyapunov stability theory, without considering the round trip time of each source. While the stability of this algorithm with considering the round trip time is analyzed as well. The analysis shows that the algorithm with only one bottleneck link accessed by several sources is also globally stable, and all trajectories described by this algorithm ultimately converge to the equilibrium point.
Synchronization and Stabilization of Chaotic Dynamics in a Quasi-1D Bose-Einstein Condensate
Directory of Open Access Journals (Sweden)
B. A. Idowu
2013-01-01
Full Text Available A nonlinear control is proposed for the exponential stabilization and synchronization of chaotic behaviour in a model of Bose-Einstein condensate (BEC. The active control technique is designed based on Lyapunov stability theory and Routh-Hurwitz criteria. The control design approach in both cases guarantees the stability of the controlled states. Whereas the synchronization of two identical BEC in their chaotic states can be realized using the scheme; a suitable controller is also capable of driving the otherwise chaotic oscillation to a stable state which could be expected in practice. The effectiveness of this technique is theoretically and numerically demonstrated.
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.
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.
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.
On the stability of KMS states in perturbative algebraic quantum field theories
Drago, Nicolo; Pinamonti, Nicola
2016-01-01
We analyze the stability properties shown by KMS states for interacting massive scalar fields propagating over Minkowski spacetime, recently constructed in the framework of perturbative algebraic quantum field theories by Fredenhagen and Lindner \\cite{FredenhagenLindner}. In particular, we prove the validity of the return to equilibrium property when the interaction Lagrangean has compact spatial support. Surprisingly, this does not hold anymore, if the adiabatic limit is considered, namely when the interaction Lagrangean is invariant under spatial translations. Consequently, an equilibrium state under the adiabatic limit for a perturbative interacting theory evolved with the free dynamics does not converge anymore to the free equilibrium state. Actually, we show that its ergodic mean converges to a non equilibrium steady state for the free theory.
Arefi, Mohammad Mehdi; Jahed-Motlagh, Mohammad Reza; Karimi, Hamid Reza
2015-08-01
In this paper, first, an adaptive neural network (NN) state-feedback controller for a class of nonlinear systems with mismatched uncertainties is proposed. By using a radial basis function NN (RBFNN), a bound of unknown nonlinear functions is approximated so that no information about the upper bound of mismatched uncertainties is required. Then, an observer-based adaptive controller based on RBFNN is designed to stabilize uncertain nonlinear systems with immeasurable states. The state-feedback and observer-based controllers are based on Lyapunov and strictly positive real-Lyapunov stability theory, respectively, and it is shown that the asymptotic convergence of the closed-loop system to zero is achieved while maintaining bounded states at the same time. The presented methods are more general than the previous approaches, handling systems with no restriction on the dimension of the system and the number of inputs. Simulation results confirm the effectiveness of the proposed methods in the stabilization of mismatched nonlinear systems.
Stabilization of parameters of asynchronous electric drive with vector control
Directory of Open Access Journals (Sweden)
N.J. Khlopenko
2015-03-01
Full Text Available A problem of stabilization of parameters of the asynchronous electric drive vector control system is considered. Usually such systems have two control channels. The synthesis of stabilizing controllers is made for every control channel. The evaluation of variables of system status is made by observer. The problem of stabilizing controllers and observer synthesis consists in calculation of state feedback intensification. Its solution is based on existing approaches form vector control theories, matrix inequalities and Lyapunov stability. Several synthesis methods of stabilizing controllers have been proposed. Structural scheme of vector control system and observer has been built. The simulation of transient processes in the vector control system is carried out with MATLAB computing environment. The most important property of obtained solution is Lyapunov stability of control loops closed-looped by state vectors. Transient processes have been investigated on the particular example. Graphs confirming stability of such processes that flow in the vector control system in minimal period of time have been plotted down.
Lyapunov-based control designs for flexible-link manipulators
Juang, Jer-Nan; Huang, Jen-Kuang; Yang, Li-Farn
1989-01-01
A feedback controller for the stabilization of closed-loop systems is proposed which is based on the Liapunov stability criterion. A feedback control law is first generated for the linear portion of the system equation using linear control theory. A feedback control is then designed for the nonlinear portion of the system equation by making negative the time derivative of a positive definite Liapunov function.
Lyapunov-based control designs for flexible-link manipulators
Juang, Jer-Nan; Huang, Jen-Kuang; Yang, Li-Farn
1989-01-01
A feedback controller for the stabilization of closed-loop systems is proposed which is based on the Liapunov stability criterion. A feedback control law is first generated for the linear portion of the system equation using linear control theory. A feedback control is then designed for the nonlinear portion of the system equation by making negative the time derivative of a positive definite Liapunov function.
Are Bred Vectors The Same As Lyapunov Vectors?
Kalnay, E.; Corazza, M.; Cai, M.
Regional loss of predictability is an indication of the instability of the underlying flow, where small errors in the initial conditions (or imperfections in the model) grow to large amplitudes in finite times. The stability properties of evolving flows have been studied using Lyapunov vectors (e.g., Alligood et al, 1996, Ott, 1993, Kalnay, 2002), singular vectors (e.g., Lorenz, 1965, Farrell, 1988, Molteni and Palmer, 1993), and, more recently, with bred vectors (e.g., Szunyogh et al, 1997, Cai et al, 2001). Bred vectors (BVs) are, by construction, closely related to Lyapunov vectors (LVs). In fact, after an infinitely long breeding time, and with the use of infinitesimal ampli- tudes, bred vectors are identical to leading Lyapunov vectors. In practical applications, however, bred vectors are different from Lyapunov vectors in two important ways: a) bred vectors are never globally orthogonalized and are intrinsically local in space and time, and b) they are finite-amplitude, finite-time vectors. These two differences are very significant in a dynamical system whose size is very large. For example, the at- mosphere is large enough to have "room" for several synoptic scale instabilities (e.g., storms) to develop independently in different regions (say, North America and Aus- tralia), and it is complex enough to have several different possible types of instabilities (such as barotropic, baroclinic, convective, and even Brownian motion). Bred vectors share some of their properties with leading LVs (Corazza et al, 2001a, 2001b, Toth and Kalnay, 1993, 1997, Cai et al, 2001). For example, 1) Bred vectors are independent of the norm used to define the size of the perturba- tion. Corazza et al. (2001) showed that bred vectors obtained using a potential enstro- phy norm were indistinguishable from bred vectors obtained using a streamfunction squared norm, in contrast with singular vectors. 2) Bred vectors are independent of the length of the rescaling period as long as the
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.
随机逼近中的Lyapunov函数%On Lyapunov Functions inStochastic Approximation
Institute of Scientific and Technical Information of China (English)
张俊华
2001-01-01
本文研究了随机逼近中满足某种条件的Lyapunov函数的存在性及如何构造Lyapunov函数的问题,讨论了算法收敛性与相应常微分方程系统的渐近稳定性之间的关系.%In this paper, we investigate existence and construction of certain Lyapunov functions instochastic approximation (SA) and discuss the relation between convergence of SA algorithms andasymptotic stability of the corresponding ordinary differential equation systems.
Yi-You Hou; Zhang-Lin Wan
2014-01-01
This paper considers the problem of the robust stability for the nonlinear system with time-varying delay and parameters uncertainties. Based on the H∞ theorem, Lyapunov-Krasovskii theory, and linear matrix inequality (LMI) optimization technique, the H∞ quasi-sliding mode controller and switching function are developed such that the nonlinear system is asymptotically stable in the quasi-sliding mode and satisfies the disturbance attenuation (H∞-norm performance). The effectiveness and accura...
