Transition to collective oscillations in finite Kuramoto ensembles

Science.gov (United States)

Peter, Franziska; Pikovsky, Arkady

2018-03-01

We present an alternative approach to finite-size effects around the synchronization transition in the standard Kuramoto model. Our main focus lies on the conditions under which a collective oscillatory mode is well defined. For this purpose, the minimal value of the amplitude of the complex Kuramoto order parameter appears as a proper indicator. The dependence of this minimum on coupling strength varies due to sampling variations and correlates with the sample kurtosis of the natural frequency distribution. The skewness of the frequency sample determines the frequency of the resulting collective mode. The effects of kurtosis and skewness hold in the thermodynamic limit of infinite ensembles. We prove this by integrating a self-consistency equation for the complex Kuramoto order parameter for two families of distributions with controlled kurtosis and skewness, respectively.

Frustration tuning and perfect phase synchronization in the Kuramoto-Sakaguchi model.

Science.gov (United States)

Brede, Markus; Kalloniatis, Alexander C

2016-06-01

We present an analysis of conditions under which the dynamics of a frustrated Kuramoto-or Kuramoto-Sakaguchi-model on sparse networks can be tuned to enhance synchronization. Using numerical optimization techniques, linear stability, and dimensional reduction analysis, a simple tuning scheme for setting node-specific frustration parameters as functions of native frequencies and degrees is developed. Finite-size scaling analysis reveals that even partial application of the tuning rule can significantly reduce the critical coupling for the onset of synchronization. In the second part of the paper, a codynamics is proposed, which allows a dynamic tuning of frustration parameters simultaneously with the ordinary Kuramoto dynamics. We find that such codynamics enhance synchronization when operating on slow time scales, and impede synchronization when operating on fast time scales relative to the Kuramoto dynamics.

Hidden physics models: Machine learning of nonlinear partial differential equations

Science.gov (United States)

Raissi, Maziar; Karniadakis, George Em

2018-03-01

While there is currently a lot of enthusiasm about "big data", useful data is usually "small" and expensive to acquire. In this paper, we present a new paradigm of learning partial differential equations from small data. In particular, we introduce hidden physics models, which are essentially data-efficient learning machines capable of leveraging the underlying laws of physics, expressed by time dependent and nonlinear partial differential equations, to extract patterns from high-dimensional data generated from experiments. The proposed methodology may be applied to the problem of learning, system identification, or data-driven discovery of partial differential equations. Our framework relies on Gaussian processes, a powerful tool for probabilistic inference over functions, that enables us to strike a balance between model complexity and data fitting. The effectiveness of the proposed approach is demonstrated through a variety of canonical problems, spanning a number of scientific domains, including the Navier-Stokes, Schrödinger, Kuramoto-Sivashinsky, and time dependent linear fractional equations. The methodology provides a promising new direction for harnessing the long-standing developments of classical methods in applied mathematics and mathematical physics to design learning machines with the ability to operate in complex domains without requiring large quantities of data.

Kuramoto model for infinite graphs with kernels

KAUST Repository

Canale, Eduardo

2015-01-07

In this paper we study the Kuramoto model of weakly coupled oscillators for the case of non trivial network with large number of nodes. We approximate of such configurations by a McKean-Vlasov stochastic differential equation based on infinite graph. We focus on circulant graphs which have enough symmetries to make the computations easier. We then focus on the asymptotic regime where an integro-partial differential equation is derived. Numerical analysis and convergence proofs of the Fokker-Planck-Kolmogorov equation are conducted. Finally, we provide numerical examples that illustrate the convergence of our method.

Continuum modeling of ion-beam eroded surfaces under normal incidence: Impact of stochastic fluctuations

International Nuclear Information System (INIS)

Dreimann, Karsten; Linz, Stefan J.

2010-01-01

Graphical abstract: Deterministic surface pattern (left) and its stochastic counterpart (right) arising in a stochastic damped Kuramoto-Sivashinsky equation that serves as a model equation for ion-beam eroded surfaces and is systematically investigated. - Abstract: Using a recently proposed field equation for the surface evolution of ion-beam eroded semiconductor target materials under normal incidence, we systematically explore the impact of additive stochastic fluctuations that are permanently present during the erosion process. Specifically, we investigate the dependence of the surface roughness, the underlying pattern forming properties and the bifurcation behavior on the strength of the fluctuations.

Lattice Boltzmann model for high-order nonlinear partial differential equations.

Science.gov (United States)

Chai, Zhenhua; He, Nanzhong; Guo, Zhaoli; Shi, Baochang

2018-01-01

In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂_{t}ϕ+∑_{k=1}^{m}α_{k}∂_{x}^{k}Π_{k}(ϕ)=0 (1≤k≤m≤6), α_{k} are constant coefficients, Π_{k}(ϕ) are some known differential functions of ϕ. As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K(n,n)-Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009)1672-179910.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009)PHYADX0378-437110.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.

Lattice Boltzmann model for high-order nonlinear partial differential equations

Science.gov (United States)

Chai, Zhenhua; He, Nanzhong; Guo, Zhaoli; Shi, Baochang

2018-01-01

In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂tϕ +∑k=1mαk∂xkΠk(ϕ ) =0 (1 ≤k ≤m ≤6 ), αk are constant coefficients, Πk(ϕ ) are some known differential functions of ϕ . As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K (n ,n ) -Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009), 10.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009), 10.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.

Hyperbolic geometry of Kuramoto oscillator networks

Science.gov (United States)

Chen, Bolun; Engelbrecht, Jan R.; Mirollo, Renato

2017-09-01

Kuramoto oscillator networks have the special property that their trajectories are constrained to lie on the (at most) 3D orbits of the Möbius group acting on the state space T N (the N-fold torus). This result has been used to explain the existence of the N-3 constants of motion discovered by Watanabe and Strogatz for Kuramoto oscillator networks. In this work we investigate geometric consequences of this Möbius group action. The dynamics of Kuramoto phase models can be further reduced to 2D reduced group orbits, which have a natural geometry equivalent to the unit disk \

Falling films on flexible inclines

Science.gov (United States)

Matar, O. K.; Craster, R. V.; Kumar, S.

2007-11-01

The nonlinear stability and dynamic behavior of falling fluid films is studied for flow over a flexible substrate. We use asymptotic methods to deduce governing equations valid in various limits. Long-wave theory is used to derive Benney-like coupled equations for the film thickness and substrate deflection. Weakly nonlinear equations are then derived from these equations that, in the limit of large wall damping and/or large wall tension, reduce to the Kuramoto-Sivashinsky equation. These models break down when inertia becomes more significant, so we also use a long-wave approximation in conjunction with integral theory to derive three strongly coupled nonlinear evolution equations for the film thickness, substrate deflection, and film volumetric flow rate valid at higher Reynolds numbers. These equations, accounting for inertia, capillary, viscous, wall tension, and damping effects, are solved over a wide range of parameters. Our results suggest that decreasing wall damping and/or wall tension can promote the development of chaos in the weakly nonlinear regime and lead to severe substrate deformations in the strongly nonlinear regime; these can give rise to situations in which the free surface and underlying substrate come into contact in finite time.

Spatio-temporal dynamics of an active, polar, viscoelastic ring.

Science.gov (United States)

Marcq, Philippe

2014-04-01

Constitutive equations for a one-dimensional, active, polar, viscoelastic liquid are derived by treating the strain field as a slow hydrodynamic variable. Taking into account the couplings between strain and polarity allowed by symmetry, the hydrodynamics of an active, polar, viscoelastic body include an evolution equation for the polarity field that generalizes the damped Kuramoto-Sivashinsky equation. Beyond thresholds of the active coupling coefficients between the polarity and the stress or the strain rate, bifurcations of the homogeneous state lead first to stationary waves, then to propagating waves of the strain, stress and polarity fields. I argue that these results are relevant to living matter, and may explain rotating actomyosin rings in cells and mechanical waves in epithelial cell monolayers.

The Kuramoto model in complex networks

Science.gov (United States)

Rodrigues, Francisco A.; Peron, Thomas K. DM.; Ji, Peng; Kurths, Jürgen

2016-01-01

Synchronization of an ensemble of oscillators is an emergent phenomenon present in several complex systems, ranging from social and physical to biological and technological systems. The most successful approach to describe how coherent behavior emerges in these complex systems is given by the paradigmatic Kuramoto model. This model has been traditionally studied in complete graphs. However, besides being intrinsically dynamical, complex systems present very heterogeneous structure, which can be represented as complex networks. This report is dedicated to review main contributions in the field of synchronization in networks of Kuramoto oscillators. In particular, we provide an overview of the impact of network patterns on the local and global dynamics of coupled phase oscillators. We cover many relevant topics, which encompass a description of the most used analytical approaches and the analysis of several numerical results. Furthermore, we discuss recent developments on variations of the Kuramoto model in networks, including the presence of noise and inertia. The rich potential for applications is discussed for special fields in engineering, neuroscience, physics and Earth science. Finally, we conclude by discussing problems that remain open after the last decade of intensive research on the Kuramoto model and point out some promising directions for future research.

PHASE CHAOS IN THE DISCRETE KURAMOTO MODEL

DEFF Research Database (Denmark)

Maistrenko, V.; Vasylenko, A.; Maistrenko, Y.

2010-01-01

The paper describes the appearance of a novel, high-dimensional chaotic regime, called phase chaos, in a time-discrete Kuramoto model of globally coupled phase oscillators. This type of chaos is observed at small and intermediate values of the coupling strength. It arises from the nonlinear...... interaction among the oscillators, while the individual oscillators behave periodically when left uncoupled. For the four-dimensional time-discrete Kuramoto model, we outline the region of phase chaos in the parameter plane and determine the regions where phase chaos coexists with different periodic...

Synchronisation under shocks: The Lévy Kuramoto model

Science.gov (United States)

Roberts, Dale; Kalloniatis, Alexander C.

2018-04-01

We study the Kuramoto model of identical oscillators on Erdős-Rényi (ER) and Barabasi-Alberts (BA) scale free networks examining the dynamics when perturbed by a Lévy noise. Lévy noise exhibits heavier tails than Gaussian while allowing for their tempering in a controlled manner. This allows us to understand how 'shocks' influence individual oscillator and collective system behaviour of a paradigmatic complex system. Skewed α-stable Lévy noise, equivalent to fractional diffusion perturbations, are considered, but overlaid by exponential tempering of rate λ. In an earlier paper we found that synchrony takes a variety of forms for identical Kuramoto oscillators subject to stable Lévy noise, not seen for the Gaussian case, and changing with α: a noise-induced drift, a smooth α dependence of the point of cross-over of synchronisation point of ER and BA networks, and a severe loss of synchronisation at low values of α. In the presence of tempering we observe both analytically and numerically a dramatic change to the α synchronisation is sustained over a larger range of values of the 'noise strength' σ, improved compared to the α > 1 tempered cases. Analytically we study the system close to the phase synchronised fixed point and solve the tempered fractional Fokker-Planck equation. There we observe that densities show stronger support in the basin of attraction at low α for fixed coupling, σ and tempering λ. We then perform numerical simulations for networks of size N = 1000 and average degree d ¯ = 10. There, we compute the order parameter r as a function of σ for fixed α and λ and observe values of r ≈ 1 over larger ranges of σ for α synchronisation down to low values of α. We propose a mechanism for this in terms of the basic shape of the tempered stable Lévy densities for various α and how it feeds into Kuramoto oscillator dynamics and illustrate this with examples of specific paths.

Stochastic Kuramoto oscillators with discrete phase states

Science.gov (United States)

Jörg, David J.

2017-09-01

We present a generalization of the Kuramoto phase oscillator model in which phases advance in discrete phase increments through Poisson processes, rendering both intrinsic oscillations and coupling inherently stochastic. We study the effects of phase discretization on the synchronization and precision properties of the coupled system both analytically and numerically. Remarkably, many key observables such as the steady-state synchrony and the quality of oscillations show distinct extrema while converging to the classical Kuramoto model in the limit of a continuous phase. The phase-discretized model provides a general framework for coupled oscillations in a Markov chain setting.

Stochastic Kuramoto oscillators with discrete phase states.

Science.gov (United States)

Jörg, David J

2017-09-01

We present a generalization of the Kuramoto phase oscillator model in which phases advance in discrete phase increments through Poisson processes, rendering both intrinsic oscillations and coupling inherently stochastic. We study the effects of phase discretization on the synchronization and precision properties of the coupled system both analytically and numerically. Remarkably, many key observables such as the steady-state synchrony and the quality of oscillations show distinct extrema while converging to the classical Kuramoto model in the limit of a continuous phase. The phase-discretized model provides a general framework for coupled oscillations in a Markov chain setting.

Connecting the Kuramoto Model and the Chimera State

Science.gov (United States)

Kotwal, Tejas; Jiang, Xin; Abrams, Daniel M.

2017-12-01

Since its discovery in 2002, the chimera state has frequently been described as a counterintuitive, puzzling phenomenon. The Kuramoto model, in contrast, has become a celebrated paradigm useful for understanding a range of phenomena related to phase transitions, synchronization, and network effects. Here we show that the chimera state can be understood as emerging naturally through a symmetry-breaking bifurcation from the Kuramoto model's partially synchronized state. Our analysis sheds light on recent observations of chimera states in laser arrays, chemical oscillators, and mechanical pendula.

Phase Chaos and Multistability in the Discrete Kuramoto Model

DEFF Research Database (Denmark)

Maistrenko, V. L.; Vasylenko, A. A.; Maistrenko, Y. L.

2008-01-01

The paper describes the appearance of a novel high-dimensional chaotic regime, called phase chaos, in the discrete Kuramoto model of globally coupled phase oscillators. This type of chaos is observed at small and intermediate values of the coupling strength. It is caused by the nonlinear interact......The paper describes the appearance of a novel high-dimensional chaotic regime, called phase chaos, in the discrete Kuramoto model of globally coupled phase oscillators. This type of chaos is observed at small and intermediate values of the coupling strength. It is caused by the nonlinear...... interaction of the oscillators, while the individual oscillators behave periodically when left uncoupled. For the four-dimensional discrete Kuramoto model, we outline the region of phase chaos in the parameter plane, distinguish the region where the phase chaos coexists with other periodic attractors...

Synchrony-optimized networks of non-identical Kuramoto oscillators

International Nuclear Information System (INIS)

Brede, Markus

2008-01-01

In this Letter we discuss a method for generating synchrony-optimized coupling architectures of Kuramoto oscillators with a heterogeneous distribution of native frequencies. The method allows us to relate the properties of the coupling network to its synchronizability. These relations were previously only established from a linear stability analysis of the identical oscillator case. We further demonstrate that the heterogeneity in the oscillator population produces heterogeneity in the optimal coupling network as well. Two rules for enhancing the synchronizability of a given network by a suitable placement of oscillators are given: (i) native frequencies of adjacent oscillators must be anti-correlated and (ii) frequency magnitudes should positively correlate with the degree of the node they are placed at

Exhaustive Classification of the Invariant Solutions for a Specific Nonlinear Model Describing Near Planar and Marginally Long-Wave Unstable Interfaces for Phase Transition

Science.gov (United States)

Ahangari, Fatemeh

2018-05-01

Problems of thermodynamic phase transition originate inherently in solidification, combustion and various other significant fields. If the transition region among two locally stable phases is adequately narrow, the dynamics can be modeled by an interface motion. This paper is devoted to exhaustive analysis of the invariant solutions for a modified Kuramoto-Sivashinsky equation in two spatial and one temporal dimensions is presented. This nonlinear partial differential equation asymptotically characterizes near planar interfaces, which are marginally long-wave unstable. For this purpose, by applying the classical symmetry method for this model the classical symmetry operators are attained. Moreover, the structure of the Lie algebra of symmetries is discussed and the optimal system of subalgebras, which yields the preliminary classification of group invariant solutions is constructed. Mainly, the Lie invariants corresponding to the infinitesimal symmetry generators as well as associated similarity reduced equations are also pointed out. Furthermore, the nonclassical symmetries of this nonlinear PDE are also comprehensively investigated.

Analysis of a power grid using a Kuramoto-like model

DEFF Research Database (Denmark)

Filatrella, G.; Nielsen, Arne Hejde; Pedersen, Niels Falsig

2008-01-01

We show that there is a link between the Kuramoto paradigm and another system of synchronized oscillators, namely an electrical power distribution grid of generators and consumers. The purpose of this work is to show both the formal analogy and some practical consequences. The mapping can be made...... quantitative, and under some necessary approximations a class of Kuramoto-like models, those with bimodal distribution of the frequencies, is most appropriate for the power-grid. In fact in the power-grid there are two kinds of oscillators: the “sources" delivering power to the “consumers"....

Explosive synchronization coexists with classical synchronization in the Kuramoto model

Energy Technology Data Exchange (ETDEWEB)

Danziger, Michael M., E-mail: michael.danziger@biu.ac.il; Havlin, Shlomo [Department of Physics, Bar-Ilan University, Ramat Gan (Israel); Moskalenko, Olga I.; Kurkin, Semen A. [Faculty of Nonlinear Processes, Saratov State University, Astrakhanskaya, 83, Saratov 410012 (Russian Federation); Saratov State Technical University, Politehnicheskaya, 77, Saratov 410054 (Russian Federation); Zhang, Xiyun [Department of Physics, East China Normal University, Shanghai 200062 (China); Boccaletti, Stefano [CNR-Institute of Complex Systems, Via Madonna del Piano 10, 50019 Sesto Fiorentino, Florence (Italy); The Italian Embassy in Israel, 25 Hamered Street, 68125 Tel Aviv (Israel)

2016-06-15

Explosive synchronization has recently been reported in a system of adaptively coupled Kuramoto oscillators, without any conditions on the frequency or degree of the nodes. Here, we find that, in fact, the explosive phase coexists with the standard phase of the Kuramoto oscillators. We determine this by extending the mean-field theory of adaptively coupled oscillators with full coupling to the case with partial coupling of a fraction f. This analysis shows that a metastable region exists for all finite values of f > 0, and therefore explosive synchronization is expected for any perturbation of adaptively coupling added to the standard Kuramoto model. We verify this theory with GPU-accelerated simulations on very large networks (N ∼ 10{sup 6}) and find that, in fact, an explosive transition with hysteresis is observed for all finite couplings. By demonstrating that explosive transitions coexist with standard transitions in the limit of f → 0, we show that this behavior is far more likely to occur naturally than was previously believed.

The WS transform for the Kuramoto model with distributed amplitudes, phase lag and time delay

Science.gov (United States)

Lohe, M. A.

2017-12-01

We apply the Watanabe-Strogatz (WS) transform to a generalized Kuramoto model with distributed parameters describing the amplitude of oscillation, phase lag, and time delay at each node of the system. The model has global coupling and identical frequencies, but allows for repulsive interactions at arbitrary nodes leading to conformist-contrarian phenomena together with variable amplitude and time-delay effects. We show how to determine the initial values of the WS system for any initial conditions for the Kuramoto system, and investigate the asymptotic behaviour of the WS variables. For the case of zero time delay the possible asymptotic configurations are determined by the sign of a single parameter μ which measures whether or not the attractive nodes dominate the repulsive nodes. If μ>0 the system completely synchronizes from general initial conditions, whereas if μ<0 one of two types of phase-locked synchronization occurs, depending on the initial values, while for μ=0 periodic solutions can occur. For the case of arbitrary non-uniform time delays we derive a stability condition for completely synchronized solutions.

Dependence of synchronization frequency of Kuramoto oscillators ...

Indian Academy of Sciences (India)

Kuramoto oscillators have been proposed earlier as a model for interacting systems that exhibit synchronization. In this article, we study the difference between networks with symmetric and asymmetric distribution of natural frequencies. We first indicate that synchronization frequency of oscillators in a completely connected ...

Generalized coupling in the Kuramoto model

DEFF Research Database (Denmark)

Filatrella, G.; Pedersen, Niels Falsig; Wiesenfeld, K.

2007-01-01

We propose a modification of the Kuramoto model to account for the effective change in the coupling constant among the oscillators, as suggested by some experiments on Josephson junction, laser arrays, and mechanical systems, where the active elements are turned on one by one. The resulting model...... with the behavior of Josephson junctions coupled via a cavity....

An approach to normal forms of Kuramoto model with distributed delays and the effect of minimal delay

Energy Technology Data Exchange (ETDEWEB)

Niu, Ben, E-mail: niubenhit@163.com [Department of Mathematics, Harbin Institute of Technology, Weihai 264209 (China); Guo, Yuxiao [Department of Mathematics, Harbin Institute of Technology, Weihai 264209 (China); Jiang, Weihua [Department of Mathematics, Harbin Institute of Technology, Harbin 150001 (China)

2015-09-25

Heterogeneous delays with positive lower bound (gap) are taken into consideration in Kuramoto model. On the Ott–Antonsen's manifold, the dynamical transitional behavior from incoherence to coherence is mediated by Hopf bifurcation. We establish a perturbation technique on complex domain, by which universal normal forms, stability and criticality of the Hopf bifurcation are obtained. Theoretically, a hysteresis loop is found near the subcritically bifurcated coherent state. With respect to Gamma distributed delay with fixed mean and variance, we find that the large gap decreases Hopf bifurcation value, induces supercritical bifurcations, avoids the hysteresis loop and significantly increases in the number of coexisting coherent states. The effect of gap is finally interpreted from the viewpoint of excess kurtosis of Gamma distribution. - Highlights: • Heterogeneously delay-coupled Kuramoto model with minimal delay is considered. • Perturbation technique on complex domain is established for bifurcation analysis. • Hysteresis phenomenon is investigated in a theoretical way. • The effect of excess kurtosis of distributed delays is discussed.

Possible influence of the Kuramoto length in a photo-catalytic water splitting reaction revealed by Poisson-Nernst-Planck equations involving ionization in a weak electrolyte

Science.gov (United States)

Suzuki, Yohichi; Seki, Kazuhiko

2018-03-01

We studied ion concentration profiles and the charge density gradient caused by electrode reactions in weak electrolytes by using the Poisson-Nernst-Planck equations without assuming charge neutrality. In weak electrolytes, only a small fraction of molecules is ionized in bulk. Ion concentration profiles depend on not only ion transport but also the ionization of molecules. We considered the ionization of molecules and ion association in weak electrolytes and obtained analytical expressions for ion densities, electrostatic potential profiles, and ion currents. We found the case that the total ion density gradient was given by the Kuramoto length which characterized the distance over which an ion diffuses before association. The charge density gradient is characterized by the Debye length for 1:1 weak electrolytes. We discuss the role of these length scales for efficient water splitting reactions using photo-electrocatalytic electrodes.

Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks.

Science.gov (United States)

Vlachas, Pantelis R; Byeon, Wonmin; Wan, Zhong Y; Sapsis, Themistoklis P; Koumoutsakos, Petros

2018-05-01

We introduce a data-driven forecasting method for high-dimensional chaotic systems using long short-term memory (LSTM) recurrent neural networks. The proposed LSTM neural networks perform inference of high-dimensional dynamical systems in their reduced order space and are shown to be an effective set of nonlinear approximators of their attractor. We demonstrate the forecasting performance of the LSTM and compare it with Gaussian processes (GPs) in time series obtained from the Lorenz 96 system, the Kuramoto-Sivashinsky equation and a prototype climate model. The LSTM networks outperform the GPs in short-term forecasting accuracy in all applications considered. A hybrid architecture, extending the LSTM with a mean stochastic model (MSM-LSTM), is proposed to ensure convergence to the invariant measure. This novel hybrid method is fully data-driven and extends the forecasting capabilities of LSTM networks.

Lyapunov functionals and stability of stochastic functional differential equations

CERN Document Server

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...

Controllability and stabilization of parabolic equations

CERN Document Server

Barbu, Viorel

2018-01-01

This monograph presents controllability and stabilization methods in control theory that solve parabolic boundary value problems. Starting from foundational questions on Carleman inequalities for linear parabolic equations, the author addresses the controllability of parabolic equations on a variety of domains and the spectral decomposition technique for representing them. This method is, in fact, designed for use in a wider class of parabolic systems that include the heat and diffusion equations. Later chapters develop another process that employs stabilizing feedback controllers with a finite number of unstable modes, with special attention given to its use in the boundary stabilization of Navier–Stokes equations for the motion of viscous fluid. In turn, these applied methods are used to explore related topics like the exact controllability of stochastic parabolic equations with linear multiplicative noise. Intended for graduate students and researchers working on control problems involving nonlinear diff...

Finding Non-Zero Stable Fixed Points of the Weighted Kuramoto model is NP-hard

OpenAIRE

Taylor, Richard

2015-01-01

The Kuramoto model when considered over the full space of phase angles [$0,2\\pi$) can have multiple stable fixed points which form basins of attraction in the solution space. In this paper we illustrate the fundamentally complex relationship between the network topology and the solution space by showing that determining the possibility of multiple stable fixed points from the network topology is NP-hard for the weighted Kuramoto Model. In the case of the unweighted model this problem is shown...

Geometry in a dynamical system without space: Hyperbolic Geometry in Kuramoto Oscillator Systems

Science.gov (United States)

Engelbrecht, Jan; Chen, Bolun; Mirollo, Renato

Kuramoto oscillator networks have the special property that their time evolution is constrained to lie on 3D orbits of the Möbius group acting on the N-fold torus TN which explains the N - 3 constants of motion discovered by Watanabe and Strogatz. The dynamics for phase models can be further reduced to 2D invariant sets in T N - 1 which have a natural geometry equivalent to the unit disk Δ with hyperbolic metric. We show that the classic Kuramoto model with order parameter Z1 (the first moment of the oscillator configuration) is a gradient flow in this metric with a unique fixed point on each generic 2D invariant set, corresponding to the hyperbolic barycenter of an oscillator configuration. This gradient property makes the dynamics especially easy to analyze. We exhibit several new families of Kuramoto oscillator models which reduce to gradient flows in this metric; some of these have a richer fixed point structure including non-hyperbolic fixed points associated with fixed point bifurcations. Work Supported by NSF DMS 1413020.

Orbital stability of solitary waves for Kundu equation

Science.gov (United States)

Zhang, Weiguo; Qin, Yinghao; Zhao, Yan; Guo, Boling

In this paper, we consider the Kundu equation which is not a standard Hamiltonian system. The abstract orbital stability theory proposed by Grillakis et al. (1987, 1990) cannot be applied directly to study orbital stability of solitary waves for this equation. Motivated by the idea of Guo and Wu (1995), we construct three invariants of motion and use detailed spectral analysis to obtain orbital stability of solitary waves for Kundu equation. Since Kundu equation is more complex than the derivative Schrödinger equation, we utilize some techniques to overcome some difficulties in this paper. It should be pointed out that the results obtained in this paper are more general than those obtained by Guo and Wu (1995). We present a sufficient condition under which solitary waves are orbitally stable for 2c+sυ1995) only considered the case 2c+sυ>0. We obtain the results on orbital stability of solitary waves for the derivative Schrödinger equation given by Colin and Ohta (2006) as a corollary in this paper. Furthermore, we obtain orbital stability of solitary waves for Chen-Lee-Lin equation and Gerdjikov-Ivanov equation, respectively.

STRICT STABILITY OF IMPULSIVE SET VALUED DIFFERENTIAL EQUATIONS

Institute of Scientific and Technical Information of China (English)

无

2011-01-01

In this paper, we develop strict stability concepts of ODE to impulsive hybrid set valued differential equations. By Lyapunov’s original method, we get some basic strict stability criteria of impulsive hybrid set valued equations.

STABILITY OF SOME KIND OF STOCHASTIC DIFFERENTIAL EQUATION

Institute of Scientific and Technical Information of China (English)

无

2011-01-01

In this paper,a kind of stochastic differential equation is investigated and the almost sure exponential stability of the equation is obtained using Gronwall's inequality.Further,we also give other noise intensity function to keep the stability of the system.

Analysis of stability for stochastic delay integro-differential equations.

Science.gov (United States)

Zhang, Yu; Li, Longsuo

2018-01-01

In this paper, we concern stability of numerical methods applied to stochastic delay integro-differential equations. For linear stochastic delay integro-differential equations, it is shown that the mean-square stability is derived by the split-step backward Euler method without any restriction on step-size, while the Euler-Maruyama method could reproduce the mean-square stability under a step-size constraint. We also confirm the mean-square stability of the split-step backward Euler method for nonlinear stochastic delay integro-differential equations. The numerical experiments further verify the theoretical results.

Handbook of functional equations stability theory

CERN Document Server

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...

On stability of Random Riccati equations

Institute of Scientific and Technical Information of China (English)

王远; 郭雷

1999-01-01

Random Riccati equations (RRE) arise frequently in filtering, estimation and control, but their stability properties are rarely rigorously explored in the literature. First a suitable stochastic observability (or excitation) condition is introduced to guarantee both the L_r-and exponential stability of RRE. Then the stability of Kalman filter is analyzed with random coefficients, and the L_r boundedness of filtering errors is established.

Fully synchronous solutions and the synchronization phase transition for the finite-N Kuramoto model

Science.gov (United States)

Bronski, Jared C.; DeVille, Lee; Jip Park, Moon

2012-09-01

We present a detailed analysis of the stability of phase-locked solutions to the Kuramoto system of oscillators. We derive an analytical expression counting the dimension of the unstable manifold associated to a given stationary solution. From this we are able to derive a number of consequences, including analytic expressions for the first and last frequency vectors to phase-lock, upper and lower bounds on the probability that a randomly chosen frequency vector will phase-lock, and very sharp results on the large N limit of this model. One of the surprises in this calculation is that for frequencies that are Gaussian distributed, the correct scaling for full synchrony is not the one commonly studied in the literature; rather, there is a logarithmic correction to the scaling which is related to the extremal value statistics of the random frequency vector.

Stabilization analysis of Euler-Bernoulli beam equation with locally distributed disturbance

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Pengcheng HAN

2017-12-01

Full Text Available In order to enrich the system stability theory of the control theories, taking Euler-Bernoulli beam equation as the research subject, the stability of Euler-Bernoulli beam equation with locally distributed disturbance is studied. A feedback controller based on output is designed to reduce the effects of the disturbances. The well-posedness of the nonlinear closed-loop system is investigated by the theory of maximal monotone operator, namely the existence and uniqueness of solutions for the closed-loop system. An appropriate state space is established, an appropriate inner product is defined, and a non-linear operator satisfying this state space is defined. Then, the system is transformed into the form of evolution equation. Based on this, the existence and uniqueness of solutions for the closed-loop system are proved. The asymptotic stability of the system is studied by constructing an appropriate Lyapunov function, which proves the asymptotic stability of the closed-loop system. The result shows that designing proper anti-interference controller is the foundation of investigating the system stability, and the research of the stability of Euler-bernoulli beam equation with locally distributed disturbance can prove the asymptotic stability of the system. This method can be extended to study the other equations such as wave equation, Timoshenko beam equation, Schrodinger equation, etc.

Kuramoto-type phase transition with metronomes

International Nuclear Information System (INIS)

Boda, Sz; Ujvári, Sz; Tunyagi, A; Néda, Z

2013-01-01

Metronomes placed on the perimeter of a disc-shaped platform, which can freely rotate in a horizontal plane, are used for a simple classroom illustration of the Kuramoto-type phase transition. The rotating platform induces a global coupling between the metronomes, and the strength of this coupling can be varied by tilting the metronomes’ swinging plane relative to the radial direction on the disc. As a function of the tilting angle, a transition from spontaneously synchronized to unsynchronized states is observable. By varying the number of metronomes on the disc, finite-size effects are also exemplified. A realistic theoretical model is introduced and used to reproduce the observed results. Computer simulations of this model allow a detailed investigation of the emerging collective behaviour in this system. (paper)

Distributional Methods for a Class of Functional Equations and Their Stabilities

Institute of Scientific and Technical Information of China (English)

Jae Young CHUNG

2007-01-01

We consider a class of n-dimensional Pompeiu equations and that of Pexider equations and their Hyers-Ulam stability problems in the spaces of Schwartz distributions. First, reducing the given distribution version of functional equations to differential equations we find their solutions. Secondly,using approximate identities we prove the Hyers-Ulam stability of the equations.

Stability of Vector Functional Differential Equations: A Survey | Gil ...

African Journals Online (AJOL)

This paper is a survey of the recent results of the author on the stability of linear and nonlinear vector differential equations with delay. Explicit conditions for the exponential and absolute stabilities are derived. Moreover, solution estimates for the considered equations are established. They provide the bounds for the regions ...

Compressible stability of growing boundary layers using parabolized stability equations

Science.gov (United States)

Chang, Chau-Lyan; Malik, Mujeeb R.; Erlebacher, Gordon; Hussaini, M. Y.

1991-01-01

The parabolized stability equation (PSE) approach is employed to study linear and nonlinear compressible stability with an eye to providing a capability for boundary-layer transition prediction in both 'quiet' and 'disturbed' environments. The governing compressible stability equations are solved by a rational parabolizing approximation in the streamwise direction. Nonparallel flow effects are studied for both the first- and second-mode disturbances. For oblique waves of the first-mode type, the departure from the parallel results is more pronounced as compared to that for the two-dimensional waves. Results for the Mach 4.5 case show that flow nonparallelism has more influence on the first mode than on the second. The disturbance growth rate is shown to be a strong function of the wall-normal distance due to either flow nonparallelism or nonlinear interactions. The subharmonic and fundamental types of breakdown are found to be similar to the ones in incompressible boundary layers.

Normal forms of Hopf-zero singularity

Science.gov (United States)

Gazor, Majid; Mokhtari, Fahimeh

2015-01-01

The Lie algebra generated by Hopf-zero classical normal forms is decomposed into two versal Lie subalgebras. Some dynamical properties for each subalgebra are described; one is the set of all volume-preserving conservative systems while the other is the maximal Lie algebra of nonconservative systems. This introduces a unique conservative-nonconservative decomposition for the normal form systems. There exists a Lie-subalgebra that is Lie-isomorphic to a large family of vector fields with Bogdanov-Takens singularity. This gives rise to a conclusion that the local dynamics of formal Hopf-zero singularities is well-understood by the study of Bogdanov-Takens singularities. Despite this, the normal form computations of Bogdanov-Takens and Hopf-zero singularities are independent. Thus, by assuming a quadratic nonzero condition, complete results on the simplest Hopf-zero normal forms are obtained in terms of the conservative-nonconservative decomposition. Some practical formulas are derived and the results implemented using Maple. The method has been applied on the Rössler and Kuramoto-Sivashinsky equations to demonstrate the applicability of our results.

Synchronisation of networked Kuramoto oscillators under stable Lévy noise

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Kalloniatis, Alexander C.; Roberts, Dale O.

2017-01-01

We study the Kuramoto model on several classes of network topologies examining the dynamics under the influence of Lévy noise. Such noise exhibits heavier tails than Gaussian and allows us to understand how 'shocks' influence the individual oscillator and collective system behaviour. Skewed α-stable Lévy noise, equivalent to fractional diffusion perturbations, are considered. We perform numerical simulations for Erdős-Rényi (ER) and Barabási-Albert (BA) scale free networks of size N = 1000 while varying the Lévy index α for the noise. We find that synchrony now assumes a surprising variety of forms, not seen for Gaussian-type noise, and changing with α: a noise-generated drift, a smooth α dependence of the point of cross-over of ER and BA networks in the degree of synchronisation, and a severe loss of synchronisation at low values of α. We also show that this robustness of the BA network across most values of α can also be understood as a consequence of the Laplacian of the graph working within the fractional Fokker-Planck equation of the linearised system, close to synchrony, with both eigenvalues and eigenvectors alternately contributing in different regimes of α.

Periodic feedback stabilization for linear periodic evolution equations

CERN Document Server

Wang, Gengsheng

2016-01-01

This book introduces a number of recent advances regarding periodic feedback stabilization for linear and time periodic evolution equations. First, it presents selected connections between linear quadratic optimal control theory and feedback stabilization theory for linear periodic evolution equations. Secondly, it identifies several criteria for the periodic feedback stabilization from the perspective of geometry, algebra and analyses respectively. Next, it describes several ways to design periodic feedback laws. Lastly, the book introduces readers to key methods for designing the control machines. Given its coverage and scope, it offers a helpful guide for graduate students and researchers in the areas of control theory and applied mathematics.

Razumikhin-Type Stability Criteria for Differential Equations with Delayed Impulses.

Science.gov (United States)

Wang, Qing; Zhu, Quanxin

2013-01-01

This paper studies stability problems of general impulsive differential equations where time delays occur in both differential and difference equations. Based on the method of Lyapunov functions, Razumikhin technique and mathematical induction, several stability criteria are obtained for differential equations with delayed impulses. Our results show that some systems with delayed impulses may be exponentially stabilized by impulses even if the system matrices are unstable. Some less restrictive sufficient conditions are also given to keep the good stability property of systems subject to certain type of impulsive perturbations. Examples with numerical simulations are discussed to illustrate the theorems. Our results may be applied to complex problems where impulses depend on both current and past states.

Razumikhin-type stability criteria for differential equations with delayed impulses

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Qing Wang

2013-01-01

Full Text Available This paper studies stability problems of general impulsive differential equations where time delays occur in both differential and difference equations. Based on the method of Lyapunov functions, Razumikhin technique and mathematical induction, several stability criteria are obtained for differential equations with delayed impulses. Our results show that some systems with delayed impulses may be exponentially stabilized by impulses even if the system matrices are unstable. Some less restrictive sufficient conditions are also given to keep the good stability property of systems subject to certain type of impulsive perturbations. Examples with numerical simulations are discussed to illustrate the theorems. Our results may be applied to complex problems where impulses depend on both current and past states.

Parabolized stability equations

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Herbert, Thorwald

1994-01-01

The parabolized stability equations (PSE) are a new approach to analyze the streamwise evolution of single or interacting Fourier modes in weakly nonparallel flows such as boundary layers. The concept rests on the decomposition of every mode into a slowly varying amplitude function and a wave function with slowly varying wave number. The neglect of the small second derivatives of the slowly varying functions with respect to the streamwise variable leads to an initial boundary-value problem that can be solved by numerical marching procedures. The PSE approach is valid in convectively unstable flows. The equations for a single mode are closely related to those of the traditional eigenvalue problems for linear stability analysis. However, the PSE approach does not exploit the homogeneity of the problem and, therefore, can be utilized to analyze forced modes and the nonlinear growth and interaction of an initial disturbance field. In contrast to the traditional patching of local solutions, the PSE provide the spatial evolution of modes with proper account for their history. The PSE approach allows studies of secondary instabilities without the constraints of the Floquet analysis and reproduces the established experimental, theoretical, and computational benchmark results on transition up to the breakdown stage. The method matches or exceeds the demonstrated capabilities of current spatial Navier-Stokes solvers at a small fraction of their computational cost. Recent applications include studies on localized or distributed receptivity and prediction of transition in model environments for realistic engineering problems. This report describes the basis, intricacies, and some applications of the PSE methodology.

Phase diagram for the Kuramoto model with van Hemmen interactions.

Science.gov (United States)

Kloumann, Isabel M; Lizarraga, Ian M; Strogatz, Steven H

2014-01-01

We consider a Kuramoto model of coupled oscillators that includes quenched random interactions of the type used by van Hemmen in his model of spin glasses. The phase diagram is obtained analytically for the case of zero noise and a Lorentzian distribution of the oscillators' natural frequencies. Depending on the size of the attractive and random coupling terms, the system displays four states: complete incoherence, partial synchronization, partial antiphase synchronization, and a mix of antiphase and ordinary synchronization.

Stability criteria for neutral delay differential-algebraic equations

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FAN Ni

2013-10-01

Full Text Available The asymptotic stability of neutral delay differential-algebraic equations is studied in this paper.Two stability criteria described by evaluating a corresponding harmonic function on the boundary of a torus region are presented.

Multiple solutions and stability of the steady transonic small-disturbance equation

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Ya Liu

2017-09-01

Full Text Available Numerical solutions of the steady transonic small-disturbance (TSD potential equation are computed using the conservative Murman−Cole scheme. Multiple solutions are discovered and mapped out for the Mach number range at zero angle of attack and the angle of attack range at Mach number 0.85 for the NACA 0012 airfoil. We present a linear stability analysis method by directly assembling and evaluating the Jacobian matrix of the nonlinear finite-difference equation of the TSD equation. The stability of all the discovered multiple solutions are then determined by the proposed eigen analysis. The relation of stability to convergence of the iterative method for solving the TSD equation is discussed. Computations and the stability analysis demonstrate the possibility of eliminating the multiple solutions and stabilizing the remaining unique solution by adding a sufficiently long splitter plate downstream the airfoil trailing edge. Finally, instability of the solution of the TSD equation is shown to be closely connected to the onset of transonic buffet by comparing with experimental data.

Equations for studies of feedback stabilization

International Nuclear Information System (INIS)

Boozer, A.H.