Absolute Stability for Lurie Control System with Unbound Time Delays
Institute of Scientific and Technical Information of China (English)
王天成; 王耀才; 洪留荣
2004-01-01
Time delay existes widely in various real engineering systems and can result in unsatisfactory performance or even an instability of control systems. Therefore, to investigate the stability for time delay systems is of vitul importance in control theory and its applications. Many researchers have studied the stability criteria of systems with constant delay or bound varying time delay, but few of them studied large time delay or unbound time delay. Large time delay existes commonly in various engineering applications. In this paper, the absolute stability of Lurie type direct control systems and indirect control systems with several time delays are discussed. Based on Lyapunov theory, the new delay dependent absolute stability criteria are derived. In our theorem, time delays can be unbound functions, which shows that the results are less conservative than that of existed criteria.
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.
Method for stability analysis based on the Floquet theory and Vidyn calculations
Energy Technology Data Exchange (ETDEWEB)
Ganander, Hans
2005-03-01
This report presents the activity 3.7 of the STEM-project Aerobig and deals with aeroelastic stability of the complete wind turbine structure at operation. As a consequence of the increase of sizes of wind turbines dynamic couplings are being more important for loads and dynamic properties. The steady ambition to increase the cost competitiveness of wind turbine energy by using optimisation methods lowers design margins, which in turn makes questions about stability of the turbines more important. The main objective of the project is to develop a general stability analysis tool, based on the VIDYN methodology regarding the turbine dynamic equations and the Floquet theory for the stability analysis. The reason for selecting the Floquet theory is that it is independent of number of blades, thus can be used for 2 as well as 3 bladed turbines. Although the latter ones are dominating on the market, the former has large potential when talking about offshore large turbines. The fact that cyclic and individual blade pitch controls are being developed as a mean for reduction of fatigue also speaks for general methods as Floquet. The first step of a general system for stability analysis has been developed, the code VIDSTAB. Together with other methods, as the snap shot method, the Coleman transformation and the use of Fourier series, eigenfrequences and modes can be analysed. It is general with no restrictions on the number of blades nor the symmetry of the rotor. The derivatives of the aerodynamic forces are calculated numerically in this first version. Later versions would include state space formulations of these forces. This would also be the case for the controllers of turbine rotation speed, yaw direction and pitch angle.
Directory of Open Access Journals (Sweden)
Fei Yu
2009-01-01
Full Text Available Based on the theory of calculus on time scales, the homeomorphism theory, Lyapunov functional method, and some analysis techniques, sufficient conditions are obtained for the existence, uniqueness, and global exponential stability of the equilibrium point of Cohen-Grossberg bidirectional associative memory (BAM neural networks with distributed delays and impulses on time scales. This is the first time applying the time-scale calculus theory to unify the discrete-time and continuous-time Cohen-Grossberg BAM neural network with impulses under the same framework.
Study on Robust Uniform Asymptotical Stability for Uncertain Linear Impulsive Delay Systems
Institute of Scientific and Technical Information of China (English)
刘斌; 刘新芝; 廖晓昕
2003-01-01
In the area of control theory the time-delay systems have been investigated. It's well known that delays often result in instability, therefore, stability analysis of time-delay systems is an important subject in control theory. As a result, many criteria for testing the stability of linear time-delay systems have been proposed. Significant progress has been made in the theory of impulsive systems and impulsive delay systems in recent years. However, the corresponding theory for uncertain impulsive systems and uncertain impulsive delay systems has not been fully developed. In this paper, robust stability criteria are established for uncertain linear delay impulsive systems by using Lyapunov function, Razumikhin techniques and the results obtained. Some examples are given to illustrate our theory.
Lateral dynamic flight stability of hovering insects:theory vs.numerical simulation
Institute of Scientific and Technical Information of China (English)
Yan-Lai Zhang; Jiang-Hao Wu; Mao Sun
2012-01-01
In the present paper,the lateral dynamic flight stability properties of two hovering model insects are predicted by an approximate theory based on the averaged model,and computed by numerical simulation that solves the complete equations of motion coupled with the Navier-Stokes equations.Comparison between the theoretical and simulational results provides a test to the validity of the assumptions made in the theory.One of the insects is a model dronefly which has relatively high wingbeat frequency (164Hz)and the other is a model hawkmoth which has relatively low wingbeat frequency (26 Hz).The following conclusion has been drawn.The theory based on the averaged model works well for the lateral motion of the dronefly.For the hawkmoth,relatively large quantitative differences exist between theory and simulation.This is because the lateral non-dimensional eigenvalues of the hawkmoth are not very small compared with the non-dimensional flapping frequency (the largest lateral non-dimensional eigenvalue is only about 10% smaller than the non-dimensional flapping frequency).Nevertheless,the theory can still correctly predict variational trends of the dynamic properties of the hawkmoth's lateral motion.
Organised structures in wall turbulence as deduced from stability theory-based methods
Indian Academy of Sciences (India)
P K Sen; S V Veeravalli; P W Carpenter; G Joshi; P S Josan
2007-02-01
In earlier work, we have explored the relevance of hydrodynamic stability theory to fully developed turbulent wall ﬂows. Using an extended Orr-Summerfeld Equation, based on an anisotropic eddy-viscosity model, it was shown that there exists a wide range of unstable wave numbers (wall modes), which mimic some of the key features of turbulent wall ﬂows. Here we present experimental conﬁrmation for the same. There is good qualitative and quantitative agreement between theory and experiment. Once the dominant coherent structure is obtained from stability theory, control of turbulence would be the next logical step. As shown, the use of a compliant wall shows considerable promise. We also present some theoretical work for bypass transition (Klebanoff/K-modes), wherein the receptivity of a laminar boundary layer to a vortex sheet in the freestream has been studied. Further, it is shown that triadic interaction between K-modes, 2D TS waves and 3D TS waves can lead to rapid algebraic growth. A similar mechanism seems to carry over to inner wall structures in wall turbulence and perhaps this is the “root cause” for sustenance of turbulence.
Saffari, Shahab; Hashemian, Mohammad; Toghraie, Davood
2017-09-01
Based on nonlocal Timoshenko beam theory, dynamic stability of functionally graded (FG) nanobeam under axial and thermal loading was investigated. Surface stress effects were implemented according to Gurtin-Murdoch continuum theory. Using power law distribution for FGM and von Karman geometric nonlinearity, governing equations were derived based on Hamilton's principle. The developed nonlocal models have the capability of interpreting small scale effects. Pasternak elastic medium was employed to represent the interaction of the FG nanobeam and the surrounding elastic medium. A parametric study was conducted to focus influences of the static load factor, temperature change, gradient index, nonlocal parameter, slenderness ratio, surface effect and springs constants of the elastic medium on the dynamic instability region (DIR) of the FG beam with simply-supported boundary conditions. It was found that differences between DIRs predicted by local and nonlocal beam theories are significant for beams with lower aspect ratio. Moreover, it was observed that in contrast to high temperature environments, at low temperatures, increasing the temperature change moves the origin of the DIR to higher excitation frequency zone and leads to further stability. Considering surface stress effects shifts the DIR of FG beam to higher frequency zone, also increasing the gradient index enhances the frequency of DIR.
Gaonkar, G.
1987-01-01
For flap lag stability of isolated rotors, experimental and analytical investigations were conducted in hover and forward flight on the adequacy of a linear quasisteady aerodynamics theory with dynamic flow. Forward flight effects on lag regressing mode were emphasized. A soft inplane hingeless rotor with three blades was tested at advance ratios as high as 0.55 and at shaft angles as high as 20 deg. The 1.62 m model rotor was untrimmed with an essentially unrestricted tilt of the tip path plane. In combination with lag natural frequencies, collective pitch settings and flap lag coupling parameters, the data base comprises nearly 1200 test points (damping and frequency) in forward flight and 200 test points in hover. By computerized symbolic manipulation, a linear model was developed in substall to predict stability margins with mode identification. To help explain the correlation between theory and data it also predicted substall and stall regions of the rotor disk from equilibrium values. The correlation showed both the strengths and weaknesses of the theory in substall ((angle of attack) equal to or less than 12 deg).
Non-probabilistic fuzzy reliability analysis of pile foundation stability by interval theory
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
Randomness and fuzziness are among the attributes of the influential factors for stability assessment of pile foundation.According to these two characteristics, the triangular fuzzy number analysis approach was introduced to determine the probability-distributed function of mechanical parameters. Then the functional function of reliability analysis was constructed based on the study of bearing mechanism of pile foundation, and the way to calculate interval values of the functional function was developed by using improved interval-truncation approach and operation rules of interval numbers. Afterwards, the non-probabilistic fuzzy reliability analysis method was applied to assessing the pile foundation, from which a method was presented for nonprobabilistic fuzzy reliability analysis of pile foundation stability by interval theory. Finally, the probability distribution curve of nonprobabilistic fuzzy reliability indexes of practical pile foundation was concluded. Its failure possibility is 0.91%, which shows that the pile foundation is stable and reliable.
George, Janine; Dronskowski, Richard
2017-02-16
Intermolecular bonds play a crucial role in the rational design of crystal structures, dubbed crystal engineering. The relatively new term tetrel bonds (TBs) describes a long-known type of such interactions presently in the focus of quantum chemical cluster calculations. Here, we energetically explore the strengths and cooperativity of these interactions in infinite chains, a possible arrangement of such tetrel bonds in extended crystals, by periodic density functional theory. In the chains, the TBs are amplified due to cooperativity by up to 60%. Moreover, we computationally take apart crystals stabilized by infinite tetrel-bonded chains and assess the importance of the TBs for the crystal stabilization. Tetrel bonds can amount to 70% of the overall interaction energy within some crystals, and they can also be energetically decisive for the taken crystal structure; their individual strengths also compete with the collective packing within the crystal structures.
Controlling chaos in power system based on finite-time stability theory
Institute of Scientific and Technical Information of China (English)
Zhao Hui; Ma Ya-Jun; Liu Si-Jia; Gao Shi-Gen; Zhong Dan
2011-01-01
Recent investigations show that a power system is a highly nonlinear system and can exhibit chaotic behaviour leading to a voltage collapse,which severely threatens the secure and stable operation of the power system.Based on the finite-time stability theory,two control strategies are presented to achieve finite-time chaos control.In addition,the problem of how to stabilize an unstable nonzero equilibrium point in a finite time is solved by coordinate transformation for the first time.Numerical simulations are presented to demonstrate the effectiveness and the robustness of the proposed scheme.The research in this paper may help to maintain the secure operation of power systems.
A review of the shale wellbore stability mechanism based on mechanical–chemical coupling theories
Directory of Open Access Journals (Sweden)
Qiangui Zhang
2015-06-01
Full Text Available Wellbore instability in hard brittle shale is a critical topic related to the effective exploitation of shale gas resources. This review first introduces the physical–chemical coupling theories applied in shale wellbore stability research, including total water absorption method, equivalent pore pressure method, elasticity incremental method of total water potential and non-equilibrium thermodynamic method. Second, the influences of water activity, membrane efficiency, clay content and drilling fluid on shale wellbore instability are summarized. Results demonstrate that shale and drilling fluid interactions can be the critical factors affecting shale wellbore stability. The effects of thermodynamics and electrochemistry may also be considered in the future, especially the microscopic reaction of shale and drilling fluid interactions. An example of this reaction is the chemical reaction between shale components and drilling fluid.
Money creation, employment and economic stability: The monetary theory of unemployment and inflation
Directory of Open Access Journals (Sweden)
Parguez Alain
2008-01-01
Full Text Available This paper by building on the general theory of the monetary circuit, proves that money-as a pure bank credit liability-exists to overcome constraints on required expenditures by firms, household and mainly the State. From this perspective the paper derives the employment function in the modern monetary economy. Thereby it is explained that full employment policy is both always possible and required. It is proven that this conclusion holds in a perfectly open economy. Ultimately it is explained that there is no trade-off between full employment and sustainable price stability.
Controlling Chaos in permanent magnet synchronous motor based on finite-time stability theory
Institute of Scientific and Technical Information of China (English)
Wei Du-Qu; Zhang So
2009-01-01
This paper reports that the performance of permanent magnet synchronous motor(PMSM)degrades due to chaos when its systemic parameters fall into a certain area.To control the undesirable chaos in PMSM,a nonlinear controller,which is simple and easy to be constructed,is presented to achieve finite-time chaos control based on the finite-time stability theory.Computer simulation results show that the proposed controller is very effective.The obtained results may help to maintain the industrial servo driven system's security operation.
Experimental test of theory for the stability of partially saturated vertical cut slopes
Morse, Michael M.; Lu, N.; Wayllace, Alexandra; Godt, Jonathan W.; Take, W.A.
2014-01-01
This paper extends Culmann's vertical-cut analysis to unsaturated soils. To test the extended theory, unsaturated sand was compacted to a uniform porosity and moisture content in a laboratory apparatus. A sliding door that extended the height of the free face of the slope was lowered until the vertical cut failed. Digital images of the slope cross section and upper surface were acquired concurrently. A recently developed particle image velocimetry (PIV) tool was used to quantify soil displacement. The PIV analysis showed strain localization at varying distances from the sliding door prior to failure. The areas of localized strain were coincident with the location of the slope crest after failure. Shear-strength and soil-water-characteristic parameters of the sand were independently tested for use in extended analyses of the vertical-cut stability and of the failure plane angle. Experimental failure heights were within 22.3% of the heights predicted using the extended theory.
Couple analysis on strength reduction theory and rheological mechanism for slope stability
Institute of Scientific and Technical Information of China (English)
刘子振; 言志信; 段建
2008-01-01
Considering the rheological properties of rock and soil body,and exploiting the merit of strength reduction technique,a theory of couple analysis is brought forward on the basis of strength reduction theory and rheological properties.Then,the concept and the calculation procedure of the safety factor are established at different time.Making use of finite element software ANSYS,the most dangerous sliding surface of the slope can be obtained through the strength reduction technique.According to the dynamic safety factor based on rheological mechanism,a good forecasting could be presented to prevent and cure the landslide.The result shows that the couple analysis reveals the process of the slope failure with the time and the important influence on the long-term stability due to the rheological parameters.
Extension of the Chern-Simons Theory: Conservation Laws, Lagrange Structures, and Stability
Kaparulin, D. S.; Karataeva, I. Yu.; Lyakhovich, S. L.
2017-03-01
We consider the class of higher derivative 3d vector field models with the wave operator being a polynomial of the Chern-Simons operator. For the nth order theory of this type, we provide a covariant procedure for constructing n-parameter family of conservation laws associated with spatiotemporal symmetries. This family includes the canonical energy that is unbounded from below, whereas others conservation laws from the family can be bounded from below for certain combinations of the Lagrangian parameters, even though higher derivatives are present in the Lagrangian. We prove that any conserved quantity bounded from below is related with invariance of the theory with respect to the time translations and ensures the stability of the model.
Extension of the Chern-Simons Theory: Conservation Laws, Lagrange Structures, and Stability
Kaparulin, D. S.; Karataeva, I. Yu.; Lyakhovich, S. L.
2017-03-01
We consider the class of higher derivative 3d vector field models with the wave operator being a polynomial of the Chern-Simons operator. For the nth order theory of this type, we provide a covariant procedure for constructing n-parameter family of conservation laws associated with spatiotemporal symmetries. This family includes the canonical energy that is unbounded from below, whereas others conservation laws from the family can be bounded from below for certain combinations of the Lagrangian parameters, even though higher derivatives are present in the Lagrangian. We prove that any conserved quantity bounded from below is related with invariance of the theory with respect to the time translations and ensures the stability of the model.