1998-01-01

Important ideal magnetohydrodynamic (MHD) instabilities grow slowly when a conducting wall surrounds a toroidal plasma. Feedback stabilization of these instabilities may be required for tokamaks and other magnetic confinement concepts to achieve adequate plasma pressure and self-driven current for practical fusion power. Equations are derived for simulating feedback stabilization, which require the minimum information about an ideal plasma for an exact analysis. The equations are solved in the approximation of one unstable mode, one wall circuit, one feedback circuit, and one sensor circuit. The analysis based on a single unstable mode is shown to be mathematically equivalent to the standard analysis of feedback of the axisymmetric vertical instability of tokamaks. Unlike that analysis, the method presented here applies to multiple modes that are coupled by the wall and to arbitrary toroidal mode numbers. copyright 1998 American Institute of Physics

STABILITY OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY

Institute of Scientific and Technical Information of China (English)

无

2009-01-01

In this paper,we obtain suffcient conditions for the stability in p-th moment of the analytical solutions and the mean square stability of a stochastic differential equation with unbounded delay proposed in [6,10] using the explicit Euler method.

Stability of Functional Differential Equations

CERN Document Server

Lemm, Jeffrey M

1986-01-01

This book provides an introduction to the structure and stability properties of solutions of functional differential equations. Numerous examples of applications (such as feedback systrems with aftereffect, two-reflector antennae, nuclear reactors, mathematical models in immunology, viscoelastic bodies, aeroautoelastic phenomena and so on) are considered in detail. The development is illustrated by numerous figures and tables.

Using machine learning to replicate chaotic attractors and calculate Lyapunov exponents from data

Science.gov (United States)

Pathak, Jaideep; Lu, Zhixin; Hunt, Brian R.; Girvan, Michelle; Ott, Edward

2017-12-01

We use recent advances in the machine learning area known as "reservoir computing" to formulate a method for model-free estimation from data of the Lyapunov exponents of a chaotic process. The technique uses a limited time series of measurements as input to a high-dimensional dynamical system called a "reservoir." After the reservoir's response to the data is recorded, linear regression is used to learn a large set of parameters, called the "output weights." The learned output weights are then used to form a modified autonomous reservoir designed to be capable of producing an arbitrarily long time series whose ergodic properties approximate those of the input signal. When successful, we say that the autonomous reservoir reproduces the attractor's "climate." Since the reservoir equations and output weights are known, we can compute the derivatives needed to determine the Lyapunov exponents of the autonomous reservoir, which we then use as estimates of the Lyapunov exponents for the original input generating system. We illustrate the effectiveness of our technique with two examples, the Lorenz system and the Kuramoto-Sivashinsky (KS) equation. In the case of the KS equation, we note that the high dimensional nature of the system and the large number of Lyapunov exponents yield a challenging test of our method, which we find the method successfully passes.

Stability analysis of impulsive functional differential equations

CERN Document Server

Stamova, Ivanka

2009-01-01

This book is devoted to impulsive functional differential equations which are a natural generalization of impulsive ordinary differential equations (without delay) and of functional differential equations (without impulses). At the present time the qualitative theory of such equationsis under rapid development. After a presentation of the fundamental theory of existence, uniqueness and continuability of solutions, a systematic development of stability theory for that class of problems is given which makes the book unique. It addresses to a wide audience such as mathematicians, applied research

The stability of quadratic-reciprocal functional equation

Science.gov (United States)

Song, Aimin; Song, Minwei

2018-04-01

A new quadratic-reciprocal functional equation f ((k +1 )x +k y )+f ((k +1 )x -k y )=2/f (x )f (y )[(k+1 ) 2f (y )+k2f (x )] [(k+1)2f (y )-k2f (x )] 2 is introduced. The Hyers-Ulam stability for the quadratic-reciprocal functional equations is proved in Banach spaces using the direct method and the fixed point method, respectively.

A Generalized Halanay Inequality for Stability of Nonlinear Neutral Functional Differential Equations

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Wansheng Wang

2010-01-01

Full Text Available This paper is devoted to generalize Halanay's inequality which plays an important rule in study of stability of differential equations. By applying the generalized Halanay inequality, the stability results of nonlinear neutral functional differential equations (NFDEs and nonlinear neutral delay integrodifferential equations (NDIDEs are obtained.

Cryostatic stability equation

International Nuclear Information System (INIS)

Sydoriak, S.G.

1976-01-01

Although criteria for cryostatic stability of superconducting magnets cooled by pool boiling of liquid helium have been widely discussed the same cannot be said for magnets cooled by natural convection or forced flow boiling in channels. Boiling in narrow channels is shown to be qualitatively superior to pool boiling because the recovery heat flux equals the breakaway flux for narrow channels, whereas the two are markedly different in pool boiling. A second advantage of channel boiling is that it is well understood and calculable; pool peak nucleate boiling heat flux has been adequately measured only for boiling from the top of an immersed heated body. Peak boiling from the bottom is much less and (probably) depends strongly on the extent of the bottom surface. Equations are presented by which one can calculate the critical boiling heat flux for parallel wall vertical channels subject to either natural convection or forced flow boiling, with one or both walls heated. The one-heated-wall forced flow equation is discussed with regard to design of a spiral wound solenoid (pancake magnet) having a slippery insulating tape between the windings

Stability of numerical method for semi-linear stochastic pantograph differential equations

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Yu Zhang

2016-01-01

Full Text Available Abstract As a particular expression of stochastic delay differential equations, stochastic pantograph differential equations have been widely used in nonlinear dynamics, quantum mechanics, and electrodynamics. In this paper, we mainly study the stability of analytical solutions and numerical solutions of semi-linear stochastic pantograph differential equations. Some suitable conditions for the mean-square stability of an analytical solution are obtained. Then we proved the general mean-square stability of the exponential Euler method for a numerical solution of semi-linear stochastic pantograph differential equations, that is, if an analytical solution is stable, then the exponential Euler method applied to the system is mean-square stable for arbitrary step-size h > 0 $h>0$ . Numerical examples further illustrate the obtained theoretical results.

An Alternative Stability Equation For Rock Armoured Rubble Mound Breakwaters

DEFF Research Database (Denmark)

Hald, Tue; Burcharth, H. F.

2000-01-01

Rubble mound breakwaters are by far the most common type of breakwater, the importance of which is clearly reflected in the vast amount of published research. Especially, the hydraulic stability of the main armour layer has been studied in order to obtain reliable design equations. It should...... equations and model test results still exists. When turning toward prototype the situation is even worse. With the objective to reduce some of the variability an alternative approach based on force considerations is presented. The paper will describe a new stability equation for rock armoured slopes derived...

Stability of the Exponential Functional Equation in Riesz Algebras

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Bogdan Batko

2014-01-01

Full Text Available We deal with the stability of the exponential Cauchy functional equation F(x+y=F(xF(y in the class of functions F:G→L mapping a group (G, + into a Riesz algebra L. The main aim of this paper is to prove that the exponential Cauchy functional equation is stable in the sense of Hyers-Ulam and is not superstable in the sense of Baker. To prove the stability we use the Yosida Spectral Representation Theorem.

On the stability of some systems of exponential difference equations

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N. Psarros

2018-01-01

Full Text Available In this paper we prove the stability of the zero equilibria of two systems of difference equations of exponential type, which are some extensions of an one-dimensional biological model. The stability of these systems is investigated in the special case when one of the eigenvalues is equal to -1 and the other eigenvalue has absolute value less than 1, using centre manifold theory. In addition, we study the existence and uniqueness of positive equilibria, the attractivity and the global asymptotic stability of these equilibria of some related systems of difference equations.

Stability analysis of solutions to nonlinear stiff Volterra functional differential equations in Banach spaces

Institute of Scientific and Technical Information of China (English)

LI Shoufu

2005-01-01

A series of stability, contractivity and asymptotic stability results of the solutions to nonlinear stiff Volterra functional differential equations (VFDEs) in Banach spaces is obtained, which provides the unified theoretical foundation for the stability analysis of solutions to nonlinear stiff problems in ordinary differential equations(ODEs), delay differential equations(DDEs), integro-differential equations(IDEs) and VFDEs of other type which appear in practice.

Conditional Stability of Solitary-Wave Solutions for Generalized Compound KdV Equation and Generalized Compound KdV-Burgers Equation

International Nuclear Information System (INIS)

Zhang Weiguo; Dong Chunyan; Fan Engui

2006-01-01

In this paper, we discuss conditional stability of solitary-wave solutions in the sense of Liapunov for the generalized compound KdV equation and the generalized compound KdV-Burgers equations. Linear stability of the exact solitary-wave solutions is proved for the above two types of equations when the small disturbance of travelling wave form satisfies some special conditions.

Computation of the stability derivatives via CFD and the sensitivity equations

Science.gov (United States)

Lei, Guo-Dong; Ren, Yu-Xin

2011-04-01

The method to calculate the aerodynamic stability derivates of aircrafts by using the sensitivity equations is extended to flows with shock waves in this paper. Using the newly developed second-order cell-centered finite volume scheme on the unstructured-grid, the unsteady Euler equations and sensitivity equations are solved simultaneously in a non-inertial frame of reference, so that the aerodynamic stability derivatives can be calculated for aircrafts with complex geometries. Based on the numerical results, behavior of the aerodynamic sensitivity parameters near the shock wave is discussed. Furthermore, the stability derivatives are analyzed for supersonic and hypersonic flows. The numerical results of the stability derivatives are found in good agreement with theoretical results for supersonic flows, and variations of the aerodynamic force and moment predicted by the stability derivatives are very close to those obtained by CFD simulation for both supersonic and hypersonic flows.

Stability of Equilibrium Points of Fractional Difference Equations with Stochastic Perturbations

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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.

Some Remarks on Stability of Generalized Equations

Czech Academy of Sciences Publication Activity Database

Outrata, Jiří; Henrion, R.; Kruger, A.Y.

2013-01-01

Roč. 159, č. 3 (2013), s. 681-697 ISSN 0022-3239 R&D Projects: GA AV ČR IAA100750802; GA ČR(CZ) GAP201/12/0671 Institutional support: RVO:67985556 Keywords : Parameterized generalized equation * Regular and limiting coderivative * Constant rank CQ * Mathematical program with equilibrium constraints Subject RIV: BA - General Mathematics Impact factor: 1.406, year: 2013 http://library.utia.cas.cz/separaty/2013/MTR/outrata-some remarks on stability of generalized equations.pdf

Stability with respect to initial time difference for generalized delay differential equations

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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.

Exponential p-stability of impulsive stochastic differential equations with delays

International Nuclear Information System (INIS)

Yang Zhiguo; Xu Daoyi; Xiang Li

2006-01-01

In this Letter, we establish a method to study the exponential p-stability of the zero solution of impulsive stochastic differential equations with delays. By establishing an L-operator inequality and using the properties of M-cone and stochastic analysis technique, we obtain some new conditions ensuring the exponential p-stability of the zero solution of impulsive stochastic differential equations with delays. Two illustrative examples have been provided to show the effectiveness of our results

Synchrony, waves and ripple in spatially coupled Kuramoto oscillators with Mexican hat connectivity.

Science.gov (United States)

Heitmann, Stewart; Ermentrout, G Bard

2015-06-01

Spatiotemporal waves of synchronized activity are known to arise in oscillatory neural networks with lateral inhibitory coupling. How such patterns respond to dynamic changes in coupling strength is largely unexplored. The present study uses analysis and simulation to investigate the evolution of wave patterns when the strength of lateral inhibition is varied dynamically. Neural synchronization was modeled by a spatial ring of Kuramoto oscillators with Mexican hat lateral coupling. Broad bands of coexisting stable wave solutions were observed at all levels of inhibition. The stability of these waves was formally analyzed in both the infinite ring and the finite ring. The broad range of multi-stability predicted hysteresis in transitions between neighboring wave solutions when inhibition is slowly varied. Numerical simulation confirmed the predicted transitions when inhibition was ramped down from a high initial value. However, non-wave solutions emerged from the uniform solution when inhibition was ramped upward from zero. These solutions correspond to spatially periodic deviations of phase that we call ripple states. Numerical continuation showed that stable ripple states emerge from synchrony via a supercritical pitchfork bifurcation. The normal form of this bifurcation was derived analytically, and its predictions compared against the numerical results. Ripple states were also found to bifurcate from wave solutions, but these were locally unstable. Simulation also confirmed the existence of hysteresis and ripple states in two spatial dimensions. Our findings show that spatial synchronization patterns can remain structurally stable despite substantial changes in network connectivity.

Orbital stability of Gausson solutions to logarithmic Schrodinger equations

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Alex H. Ardila

2016-12-01

Full Text Available In this article we prove of the orbital stability of the ground state for logarithmic Schrodinger equation in any dimension and under nonradial perturbations. This general stability result was announced by Cazenave and Lions [9, Remark II.3], but no details were given there.

On Robust Stability of Systems of Differential-Algebraic Equations

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A. Shcheglova

2016-06-01

The sufficient conditions of robust stability for index-one and index-two systems are obtained. We use the values of real and complex stability radii obtained for system of ordinary differential equations solved with respect to the derivatives. We consider the example illustrating the obtained results.

Transitions amongst synchronous solutions in the stochastic Kuramoto model

Science.gov (United States)

DeVille, Lee

2012-05-01

We consider the Kuramoto model of coupled oscillators with nearest-neighbour coupling and additive white noise. We show that synchronous solutions which are stable without the addition of noise become metastable and that we have transitions amongst synchronous solutions on long timescales. We compute these timescales and, moreover, compute the most likely path in phase space that transitions will follow. We show that these transition timescales do not increase as the number of oscillators in the system increases, and are roughly constant in the system size. Finally, we show that the transitions correspond to a splitting of one synchronous solution into two communities which move independently for some time and which rejoin to form a different synchronous solution.

Phase-locked patterns of the Kuramoto model on 3-regular graphs

Science.gov (United States)

DeVille, Lee; Ermentrout, Bard

2016-09-01

We consider the existence of non-synchronized fixed points to the Kuramoto model defined on sparse networks: specifically, networks where each vertex has degree exactly three. We show that "most" such networks support multiple attracting phase-locked solutions that are not synchronized and study the depth and width of the basins of attraction of these phase-locked solutions. We also show that it is common in "large enough" graphs to find phase-locked solutions where one or more of the links have angle difference greater than π/2.

A New Pseudoinverse Matrix Method For Balancing Chemical Equations And Their Stability

International Nuclear Information System (INIS)

Risteski, Ice B.

2008-01-01

In this work is given a new pseudoniverse matrix method for balancing chemical equations. Here offered method is founded on virtue of the solution of a Diophantine matrix equation by using of a Moore-Penrose pseudoinverse matrix. The method has been tested on several typical chemical equations and found to be very successful for the all equations in our extensive balancing research. This method, which works successfully without any limitations, also has the capability to determine the feasibility of a new chemical reaction, and if it is feasible, then it will balance the equation. Chemical equations treated here possess atoms with fractional oxidation numbers. Also, in the present work are introduced necessary and sufficient criteria for stability of chemical equations over stability of their extended matrices

Stability and instability of stationary solutions for sublinear parabolic equations

Science.gov (United States)

Kajikiya, Ryuji

2018-01-01

In the present paper, we study the initial boundary value problem of the sublinear parabolic equation. We prove the existence of solutions and investigate the stability and instability of stationary solutions. We show that a unique positive and a unique negative stationary solutions are exponentially stable and give the exact exponent. We prove that small stationary solutions are unstable. For one space dimensional autonomous equations, we elucidate the structure of stationary solutions and study the stability of all stationary solutions.

Stability of generalized Runge-Kutta methods for stiff kinetics coupled differential equations

International Nuclear Information System (INIS)

Aboanber, A E

2006-01-01

A stability and efficiency improved class of generalized Runge-Kutta methods of order 4 are developed for the numerical solution of stiff system kinetics equations for linear and/or nonlinear coupled differential equations. The determination of the coefficients required by the method is precisely obtained from the so-called equations of condition which in turn are derived by an approach based on Butcher series. Since the equations of condition are fewer in number, free parameters can be chosen for optimizing any desired feature of the process. A further related coefficient set with different values of these parameters and the region of absolute stability of the method have been introduced. In addition, the A(α) stability properties of the method are investigated. Implementing the method in a personal computer estimated the accuracy and speed of calculations and verified the good performances of the proposed new schemes for several sample problems of the stiff system point kinetics equations with reactivity feedback

Lyapunov functionals and stability of stochastic difference equations

CERN Document Server

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.

Maintaining the stability of nonlinear differential equations by the enhancement of HPM

International Nuclear Information System (INIS)

Hosein Nia, S.H.; Ranjbar, A.N.; Ganji, D.D.; Soltani, H.; Ghasemi, J.

2008-01-01

Homotopy perturbation method is an effective method to find a solution of a nonlinear differential equation. In this method, a nonlinear complex differential equation is transformed to a series of linear and nonlinear parts, almost simpler differential equations. These sets of equations are then solved iteratively. Finally, a linear series of the solutions completes the answer if the convergence is maintained. In this Letter, the need for stability verification is shown through some examples. Consequently, HPM is enhanced by a preliminary assumption. The idea is to keep the inherent stability of nonlinear dynamic, even the selected linear part is not

Stability and bifurcation analysis of a generalized scalar delay differential equation.

Science.gov (United States)

Bhalekar, Sachin

2016-08-01

This paper deals with the stability and bifurcation analysis of a general form of equation D(α)x(t)=g(x(t),x(t-τ)) involving the derivative of order α ∈ (0, 1] and a constant delay τ ≥ 0. The stability of equilibrium points is presented in terms of the stability regions and critical surfaces. We provide a necessary condition to exist chaos in the system also. A wide range of delay differential equations involving a constant delay can be analyzed using the results proposed in this paper. The illustrative examples are provided to explain the theory.

Fractional Differential Equations in Terms of Comparison Results and Lyapunov Stability with Initial Time Difference

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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.

On Robust Stability of Differential-Algebraic Equations with Structured Uncertainty

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A. Kononov

2018-03-01

Full Text Available We consider a linear time-invariant system of differential-algebraic equations (DAE, which can be written as a system of ordinary differential equations with non-invertible coefficients matrices. An important characteristic of DAE is the unsolvability index, which reflects the complexity of the internal structure of the system. The question of the asymptotic stability of DAE containing the uncertainty given by the matrix norm is investigated. We consider a perturbation in the structured uncertainty case. It is assumed that the initial nominal system is asymptotically stable. For the analysis, the original equation is reduced to the structural form, in which the differential and algebraic subsystems are separated. This structural form is equivalent to the input system in the sense of coincidence of sets of solutions, and the operator transforming the DAE into the structural form possesses the inverse operator. The conversion to structural form does not use a change of variables. Regularity of matrix pencil of the source equation is the necessary and sufficient condition of structural form existence. Sufficient conditions have been obtained that perturbations do not break the internal structure of the nominal system. Under these conditions robust stability of the DAE with structured uncertainty is investigated. Estimates for the stability radius of the perturbed DAE system are obtained. The text of the article is from the simpler case, in which the perturbation is present only for an unknown function, to a more complex one, under which the perturbation is also present in the derivative of the unknown function. We used values of the real and the complex stability radii of explicit ordinary differential equations for obtaining the results. We consider the example illustrating the obtained results.

Stability, bifurcation and a new chaos in the logistic differential equation with delay

International Nuclear Information System (INIS)

Jiang Minghui; Shen Yi; Jian Jigui; Liao Xiaoxin

2006-01-01

This Letter is concerned with bifurcation and chaos in the logistic delay differential equation with a parameter r. The linear stability of the logistic equation is investigated by analyzing the associated characteristic transcendental equation. Based on the normal form approach and the center manifold theory, the formula for determining the direction of Hopf bifurcation and the stability of bifurcation periodic solution in the first bifurcation values is obtained. By theoretical analysis and numerical simulation, we found a new chaos in the logistic delay differential equation

Stability test for a parabolic partial differential equation

NARCIS (Netherlands)

Vajta, Miklos

2001-01-01

The paper describes a stability test applied to coupled parabolic partial differential equations. The PDE's describe the temperature distribution of composite structures with linear inner heat sources. The distributed transfer functions are developed based on the transmission matrix of each layer.

Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations

KAUST Repository

Abdulle, Assyr

2013-01-01

We introduce a new family of explicit integrators for stiff Itô stochastic differential equations (SDEs) of weak order two. These numerical methods belong to the class of one-step stabilized methods with extended stability domains and do not suffer from the step size reduction faced by standard explicit methods. The family is based on the standard second order orthogonal Runge-Kutta-Chebyshev (ROCK2) methods for deterministic problems. The convergence, meansquare, and asymptotic stability properties of the methods are analyzed. Numerical experiments, including applications to nonlinear SDEs and parabolic stochastic partial differential equations are presented and confirm the theoretical results. © 2013 Society for Industrial and Applied Mathematics.