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...
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).
On Controllability and Observability of Fuzzy Dynamical Matrix Lyapunov Systems
Directory of Open Access Journals (Sweden)
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.
Detecting Epileptic Seizure from Scalp EEG Using Lyapunov Spectrum
Directory of Open Access Journals (Sweden)
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.
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 stability conditions for theories of modified gravity in the presence of matter fields
De Felice, Antonio; Frusciante, Noemi; Papadomanolakis, Georgios
2017-03-01
We present a thorough stability analysis of modified gravity theories in the presence of matter fields. We use the Effective Field Theory framework for Dark Energy and Modified Gravity to retain a general approach for the gravity sector and a Sorkin-Schutz action for the matter one. Then, we work out the proper viability conditions to guarantee in the scalar sector the absence of ghosts, gradient and tachyonic instabilities. The absence of ghosts can be achieved by demanding a positive kinetic matrix, while the lack of a gradient instability is ensured by imposing a positive speed of propagation for all the scalar modes. In case of tachyonic instability, the mass eigenvalues have been studied and we work out the appropriate expressions. For the latter, an instability occurs only when the negative mass eigenvalue is much larger, in absolute value, than the Hubble parameter. We discuss the results for the minimally coupled quintessence model showing for a particular set of parameters two typical behaviours which in turn lead to a stable and an unstable configuration. Moreover, we find that the speeds of propagation of the scalar modes strongly depend on matter densities, for the beyond Horndeski theories. Our findings can be directly employed when testing modified gravity theories as they allow to identify the correct viability space.
Song, Qiankun; Yan, Huan; Zhao, Zhenjiang; Liu, Yurong
2016-09-01
This paper investigates the stability problem for a class of impulsive complex-valued neural networks with both asynchronous time-varying and continuously distributed delays. By employing the idea of vector Lyapunov function, M-matrix theory and inequality technique, several sufficient conditions are obtained to ensure the global exponential stability of equilibrium point. When the impulsive effects are not considered, several sufficient conditions are also given to guarantee the existence, uniqueness and global exponential stability of equilibrium point. Two examples are given to illustrate the effectiveness and lower level of conservatism of the proposed criteria in comparison with some existing results.
Institute of Scientific and Technical Information of China (English)
曾庆山; 曹广益; 朱新坚
2004-01-01
The definitions of controllability, observability and stability were presented for fractional-order linear systems. Using the Cayley-Hamilton theorem and Mittag-Leffler function in two parameters, the sufficient and necessary conditions of controllability and observability for such systems were derived. In terms of Lyapunov's stability theory, using the theorems of Mittage-Leffler function in two parameters this paper directly derived the sufficient and necessary condition of stability for such systems. The results obtained are useful for the analysis and synthesis of fractional-order linear control systems.
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.
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.
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.
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.
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.
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.
Shah, Neerav
2011-01-01
The Magnetospheric MultiScale Mission (MMS) is scheduled to launch in late 2014. Its primary goal is to discover the fundamental plasma physics processes of reconnection in the Earth's magnetosphere. Each of the four MMS spacecraft is spin-stabilized at a nominal rate of 3 RPM. Traditional spin-stabilized spacecraft have used a number of separate modes to control nutation, spin rate, and precession. To reduce the number of modes and simplify operations, the Delta-H control mode is designed to accomplish nutation control, spin rate control, and precession control simultaneously. A nonlinear design technique, Lyapunov's method, is used to design the Delta-H control mode. A global spin rate controller selected as the baseline controller for MMS, proved to be insufficient due to an ambiguity in the attitude. Lyapunov's design method was used to solve this ambiguity, resulting in a controller that meets the design goals. Simulation results show the advantage of the pointing and rate controller for maneuvers larger than 90 deg and provide insight into the performance of this controller.
Nonlinear stability of cylindrical shells subjected to axial flow: Theory and experiments
Karagiozis, K. N.; Païdoussis, M. P.; Amabili, M.; Misra, A. K.
2008-01-01
This paper, is concerned with the nonlinear dynamics and stability of thin circular cylindrical shells clamped at both ends and subjected to axial fluid flow. In particular, it describes the development of a nonlinear theoretical model and presents theoretical results displaying the nonlinear behaviour of the clamped shell subjected to flowing fluid. The theoretical model employs the Donnell nonlinear shallow shell equations to describe the geometrically nonlinear structure. The clamped beam eigenfunctions are used to describe the axial variations of the shell deformation, automatically satisfying the boundary conditions and the circumferential continuity condition exactly. The fluid is assumed to be incompressible and inviscid, and the fluid-structure interaction is described by linear potential flow theory. The partial differential equation of motion is discretized using the Galerkin method and the final set of ordinary differential equations are integrated numerically using a pseudo-arclength continuation and collocation techniques and the Gear backward differentiation formula. A theoretical model for shells with simply supported ends is presented as well. Experiments are also described for (i) elastomer shells subjected to annular (external) air-flow and (ii) aluminium and plastic shells with internal water flow. The experimental results along with the theoretical ones indicate loss of stability by divergence with a subcritical nonlinear behaviour. Finally, theory and experiments are compared, showing good qualitative and reasonable quantitative agreement.
一类线性脉冲微分系统的变差稳定性%Variational Stability for a Class of Linear Differential Systems with Impulses
Institute of Scientific and Technical Information of China (English)
李宝麟; 王倩倩
2012-01-01
利用Henstock积分、Lyapunov函数以及脉冲微分系统理论,讨论了一类带脉冲效应的线性微分系统有界变差解的稳定性,并建立了有界变差解的变差稳定性和渐近变差稳定性的Lyapunov型定理.%By using the Henstock integral, Lyapunov function and the theory of impulsive differential system , the variational stability of bounded variation solutions for a class of linear differential systems with impulses was discussed, and trie Lyapunov type theorems for variational stability and asymptotically variational stability of bounded variation solutions were established.
The Stability Analysis Method of the Cohesive Granular Slope on the Basis of Graph Theory.
Guan, Yanpeng; Liu, Xiaoli; Wang, Enzhi; Wang, Sijing
2017-02-27
This paper attempted to provide a method to calculate progressive failure of the cohesivefrictional granular geomaterial and the spatial distribution of the stability of the cohesive granular slope. The methodology can be divided into two parts: the characterization method of macro-contact and the analysis of the slope stability. Based on the graph theory, the vertexes, the edges and the edge sequences are abstracted out to characterize the voids, the particle contact and the macro-contact, respectively, bridging the gap between the mesoscopic and macro scales of granular materials. This paper adopts this characterization method to extract a graph from a granular slope and characterize the macro sliding surface, then the weighted graph is analyzed to calculate the slope safety factor. Each edge has three weights representing the sliding moment, the anti-sliding moment and the braking index of contact-bond, respectively, . The safety factor of the slope is calculated by presupposing a certain number of sliding routes and reducing Weight repeatedly and counting the mesoscopic failure of the edge. It is a kind of slope analysis method from mesoscopic perspective so it can present more detail of the mesoscopic property of the granular slope. In the respect of macro scale, the spatial distribution of the stability of the granular slope is in agreement with the theoretical solution.