Non-uniform Erosion and Surface Evolution of Plasma-Facing Materials for Electric Propulsion

Science.gov (United States)

Matthes, Christopher Stanley Rutter

A study regarding the surface evolution of plasma-facing materials is presented. Experimental efforts were performed in the UCLA Pi Facility, designed to explore the physics of plasma-surface interactions. The influence of micro-architectured surfaces on the effects of plasma sputtering is compared with the response of planar samples. Ballistic deposition of sputtered atoms as a result of geometric re-trapping is observed. This provides a self-healing mechanism of micro-architectured surfaces during plasma exposure. This result is quantified using a QCM to demonstrate the evolution of surface features and the corresponding influence on the instantaneous sputtering yield. The sputtering yield of textured molybdenum samples exposed to 300 eV Ar plasma is found to be roughly 1 of the 2 corresponding value of flat samples, and increases with ion fluence. Mo samples exhibited a sputtering yield initially as low as 0.22+/-8%, converging to 0.4+/-8% at high fluence. Although the yield is dependent on the initial surface structure, it is shown to be transient, reaching a steady-state value that is independent of initial surface conditions. A continuum model of surface evolution resulting from sputtering, deposition and surface diffusion is also derived to resemble the damped Kuramoto-Sivashinsky (KS) equation of non-linear dynamics. Linear stability analysis of the evolution equation provides an estimate of the selected wavelength, and its dependence on the ion energy and angle of incidence. The analytical results are confirmed by numerical simulations of the equation with a Fast Fourier Transform method. It is shown that for an initially flat surface, small perturbations lead to the evolution of a selected surface pattern that has nano- scale wavelength. When the surface is initially patterned by other means, the final resulting pattern is a competition between the "templated" pattern and the "self-organized" structure. Potential future routes of research are also

Stability of the Filter Equation for a Time-Dependent Signal on Rd

International Nuclear Information System (INIS)

Stannat, Wilhelm

2005-01-01

Stability of the pathwise filter equation for a time-dependent signal process induced by a d-dimensional stochastic differential equation and a linear observation is studied, using a variational approach. A lower bound for the rate of stability is identified in terms of the mass-gap of a parabolic ground state transform associated with the generator of the signal process and the square of the observation. The lower bound can be easily calculated a priori and provides hints on how precisely to measure the signal in order to reach a certain rate of stability. Ergodicity of the signal process is not needed

Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations

KAUST Repository

Abdulle, Assyr; Vilmart, Gilles; Zygalakis, Konstantinos C.

2013-01-01

We introduce a new family of explicit integrators for stiff Itô stochastic differential equations (SDEs) of weak order two. These numerical methods belong to the class of one-step stabilized methods with extended stability domains and do not suffer

TOPICAL REVIEW: The stability for the Cauchy problem for elliptic equations

Science.gov (United States)

Alessandrini, Giovanni; Rondi, Luca; Rosset, Edi; Vessella, Sergio

2009-12-01

We discuss the ill-posed Cauchy problem for elliptic equations, which is pervasive in inverse boundary value problems modeled by elliptic equations. We provide essentially optimal stability results, in wide generality and under substantially minimal assumptions. As a general scheme in our arguments, we show that all such stability results can be derived by the use of a single building brick, the three-spheres inequality. Due to the current absence of research funding from the Italian Ministry of University and Research, this work has been completed without any financial support.

Exponential stability in a scalar functional differential equation

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Pituk Mihály

2006-01-01

Full Text Available We establish a criterion for the global exponential stability of the zero solution of the scalar retarded functional differential equation whose linear part generates a monotone semiflow on the phase space with respect to the exponential ordering, and the nonlinearity has at most linear growth.

There is no non-zero stable fixed point for dense networks in the homogeneous Kuramoto model

International Nuclear Information System (INIS)

Taylor, Richard

2012-01-01

This paper is concerned with the existence of multiple stable fixed point solutions of the homogeneous Kuramoto model. We develop a necessary condition for the existence of stable fixed points for the general network Kuramoto model. This condition is applied to show that for sufficiently dense n-node networks, with node degrees at least 0.9395(n−1), the homogeneous (equal frequencies) model has only one stable fixed point solution over the full space of phase angles in the range −π to π. This is the zero fixed point solution defined by all phase angle differences being zero. This result, together with existing research, proves a conjecture of Verwoerd and Mason (2007 Proc. of the American Control Conf. pp 4613–8) that for the complete network and the homogeneous model, the zero fixed point has a basin of attraction consisting of the entire space minus a set of measure zero. The necessary conditions are also tested to see how close to sufficiency they might be by applying them to a class of regular degree networks studied by Wiley et al (2006 Chaos 16 015103). (paper)

Lyapunov stability and its application to systems of ordinary differential equations

Science.gov (United States)

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.

Stability analysis of a class of fractional delay differential equations

Indian Academy of Sciences (India)

In this paper we analyse stability of nonlinear fractional order delay differential equations of the form D y ( t ) = a f ( y ( t − ) ) − by ( t ) , where D is a Caputo fractional derivative of order 0 < ≤ 1. We describe stability regions using critical curves. To explain the proposed theory, we discuss fractional order logistic ...

Stability theory for dynamic equations on time scales

CERN Document Server

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...

MHD stability properties of a system of reduced toroidal MHD equations

International Nuclear Information System (INIS)

Maschke, E.K.; Morros Tosas, J.; Urquijo, G.

1993-01-01

A system of reduced toroidal magneto-hydrodynamic (MHD) equations is derived from a general scalar representation of the complete MHD system, using an ordering in terms of the inverse aspect ratio ε of a toroidal plasma. It is shown that the energy principle for the reduced equations is identical with the usual energy principle of the complete MHD system, to the appropriate order in ε. Thus, the reduced equations have the same ideal MHD stability limits as the full MHD equations. (authors). 6 refs

Stability in terms of two measures for a class of semilinear impulsive parabolic equations

International Nuclear Information System (INIS)

Dvirnyj, Aleksandr I; Slyn'ko, Vitalij I

2013-01-01

The problem of stability in terms of two measures is considered for semilinear impulsive parabolic equations. A new version of the comparison method is proposed, and sufficient conditions for stability in terms of two measures are obtained on this basis. An example of a hybrid impulsive system formed by a system of ordinary differential equations coupled with a partial differential equation of parabolic type is given. The efficiency of the described approaches is demonstrated. Bibliography: 24 titles.

Hyers-Ulam stability for second-order linear differential equations with boundary conditions

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Pasc Gavruta

2011-06-01

Full Text Available We prove the Hyers-Ulam stability of linear differential equations of second-order with boundary conditions or with initial conditions. That is, if y is an approximate solution of the differential equation $y''+ eta (x y = 0$ with $y(a = y(b =0$, then there exists an exact solution of the differential equation, near y.

Stabilizing local boundary conditions for two-dimensional shallow water equations

KAUST Repository

Dia, Ben Mansour

2018-03-27

In this article, we present a sub-critical two-dimensional shallow water flow regulation. From the energy estimate of a set of one-dimensional boundary stabilization problems, we obtain a set of polynomial equations with respect to the boundary values as a requirement for the energy decrease. Using the Riemann invariant analysis, we build stabilizing local boundary conditions that guarantee the stability of the hydrodynamical state around a given steady state. Numerical results for the controller applied to the nonlinear problem demonstrate the performance of the method.

Stability of time-dependent particle-like solutions of some wave equations

International Nuclear Information System (INIS)

Voronov, N.A.

1978-01-01

The proof of the nonstability of the one-dimensional periodical localized solutions of the equation with a spontaneously broken symmetry is given. The stability of the one-dimensional oscillating solutions of the sine-Gordon equation was also considered with regard to such perturbations. As it was expected these solutions proved to be stable

Remarks on the stability of some quadratic functional equations

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Zygfryd Kominek

2008-01-01

Full Text Available Stability problems concerning the functional equations of the form \\[f(2x+y=4f(x+f(y+f(x+y-f(x-y,\\tag{1}\\] and \\[f(2x+y+f(2x-y=8f(x+2f(y\\tag{2}\\] are investigated. We prove that if the norm of the difference between the LHS and the RHS of one of equations \\((1\\ or \\((2\\, calculated for a function \\(g\\ is say, dominated by a function \\(\\varphi\\ in two variables having some standard properties then there exists a unique solution \\(f\\ of this equation and the norm of the difference between \\(g\\ and \\(f\\ is controlled by a function depending on \\(\\varphi\\.

Stability of a simple Levi–Civitá functional equation on non-unital ...

Indian Academy of Sciences (India)

Ulam stability; Levi–Civitá equation; non-unital semigroup; 2-divisible group. 2010 Mathematics Subject Classification. Primary: 39B82. 1. Introduction ... for all x ∈ G? These kind of questions form the material for the stability theory of func-.

Stability of negative solitary waves for an integrable modified Camassa-Holm equation

International Nuclear Information System (INIS)

Yin Jiuli; Tian Lixin; Fan Xinghua

2010-01-01

In this paper, we prove that the modified Camassa-Holm equation is Painleve integrable. We also study the orbital stability problem of negative solitary waves for this integrable equation. It is shown that the negative solitary waves are stable for arbitrary wave speed of propagation.

Control and Stabilization of the Benjamin-Ono Equation in {L^2({{T})}}

Science.gov (United States)

Laurent, Camille; Linares, Felipe; Rosier, Lionel

2015-12-01

We study the control and stabilization of the Benjamin-Ono equation in {L^2({T})}, the lowest regularity where the initial value problem is well-posed. This problem was already initiated in Linares and Rosier (Trans Am Math Soc 367:4595-4626, 2015) where a stronger stabilization term was used (that makes the equation of parabolic type in the control zone). Here we employ a more natural stabilization term related to the L 2-norm. Moreover, by proving a theorem of controllability in L 2, we manage to prove the global controllability in large time. Our analysis relies strongly on the bilinear estimates proved in Molinet and Pilod (Anal PDE 5:365-395, 2012) and some new extension of these estimates established here.

Stabilization of solutions to higher-order nonlinear Schrodinger equation with localized damping

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Eleni Bisognin

2007-01-01

Full Text Available We study the stabilization of solutions to higher-order nonlinear Schrodinger equations in a bounded interval under the effect of a localized damping mechanism. We use multiplier techniques to obtain exponential decay in time of the solutions of the linear and nonlinear equations.

A new design equation for drained stability of conical slopes in cohesive-frictional soils

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Boonchai Ukritchon

2018-04-01

Full Text Available New plasticity solutions to the drained stability of conical slopes in homogeneous cohesive-frictional soils were investigated by axisymmetric finite element limit analysis. Three parameters were studied, i.e. excavated height ratios, slope inclination angles, and soil friction angles. The influences of these parameters on the stability factor and predicted failure mechanism of conical slopes were discussed. A new design equation developed from a nonlinear regression of the lower bound solution was proposed for drained stability analyses of a conical slope in practice. Numerical examples were given to demonstrate a practical application of the proposed equation to stability evaluations of conical slopes with both associated and non-associated flow rules. Keywords: Limit analysis, Slope stability, Conical slope, Unsupported excavation, Cohesive-frictional soils

Stability analysis of a boundary layer over a hump using parabolized stability equations

Energy Technology Data Exchange (ETDEWEB)

Gao, B; Park, D H; Park, S O, E-mail: sopark@kaist.ac.kr [Division of Aerospace Engineering, Korea Advanced Institute of Science and Technology, Gusong-dong, Yusong-gu, Daejeon 305-701 (Korea, Republic of)

2011-10-15

Parabolized stability equations (PSEs) were used to investigate the stability of boundary layer flows over a small hump. The applicability of PSEs to flows with a small separation bubble was examined by comparing the result with DNS data. It was found that PSEs can efficiently track the disturbance waves with an acceptable accuracy in spite of a small separation bubble. A typical evolution scenario of Tollmien-Schlichting (TS) wave is presented. The adverse pressure gradient and the flow separation due to the hump have a strong effect on the amplification of the disturbances. The effect of hump width and height is also examined. When the width of the hump is reduced, the amplification factor is increased. The height of the hump is found to obviously influence the stability only when it is greater than the critical layer thickness.

Stability analysis of a boundary layer over a hump using parabolized stability equations

International Nuclear Information System (INIS)

Gao, B; Park, D H; Park, S O

2011-01-01

Parabolized stability equations (PSEs) were used to investigate the stability of boundary layer flows over a small hump. The applicability of PSEs to flows with a small separation bubble was examined by comparing the result with DNS data. It was found that PSEs can efficiently track the disturbance waves with an acceptable accuracy in spite of a small separation bubble. A typical evolution scenario of Tollmien-Schlichting (TS) wave is presented. The adverse pressure gradient and the flow separation due to the hump have a strong effect on the amplification of the disturbances. The effect of hump width and height is also examined. When the width of the hump is reduced, the amplification factor is increased. The height of the hump is found to obviously influence the stability only when it is greater than the critical layer thickness.

The Local Stability of Solutions for a Nonlinear Equation

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Haibo Yan

2014-01-01

Full Text Available The approach of Kruzkov’s device of doubling the variables is applied to establish the local stability of strong solutions for a nonlinear partial differential equation in the space L1(R by assuming that the initial value only lies in the space L1(R∩L∞(R.

Stability analysis of Runge-Kutta methods for nonlinear neutral delay integro-differential equations

Institute of Scientific and Technical Information of China (English)

2007-01-01

The sufficient conditions for the stability and asymptotic stability of Runge-Kutta methods for nonlinear neutral delay integro-differential equations are derived. A numerical test that confirms the theoretical results is given in the end.

Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators

International Nuclear Information System (INIS)

Giacomin, Giambattista; Pakdaman, Khashayar; Pellegrin, Xavier

2012-01-01

We study the dynamics of the large N limit of the Kuramoto model of coupled phase oscillators, subject to white noise. We introduce the notion of shadow inertial manifold and we prove their existence for this model, supporting the fact that the long-term dynamics of this model is finite dimensional. Following this, we prove that the global attractor of this model takes one of two forms. When coupling strength is below a critical value, the global attractor is a single equilibrium point corresponding to an incoherent state. Otherwise, when coupling strength is beyond this critical value, the global attractor is a two-dimensional disc composed of radial trajectories connecting a saddle-point equilibrium (the incoherent state) to an invariant closed curve of locally stable equilibria (partially synchronized state). Our analysis hinges, on the one hand, upon sharp existence and uniqueness results and their consequence for the existence of a global attractor, and, on the other hand, on the study of the dynamics in the vicinity of the incoherent and coherent (or synchronized) equilibria. We prove in particular nonlinear stability of each synchronized equilibrium, and normal hyperbolicity of the set of such equilibria. We explore mathematically and numerically several properties of the global attractor, in particular we discuss the limit of this attractor as noise intensity decreases to zero

Finite difference method for inner-layer equations in the resistive MagnetoHydroDynamic stability analysis

International Nuclear Information System (INIS)

Tokuda, Shinji; Watanabe, Tomoko.

1996-08-01

The matching problem in resistive MagnetoHydroDynamic stability analysis by the asymptotic matching method has been reformulated as an initial-boundary value problem for the inner-layer equations describing the plasma dynamics in the thin layer around a rational surface. The third boundary conditions at boundaries of a finite interval are imposed on the inner layer equations in the formulation instead of asymptotic conditions at infinities. The finite difference method for this problem has been applied to model equations whose solutions are known in a closed form. It has been shown that the initial value problem and the associated eigenvalue problem for the model equations can be solved by the finite difference method with numerical stability. The formulation presented here enables the asymptotic matching method to be a practical method for the resistive MHD stability analysis. (author)

Stabilizing local boundary conditions for two-dimensional shallow water equations

KAUST Repository

Dia, Ben Mansour; Oppelstrup, Jesper

2018-01-01

In this article, we present a sub-critical two-dimensional shallow water flow regulation. From the energy estimate of a set of one-dimensional boundary stabilization problems, we obtain a set of polynomial equations with respect to the boundary

W-transform for exponential stability of second order delay differential equations without damping terms.

Science.gov (United States)

Domoshnitsky, Alexander; Maghakyan, Abraham; Berezansky, Leonid

2017-01-01

In this paper a method for studying stability of the equation [Formula: see text] not including explicitly the first derivative is proposed. We demonstrate that although the corresponding ordinary differential equation [Formula: see text] is not exponentially stable, the delay equation can be exponentially stable.

Role of secondary instability theory and parabolized stability equations in transition modeling

Science.gov (United States)

El-Hady, Nabil M.; Dinavahi, Surya P.; Chang, Chau-Lyan; Zang, Thomas A.

1993-01-01

In modeling the laminar-turbulent transition region, the designer depends largely on benchmark data from experiments and/or direct numerical simulations that are usually extremely expensive. An understanding of the evolution of the Reynolds stresses, turbulent kinetic energy, and quantifies in the transport equations like the dissipation and production is essential in the modeling process. The secondary instability theory and the parabolized stability equations method are used to calculate these quantities, which are then compared with corresponding quantities calculated from available direct numerical simulation data for the incompressible boundary-layer flow of laminar-turbulent transition conditions. The potential of the secondary instability theory and the parabolized stability equations approach in predicting these quantities is discussed; results indicate that inexpensive data that are useful for transition modeling in the early stages of the transition region can be provided by these tools.

Analysis of stability and Hopf bifurcation for a delayed logistic equation

International Nuclear Information System (INIS)

Sun Chengjun; Han Maoan; Lin Yiping

2007-01-01

The dynamics of a logistic equation with discrete delay are investigated, together with the local and global stability of the equilibria. In particular, the conditions under which a sequence of Hopf bifurcations occur at the positive equilibrium are obtained. Explicit algorithm for determining the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation are derived by using the theory of normal form and center manifold [Hassard B, Kazarino D, Wan Y. Theory and applications of Hopf bifurcation. Cambridge: Cambridge University Press; 1981.]. Global existence of periodic solutions is also established by using a global Hopf bifurcation result of Wu [Symmetric functional differential equations and neural networks with memory. Trans Amer Math Soc 350:1998;4799-38.

STABILITY OF A FUNCTIONAL EQUATION IN COMPLEX BANACH SPACES

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PRATAP MONDAL

2016-12-01

Full Text Available Using fixed point technique, in the present paper , we wish to examine gen- eralization of the Hyers-Ulam-Rassias stability theorem for the functional equations f ( 2 x + i y + f ( x + 2 i y = 4 f ( x + i y + f ( x + f ( y (0.1 and f ( 2 x + i y .

Hyers-Ulam stability of linear second-order differential equations in complex Banach spaces

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Yongjin Li

2013-08-01

Full Text Available We prove the Hyers-Ulam stability of linear second-order differential equations in complex Banach spaces. That is, if y is an approximate solution of the differential equation $y''+ alpha y'(t +eta y = 0$ or $y''+ alpha y'(t +eta y = f(t$, then there exists an exact solution of the differential equation near to y.

Stability properties of solitary waves for fractional KdV and BBM equations

Science.gov (United States)

Angulo Pava, Jaime

2018-03-01

This paper sheds new light on the stability properties of solitary wave solutions associated with Korteweg-de Vries-type models when the dispersion is very low. Using a compact, analytic approach and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so a criterium of spectral instability of solitary waves is obtained for both models. Moreover, the nonlinear stability and spectral instability of the ground state solutions for both models is obtained for some specific regimen of parameters. Via a Lyapunov strategy and a variational analysis, we obtain the stability of the blow-up of solitary waves for the critical fractional KdV equation. The arguments presented in this investigation show promise for use in the study of the instability of traveling wave solutions of other nonlinear evolution equations.

Derivation of the physical equations solved in the inertial confinement stability code DOC. Informal report

International Nuclear Information System (INIS)

Scannapieco, A.J.; Cranfill, C.W.

1978-11-01

There now exists an inertial confinement stability code called DOC, which runs as a postprocessor. DOC (a code that has evolved from a previous code, PANSY) is a spherical harmonic linear stability code that integrates, in time, a set of Lagrangian perturbation equations. Effects due to real equations of state, asymmetric energy deposition, thermal conduction, shock propagation, and a time-dependent zeroth-order state are handled in the code. We present here a detailed derivation of the physical equations that are solved in the code

Derivation of the physical equations solved in the inertial confinement stability code DOC. Informal report

Energy Technology Data Exchange (ETDEWEB)

Scannapieco, A.J.; Cranfill, C.W.

1978-11-01

There now exists an inertial confinement stability code called DOC, which runs as a postprocessor. DOC (a code that has evolved from a previous code, PANSY) is a spherical harmonic linear stability code that integrates, in time, a set of Lagrangian perturbation equations. Effects due to real equations of state, asymmetric energy deposition, thermal conduction, shock propagation, and a time-dependent zeroth-order state are handled in the code. We present here a detailed derivation of the physical equations that are solved in the code.

Numerical and analytical investigation of the chimera state excitation conditions in the Kuramoto-Sakaguchi oscillator network

Science.gov (United States)

Frolov, Nikita S.; Goremyko, Mikhail V.; Makarov, Vladimir V.; Maksimenko, Vladimir A.; Hramov, Alexander E.

2017-03-01

In this paper we study the conditions of chimera states excitation in ensemble of non-locally coupled Kuramoto-Sakaguchi (KS) oscillators. In the framework of current research we analyze the dynamics of the homogeneous network containing identical oscillators. We show the chimera state formation process is sensitive to the parameters of coupling kernel and to the KS network initial state. To perform the analysis we have used the Ott-Antonsen (OA) ansatz to consider the behavior of infinitely large KS network.

Fixed Point Methods in the Stability of the Cauchy Functional Equations

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Z. Dehvari

2013-03-01

Full Text Available By using the fixed point methods, we prove some generalized Hyers-Ulam stability of homomorphisms for Cauchy and CauchyJensen functional equations on the product algebras and on the triple systems.

Stability analysis for neutral stochastic differential equation of second order driven by Poisson jumps

Science.gov (United States)

Chadha, Alka; Bora, Swaroop Nandan

2017-11-01

This paper studies the existence, uniqueness, and exponential stability in mean square for the mild solution of neutral second order stochastic partial differential equations with infinite delay and Poisson jumps. By utilizing the Banach fixed point theorem, first the existence and uniqueness of the mild solution of neutral second order stochastic differential equations is established. Then, the mean square exponential stability for the mild solution of the stochastic system with Poisson jumps is obtained with the help of an established integral inequality.

Conformists and contrarians in a Kuramoto model with identical natural frequencies

Science.gov (United States)

Hong, Hyunsuk; Strogatz, Steven H.

2011-10-01

We consider a variant of the Kuramoto model in which all the oscillators are now assumed to have the same natural frequency, but some of them are negatively coupled to the mean field. These contrarian oscillators tend to align in antiphase with the mean field, whereas, the positively coupled conformist oscillators favor an in-phase relationship. The interplay between these effects can lead to rich dynamics. In addition to a splitting of the population into two diametrically opposed factions, the system can also display traveling waves, complete incoherence, and a blurred version of the two-faction state. Exact solutions for these states and their bifurcations are obtained by means of the Watanabe-Strogatz transformation and the Ott-Antonsen ansatz. Curiously, this system of oscillators with identical frequencies turns out to exhibit more complicated dynamics than its counterpart with heterogeneous natural frequencies.

Stability of the trivial solution for linear stochastic differential equations with Poisson white noise

International Nuclear Information System (INIS)

Grigoriu, Mircea; Samorodnitsky, Gennady

2004-01-01

Two methods are considered for assessing the asymptotic stability of the trivial solution of linear stochastic differential equations driven by Poisson white noise, interpreted as the formal derivative of a compound Poisson process. The first method attempts to extend a result for diffusion processes satisfying linear stochastic differential equations to the case of linear equations with Poisson white noise. The developments for the method are based on Ito's formula for semimartingales and Lyapunov exponents. The second method is based on a geometric ergodic theorem for Markov chains providing a criterion for the asymptotic stability of the solution of linear stochastic differential equations with Poisson white noise. Two examples are presented to illustrate the use and evaluate the potential of the two methods. The examples demonstrate limitations of the first method and the generality of the second method

Ensemble/Variational Estimation (EnVE) and its application to canonical turbulent flow realizations

Science.gov (United States)

Colburn, Christopher; Cessna, Joseph; Bewley, Thomas

2008-11-01

The recently-developed hybrid EnVE method for data assimilation incorporates successive adjoint optimizations to update the initial conditions of a flow model, over various horizons of interest, in order to reconcile this model with recent measurements. Such adjoint optimizations typically require the trajectory to be saved over the entire interval over which the optimization is performed; in high-dimensional systems, this can lead to significant storage problems, which can be partially alleviated via checkpointing. In the EnVE framework, this requirement is eliminated, and supplanted by a requirement to march the state of the system backward in time simultaneously with the adjoint. If the system is derived from a PDE with a diffusive component, this backward-in-time state march is ill conditioned, and requires regularization/smoothing to prevent errors from accumulating rapidly at the small scales. The present talk focuses on this peculiar requirement of the EnVE algorithm. As the forecasting problem may itself be considered as a smoothing problem, it is, in fact, expected to find a ``smoothing'' ingredient at the heart of an algorithm of this sort. Various strategies are proposed and tested for accomplishing the required smoothing in the EnVE setting, and are tested on both a chaotic 1D PDE (the Kuramoto-Sivashinsky equation) as well as our in-house spectral 3D DNS/LES code, diablo.

Ulam-Hyers Stability of Trigonometric Functional Equation with Involution

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Jaeyoung Chung

2015-01-01

Full Text Available Let S and G be a commutative semigroup and a commutative group, respectively, C and R+ the sets of complex numbers and nonnegative real numbers, respectively, and σ:S→S or σ:G→G an involution. In this paper, we first investigate general solutions of the functional equation f(x+σy=f(xg(y-g(xf(y for all x,y∈S, where f,g:S→C. We then prove the Hyers-Ulam stability of the functional equation; that is, we study the functional inequality |f(x+σy-f(xg(y+g(xf(y|≤ψ(y for all x,y∈G, where f,g:G→C and ψ:G→R+.

Conformist-contrarian interactions and amplitude dependence in the Kuramoto model

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Lohe, M. A.

2014-11-01

We derive exact formulas for the frequency of synchronized oscillations in Kuramoto models with conformist-contrarian interactions, and determine necessary conditions for synchronization to occur. Numerical computations show that for certain parameters repulsive nodes behave as conformists, and that in other cases attractive nodes can display frustration, being neither conformist nor contrarian. The signs of repulsive couplings can be placed equivalently outside the sum, as proposed in Hong and Strogatz (2011 Phys. Rev. Lett. 106 054102), or inside the sum as in Hong and Strogatz (2012 Phys. Rev. E 85 056210), but the two models have different characteristics for small magnitudes of the coupling constants. In the latter case we show that the distributed coupling constants can be viewed as oscillator amplitudes which are constant in time, with the property that oscillators of small amplitude couple only weakly to connected nodes. Such models provide a means of investigating the effect of amplitude variations on synchronization properties.

Conformist–contrarian interactions and amplitude dependence in the Kuramoto model

International Nuclear Information System (INIS)

Lohe, M A

2014-01-01

We derive exact formulas for the frequency of synchronized oscillations in Kuramoto models with conformist–contrarian interactions, and determine necessary conditions for synchronization to occur. Numerical computations show that for certain parameters repulsive nodes behave as conformists, and that in other cases attractive nodes can display frustration, being neither conformist nor contrarian. The signs of repulsive couplings can be placed equivalently outside the sum, as proposed in Hong and Strogatz (2011 Phys. Rev. Lett. 106 054102), or inside the sum as in Hong and Strogatz (2012 Phys. Rev. E 85 056210), but the two models have different characteristics for small magnitudes of the coupling constants. In the latter case we show that the distributed coupling constants can be viewed as oscillator amplitudes which are constant in time, with the property that oscillators of small amplitude couple only weakly to connected nodes. Such models provide a means of investigating the effect of amplitude variations on synchronization properties. (paper)

Stabilized linear semi-implicit schemes for the nonlocal Cahn-Hilliard equation

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Du, Qiang; Ju, Lili; Li, Xiao; Qiao, Zhonghua

2018-06-01

Comparing with the well-known classic Cahn-Hilliard equation, the nonlocal Cahn-Hilliard equation is equipped with a nonlocal diffusion operator and can describe more practical phenomena for modeling phase transitions of microstructures in materials. On the other hand, it evidently brings more computational costs in numerical simulations, thus efficient and accurate time integration schemes are highly desired. In this paper, we propose two energy-stable linear semi-implicit methods with first and second order temporal accuracies respectively for solving the nonlocal Cahn-Hilliard equation. The temporal discretization is done by using the stabilization technique with the nonlocal diffusion term treated implicitly, while the spatial discretization is carried out by the Fourier collocation method with FFT-based fast implementations. The energy stabilities are rigorously established for both methods in the fully discrete sense. Numerical experiments are conducted for a typical case involving Gaussian kernels. We test the temporal convergence rates of the proposed schemes and make a comparison of the nonlocal phase transition process with the corresponding local one. In addition, long-time simulations of the coarsening dynamics are also performed to predict the power law of the energy decay.

Dependence of synchronization frequency of Kuramoto oscillators on symmetry of intrinsic frequency in ring network

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Saha, Arindam; Amritkar, R. E.

2014-12-01

Kuramoto oscillators have been proposed earlier as a model for interacting systems that exhibit synchronisation. In this article we study the difference between networks with symmetric and asymmetric distribution of natural frequencies. We first indicate that the synchronisation frequency of the oscillators is independent of the natural frequency distribution for a completely connected network. Further we analyse the case of oscillators in a directed ring-network where asymmetry in the natural frequency distribution is seen to shift the synchronisation frequency of the network. We also present an estimate of the shift in the frequencies for slightly asymmetric distributions.

One-way spatial integration of Navier-Stokes equations: stability of wall-bounded flows

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Rigas, Georgios; Colonius, Tim; Towne, Aaron; Beyar, Michael

2016-11-01

For three-dimensional flows, questions of stability, receptivity, secondary flows, and coherent structures require the solution of large partial-derivative eigenvalue problems. Reduced-order approximations are thus required for engineering prediction since these problems are often computationally intractable or prohibitively expensive. For spatially slowly evolving flows, such as jets and boundary layers, a regularization of the equations of motion sometimes permits a fast spatial marching procedure that results in a huge reduction in computational cost. Recently, a novel one-way spatial marching algorithm has been developed by Towne & Colonius. The new method overcomes the principle flaw observed in Parabolized Stability Equations (PSE), namely the ad hoc regularization that removes upstream propagating modes. The one-way method correctly parabolizes the flow equations based on estimating, in a computationally efficient way, the local spectrum in each cross-stream plane and an efficient spectral filter eliminates modes with upstream group velocity. Results from the application of the method to wall-bounded flows will be presented and compared with predictions from the full linearized compressible Navier-Stokes equations and PSE.

Stabilization and asymptotic behavior of a generalized telegraph equation

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Nicaise, Serge

2015-12-01

We analyze the stability of different models of the telegraph equation set in a real interval. They correspond to the coupling between a first-order hyperbolic system and a first-order differential equation of parabolic type. We show that some models have an exponential decay rate, while other ones are only polynomially stable. When the parameters are constant, we show that the obtained polynomial decay is optimal and in the case of an exponential decay that the decay rate is equal to the spectral abscissa. These optimality results are based on a careful spectral analysis of the operator. In particular, we characterize its full spectrum that is made of a discrete set of eigenvalues and an essential spectrum reduced to one point.

Stability estimates for solution of IBVP to fractional parabolic differential and difference equations

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Ashyralyev, Allaberen; Cakir, Zafer

2016-08-01

In this work, we investigate initial-boundary value problems for fractional parabolic equations with the Neumann boundary condition. Stability estimates for the solution of this problem are established. Difference schemes for approximate solution of initial-boundary value problem are constructed. Furthermore, we give theorem on coercive stability estimates for the solution of the difference schemes.

Stabilization of the Wave Equation with Boundary Time-Varying Delay

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Hao Li

2014-01-01

Full Text Available We study the stabilization of the wave equation with variable coefficients in a bounded domain and a time-varying delay term in the time-varying, weakly nonlinear boundary feedbacks. By the Riemannian geometry methods and a suitable assumption of nonlinearity, we obtain the uniform decay of the energy of the closed loop system.

Stability of Jensen functional equation in intuitionistic fuzzy normed space

International Nuclear Information System (INIS)

Mohiuddine, S.A.

2009-01-01

In this paper, we determine some stability results concerning the Jensen functional equation 2f((x+y)/2)=f(x)+f(y) in intuitionistic fuzzy normed spaces (IFNS). We define the intuitionistic fuzzy continuity of the Jensen mappings and prove that the existence of a solution for any approximately Jensen mapping implies the completeness of IFNS.

Stability of a nonlinear second order equation under parametric bounded noise excitation

International Nuclear Information System (INIS)

Wiebe, Richard; Xie, Wei-Chau

2016-01-01

The motivation for the following work is a structural column under dynamic axial loads with both deterministic (harmonic transmitted forces from the surrounding structure) and random (wind and/or earthquake) loading components. The bounded noise used herein is a sinusoid with an argument composed of a random (Wiener) process deviation about a mean frequency. By this approach, a noise parameter may be used to investigate the behavior through the spectrum from simple harmonic forcing, to a bounded random process with very little harmonic content. The stability of both the trivial and non-trivial stationary solutions of an axially-loaded column (which is modeled as a second order nonlinear equation) under parametric bounded noise excitation is investigated by use of Lyapunov exponents. Specifically the effect of noise magnitude, amplitude of the forcing, and damping on stability of a column is investigated. First order averaging is employed to obtain analytical approximations of the Lyapunov exponents of the trivial solution. For the non-trivial stationary solution however, the Lyapunov exponents are obtained via Monte Carlo simulation as the stability equations become analytically intractable. (paper)

Stability of oscillatory solutions of differential equations with a general piecewise constant argument

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Kuo-Shou Chiu

2011-11-01

Full Text Available We examine scalar differential equations with a general piecewise constant argument, in short DEPCAG, that is, the argument is a general step function. Criteria of existence of the oscillatory and nonoscillatory solutions of such equations are proposed. Necessary and sufficient conditions for stability of the zero solution are obtained. Appropriate examples are given to show our results.

Contractivity and Exponential Stability of Solutions to Nonlinear Neutral Functional Differential Equations in Banach Spaces

Institute of Scientific and Technical Information of China (English)

Wan-sheng WANG; Shou-fu LI; Run-sheng YANG

2012-01-01

A series of contractivity and exponential stability results for the solutions to nonlinear neutral functional differential equations (NFDEs) in Banach spaces are obtained,which provide unified theoretical foundation for the contractivity analysis of solutions to nonlinear problems in functional differential equations (FDEs),neutral delay differential equations (NDDEs) and NFDEs of other types which appear in practice.

Stability of the equation of homomorphism and completeness of the underlying space

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Zenon Moszner

2008-01-01

Full Text Available We prove that all assumptions of a Theorem of Forti and Schwaiger (cf. [G. L. Forti, J. Schwaiger, Stability of homomorphisms and completeness, C. R. Math. Rep. Acad. Sci. Canada 11 (1989, 215–220] on the coherence of stability of the equation of homomorphism with the completeness of the space of values of all these homomorphisms, are essential. We give some generalizations of this theorem and certain examples of applications.

Stability of abstract nonlinear nonautonomous differential-delay equations with unbounded history-responsive operators

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Gil', M. I.

2005-08-01

We consider a class of nonautonomous functional-differential equations in a Banach space with unbounded nonlinear history-responsive operators, which have the local Lipshitz property. Conditions for the boundedness of solutions, Lyapunov stability, absolute stability and input-output one are established. Our approach is based on a combined usage of properties of sectorial operators and spectral properties of commuting operators.

Stability and square integrability of solutions of nonlinear fourth order differential equations

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Moussadek Remili

2016-05-01

Full Text Available The aim of the present paper is to establish a new result, which guarantees the asymptotic stability of zero solution and square integrability of solutions and their derivatives to nonlinear differential equations of fourth order.

Nonlinear stability of source defects in the complex Ginzburg–Landau equation

International Nuclear Information System (INIS)

Beck, Margaret; Nguyen, Toan T; Sandstede, Björn; Zumbrun, Kevin

2014-01-01

In an appropriate moving coordinate frame, source defects are time-periodic solutions to reaction–diffusion equations that are spatially asymptotic to spatially periodic wave trains whose group velocities point away from the core of the defect. In this paper, we rigorously establish nonlinear stability of spectrally stable source defects in the complex Ginzburg–Landau equation. Due to the outward transport at the far field, localized perturbations may lead to a highly non-localized response even on the linear level. To overcome this, we first investigate in detail the dynamics of the solution to the linearized equation. This allows us to determine an approximate solution that satisfies the full equation up to and including quadratic terms in the nonlinearity. This approximation utilizes the fact that the non-localized phase response, resulting from the embedded zero eigenvalues, can be captured, to leading order, by the nonlinear Burgers equation. The analysis is completed by obtaining detailed estimates for the resolvent kernel and pointwise estimates for Green's function, which allow one to close a nonlinear iteration scheme. (paper)

Stability of Pexiderized Quadratic Functional Equation in Random 2-Normed Spaces

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Mohammed A. Alghamdi

2015-01-01

Full Text Available The aim of this paper is to investigate the stability of Hyers-Ulam-Rassias type theorems by considering the pexiderized quadratic functional equation in the setting of random 2-normed spaces (RTNS, while the concept of random 2-normed space has been recently studied by Goleţ (2005.

PEAK COVARIANCE STABILITY OF A RANDOM RICCATI EQUATION ARISING FROM KALMAN FILTERING WITH OBSERVATION LOSSES

Institute of Scientific and Technical Information of China (English)

Li XIE; Lihua XIE

2007-01-01

We consider the stability of a random Riccati equation with a Markovian binary jump coefficient. More specifically, we are concerned with the boundedness of the solution of a random Riccati difference equation arising from Kalman filtering with measurement losses. A sufficient condition for the peak covariance stability is obtained which has a simpler form and is shown to be less conservative in some cases than a very recent result in existing literature. Furthermore, we show that a known sufficient condition is also necessary when the observability index equals one.

Principles and equations for measuring and interpreting protein stability: From monomer to tetramer.

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Bedouelle, Hugues

2016-02-01

The ability to measure the thermodynamic stability of proteins with precision is important for both academic and applied research. Such measurements rely on mathematical models of the protein denaturation profile, i.e. the relation between a global protein signal, corresponding to the folding states in equilibrium, and the variable value of a denaturing agent, either heat or a chemical molecule, e.g. urea or guanidinium hydrochloride. In turn, such models rely on a handful of physical laws: the laws of mass action and conservation, the law that relates the protein signal and concentration, and the one that relates stability and denaturant value. So far, equations have been derived mainly for the denaturation profiles of homomeric proteins. Here, we review the underlying basic physical laws and show in detail how to derive model equations for the unfolding equilibria of homomeric or heteromeric proteins up to trimers and potentially tetramers, with or without folding intermediates, and give full demonstrations. We show that such equations cannot be derived for pentamers or higher oligomers except in special degenerate cases. We expand the method to signals that do not correspond to extensive protein properties. We review and expand methods for uncovering hidden intermediates of unfolding. Finally, we review methods for comparing and interpreting the thermodynamic parameters that derive from stability measurements for cognate wild-type and mutant proteins. This work should provide a robust theoretical basis for measuring the stability of complex proteins. Copyright © 2015 Elsevier B.V. and Société Française de Biochimie et Biologie Moléculaire (SFBBM). All rights reserved.

Existence and Stability of Traveling Waves for Degenerate Reaction-Diffusion Equation with Time Delay

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Huang, Rui; Jin, Chunhua; Mei, Ming; Yin, Jingxue

2018-01-01

This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction-diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of c≥c^* for the degenerate reaction-diffusion equation without delay, where c^*>0 is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay τ >0 . Furthermore, we prove the global existence and uniqueness of C^{α ,β } -solution to the time-delayed degenerate reaction-diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted L^1 -space. The exponential convergence rate is also derived.

Existence and Stability of Traveling Waves for Degenerate Reaction-Diffusion Equation with Time Delay

Science.gov (United States)

Huang, Rui; Jin, Chunhua; Mei, Ming; Yin, Jingxue

2018-06-01

This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction-diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of c≥c^* for the degenerate reaction-diffusion equation without delay, where c^*>0 is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay τ >0. Furthermore, we prove the global existence and uniqueness of C^{α ,β }-solution to the time-delayed degenerate reaction-diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted L^1-space. The exponential convergence rate is also derived.

Almost Surely Asymptotic Stability of Exact and Numerical Solutions for Neutral Stochastic Pantograph Equations

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Zhanhua Yu

2011-01-01

Full Text Available We study the almost surely asymptotic stability of exact solutions to neutral stochastic pantograph equations (NSPEs, and sufficient conditions are obtained. Based on these sufficient conditions, we show that the backward Euler method (BEM with variable stepsize can preserve the almost surely asymptotic stability. Numerical examples are demonstrated for illustration.

Stabilization of wave equations with variable coefficient and delay in the dynamical boundary feedback

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Dandan Guo

2017-08-01

Full Text Available In this article we consider the boundary stabilization of a wave equation with variable coefficients. This equation has an acceleration term and a delayed velocity term on the boundary. Under suitable geometric conditions, we obtain the exponential decay for the solutions. Our proof relies on the geometric multiplier method and the Lyapunov approach.