The Stability Analysis Method of the Cohesive Granular Slope on the Basis of Graph Theory
Directory of Open Access Journals (Sweden)
Yanpeng Guan
2017-02-01
Full Text Available This paper attempted to provide a method to calculate progressive failure of the cohesivefrictional granular geomaterial and the spatial distribution of the stability of the cohesive granular slope. The methodology can be divided into two parts: the characterization method of macro-contact and the analysis of the slope stability. Based on the graph theory, the vertexes, the edges and the edge sequences are abstracted out to characterize the voids, the particle contact and the macro-contact, respectively, bridging the gap between the mesoscopic and macro scales of granular materials. This paper adopts this characterization method to extract a graph from a granular slope and characterize the macro sliding surface, then the weighted graph is analyzed to calculate the slope safety factor. Each edge has three weights representing the sliding moment, the anti-sliding moment and the braking index of contact-bond, respectively, . The safety factor of the slope is calculated by presupposing a certain number of sliding routes and reducing Weight repeatedly and counting the mesoscopic failure of the edge. It is a kind of slope analysis method from mesoscopic perspective so it can present more detail of the mesoscopic property of the granular slope. In the respect of macro scale, the spatial distribution of the stability of the granular slope is in agreement with the theoretical solution.
EVENTUAL STABILITY OF IMPULSIVE DIFFERENTIAL SYSTEMS
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
In this article, criteria of eventual stability are established for impulsive differential systems using piecewise continuous Lyapunov functions. The sufficient conditions that are obtained significantly depend on the moments of impulses. An example is discussed to illustrate the theorem.
Global stabilization of nonlinear systems with uncertain structure
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
The global stabilization problem of nonlinear systems with uncertain structure is dealt with. Based on control Lyapunov function (CLF), a sufficient and necessary condition for Lyapunov stabilization is given. From the condition,several simply sufficient conditions for the globally asymptotical stability are deduced. A state feedback control law is designed to globally asymptotically stabilize the equilibrium of the closed system. Last, a simulation shows the effectiveness of the method.
Dilaton stabilization and composite dark matter in the string frame of heterotic-M-theory
Zanzi, Andrea
2012-01-01
In this paper we further elaborate on our recently proposed solution to the cosmological constant problem - Phys. Rev. D82 (2010) 044006. One of the elements of the solution is the chameleonic behaviour of the Einstein frame dilaton: the mass of the dilaton is an increasing function of the matter density. In that model, a proper structure of the string-frame form factors in the strong coupling region of string theory is assumed. In particular, a stabilizing potential of the string frame dilaton is present and it is supposed to be the result of a quantum calculation. Our main purpose in this article is to point out that the lagrangian of the chameleonic model for the dilaton can be embedded, to a large extent, in heterotic-M-theory. We illustrate some theoretical grounds that support the ansatz about the form factors. In this paper, we break bulk supersymmetry with a massive sterile spinor field (i.e. a bulk neutrino field) and, under certain assumptions about the full M-theory action, we point out the Casimir...
Random matrix theory filters in portfolio optimisation: A stability and risk assessment
Daly, J.; Crane, M.; Ruskin, H. J.
2008-07-01
Random matrix theory (RMT) filters, applied to covariance matrices of financial returns, have recently been shown to offer improvements to the optimisation of stock portfolios. This paper studies the effect of three RMT filters on the realised portfolio risk, and on the stability of the filtered covariance matrix, using bootstrap analysis and out-of-sample testing. We propose an extension to an existing RMT filter, (based on Krzanowski stability), which is observed to reduce risk and increase stability, when compared to other RMT filters tested. We also study a scheme for filtering the covariance matrix directly, as opposed to the standard method of filtering correlation, where the latter is found to lower the realised risk, on average, by up to 6.7%. We consider both equally and exponentially weighted covariance matrices in our analysis, and observe that the overall best method out-of-sample was that of the exponentially weighted covariance, with our Krzanowski stability-based filter applied to the correlation matrix. We also find that the optimal out-of-sample decay factors, for both filtered and unfiltered forecasts, were higher than those suggested by Riskmetrics [J.P. Morgan, Reuters, Riskmetrics technical document, Technical Report, 1996. http://www.riskmetrics.com/techdoc.html], with those for the latter approaching a value of α=1. In conclusion, RMT filtering reduced the realised risk, on average, and in the majority of cases when tested out-of-sample, but increased the realised risk on a marked number of individual days-in some cases more than doubling it.
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.
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.
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.
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.
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.
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...
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.
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...
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.
Stability and bifurcation analysis of a vector-bias model of malaria transmission.
Buonomo, Bruno; Vargas-De-León, Cruz
2013-03-01
The vector-bias model of malaria transmission, recently proposed by Chamchod and Britton, is considered. Nonlinear stability analysis is performed by means of the Lyapunov theory and the LaSalle Invariance Principle. The classical threshold for the basic reproductive number, R(0), is obtained: if R(0)>1, then the disease will spread and persist within its host population. If R(0)1, the endemic persistence of the disease has been proved to hold also for the extended model. This last result is obtained by means of the geometric approach to global stability. Copyright © 2012 Elsevier Inc. All rights reserved.
Nonlinear model predictive control with guaraneed stability based on pesudolinear neural networks
Institute of Scientific and Technical Information of China (English)
WANG Yongji; WANG Hong
2004-01-01
A nonlinear model predictive control problem based on pseudo-linear neural network (PNN) is discussed, in which the second order on-line optimization method is adopted. The recursive computation of Jacobian matrix is investigated. The stability of the closed loop model predictive control system is analyzed based on Lyapunov theory to obtain the sufficient condition for the asymptotical stability of the neural predictive control system. A simulation was carried out for an exothermic first-order reaction in a continuous stirred tank reactor. It is demonstrated that the proposed control strategy is applicable to some of nonlinear systems.
Institute of Scientific and Technical Information of China (English)
阮炯; 王军平; 郭德典
2004-01-01
In this paper, we first introduce the model of discrete-time neural networks with generalized input-output function and present a proof of the existence of a fixed point by Schauder fixed-point principle. Secondly, we study the uniformly asymptotical stability of equilibrium in non-autonomous discrete-time neural networks and give some sufficient conditions that guarantee the stability of it by using the converse theorem of Lyapunov function. Finally, several examples and numerical simulations are given to illustrate and reinforce our theories.
Lyapunov matrices approach to the parametric optimization of time-delay systems
Directory of Open Access Journals (Sweden)
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.
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
Stability Analysis for Stochastic Delayed High-order Neural Networks
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
In this paper, the global asymptotic stability analysis problem is considered for a class of stochastic high-order neural networks with time-delays. Based on a Lyapunov-Krasovskii functional and the stochastic stability analysis theory, several sufficient conditions are derived in order to guarantee the global asymptotic convergence of the equilibrium point in the mean square. Investigation shows that the addressed stochastic highorder delayed neural networks are globally asymptotically stable in the mean square if there are solutions to some linear matrix inequalities (LMIs). Hence, the global asymptotic stability of the studied stochastic high-order delayed neural networks can be easily checked by the Matlab LMI toolbox. A numerical example is given to demonstrate the usefulness of the proposed global stability criteria.
Directory of Open Access Journals (Sweden)
Wu Huaiqin
2009-01-01
Full Text Available This paper considers a new class of additive neural networks where the neuron activations are modelled by discontinuous functions with nonlinear growth. By Leray-Schauder alternative theorem in differential inclusion theory, matrix theory, and generalized Lyapunov approach, a general result is derived which ensures the existence and global asymptotical stability of a unique periodic solution for such neural networks. The obtained results can be applied to neural networks with a broad range of activation functions assuming neither boundedness nor monotonicity, and also show that Forti's conjecture for discontinuous neural networks with nonlinear growth activations is true.
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
The H∞ output feedback control problem for uncertain discrete-time switched systems is reasearched. A new characterization of stability and H∞ performance for the switched system under arbitrary switching is obtained by using switched Lyapunov function.Then,based on the characterization,a linear matrix inequality (LMI)approach is developed to design a switched output feedback controller which guarantees the stability and H∞ performance of the closed-loop system.A numerical example is presented to demonstrate the application of the proposed method.