Stability of stationary states of non-local equations with singular interaction potentials

KAUST Repository

Fellner, Klemens

2011-04-01

We study the large-time behaviour of a non-local evolution equation for the density of particles or individuals subject to an external and an interaction potential. In particular, we consider interaction potentials which are singular in the sense that their first derivative is discontinuous at the origin.For locally attractive singular interaction potentials we prove under a linear stability condition local non-linear stability of stationary states consisting of a finite sum of Dirac masses. For singular repulsive interaction potentials we show the stability of stationary states of uniformly bounded solutions under a convexity condition.Finally, we present numerical simulations to illustrate our results. © 2010 Elsevier Ltd.

Integral equation based stability analysis of short wavelength drift modes in tokamaks

International Nuclear Information System (INIS)

Hirose, A.; Elia, M.

2003-01-01

Linear stability of electron skin-size drift modes in collisionless tokamak discharges has been investigated in terms of electromagnetic, kinetic integral equations in which neither ions nor electrons are assumed to be adiabatic. A slab-like ion temperature gradient mode persists in such a short wavelength regime. However, toroidicity has a strong stabilizing influence on this mode. In the electron branch, the toroidicity induced skin-size drift mode previously predicted in terms of local kinetic analysis has been recovered. The mode is driven by positive magnetic shear and strongly stabilized for negative shear. The corresponding mixing length anomalous thermal diffusivity exhibits favourable isotope dependence. (author)

Stabilization of a semilinear parabolic equation in the exterior of a bounded domain by means of boundary controls

International Nuclear Information System (INIS)

Gorshkov, A V

2003-01-01

The problem of the stabilization of a semilinear equation in the exterior of a bounded domain is considered. In view of the impossibility of an exponential stabilization of the form e -σt of the solution of a parabolic equation in an unbounded domain no matter what the boundary control is, one poses the problem of power-like stabilization by means of a boundary control. For a fixed initial condition and parameter k>0 of the rate of stabilization the existence of a boundary control such that the solution approaches zero at the rate 1/t k is demonstrated

Higher-dimensional generalizations of the Watanabe–Strogatz transform for vector models of synchronization

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Lohe, M. A.

2018-06-01

We generalize the Watanabe–Strogatz (WS) transform, which acts on the Kuramoto model in d  =  2 dimensions, to a higher-dimensional vector transform which operates on vector oscillator models of synchronization in any dimension , for the case of identical frequency matrices. These models have conserved quantities constructed from the cross ratios of inner products of the vector variables, which are invariant under the vector transform, and have trajectories which lie on the unit sphere S d‑1. Application of the vector transform leads to a partial integration of the equations of motion, leaving independent equations to be solved, for any number of nodes N. We discuss properties of complete synchronization and use the reduced equations to derive a stability condition for completely synchronized trajectories on S d‑1. We further generalize the vector transform to a mapping which acts in and in particular preserves the unit ball , and leaves invariant the cross ratios constructed from inner products of vectors in . This mapping can be used to partially integrate a system of vector oscillators with trajectories in , and for d  =  2 leads to an extension of the Kuramoto system to a system of oscillators with time-dependent amplitudes and trajectories in the unit disk. We find an inequivalent generalization of the Möbius map which also preserves but leaves invariant a different set of cross ratios, this time constructed from the vector norms. This leads to a different extension of the Kuramoto model with trajectories in the complex plane that can be partially integrated by means of fractional linear transformations.

Stability equation and two-component Eigenmode for domain walls in scalar potential model

International Nuclear Information System (INIS)

Dias, G.S.; Graca, E.L.; Rodrigues, R. de Lima

2002-08-01

Supersymmetric quantum mechanics involving a two-component representation and two-component eigenfunctions is applied to obtain the stability equation associated to a potential model formulated in terms of two coupled real scalar fields. We investigate the question of stability by introducing an operator technique for the Bogomol'nyi-Prasad-Sommerfield (BPS) and non-BPS states on two domain walls in a scalar potential model with minimal N 1-supersymmetry. (author)

New stability and boundedness results to Volterra integro-differential equations with delay

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Cemil Tunç

2016-04-01

Full Text Available In this paper, we consider a certain non-linear Volterra integro-differential equations with delay. We study stability and boundedness of solutions. The technique of proof involves defining suitable Lyapunov functionals. Our results improve and extend the results obtained in literature.

Stability analysis of explicit entropy viscosity methods for non-linear scalar conservation equations

KAUST Repository

Bonito, Andrea; Guermond, Jean-Luc; Popov, Bojan

2013-01-01

We establish the L2-stability of an entropy viscosity technique applied to nonlinear scalar conservation equations. First-and second-order explicit time-stepping techniques using continuous finite elements in space are considered. The method

Stability by fixed point theory for functional differential equations

CERN Document Server

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

Stability of line solitons for the KP-II equation in R2

CERN Document Server

Mizumachi, Tetsu

2015-01-01

The author proves nonlinear stability of line soliton solutions of the KP-II equation with respect to transverse perturbations that are exponentially localized as x\\to\\infty. He finds that the amplitude of the line soliton converges to that of the line soliton at initial time whereas jumps of the local phase shift of the crest propagate in a finite speed toward y=\\pm\\infty. The local amplitude and the phase shift of the crest of the line solitons are described by a system of 1D wave equations with diffraction terms.

Cycle expansions: From maps to turbulence

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Lan, Y.

2010-03-01

We present a derivation, a physical explanation and applications of cycle expansions in different dynamical systems, ranging from simple one-dimensional maps to turbulence in fluids. Cycle expansion is a newly devised powerful tool for computing averages of physical observables in nonlinear chaotic systems which combines many innovative ideas developed in dynamical systems, such as hyperbolicity, invariant manifolds, symbolic dynamics, measure theory and thermodynamic formalism. The concept of cycle expansion has a deep root in theoretical physics, bearing a close analogy to cumulant expansion in statistical physics and effective action functional in quantum field theory, the essence of which is to represent a physical system in a hierarchical way by utilizing certain multiplicative structures such that the dominant parts of physical observables are captured by compact, maneuverable objects while minor detailed variations are described by objects with a larger space and time scale. The technique has been successfully applied to many low-dimensional dynamical systems and much effort has recently been made to extend its use to spatially extended systems. For one-dimensional systems such as the Kuramoto-Sivashinsky equation, the method turns out to be very effective while for more complex real-world systems including the Navier-Stokes equation, the method is only starting to yield its first fruits and much more work is needed to enable practical computations. However, the experience and knowledge accumulated so far is already very useful to a large set of research problems. Several such applications are briefly described in what follows. As more research effort is devoted to the study of complex dynamics of nonlinear systems, cycle expansion will undergo a fast development and find wide applications.

Full-order optimal compensators for flow control: the multiple inputs case

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Semeraro, Onofrio; Pralits, Jan O.

2018-03-01

Flow control has been the subject of numerous experimental and theoretical works. We analyze full-order, optimal controllers for large dynamical systems in the presence of multiple actuators and sensors. The full-order controllers do not require any preliminary model reduction or low-order approximation: this feature allows us to assess the optimal performance of an actuated flow without relying on any estimation process or further hypothesis on the disturbances. We start from the original technique proposed by Bewley et al. (Meccanica 51(12):2997-3014, 2016. https://doi.org/10.1007/s11012-016-0547-3), the adjoint of the direct-adjoint (ADA) algorithm. The algorithm is iterative and allows bypassing the solution of the algebraic Riccati equation associated with the optimal control problem, typically infeasible for large systems. In this numerical work, we extend the ADA iteration into a more general framework that includes the design of controllers with multiple, coupled inputs and robust controllers (H_{∞} methods). First, we demonstrate our results by showing the analytical equivalence between the full Riccati solutions and the ADA approximations in the multiple inputs case. In the second part of the article, we analyze the performance of the algorithm in terms of convergence of the solution, by comparing it with analogous techniques. We find an excellent scalability with the number of inputs (actuators), making the method a viable way for full-order control design in complex settings. Finally, the applicability of the algorithm to fluid mechanics problems is shown using the linearized Kuramoto-Sivashinsky equation and the Kármán vortex street past a two-dimensional cylinder.

Stability of Nonlinear Neutral Stochastic Functional Differential Equations

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Minggao Xue

2010-01-01

Full Text Available Neutral stochastic functional differential equations (NSFDEs have recently been studied intensively. The well-known conditions imposed for the existence and uniqueness and exponential stability of the global solution are the local Lipschitz condition and the linear growth condition. Therefore, the existing results cannot be applied to many important nonlinear NSFDEs. The main aim of this paper is to remove the linear growth condition and establish a Khasminskii-type test for nonlinear NSFDEs. New criteria not only cover a wide class of highly nonlinear NSFDEs but they can also be verified much more easily than the classical criteria. Finally, several examples are given to illustrate main results.

Orbital stability of periodic traveling-wave solutions for the log-KdV equation

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Natali, Fábio; Pastor, Ademir; Cristófani, Fabrício

2017-09-01

In this paper we establish the orbital stability of periodic waves related to the logarithmic Korteweg-de Vries equation. Our motivation is inspired in the recent work [3], in which the authors established the well-posedness and the linear stability of Gaussian solitary waves. By using the approach put forward recently in [20] to construct a smooth branch of periodic waves as well as to get the spectral properties of the associated linearized operator, we apply the abstract theories in [13] and [25] to deduce the orbital stability of the periodic traveling waves in the energy space.

Improved algorithm for solving nonlinear parabolized stability equations

International Nuclear Information System (INIS)

Zhao Lei; Zhang Cun-bo; Liu Jian-xin; Luo Ji-sheng

2016-01-01

Due to its high computational efficiency and ability to consider nonparallel and nonlinear effects, nonlinear parabolized stability equations (NPSE) approach has been widely used to study the stability and transition mechanisms. However, it often diverges in hypersonic boundary layers when the amplitude of disturbance reaches a certain level. In this study, an improved algorithm for solving NPSE is developed. In this algorithm, the mean flow distortion is included into the linear operator instead of into the nonlinear forcing terms in NPSE. An under-relaxation factor for computing the nonlinear terms is introduced during the iteration process to guarantee the robustness of the algorithm. Two case studies, the nonlinear development of stationary crossflow vortices and the fundamental resonance of the second mode disturbance in hypersonic boundary layers, are presented to validate the proposed algorithm for NPSE. Results from direct numerical simulation (DNS) are regarded as the baseline for comparison. Good agreement can be found between the proposed algorithm and DNS, which indicates the great potential of the proposed method on studying the crossflow and streamwise instability in hypersonic boundary layers. (paper)

Method of Lyapunov functions in problems of stability of solutions of systems of differential equations with impulse action

International Nuclear Information System (INIS)

Ignat'yev, A O

2003-01-01

A system of ordinary differential equations with impulse action at fixed moments of time is considered. The system is assumed to have the zero solution. It is shown that the existence of a corresponding Lyapunov function is a necessary and sufficient condition for the uniform asymptotic stability of the zero solution. Restrictions on perturbations of the right-hand sides of differential equations and impulse actions are obtained under which the uniform asymptotic stability of the zero solution of the 'unperturbed' system implies the uniform asymptotic stability of the zero solution of the 'perturbed' system

Stability and bifurcation of numerical discretization of a second-order delay differential equation with negative feedback

International Nuclear Information System (INIS)

Ding Xiaohua; Su Huan; Liu Mingzhu

2008-01-01

The paper analyzes a discrete second-order, nonlinear delay differential equation with negative feedback. The characteristic equation of linear stability is solved, as a function of two parameters describing the strength of the feedback and the damping in the autonomous system. The existence of local Hopf bifurcations is investigated, and the direction and stability of periodic solutions bifurcating from the Hopf bifurcation of the discrete model are determined by the Hopf bifurcation theory of discrete system. Finally, some numerical simulations are performed to illustrate the analytical results found

Theoretical stability in coefficient inverse problems for general hyperbolic equations with numerical reconstruction

Science.gov (United States)

Yu, Jie; Liu, Yikan; Yamamoto, Masahiro

2018-04-01

In this article, we investigate the determination of the spatial component in the time-dependent second order coefficient of a hyperbolic equation from both theoretical and numerical aspects. By the Carleman estimates for general hyperbolic operators and an auxiliary Carleman estimate, we establish local Hölder stability with either partial boundary or interior measurements under certain geometrical conditions. For numerical reconstruction, we minimize a Tikhonov functional which penalizes the gradient of the unknown function. Based on the resulting variational equation, we design an iteration method which is updated by solving a Poisson equation at each step. One-dimensional prototype examples illustrate the numerical performance of the proposed iteration.

Volume-preserving normal forms of Hopf-zero singularity

Science.gov (United States)

Gazor, Majid; Mokhtari, Fahimeh

2013-10-01

A practical method is described for computing the unique generator of the algebra of first integrals associated with a large class of Hopf-zero singularity. The set of all volume-preserving classical normal forms of this singularity is introduced via a Lie algebra description. This is a maximal vector space of classical normal forms with first integral; this is whence our approach works. Systems with a nonzero condition on their quadratic parts are considered. The algebra of all first integrals for any such system has a unique (modulo scalar multiplication) generator. The infinite level volume-preserving parametric normal forms of any nondegenerate perturbation within the Lie algebra of any such system is computed, where it can have rich dynamics. The associated unique generator of the algebra of first integrals are derived. The symmetry group of the infinite level normal forms are also discussed. Some necessary formulas are derived and applied to appropriately modified Rössler and generalized Kuramoto-Sivashinsky equations to demonstrate the applicability of our theoretical results. An approach (introduced by Iooss and Lombardi) is applied to find an optimal truncation for the first level normal forms of these examples with exponentially small remainders. The numerically suggested radius of convergence (for the first integral) associated with a hypernormalization step is discussed for the truncated first level normal forms of the examples. This is achieved by an efficient implementation of the results using Maple.

Energy Stability Analysis of Some Fully Discrete Numerical Schemes for Incompressible Navier–Stokes Equations on Staggered Grids

KAUST Repository

Chen, Huangxin

2017-09-01

In this paper we consider the energy stability estimates for some fully discrete schemes which both consider time and spatial discretizations for the incompressible Navier–Stokes equations. We focus on three kinds of fully discrete schemes, i.e., the linear implicit scheme for time discretization with the finite difference method (FDM) on staggered grids for spatial discretization, pressure-correction schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations, and pressure-stabilization schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations. The energy stability estimates are obtained for the above each fully discrete scheme. The upwind scheme is used in the discretization of the convection term which plays an important role in the design of unconditionally stable discrete schemes. Numerical results are given to verify the theoretical analysis.

A remark on the stability and boundedness criteria in retarded Volterra integro-differential equations

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Cemil Tunç

2017-10-01

Full Text Available In this article, the authors obtain some clear assumptions for the asymptotic stability (AS and boundedness (B of solutions of non-linear retarded Volterra integro-differential equations (VIDEs of first order by constructing a new Lyapunov functional (LF. The results obtained are new and differ from those found in the literature, and they also contain and improve a result found in the literature under more less restrictive conditions. We establish an example and give a discussion to indicate the applicability of the weaker conditions obtained. We also employ MATLAB-Simulink to display the behaviors of the orbits of the (VIDEs considered. Keywords: Nonlinear, Volterra integro-differential equations, First order, Asymptotic stability, Boundedness, Lyapunov functional, MSC: 34D05, 34K20, 45J05

Well-posedness and exponential stability for a wave equation with nonlocal time-delay condition

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Carlos Alberto Raposo

2017-11-01

Full Text Available Well-posedness and exponential stability of nonlocal time-delayed of a wave equation with a integral conditions of the 1st kind forms the center of this work. Through semigroup theory we prove the well-posedness by the Hille-Yosida theorem and the exponential stability exploring the dissipative properties of the linear operator associated to damped model using the Gearhart-Huang-Pruss theorem.

Stability of finite difference schemes for generalized von Foerster equations with renewal

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Henryk Leszczyński

2014-01-01

Full Text Available We consider a von Foerster-type equation describing the dynamics of a population with the production of offsprings given by the renewal condition. We construct a finite difference scheme for this problem and give sufficient conditions for its stability with respect to \\(l^1\\ and \\(l^\\infty\\ norms.

MAIA, Eigenvalues for MHD Equation of Tokamak Plasma Stability Problems

International Nuclear Information System (INIS)

Tanaka, Y.; Azumi, M.; Kurita, G.; Tsunematsu, T.; Takeda, T.

1986-01-01

1 - Description of program or function: This program solves an eigenvalue problem zBx=Ax where A and B are real block tri-diagonal matrices. This eigenvalue problem is derived from a reduced set of linear resistive MHD equations which is often employed to study tokamak plasma stability problem. 2 - Method of solution: Both the determinant and inverse iteration methods are employed. 3 - Restrictions on the complexity of the problem: The eigenvalue z must be real

Stability analysis of implicit time discretizations for the Compton-scattering Fokker-Planck equation

International Nuclear Information System (INIS)

Densmore, Jeffery D.; Warsa, James S.; Lowrie, Robert B.; Morel, Jim E.

2009-01-01

The Fokker-Planck equation is a widely used approximation for modeling the Compton scattering of photons in high energy density applications. In this paper, we perform a stability analysis of three implicit time discretizations for the Compton-Scattering Fokker-Planck equation. Specifically, we examine (i) a Semi-Implicit (SI) scheme that employs backward-Euler differencing but evaluates temperature-dependent coefficients at their beginning-of-time-step values, (ii) a Fully Implicit (FI) discretization that instead evaluates temperature-dependent coefficients at their end-of-time-step values, and (iii) a Linearized Implicit (LI) scheme, which is developed by linearizing the temperature dependence of the FI discretization within each time step. Our stability analysis shows that the FI and LI schemes are unconditionally stable and cannot generate oscillatory solutions regardless of time-step size, whereas the SI discretization can suffer from instabilities and nonphysical oscillations for sufficiently large time steps. With the results of this analysis, we present time-step limits for the SI scheme that prevent undesirable behavior. We test the validity of our stability analysis and time-step limits with a set of numerical examples.

Stability analysis of implicit time discretizations for the Compton-scattering Fokker-Planck equation

Energy Technology Data Exchange (ETDEWEB)

Densmore, Jeffery D [Los Alamos National Laboratory; Warsa, James S [Los Alamos National Laboratory; Lowrie, Robert B [Los Alamos National Laboratory; Morel, Jim E [TEXAS A& M UNIV

2008-01-01

The Fokker-Planck equation is a widely used approximation for modeling the Compton scattering of photons in high energy density applications. In this paper, we perform a stability analysis of three implicit time discretizations for the Compton-Scattering Fokker-Planck equation. Specifically, we examine (i) a Semi-Implicit (SI) scheme that employs backward-Euler differencing but evaluates temperature-dependent coefficients at their beginning-of-time-step values, (ii) a Fully Implicit (FI) discretization that instead evaluates temperature-dependent coefficients at their end-of-time-step values, and (iii) a Linearized Implicit (LI) scheme, which is developed by linearizing the temperature dependence of the FI discretization within each time step. Our stability analysis shows that the FI and LI schemes are unconditionally stable and cannot generate oscillatory solutions regardless of time-step size, whereas the SI discretization can suffer from instabilities and nonphysical oscillations for sufficiently large time steps. With the results of this analysis, we present time-step limits for the SI scheme that prevent undesirable behavior. We test the validity of our stability analysis and time-step limits with a set of numerical examples.

Stability analysis of implicit time discretizations for the Compton-scattering Fokker-Planck equation

Science.gov (United States)

Densmore, Jeffery D.; Warsa, James S.; Lowrie, Robert B.; Morel, Jim E.

2009-09-01

The Fokker-Planck equation is a widely used approximation for modeling the Compton scattering of photons in high energy density applications. In this paper, we perform a stability analysis of three implicit time discretizations for the Compton-Scattering Fokker-Planck equation. Specifically, we examine (i) a Semi-Implicit (SI) scheme that employs backward-Euler differencing but evaluates temperature-dependent coefficients at their beginning-of-time-step values, (ii) a Fully Implicit (FI) discretization that instead evaluates temperature-dependent coefficients at their end-of-time-step values, and (iii) a Linearized Implicit (LI) scheme, which is developed by linearizing the temperature dependence of the FI discretization within each time step. Our stability analysis shows that the FI and LI schemes are unconditionally stable and cannot generate oscillatory solutions regardless of time-step size, whereas the SI discretization can suffer from instabilities and nonphysical oscillations for sufficiently large time steps. With the results of this analysis, we present time-step limits for the SI scheme that prevent undesirable behavior. We test the validity of our stability analysis and time-step limits with a set of numerical examples.

Oscillation and asymptotic stability of a delay differential equation with Richard's nonlinearity

Directory of Open Access Journals (Sweden)

Leonid Berezansky

2005-04-01

Full Text Available We obtain sufficient conditions for oscillation of solutions, and for asymptotical stability of the positive equilibrium, of the scalar nonlinear delay differential equation $$ frac{dN}{dt} = r(tN(tBig[a-Big(sum_{k=1}^m b_k N(g_k(tBig^{gamma}Big], $$ where $ g_k(tleq t$.