Analysis of the Evolution of Tannic Acid Stabilized Gold Nanoparticles Using Mie Theory
Directory of Open Access Journals (Sweden)
Assia Rachida Senoudi
2014-01-01
Full Text Available Spherical gold nanoparticles (GNPs have been synthesized in aqueous solutions using sodium citrate (SC and tannic acid (TA as reducing and stabilizing agents. Upon addition of TA and compared to the GNP TA-free aqueous solutions, a reduction of the GNPs size and consequently a dramatic change of their optical properties have been observed and quantitatively analyzed using Mie theory. An increase in the concentration of TA reveals a modification of the colloidal solution refractive index that is evidenced by the shift in the peak position of the localized surface plasmon resonance (LSPR band. The variations of the peak absorbance with the TA concentration are examined in the low and high concentration regimes.
A flux-scaling scenario for high-scale moduli stabilization in string theory
Directory of Open Access Journals (Sweden)
Ralph Blumenhagen
2015-08-01
Full Text Available Tree-level moduli stabilization via geometric and non-geometric fluxes in type IIB orientifolds on Calabi–Yau manifolds is investigated. The focus is on stable non-supersymmetric minima, where all moduli are fixed except for some massless axions. The scenario includes the purely axionic orientifold-odd moduli. A set of vacua allowing for parametric control over the moduli vacuum expectation values and their masses is presented, featuring a specific scaling with the fluxes. Uplift mechanisms and supersymmetry breaking soft masses on MSSM-like D7-branes are discussed as well. This scenario provides a complete effective framework for realizing the idea of F-term axion monodromy inflation in string theory. It is argued that, with all masses close to the Planck and GUT scales, one is confronted with working at the threshold of controlling all mass hierarchies.
The troublesome birth of hydrodynamic stability theory: Sommerfeld and the turbulence problem
Eckert, M.
2010-07-01
More than a hundred years ago William McFadden Orr and Arnold Sommerfeld conceived an approach to account for the transition from laminar to turbulent flow in terms of hydrodynamic stability theory. But the “turbulence problem”, as this challenge became notoriously famous, could not be solved by this method. By 1920, it was widely recognized as an outstanding riddle. Although famous theoretical physicists like Werner Heisenberg dedicated a considerable effort to this problem, the “Orr-Sommerfeld method” has never found the attention of historians of science. This article describes its early perception and development in Germany, and how the “turbulence problem” reached center stage after the First World war as a major challenge for theorists with different perspectives.
A flux-scaling scenario for high-scale moduli stabilization in string theory
Energy Technology Data Exchange (ETDEWEB)
Blumenhagen, Ralph [Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805 München (Germany); Font, Anamaría [Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805 München (Germany); Arnold Sommerfeld Center for Theoretical Physics, LMU, Theresienstr. 37, 80333 München (Germany); Fuchs, Michael [Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805 München (Germany); Herschmann, Daniela, E-mail: herschma@mpp.mpg.de [Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805 München (Germany); Plauschinn, Erik [Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università di Padova, Via Marzolo 8, 35131 Padova (Italy); INFN, Sezione di Padova, Via Marzolo 8, 35131 Padova (Italy); Sekiguchi, Yuta; Wolf, Florian [Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805 München (Germany); Arnold Sommerfeld Center for Theoretical Physics, LMU, Theresienstr. 37, 80333 München (Germany)
2015-08-15
Tree-level moduli stabilization via geometric and non-geometric fluxes in type IIB orientifolds on Calabi–Yau manifolds is investigated. The focus is on stable non-supersymmetric minima, where all moduli are fixed except for some massless axions. The scenario includes the purely axionic orientifold-odd moduli. A set of vacua allowing for parametric control over the moduli vacuum expectation values and their masses is presented, featuring a specific scaling with the fluxes. Uplift mechanisms and supersymmetry breaking soft masses on MSSM-like D7-branes are discussed as well. This scenario provides a complete effective framework for realizing the idea of F-term axion monodromy inflation in string theory. It is argued that, with all masses close to the Planck and GUT scales, one is confronted with working at the threshold of controlling all mass hierarchies.
Cherif, Alhaji; Barley, Kamal
2010-12-29
Quantification of historical sociological processes have recently gained attention among theoreticians in the effort of providing a solid theoretical understanding of the behaviors and regularities present in socio-political dynamics. Here we present a reliability theory of polity processes with emphases on individual political dynamics of African countries. We found that the structural properties of polity failure rates successfully capture the risk of political vulnerability and instabilities in which , , , and of the countries with monotonically increasing, unimodal, U-shaped and monotonically decreasing polity failure rates, respectively, have high level of state fragility indices. The quasi-U-shape relationship between average polity duration and regime types corroborates historical precedents and explains the stability of the autocracies and democracies.
Lane, S; Marsiglio, F; Zhi, Y; Meldrum, A
2015-02-20
Fluorescent-core microcapillaries (FCMs) present a robust basis for the application of optical whispering gallery modes toward refractometric sensing. An important question concerns whether these devices can be rendered insensitive to local temperature fluctuations, which may otherwise limit their refractometric detection limits, mainly as a result of thermorefractive effects. Here, we first use a standard cylindrical cavity formalism to develop the refractometric and thermally limited detection limits for the FCM structure. We then measure the thermal response of a real device with different analytes in the channel and compare the result to the theory. Good stability against temperature fluctuations was obtained for an ethanol solvent, with a near-zero observed thermal shift for the transverse magnetic modes. Similarly good results could in principle be obtained for any other solvent (e.g., water), if the thickness of the fluorescent layer can be sufficiently well controlled.
Well-posedness and stability analysis of hybrid feedback systems using Shkalikov's theory
Directory of Open Access Journals (Sweden)
Piotr Grabowski
2006-01-01
Full Text Available The modern method of analysis of the distributed parameter systems relies on the transformation of the dynamical model to an abstract differential equation on an appropriately chosen Banach or, if possible, Hilbert space. A linear dynamical model in the form of a first order abstract differential equation is considered to be well-posed if its right-hand side generates a strongly continuous semigroup. Similarly, a dynamical model in the form of a second order abstract differential equation is well-posed if its right-hand side generates a strongly continuous cosine family of operators. Unfortunately, the presence of a feedback leads to serious complications or even excludes a direct verification of assumptions of the Hille-Phillips-Yosida and/or the Sova-Fattorini Theorems. The class of operators which are similar to a normal discrete operator on a Hilbert space describes a wide variety of linear operators. In the papers [Grabowski P., Well–posedness and stability analysis of hybrid feedback systems, Journal of Mathematical Systems, Estimation and Control 6 (1996, 121–124 (summary, full electronic manuscript – retrieval code 15844, Grabowski P., Spectral approach to well–posedness and stability analysis of hybrid feedback systems, In: Wajs W., Grabowski P. (Eds., Studies in Automatics, 1996, Kraków, Wydawnictwa AGH, 104–139] two groups of similarity criteria for a given hybrid closed-lop system operator are given. The criteria of the first group are based on some perturbation results, and of the second, on the application of Shkalikov's theory of the Sturm-Liouville eigenproblems with a spectral parameter in the boundary conditions. In the present paper we continue those investigations showing certain advanced applications of the Shkalikov's theory. The results are illustrated by feedback control systems examples governed by wave and beam equations with increasing degree of complexity of the boundary conditions.
Stability of Nonlinear Stochastic Discrete-Time Systems
2013-01-01
This paper studies the stability for nonlinear stochastic discrete-time systems. First of all, several definitions on stability are introduced, such as stability, asymptotical stability, and pth moment exponential stability. Moreover, using the method of the Lyapunov functionals, some efficient criteria for stochastic stability are obtained. Some examples are presented to illustrate the effectiveness of the proposed theoretical results.