Lyapunov stability and poisson structure of the thermal TDHF and RPA equations

International Nuclear Information System (INIS)

Balian, R.; Veneroni, M.

1989-01-01

The thermal TDHF equation is analyzed in the Liouville representation of quantum mechanics, where the matrix elements of the single-particle (s.p) density ρ behave as classical dynamical variables. By introducing the Lie--Poisson bracket associated with the unitary group of the s.p. Hilbert space, we show that TDHF has a Hamiltonian, but non-canonical, classical form. Within this Poisson structure, either the s.p. energy or the s.p. grand potential Ω(ρ) act as a Hamilton function. The Lyapunov stability of both the TDHF and RPA equations around a HF state then follows, since the HF approximation for thermal equilibrium is determined by minimizing Ω(ρ). The RPA matrix in the Liouville space is expressed as the product of the Poisson tensor with the HF stability matrix, interpreted as a metric tensor generated by the entropy. This factorization displays the roles of the energy and entropy terms arising from Ω(ρ) in the RPA dynamics, and it helps to construct the RPA modes. Several extensions are considered. copyright 1989 Academic Press, Inc

Lyapunov stability and Poisson structure of the thermal TDHF and RPA equations

International Nuclear Information System (INIS)

Veneroni, M.; Balian, R.

1989-01-01

The thermal TDHF equation is analyzed in the Liouville representation of quantum mechanics, where the matrix elements of the single-particle (s.p.) density ρ behave as classical dynamical variables. By introducing the Lie-Poisson bracket associated with the unitary group of the s.p. Hilbert space, we show that TDHF has a hamiltonian, but non-canonical, classical form. Within this Poisson structure, either the s.p. energy or the s.p. grand potential Ω(ρ) act as a Hamilton function. The Lyapunov stability of both the TDHF and RPA equations around a HF state then follows, since the HF approximation for thermal equilibrium is determined by minimizing Ω(ρ). The RPA matrix in the Liouville space is expressed as the product of the Poisson tensor with the HF stability matrix, interpreted as a metric tensor generated by the entropy. This factorization displays the roles of the energy and entropy terms arising from Ω(ρ) in the RPA dynamics, and it helps to construct the RPA modes. Several extensions are considered

The stability of coupled renewal-differential equations with econometric applications

Science.gov (United States)

Rhoten, R. P.; Aggarwal, J. K.

1969-01-01

Concepts and results are presented in the fields of mathematical modeling, economics, and stability analysis. A coupled renewal-differential equation structure is presented as a modeling form for systems possessing hereditary characteristics, and this structure is applied to a model of the Austrian theory of business cycles. For realistic conditions, the system is shown to have an infinite number of poles, and conditions are presented which are both necessary and sufficient for all poles to lie strictly in the left half plane.

Improved algorithm for solving nonlinear parabolized stability equations

Science.gov (United States)

Zhao, Lei; Zhang, Cun-bo; Liu, Jian-xin; Luo, Ji-sheng

2016-08-01

Due to its high computational efficiency and ability to consider nonparallel and nonlinear effects, nonlinear parabolized stability equations (NPSE) approach has been widely used to study the stability and transition mechanisms. However, it often diverges in hypersonic boundary layers when the amplitude of disturbance reaches a certain level. In this study, an improved algorithm for solving NPSE is developed. In this algorithm, the mean flow distortion is included into the linear operator instead of into the nonlinear forcing terms in NPSE. An under-relaxation factor for computing the nonlinear terms is introduced during the iteration process to guarantee the robustness of the algorithm. Two case studies, the nonlinear development of stationary crossflow vortices and the fundamental resonance of the second mode disturbance in hypersonic boundary layers, are presented to validate the proposed algorithm for NPSE. Results from direct numerical simulation (DNS) are regarded as the baseline for comparison. Good agreement can be found between the proposed algorithm and DNS, which indicates the great potential of the proposed method on studying the crossflow and streamwise instability in hypersonic boundary layers. Project supported by the National Natural Science Foundation of China (Grant Nos. 11332007 and 11402167).

Conditional stability in determination of initial data for stochastic parabolic equations

International Nuclear Information System (INIS)

Yuan, Ganghua

2017-01-01

In this paper, we solve two kinds of inverse problems in determination of the initial data for stochastic parabolic equations. One is determination of the initial data by lateral boundary observation on arbitrary portion of the boundary, the second one is determination of the initial data by internal observation in a subregion inside the domain. We obtain conditional stability for the two kinds of inverse problems. To prove the results, we estimate the initial data by a terminal observation near the initial time, then we estimate this terminal observation by lateral boundary observation on arbitrary portion of the boundary or internal observation in a subregion inside the domain. To achieve those goals, we derive several new Carleman estimates for stochastic parabolic equations in this paper. (paper)

Conditional stability in determination of initial data for stochastic parabolic equations

Science.gov (United States)

Yuan, Ganghua

2017-03-01

In this paper, we solve two kinds of inverse problems in determination of the initial data for stochastic parabolic equations. One is determination of the initial data by lateral boundary observation on arbitrary portion of the boundary, the second one is determination of the initial data by internal observation in a subregion inside the domain. We obtain conditional stability for the two kinds of inverse problems. To prove the results, we estimate the initial data by a terminal observation near the initial time, then we estimate this terminal observation by lateral boundary observation on arbitrary portion of the boundary or internal observation in a subregion inside the domain. To achieve those goals, we derive several new Carleman estimates for stochastic parabolic equations in this paper.

Approximation of the unsteady Brinkman-Forchheimer equations by the pressure stabilization method

KAUST Repository

Louaked, Mohammed; Seloula, Nour; Trabelsi, Saber

2017-01-01

In this work, we propose and analyze the pressure stabilization method for the unsteady incompressible Brinkman-Forchheimer equations. We present a time discretization scheme which can be used with any consistent finite element space approximation. Second-order error estimate is proven. Some numerical results are also given.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

Approximation of the unsteady Brinkman-Forchheimer equations by the pressure stabilization method

KAUST Repository

Louaked, Mohammed

2017-07-20

In this work, we propose and analyze the pressure stabilization method for the unsteady incompressible Brinkman-Forchheimer equations. We present a time discretization scheme which can be used with any consistent finite element space approximation. Second-order error estimate is proven. Some numerical results are also given.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

On partial stabilization of a system of the Euler-Bernoulli beam equations

International Nuclear Information System (INIS)

Zuyev, Alexander

2003-11-01

The paper is focused on the stabilization problem for the following system of differential equations ∂ 2 (t) = v, t ≥ 0, (∂ 2 ω i (x,t))/∂t 2 + c 2 (∂ 4 ω i (x,t))/∂x 4 = ∂ 2 (t)ω i (x,t) - (x+d)v, x is an element of [0,l], i = 1,2,...,k, where v is an element of R is the control parameter. The above system describes a rotating rigid body endowed with a number of elastic beams. To solve the stabilization problem, we prove a sufficient condition for partial strong asymptotic stability which is valid for general nonlinear dynamical systems in a Banach space. This result is applied to deriving a feedback control explicitly. In addition, we prove strong (non-asymptotic) stability in the sense of Lyapunov as well as precompacness of the trajectories for the corresponding nonlinear semigroup. Some simulation results are given in conclusion. (author)

ON THE BOUNDEDNESS AND THE STABILITY OF SOLUTION TO THIRD ORDER NON-LINEAR DIFFERENTIAL EQUATIONS

Institute of Scientific and Technical Information of China (English)

无

2008-01-01

In this paper we investigate the global asymptotic stability,boundedness as well as the ultimate boundedness of solutions to a general third order nonlinear differential equation,using complete Lyapunov function.

Stability of a modified Peaceman–Rachford method for the paraxial Helmholtz equation on adaptive grids

Energy Technology Data Exchange (ETDEWEB)

Sheng, Qin, E-mail: Qin_Sheng@baylor.edu [Department of Mathematics and Center for Astrophysics, Space Physics and Engineering Research, Baylor University, One Bear Place, Waco, TX 76798-7328 (United States); Sun, Hai-wei, E-mail: hsun@umac.mo [Department of Mathematics, University of Macau (Macao)

2016-11-15

This study concerns the asymptotic stability of an eikonal, or ray, transformation based Peaceman–Rachford splitting method for solving the paraxial Helmholtz equation with high wave numbers. Arbitrary nonuniform grids are considered in transverse and beam propagation directions. The differential equation targeted has been used for modeling propagations of high intensity laser pulses over a long distance without diffractions. Self-focusing of high intensity beams may be balanced with the de-focusing effect of created ionized plasma channel in the situation, and applications of grid adaptations are frequently essential. It is shown rigorously that the fully discretized oscillation-free decomposition method on arbitrary adaptive grids is asymptotically stable with a stability index one. Simulation experiments are carried out to illustrate our concern and conclusions.

Stability of phase locking in a ring of unidirectionally coupled oscillators

International Nuclear Information System (INIS)

Rogge, J A; Aeyels, D

2004-01-01

We discuss the dynamic behaviour of a finite group of phase oscillators unidirectionally coupled in a ring. The dynamics are based on the Kuramoto model. In the case of identical oscillators, all phase locking solutions and their stability properties are obtained. For nonidentical oscillators it is proven that there exist phase locking solutions for sufficiently strong coupling. An algorithm to obtain all phase locking solutions is proposed. These solutions can be classified into classes, each with its own stability properties. The stability properties are obtained by means of a novel extension of Gershgorin's theorem. One class of stable solutions has the property that all phase differences between neighbouring cells are contained in (-π/2, π/2). Contrary to intuition, a second class of stable solutions is established with exactly one of the phase differences contained in (π/2, 3π/2). The stability results are extended from sinusoidal interconnections to a class of odd functions. To conclude, a connection with the field of active antenna arrays is made, generalizing some results earlier obtained in this field

On the calculation of linear stability with the aid of asymptotic solutions of Orr-Sommerfeld equation, 1

International Nuclear Information System (INIS)

Fujimura, Kaoru

1980-11-01

The numerical treatment of Orr-Sommerfeld equation which is the fundamental equation of linear hydrodynamic stability theory is described. Present calculation procedure is applied to the two-dimensional quasi-parallel flow for which linearized disturbance equation (Orr-Sommerfeld equation) contains one simple turning point and αR >> 1. The numerical procedure for this problem and one numerical example for Jeffery-Hamel flow (J-H III 1 ) are presented. These treatment can be extended to the other velocity profiles by slight midifications. (author)

Asymptotic estimates and exponential stability for higher-order monotone difference equations

Directory of Open Access Journals (Sweden)

Pituk Mihály

2005-01-01

Full Text Available Asymptotic estimates are established for higher-order scalar difference equations and inequalities the right-hand sides of which generate a monotone system with respect to the discrete exponential ordering. It is shown that in some cases the exponential estimates can be replaced with a more precise limit relation. As corollaries, a generalization of discrete Halanay-type inequalities and explicit sufficient conditions for the global exponential stability of the zero solution are given.

Asymptotic estimates and exponential stability for higher-order monotone difference equations

Directory of Open Access Journals (Sweden)

Mihály Pituk

2005-03-01

Full Text Available Asymptotic estimates are established for higher-order scalar difference equations and inequalities the right-hand sides of which generate a monotone system with respect to the discrete exponential ordering. It is shown that in some cases the exponential estimates can be replaced with a more precise limit relation. As corollaries, a generalization of discrete Halanay-type inequalities and explicit sufficient conditions for the global exponential stability of the zero solution are given.

Stability properties of a heat equation with state-dependent parameters and asymmetric boundary conditions

DEFF Research Database (Denmark)

Backi, Christoph Josef; Bendtsen, Jan Dimon; Leth, John

2015-01-01

In this work the stability properties of a partial differential equation (PDE) with state-dependent parameters and asymmetric boundary conditions are investigated. The PDE describes the temperature distribution inside foodstuff, but can also hold for other applications and phenomena. We show...

Stochastic modeling of mode interactions via linear parabolized stability equations

Science.gov (United States)

Ran, Wei; Zare, Armin; Hack, M. J. Philipp; Jovanovic, Mihailo

2017-11-01

Low-complexity approximations of the Navier-Stokes equations have been widely used in the analysis of wall-bounded shear flows. In particular, the parabolized stability equations (PSE) and Floquet theory have been employed to capture the evolution of primary and secondary instabilities in spatially-evolving flows. We augment linear PSE with Floquet analysis to formally treat modal interactions and the evolution of secondary instabilities in the transitional boundary layer via a linear progression. To this end, we leverage Floquet theory by incorporating the primary instability into the base flow and accounting for different harmonics in the flow state. A stochastic forcing is introduced into the resulting linear dynamics to model the effect of nonlinear interactions on the evolution of modes. We examine the H-type transition scenario to demonstrate how our approach can be used to model nonlinear effects and capture the growth of the fundamental and subharmonic modes observed in direct numerical simulations and experiments.

Ulam stability for fractional differential equations in the sense of Caputo operator

Directory of Open Access Journals (Sweden)

Rabha W. Ibrahim

2012-12-01

Full Text Available In this paper, we consider the Hyers-Ulam stability for the following fractional differential equations, in the sense ofcomplex Caputo fractional derivative defined, in the unit disk: cDßzf(z=G(f(z, cDázf(z,zf‘(z;z 0<á<1<ß<2 . Furthermore,a generalization of the admissible functions in complex Banach spaces is imposed and applications are illustrated.

Fixed Points and Fuzzy Stability of Functional Equations Related to Inner Product

Directory of Open Access Journals (Sweden)

Hassan Azadi Kenary

2012-04-01

Full Text Available In , Th.M. Rassias introduced the following equality sum_{i,j=1}^m |x_i - x_j |^2 = 2m sum_{i=1}^m|x_i|^2, qquad sum_{i=1}^m x_i =0 for a fixed integer $m ge 3$. Let $V, W$ be real vector spaces. It is shown that if a mapping $f : V ightarrow W$ satisfies sum_{i,j=1}^m f(x_i - x_j = 2m sum_{i=1}^m f(x_i for all $x_1, ldots, x_{m} in V$ with $sum_{i=1}^m x_i =0$, then the mapping $f : V ightarrow W$ is realized as the sum of an additive mapping and a quadratic mapping. From the above equality we can define the functional equation f(x-y +f(2x+y + f(x+2y= 3f(x+ 3f(y + 3f(x+y , which is called a {it quadratic functional equation}. Every solution of the quadratic functional equation is said to be a {it quadratic mapping}. Using fixed point theorem we prove the Hyers-Ulam stability of the functional equation ( in fuzzy Banach spaces.

On the Gross–Pitaevskii Equation with Pumping and Decay: Stationary States and Their Stability

KAUST Repository

Sierra Nunez, Jesus Alfredo; Kasimov, Aslan R.; Markowich, Peter A.; Weishä upl, Rada Maria

2015-01-01

We investigate the behavior of solutions of the complex Gross–Pitaevskii equation, a model that describes the dynamics of pumped decaying Bose–Einstein condensates. The stationary radially symmetric solutions of the equation are studied, and their linear stability with respect to two-dimensional perturbations is analyzed. Using numerical continuation, we calculate not only the ground state of the system, but also a number of excited states. Accurate numerical integration is employed to study the general nonlinear evolution of the system from the unstable stationary solutions to the formation of stable vortex patterns.

On the Gross–Pitaevskii Equation with Pumping and Decay: Stationary States and Their Stability

KAUST Repository

Sierra Nunez, Jesus Alfredo

2015-02-11

We investigate the behavior of solutions of the complex Gross–Pitaevskii equation, a model that describes the dynamics of pumped decaying Bose–Einstein condensates. The stationary radially symmetric solutions of the equation are studied, and their linear stability with respect to two-dimensional perturbations is analyzed. Using numerical continuation, we calculate not only the ground state of the system, but also a number of excited states. Accurate numerical integration is employed to study the general nonlinear evolution of the system from the unstable stationary solutions to the formation of stable vortex patterns.

Stability analysis of explicit entropy viscosity methods for non-linear scalar conservation equations

KAUST Repository

Bonito, Andrea

2013-10-03

We establish the L2-stability of an entropy viscosity technique applied to nonlinear scalar conservation equations. First-and second-order explicit time-stepping techniques using continuous finite elements in space are considered. The method is shown to be stable independently of the polynomial degree of the space approximation under the standard CFL condition. © 2013 American Mathematical Society.

A collective variable approach and stabilization for dispersion-managed optical solitons in the quintic complex Ginzburg-Landau equation as perturbations of the nonlinear Schroedinger equation

International Nuclear Information System (INIS)

Fewo, S I; Kenfack-Jiotsa, A; Kofane, T C

2006-01-01

With the help of the one-dimensional quintic complex Ginzburg-Landau equation (CGLE) as perturbations of the nonlinear Schroedinger equation (NLSE), we derive the equations of motion of pulse parameters called collective variables (CVs), of a pulse propagating in dispersion-managed (DM) fibre optic links. The equations obtained are investigated numerically in order to view the evolution of pulse parameters along the propagation distance, and also to analyse effects of initial amplitude and width on the propagating pulse. Nonlinear gain is shown to be beneficial in stabilizing DM solitons. A fully numerical simulation of the one-dimensional quintic CGLE as perturbations of NLSE finally tests the results of the CV theory. A good agreement is observed between both methods

On the construction of inertial manifolds under symmetry constraints II: O(2) constraint and inertial manifolds on thin domains

International Nuclear Information System (INIS)

Rodriguez-Bernal, A.

1993-01-01

On a model example, the Kuramoto-Velarde equation, which includes the Kuramoto-Sivashin-sky and the Cahn-Hilliard models, and under suitable and reasonable hypothesis, we show the dimension and determining modes of inertial manifolds for several classes of solutions. We also give bounds for the dimensions of inertial manifolds of the full system as a parameter is varied. The results are pointed out to be almost model-independent. The same ideas are also applied to a class of parabolic equations in higher space dimension, obtaining results about inertial manifolds on thin and small domains. (Author). 30 refs

Hysteretic transitions in the Kuramoto model with inertia.

Science.gov (United States)

Olmi, Simona; Navas, Adrian; Boccaletti, Stefano; Torcini, Alessandro

2014-10-01

We report finite-size numerical investigations and mean-field analysis of a Kuramoto model with inertia for fully coupled and diluted systems. In particular, we examine, for a gaussian distribution of the frequencies, the transition from incoherence to coherence for increasingly large system size and inertia. For sufficiently large inertia the transition is hysteretic, and within the hysteretic region clusters of locked oscillators of various sizes and different levels of synchronization coexist. A modification of the mean-field theory developed by Tanaka, Lichtenberg, and Oishi [Physica D 100, 279 (1997)] allows us to derive the synchronization curve associated to each of these clusters. We have also investigated numerically the limits of existence of the coherent and of the incoherent solutions. The minimal coupling required to observe the coherent state is largely independent of the system size, and it saturates to a constant value already for moderately large inertia values. The incoherent state is observable up to a critical coupling whose value saturates for large inertia and for finite system sizes, while in the thermodinamic limit this critical value diverges proportionally to the mass. By increasing the inertia the transition becomes more complex, and the synchronization occurs via the emergence of clusters of whirling oscillators. The presence of these groups of coherently drifting oscillators induces oscillations in the order parameter. We have shown that the transition remains hysteretic even for randomly diluted networks up to a level of connectivity corresponding to a few links per oscillator. Finally, an application to the Italian high-voltage power grid is reported, which reveals the emergence of quasiperiodic oscillations in the order parameter due to the simultaneous presence of many competing whirling clusters.

Existence and global exponential stability of periodic solutions for n-dimensional neutral dynamic equations on time scales.

Science.gov (United States)

Li, Bing; Li, Yongkun; Zhang, Xuemei

2016-01-01

In this paper, by using the existence of the exponential dichotomy of linear dynamic equations on time scales and the theory of calculus on time scales, we study the existence and global exponential stability of periodic solutions for a class of n-dimensional neutral dynamic equations on time scales. We also present an example to illustrate the feasibility of our results. The results of this paper are completely new and complementary to the previously known results even in both the case of differential equations (time scale [Formula: see text]) and the case of difference equations (time scale [Formula: see text]).

Extended dynamic mode decomposition with dictionary learning: A data-driven adaptive spectral decomposition of the Koopman operator.