Effect of graph generation on slope stability analysis based on graph theory
Institute of Scientific and Technical Information of China (English)
Enpu Li; Xiaoying Zhuang; Wenbo Zheng; Yongchang Cai
2014-01-01
Limit equilibrium method (LEM) and strength reduction method (SRM) are the most widely used methods for slope stability analysis. However, it can be noted that they both have some limitations in practical application. In the LEM, the constitutive model cannot be considered and many assumptions are needed between slices of soil/rock. The SRM requires iterative calculations and does not give the slip surface directly. A method for slope stability analysis based on the graph theory is recently developed to directly calculate the minimum safety factor and potential critical slip surface according to the stress results of numerical simulation. The method is based on current stress state and can overcome the disadvantages mentioned above in the two traditional methods. The influences of edge generation and mesh geometry on the position of slip surface and the safety factor of slope are studied, in which a new method for edge generation is proposed, and reasonable mesh size is suggested. The results of bench-mark examples and a rock slope show good accuracy and efficiency of the presented method.
On the stability conditions for theories of modified gravity coupled to matter fields
De Felice, Antonio; Papadomanolakis, Georgios
2016-01-01
We present a thorough stability analysis of modified gravity theories when the coupling to matter fields is considered. We use the Effective Field Theory framework for Dark Energy and Modified Gravity to retain a general approach for the gravity sector and a Sorkin-Schutz action for the matter one. Then, we work out the proper viability conditions to guarantee in the scalar sector the absence of ghosts, gradient and tachyonic instabilities. The absence of ghosts can be achieved by demanding a positive kinetic matrix, while the lack of a gradient instability is ensured by imposing a positive speed of propagation for all the scalar modes. In case of tachyonic instability, the mass eigenvalues have been studied and we work out the appropriate expressions. For the latter, an instability occurs only when the negative mass eigenvalue is much larger, in absolute value, than the Hubble parameter. We discuss the results for the minimally coupled quintessence model showing for a particular set of parameters two typical...
Mirage Models Confront the LHC: I. Kahler-Stabilized Heterotic String Theory
Kaufman, Bryan L; Gaillard, Mary K
2013-01-01
We begin the study of a class of string-motivated effective supergravity theories in light of current data from the CERN Large Hadron Collider (LHC). The case of heterotic string theory, in which the dilaton is stabilized via non-perturbative corrections to the Kahler metric, will be considered first. This model is highly constrained and therefore predictive. We find that much of the reasonable parameter space afforded to the model -- representing the strong dynamics of a presumed gaugino condensation in the hidden sector -- is now observationally disfavored by the LHC results. Most of the theoretically-motivated parameter space that remains can be probed with data that has already been collected, and most of the remainder will be definitively explored within the first year of operation at center of mass energy of 13 TeV. Expected signatures for a number of benchmark points are discussed. We find that the surviving space of the model makes a precise prediction as to the relation of many superpartner masses, a...
Phase stability in heavy f-electron metals from first-principles theory
Energy Technology Data Exchange (ETDEWEB)
Soderlind, P
2005-11-17
The structural phase stability of heavy f-electron metals is studied by means of density-functional theory (DFT). These include temperature-induced transitions in plutonium metal as well as pressure-induced transitions in the trans-plutonium metals Am, Cm, Bk, and Cf. The early actinides (Th-Np) display phases that could be rather well understood from the competition of a crystal-symmetry breaking mechanism (Peierls distortion) of the 5f states and electrostatic forces, while for the trans-plutonium metals (Am-Cf) the ground-state structures are governed by 6d bonding. We show in this paper that new physics is needed to understand the phases of the actinides in the volume range of about 15-30 {angstrom}{sup 3}. At these volumes one would expect, from theoretical arguments made in the past, to encounter highly complex crystal phases due to a Peierls distortion. Here we argue that the symmetry reduction associated with spin polarization can make higher symmetry phases competitive. Taking this into account, DFT is shown to describe the well-known phase diagram of plutonium and also the recently discovered complex and intriguing high-pressure phase diagrams of Am and Cm. The theory is further applied to investigate the behaviors of Bk and Cf under compression.
Hossienkhani, Hossien
2016-01-01
A spatially homogeneous and anisotropic Bianchi type I universe has been studied with the ghost dark energy (GDE) in the framework of Brans-Dicke theory. For this purpose, we use the squared sound speed $v_s^2$ whose sign determines the stability of the model. At first, we obtain the equation of state parameter, $\\omega_\\Lambda$, the deceleration parameter $q$ and the evolution equation of the ghost dark energy. Then, we extend our study to the case of ghost dark energy in a non-isotropic and Brans-Dicke framework and find out that the transition of $\\omega_\\Lambda$ to the phantom regime can be more easily accounted for than when it is restored into the Einstein field equations. Our numerical result show the effects of the interaction and anisotropic on the evolutionary behaviour the ghost dark energy models. In conclusion, we find evidence that the ghost dark energy in BD theory can lead to a stable universe favored by observations at the present time.
Tanveer, S.
1989-01-01
An asymptotic theory is presented for the determination of velocity and linear stability of a steady symmetric bubble in a Hele-Shaw cell for small surface tension. First the bubble velocity relative to the fluid velocity at infinity is determined for small surface tension by means of a transcendentally small correction to the asymptotic series solution. In addition, a linear stability analysis shows that only the solution branch corresponding to the largest possible bubble velocity for given surface tension is stable, while all the others are unstable.
LYAPUNOV-Based Sensor Failure Detection and Recovery for the Reverse Water Gas Shift Process
Haralambous, Michael G.
2002-01-01
Livingstone, a model-based AI software system, is planned for use in the autonomous fault diagnosis, reconfiguration, and control of the oxygen-producing reverse water gas shift (RWGS) process test-bed located in the Applied Chemistry Laboratory at KSC. In this report the RWGS process is first briefly described and an overview of Livingstone is given. Next, a Lyapunov-based approach for detecting and recovering from sensor failures, differing significantly from that used by Livingstone, is presented. In this new method, models used are in t e m of the defining differential equations of system components, thus differing from the qualitative, static models used by Livingstone. An easily computed scalar inequality constraint, expressed in terms of sensed system variables, is used to determine the existence of sensor failures. In the event of sensor failure, an observer/estimator is used for determining which sensors have failed. The theory underlying the new approach is developed. Finally, a recommendation is made to use the Lyapunov-based approach to complement the capability of Livingstone and to use this combination in the RWGS process.
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...
LYAPUNOV-Based Sensor Failure Detection and Recovery for the Reverse Water Gas Shift Process
Haralambous, Michael G.
2002-01-01
Livingstone, a model-based AI software system, is planned for use in the autonomous fault diagnosis, reconfiguration, and control of the oxygen-producing reverse water gas shift (RWGS) process test-bed located in the Applied Chemistry Laboratory at KSC. In this report the RWGS process is first briefly described and an overview of Livingstone is given. Next, a Lyapunov-based approach for detecting and recovering from sensor failures, differing significantly from that used by Livingstone, is presented. In this new method, models used are in t e m of the defining differential equations of system components, thus differing from the qualitative, static models used by Livingstone. An easily computed scalar inequality constraint, expressed in terms of sensed system variables, is used to determine the existence of sensor failures. In the event of sensor failure, an observer/estimator is used for determining which sensors have failed. The theory underlying the new approach is developed. Finally, a recommendation is made to use the Lyapunov-based approach to complement the capability of Livingstone and to use this combination in the RWGS process.
Stochastic stability of mechanical systems under renewal jump process parametric excitation
DEFF Research Database (Denmark)
Iwankiewicz, R.; Nielsen, Søren R.K.; Larsen, Jesper Winther
2005-01-01
if the state space is augmented by the products of the original state variables and the excitation variable. Asymptotic mean and mean-square stability as well as asymptotic sample (Lyapunov) stability with probability 1 are investigated. The Lyapunov exponents have been evaluated both by the direct simulation...
STABILITY OF SOLUTIONS TO CERTAIN FOURTH-ORDER DELAY DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
By the Lyapunov functional approach, some better results on the asymptotic stabiBy the Lyapunov functional approach, some better results on the asymptotic stability and global asymptotic stability of zero solution to a certain fourth-order non-linear differential equation with delay are obtained.
Institute of Scientific and Technical Information of China (English)
Xianming ZHANG; Min WU; Jinhua SHE; Dongsheng HAN
2007-01-01
This paper addresses the problems of the robust stability and robust stabilization of a discrete-time system with polytopic uncertainties.A new and simple method is presented to directly decouple the Lyapunov matrix and the system dynamic matrix.Combining this method with the parameter-dependent Lyapunov function approach yields new criteria that include some existing ones as special cases.A numerical example illustrates the improvement over the existing ones.
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.
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.
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.
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.
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...
Improved Stabilization Criteria for Neutral Time-Delay Systems
Lianglin Xiong; Haiyang Zhang; Yongkun Li; Zixin Liu
2016-01-01
This paper addresses the stabilization conditions for neutral systems with mixed time delays. By constructing a novel class of Lyapunov functionals which contains an augmented Lyapunov functional, using a new class of improved Jensen’s like inequalities, two improved delay-dependent stability criteria are firstly established. Next, state feedback controllers are designed according to the stability conditions in different cases. Finally, five numerical examples are provided to demonstrate the ...
Tian, Li-Ping; Shi, Zhong-Ke; Liu, Li-Zhi; Wu, Fang-Xiang
2013-10-01
Stability is essential for designing and controlling any dynamic systems. Recently, the stability of genetic regulatory networks has been widely studied by employing linear matrix inequality (LMI) approach, which results in checking the existence of feasible solutions to high-dimensional LMIs. In the previous study, the authors present several stability conditions for genetic regulatory networks with time-varying delays, based on M-matrix theory and using the non-smooth Lyapunov function, which results in determining whether a low-dimensional matrix is a non-singular M-matrix. However, the previous approach cannot be applied to analyse the stability of genetic regulatory networks with noise perturbations. Here, the authors design a smooth Lyapunov function quadratic in state variables and employ M-matrix theory to derive new stability conditions for genetic regulatory networks with time-varying delays. Theoretically, these conditions are less conservative than existing ones in some genetic regulatory networks. Then the results are extended to genetic regulatory networks with time-varying delays and noise perturbations. For genetic regulatory networks with n genes and n proteins, the derived conditions are to check if an n × n matrix is a non-singular M-matrix. To further present the new theories proposed in this study, three example regulatory networks are analysed.
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.
Global Stability and Hopf Bifurcation of a Predator-Prey Model with Time Delay and Stage Structure
Directory of Open Access Journals (Sweden)
Lingshu Wang
2014-01-01
Full Text Available A delayed predator-prey system with Holling type II functional response and stage structure for both the predator and the prey is investigated. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the system is addressed and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By means of persistence theory on infinite dimensional systems, it is proved that the system is permanent. By using Lyapunov functions and the LaSalle invariant principle, the global stability of each of the feasible equilibria of the model is discussed. Numerical simulations are carried out to illustrate the main theoretical results.
Directory of Open Access Journals (Sweden)
Yingwei Li
2013-01-01
Full Text Available The global exponential stability issues are considered for almost periodic solution of the neural networks with mixed time-varying delays and discontinuous neuron activations. Some sufficient conditions for the existence, uniqueness, and global exponential stability of almost periodic solution are achieved in terms of certain linear matrix inequalities (LMIs, by applying differential inclusions theory, matrix inequality analysis technique, and generalized Lyapunov functional approach. In addition, the existence and asymptotically almost periodic behavior of the solution of the neural networks are also investigated under the framework of the solution in the sense of Filippov. Two simulation examples are given to illustrate the validity of the theoretical results.
Institute of Scientific and Technical Information of China (English)
Shu-jun Liu; Ji-feng Zhang; Zhong-ping Jiang
2008-01-01
In this paper, the property of practical input-to-state stability and its application to stability of cascaded nonlinear systems are investigated in the stochastic framework. Firstly, the notion of (practical)stochastic input-to-state stability with respect to a stochastic input is introduced, and then by the method of changing supply functions, (a) an (practical) SISS-Lyapunov function for the overall system is obtained from the corresponding Lyapunov functions for cascaded (practical) SISS subsystems.
Offshore wellbore stability analysis based on fully coupled poro-thermo-elastic theory
Cao, Wenke; Deng, Jingen; Yu, Baohua; Liu, Wei; Tan, Qiang
2017-03-01
Drilling-induced tensile fractures are usually caused when the weight of mud is too high, and the effective tangential stress becomes tensile. It is thus hard to explain why tensile fractures are distributed along the lower part of a hole in an offshore exploration well when the mud weight is low. According to analysis, the reason could be the thermal effect, which cannot be ignored because of the drilling fluid and the cooling action of sea water during circulation. A heat transfer model is set up to obtain the temperature distribution of the wellbore and its formation by the finite difference method. Then, fully coupled poro-thermo-elastic theory is used to study the pore pressure and effective stress around the wellbore. By comparing it with both poroelastic and elastic models, it is indicated that the poroelastic effect is dominant at the beginning of circulation and inhibits tensile fractures from forming; then, the thermal effect becomes more important and decreases the effective tangential stress with the passing of time, so the drilling fluid and the cooling effect of sea water can cause tensile fractures to happen. Meanwhile, tensile fractures are shallow and not likely to lead to mud leakage with lower mud weight, which agrees with the actual drilling process. On the other hand, the fluid cooling effect could increase the strength of the rock and reduce the likelihood of shear failure, which would be beneficial for wellbore stability. So, the thermal effect cannot be neglected in offshore wellbore stability analysis, and mud weight and borehole exposure time should be controlled in the case of mud loss.
Sakamoto, Noboru; Schaft, Arjan J. van der
2007-01-01
In this paper, an analytical approximation approach for the stabilizing solution of the Hamilton-Jacobi equation using stable manifold theory is proposed. The proposed method gives approximated flows on the stable manifold of the associated Hamiltonian system and provides approximations of the
Sakamoto, Noboru; Schaft, Arjan J. van der
2007-01-01
In this paper, an analytical approximation approach for the stabilizing solution of the Hamilton-Jacobi equation using stable manifold theory is proposed. The proposed method gives approximated flows on the stable manifold of the associated Hamiltonian system and provides approximations of the stabl
Lettow, B. van; Vries, H. de; Burdorf, A.; Conner, M.; Empelen, P. van
2014-01-01
Objectives: Prototypes (i.e., social images) predict health-related behaviours and intentions within the context of the Theory of Planned Behaviour (TPB). This study tested the moderating role of temporal stability of drinker prototype perceptions on prototype-intentions and prototype-behaviour rela
A Lyapunov-Based Extension to Particle Swarm Dynamics for Continuous Function Optimization
Bhattacharya, Sayantani; Konar, Amit; Das, Swagatam; Han, Sang Yong
2009-01-01
The paper proposes three alternative extensions to the classical global-best particle swarm optimization dynamics, and compares their relative performance with the standard particle swarm algorithm. The first extension, which readily follows from the well-known Lyapunov's stability theorem, provides a mathematical basis of the particle dynamics with a guaranteed convergence at an optimum. The inclusion of local and global attractors to this dynamics leads to faster convergence speed and better accuracy than the classical one. The second extension augments the velocity adaptation equation by a negative randomly weighted positional term of individual particle, while the third extension considers the negative positional term in place of the inertial term. Computer simulations further reveal that the last two extensions outperform both the classical and the first extension in terms of convergence speed and accuracy. PMID:22303158
Detection of the onset of numerical chaotic instabilities by lyapunov exponents
Directory of Open Access Journals (Sweden)
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.