Science.gov (United States)

Li, Qianxiao; Dietrich, Felix; Bollt, Erik M; Kevrekidis, Ioannis G

2017-10-01

Numerical approximation methods for the Koopman operator have advanced considerably in the last few years. In particular, data-driven approaches such as dynamic mode decomposition (DMD) 51 and its generalization, the extended-DMD (EDMD), are becoming increasingly popular in practical applications. The EDMD improves upon the classical DMD by the inclusion of a flexible choice of dictionary of observables which spans a finite dimensional subspace on which the Koopman operator can be approximated. This enhances the accuracy of the solution reconstruction and broadens the applicability of the Koopman formalism. Although the convergence of the EDMD has been established, applying the method in practice requires a careful choice of the observables to improve convergence with just a finite number of terms. This is especially difficult for high dimensional and highly nonlinear systems. In this paper, we employ ideas from machine learning to improve upon the EDMD method. We develop an iterative approximation algorithm which couples the EDMD with a trainable dictionary represented by an artificial neural network. Using the Duffing oscillator and the Kuramoto Sivashinsky partical differential equation as examples, we show that our algorithm can effectively and efficiently adapt the trainable dictionary to the problem at hand to achieve good reconstruction accuracy without the need to choose a fixed dictionary a priori. Furthermore, to obtain a given accuracy, we require fewer dictionary terms than EDMD with fixed dictionaries. This alleviates an important shortcoming of the EDMD algorithm and enhances the applicability of the Koopman framework to practical problems.

Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations

NARCIS (Netherlands)

Jarlebring, E.; Hochstenbach, M.E.

2009-01-01

Several recent methods used to analyze asymptotic stability of delay-differential equations (DDEs) involve determining the eigenvalues of a matrix, a matrix pencil or a matrix polynomial constructed by Kronecker products. Despite some similarities between the different types of these so-called

Stability analysis of the soliton solutions for the generalized quintic derivative nonlinear Schrödinger equation

Directory of Open Access Journals (Sweden)

Chen Yue

Full Text Available The propagation of hydrodynamic wave packets and media with negative refractive index is studied in a quintic derivative nonlinear Schrödinger (DNLS equation. The quintic DNLS equation describe the wave propagation on a discrete electrical transmission line. We obtain a Lagrangian and the invariant variational principle for quintic DNLS equation. By using a class of ordinary differential equation, we found four types of exact solutions of the quintic DNLS equation, which are kink-type solitary wave solution, antikink-type solitary wave solution, sinusoidal solitary wave solution, bell-type solitary wave solution. By applying the modulation instability to discuss stability analysis of the obtained solutions. Modulation instabilities of continuous waves and localized solutions on a zero background have been investigated. Keywords: Quintic derivative NLS equation, Solitary wave solutions, Mathematical physics methods, 2000 MR Subject Classification: 35G20, 35Q53, 37K10, 49S05, 76A60

A fixed point approach towards stability of delay differential equations with applications to neural networks

NARCIS (Netherlands)

Chen, Guiling

2013-01-01

This thesis studies asymptotic behavior and stability of determinsitic and stochastic delay differential equations. The approach used in this thesis is based on fixed point theory, which does not resort to any Liapunov function or Liapunov functional. The main contribution of this thesis is to study

Impulsive moving mirror model and the stability of Schroedinger equation with impulse effect in a Banach space

International Nuclear Information System (INIS)

Kostadinov, S.I.; Petrov, G.

1992-01-01

From a special class of systems has been used a Schroedinger equation with impulse effect in Minkowski space field theory with time dependent boundary conditions, i.e. those of moving mirrors. The field theoretical approach for studying the properties of the vacuum starts from an analysis of the behaviour of local field quantities in Minkowski space with uniformly moving mirrors. For the impulsive moving mirror model is the real process of interaction between the quantum field and the external mirror a subject to disturbances in its evolution acting in time very short compared with the entire duration of the process. So the stability of the solution of the Schroedinger evolution equation for the process in the stability of the vacuum of Casimir. 8 refs

Kuramoto model of coupled oscillators with positive and negative coupling parameters: an example of conformist and contrarian oscillators.

Science.gov (United States)

Hong, Hyunsuk; Strogatz, Steven H

2011-02-04

We consider a generalization of the Kuramoto model in which the oscillators are coupled to the mean field with random signs. Oscillators with positive coupling are "conformists"; they are attracted to the mean field and tend to synchronize with it. Oscillators with negative coupling are "contrarians"; they are repelled by the mean field and prefer a phase diametrically opposed to it. The model is simple and exactly solvable, yet some of its behavior is surprising. Along with the stationary states one might have expected (a desynchronized state, and a partially-synchronized state, with conformists and contrarians locked in antiphase), it also displays a traveling wave, in which the mean field oscillates at a frequency different from the population's mean natural frequency.

Chimera patterns in the Kuramoto–Battogtokh model

International Nuclear Information System (INIS)

Smirnov, Lev; Osipov, Grigory; Pikovsky, Arkady

2017-01-01

Kuramoto and Battogtokh (2002 Nonlinear Phenom. Complex Syst . 5 380) discovered chimera states represented by stable coexisting synchrony and asynchrony domains in a lattice of coupled oscillators. After a reformulation in terms of a local order parameter, the problem can be reduced to partial differential equations. We find uniformly rotating, spatially periodic chimera patterns as solutions of a reversible ordinary differential equation, and demonstrate a plethora of such states. In the limit of neutral coupling they reduce to analytical solutions in the form of one- and two-point chimera patterns as well as localized chimera solitons. Patterns at weakly attracting coupling are characterized by virtue of a perturbative approach. Stability analysis reveals that only the simplest chimeras with one synchronous region are stable. (letter)

Constitutive equations for extensional flow of wormlike micelles : stability analysis of the Bautista-Manero model

NARCIS (Netherlands)

Boek, E.S.; Padding, J.T.; Anderson, V.J.; Tardy, P.M.J.; Crawshaw, J.P.; Pearson, J.R.A.

2005-01-01

We carry out a stability analysis of the Bautista-Manero (B-M) constitutive equations for extensional flow of wormlike micelles. We show that all solutions for the steady-state extensional viscosity ¿E are unstable when the elongational rates e exceed some critical value. In some cases the only real

Macroscopic plasma properties and stability theory

International Nuclear Information System (INIS)

Sakanaka, P.H.

1981-01-01

1. Two-fluid equations: (a) Boltzmann equation: complete set of equations; collision models - Vlasov, BGK, Fokker-Planck-Landau, Boltzmann. (b) Moments of the Boltzmann equation: problem of closure. (c) Two-fluid equations. 2. One-fluid equation: (a) One-fluid variables. (b) One-fluid equations: quasi-neutrality. (c) Resistive MHD equations. (d) Ideal MHD equations: one-adiabatic approximation; double-adiabatic approximation - CGL. 3. MHD stability problem - energy principle: (a) Linearized ideal MHD equations: force-operator equation. (b) Boundary conditions. (c) Self-adjointness of force operator. (d) The energy principle. 4. Stability problems: application of the energy principle; stability of sharp-boundary plasmas. 5. Thermodynamic approach for stability of plasmas: Newcomb and Rosenbluth's stability criteria. (author)

Global existence and uniform stabilization of a generalized dissipative Klein-Gordon equation type with boundary damping

International Nuclear Information System (INIS)

Zhang Zaiyun; Miao Xiujin; Chen Yuezhong; Liu Zhenhai

2011-01-01

In this paper, we prove the existence, uniqueness, and uniform stability of strong and weak solutions of the nonlinear generalized Klein-Gordon equation (1.1) 1 (see Sec. I) in bounded domains with nonlinear damped boundary conditions given by (1.1) 3 (see Sec. I) with some restrictions on function f(u), h(∇u), g(u t ), and b(x), we prove the existence and uniqueness by means of nonlinear semigroup method and obtain the uniform stabilization by using the multiplier technique.

Exponential mean-square stability of two classes of theta Milstein methods for stochastic delay differential equations

Science.gov (United States)

Rouz, Omid Farkhondeh; Ahmadian, Davood; Milev, Mariyan

2017-12-01

This paper establishes exponential mean square stability of two classes of theta Milstein methods, namely split-step theta Milstein (SSTM) method and stochastic theta Milstein (STM) method, for stochastic differential delay equations (SDDEs). We consider the SDDEs problem under a coupled monotone condition on drift and diffusion coefficients, as well as a necessary linear growth condition on the last term of theta Milstein method. It is proved that the SSTM method with θ ∈ [0, ½] can recover the exponential mean square stability of the exact solution with some restrictive conditions on stepsize, but for θ ∈ (½, 1], we proved that the stability results hold for any stepsize. Then, based on the stability results of SSTM method, we examine the exponential mean square stability of the STM method and obtain the similar stability results to that of the SSTM method. In the numerical section the figures show thevalidity of our claims.

Stability study of pre-stack seismic inversion based on the full Zoeppritz equation

Science.gov (United States)

Liang, Lifeng; Zhang, Hongbing; Guo, Qiang; Saeed, Wasif; Shang, Zuoping; Huang, Guojiao

2017-10-01

Pre-stack seismic inversion is highly important and complicated. Its result is non-unique, and the process is unstable because pre-stack seismic inversion is an ill-posed problem that simultaneously obtains the results of multiple parameters. Combining the full Zoeppritz equation and additional assumptions with edge-preserving regularization (EPR) can help mitigate the problem. To achieve this combination, we developed an inversion method by constructing a new objective function, which includes the EPR and the Markov random field. The method directly gains reflectivity R PP by the full Zoeppritz equation instead of its approximations and effectively controls the stability of simultaneous inversion by two additional assumptions: the sectional constant V S/V P and the generalized Gardner equation. Thus, the simultaneous inversion of multiple parameters is directed toward to V P, ΔL S (the fitting deviation of V S) and density, and the generalized Gardner equation is regarded as a constraint from which the fitting relationship is derived. We applied the fast simulated annealing algorithm to solve the nonlinear optimization problem. The test results on 2D synthetic data indicated that the stability of simultaneous inversion for V P, ΔL S and density is better than these for V P, V S, and density. The inverted result of density gradually worsens as the deviation ΔL D (the fitting deviation of the density) increases. Moreover, the inverted results were acceptable when using the fitting relationships with error, although they showed varying degrees of influence. We constructed time-varying and space-varying fitting relationships using the logging data in pre-stack inversion of the field seismic data. This improved the inverted results of the simultaneous inversion for complex geological models. Finally, the inverted results of the field data distinctly revealed more detailed information about the layers and matched well with the logging data along the wells over most

Kuramoto model with uniformly spaced frequencies: Finite-N asymptotics of the locking threshold.

Science.gov (United States)

Ottino-Löffler, Bertrand; Strogatz, Steven H

2016-06-01

We study phase locking in the Kuramoto model of coupled oscillators in the special case where the number of oscillators, N, is large but finite, and the oscillators' natural frequencies are evenly spaced on a given interval. In this case, stable phase-locked solutions are known to exist if and only if the frequency interval is narrower than a certain critical width, called the locking threshold. For infinite N, the exact value of the locking threshold was calculated 30 years ago; however, the leading corrections to it for finite N have remained unsolved analytically. Here we derive an asymptotic formula for the locking threshold when N≫1. The leading correction to the infinite-N result scales like either N^{-3/2} or N^{-1}, depending on whether the frequencies are evenly spaced according to a midpoint rule or an end-point rule. These scaling laws agree with numerical results obtained by Pazó [D. Pazó, Phys. Rev. E 72, 046211 (2005)PLEEE81539-375510.1103/PhysRevE.72.046211]. Moreover, our analysis yields the exact prefactors in the scaling laws, which also match the numerics.

Energy Stability Analysis of Some Fully Discrete Numerical Schemes for Incompressible Navier–Stokes Equations on Staggered Grids

KAUST Repository

Chen, Huangxin; Sun, Shuyu; Zhang, Tao

2017-01-01

In this paper we consider the energy stability estimates for some fully discrete schemes which both consider time and spatial discretizations for the incompressible Navier–Stokes equations. We focus on three kinds of fully discrete schemes, i

Stability analysis solutions and optical solitons in extended nonlinear Schrödinger equation with higher-order odd and even terms

Science.gov (United States)

Peng, Wei-Qi; Tian, Shou-Fu; Zou, Li; Zhang, Tian-Tian

2018-01-01

In this paper, the extended nonlinear Schrödinger equation with higher-order odd (third order) and even (fourth order) terms is investigated, whose particular cases are the Hirota equation, the Sasa-Satsuma equation and Lakshmanan-Porsezian-Daniel equation by selecting some specific values on the parameters of higher-order terms. We first study the stability analysis of the equation. Then, using the ansatz method, we derive its bright, dark solitons and some constraint conditions which can guarantee the existence of solitons. Moreover, the Ricatti equation extension method is employed to derive some exact singular solutions. The outstanding characteristics of these solitons are analyzed via several diverting graphics.

Developments in functional equations and related topics

CERN Document Server

Ciepliński, Krzysztof; Rassias, Themistocles

2017-01-01

This book presents current research on Ulam stability for functional equations and inequalities. Contributions from renowned scientists emphasize fundamental and new results, methods and techniques. Detailed examples are given to theories to further understanding at the graduate level for students in mathematics, physics, and engineering. Key topics covered in this book include: Quasi means Approximate isometries Functional equations in hypergroups Stability of functional equations Fischer-Muszély equation Haar meager sets and Haar null sets Dynamical systems Functional equations in probability theory Stochastic convex ordering Dhombres functional equation Nonstandard analysis and Ulam stability This book is dedicated in memory of Staniłsaw Marcin Ulam, who posed the fundamental problem concerning approximate homomorphisms of groups in 1940; which has provided the stimulus for studies in the stability of functional equations and inequalities.

THE EXISTENCE OF THE STABILIZING SOLUTION OF THE RICCATI EQUATION ARISING IN DISCRETE-TIME STOCHASTIC ZERO SUM LQ DYNAMIC GAMES WITH PERIODIC COEFFICIENTS

Directory of Open Access Journals (Sweden)

Vasile Dr ̆agan

2017-06-01

Full Text Available We investigate the problem for solving a discrete-time periodic gen- eralized Riccati equation with an indefinite sign of the quadratic term. A necessary condition for the existence of bounded and stabilizing solution of the discrete-time Riccati equation with an indefinite quadratic term is derived. The stabilizing solution is positive semidefinite and satisfies the introduced sign conditions. The proposed condition is illustrated via a numerical example.

Convergence and stability of the exponential Euler method for semi-linear stochastic delay differential equations.

Science.gov (United States)

Zhang, Ling

2017-01-01

The main purpose of this paper is to investigate the strong convergence and exponential stability in mean square of the exponential Euler method to semi-linear stochastic delay differential equations (SLSDDEs). It is proved that the exponential Euler approximation solution converges to the analytic solution with the strong order [Formula: see text] to SLSDDEs. On the one hand, the classical stability theorem to SLSDDEs is given by the Lyapunov functions. However, in this paper we study the exponential stability in mean square of the exact solution to SLSDDEs by using the definition of logarithmic norm. On the other hand, the implicit Euler scheme to SLSDDEs is known to be exponentially stable in mean square for any step size. However, in this article we propose an explicit method to show that the exponential Euler method to SLSDDEs is proved to share the same stability for any step size by the property of logarithmic norm.

Convergence and stability of the exponential Euler method for semi-linear stochastic delay differential equations

Directory of Open Access Journals (Sweden)

Ling Zhang

2017-10-01

Full Text Available Abstract The main purpose of this paper is to investigate the strong convergence and exponential stability in mean square of the exponential Euler method to semi-linear stochastic delay differential equations (SLSDDEs. It is proved that the exponential Euler approximation solution converges to the analytic solution with the strong order 1 2 $\\frac{1}{2}$ to SLSDDEs. On the one hand, the classical stability theorem to SLSDDEs is given by the Lyapunov functions. However, in this paper we study the exponential stability in mean square of the exact solution to SLSDDEs by using the definition of logarithmic norm. On the other hand, the implicit Euler scheme to SLSDDEs is known to be exponentially stable in mean square for any step size. However, in this article we propose an explicit method to show that the exponential Euler method to SLSDDEs is proved to share the same stability for any step size by the property of logarithmic norm.

Stability and square integrability of derivatives of solutions of nonlinear fourth order differential equations with delay.

Science.gov (United States)

Korkmaz, Erdal

2017-01-01

In this paper, we give sufficient conditions for the boundedness, uniform asymptotic stability and square integrability of the solutions to a certain fourth order non-autonomous differential equations with delay by using Lyapunov's second method. The results obtained essentially improve, include and complement the results in the literature.

Integral φ0-Stability in terms of Two Measures for Impulsive Differential Equations with “Supremum”

Directory of Open Access Journals (Sweden)

Peiguang Wang

2014-01-01

Full Text Available This paper establishes a criterion on integral φ0-stability in terms of two measures for impulsive differential equations with “supremum” by using the cone-valued piecewise continuous Lyapunov functions, Razumikhin method, and comparative method. Meantime, an example is given to illustrate our result.

Deciphering the imprint of topology on nonlinear dynamical network stability

International Nuclear Information System (INIS)

Nitzbon, J; Schultz, P; Heitzig, J; Kurths, J; Hellmann, F

2017-01-01

Coupled oscillator networks show complex interrelations between topological characteristics of the network and the nonlinear stability of single nodes with respect to large but realistic perturbations. We extend previous results on these relations by incorporating sampling-based measures of the transient behaviour of the system, its survivability, as well as its asymptotic behaviour, its basin stability. By combining basin stability and survivability we uncover novel, previously unknown asymptotic states with solitary, desynchronized oscillators which are rotating with a frequency different from their natural one. They occur almost exclusively after perturbations at nodes with specific topological properties. More generally we confirm and significantly refine the results on the distinguished role tree-shaped appendices play for nonlinear stability. We find a topological classification scheme for nodes located in such appendices, that exactly separates them according to their stability properties, thus establishing a strong link between topology and dynamics. Hence, the results can be used for the identification of vulnerable nodes in power grids or other coupled oscillator networks. From this classification we can derive general design principles for resilient power grids. We find that striving for homogeneous network topologies facilitates a better performance in terms of nonlinear dynamical network stability. While the employed second-order Kuramoto-like model is parametrised to be representative for power grids, we expect these insights to transfer to other critical infrastructure systems or complex network dynamics appearing in various other fields. (paper)

Variational Approach to the Orbital Stability of Standing Waves of the Gross-Pitaevskii Equation

KAUST Repository

Hadj Selem, Fouad

2014-08-26

This paper is concerned with the mathematical analysis of a masssubcritical nonlinear Schrödinger equation arising from fiber optic applications. We show the existence and symmetry of minimizers of the associated constrained variational problem. We also prove the orbital stability of such solutions referred to as standing waves and characterize the associated orbit. In the last section, we illustrate our results with few numerical simulations. © 2014 Springer Basel.

Stability, accuracy and numerical diffusion analysis of nodal expansion method for steady convection diffusion equation

International Nuclear Information System (INIS)

Zhou, Xiafeng; Guo, Jiong; Li, Fu

2015-01-01

Highlights: • NEMs are innovatively applied to solve convection diffusion equation. • Stability, accuracy and numerical diffusion for NEM are analyzed for the first time. • Stability and numerical diffusion depend on the NEM expansion order and its parity. • NEMs have higher accuracy than both second order upwind and QUICK scheme. • NEMs with different expansion orders are integrated into a unified discrete form. - Abstract: The traditional finite difference method or finite volume method (FDM or FVM) is used for HTGR thermal-hydraulic calculation at present. However, both FDM and FVM require the fine mesh sizes to achieve the desired precision and thus result in a limited efficiency. Therefore, a more efficient and accurate numerical method needs to be developed. Nodal expansion method (NEM) can achieve high accuracy even on the coarse meshes in the reactor physics analysis so that the number of spatial meshes and computational cost can be largely decreased. Because of higher efficiency and accuracy, NEM can be innovatively applied to thermal-hydraulic calculation. In the paper, NEMs with different orders of basis functions are successfully developed and applied to multi-dimensional steady convection diffusion equation. Numerical results show that NEMs with three or higher order basis functions can track the reference solutions very well and are superior to second order upwind scheme and QUICK scheme. However, the false diffusion and unphysical oscillation behavior are discovered for NEMs. To explain the reasons for the above-mentioned behaviors, the stability, accuracy and numerical diffusion properties of NEM are analyzed by the Fourier analysis, and by comparing with exact solutions of difference and differential equation. The theoretical analysis results show that the accuracy of NEM increases with the expansion order. However, the stability and numerical diffusion properties depend not only on the order of basis functions but also on the parity of

Stability, accuracy and numerical diffusion analysis of nodal expansion method for steady convection diffusion equation

Energy Technology Data Exchange (ETDEWEB)

Zhou, Xiafeng, E-mail: zhou-xf11@mails.tsinghua.edu.cn; Guo, Jiong, E-mail: guojiong12@tsinghua.edu.cn; Li, Fu, E-mail: lifu@tsinghua.edu.cn

2015-12-15

Highlights: • NEMs are innovatively applied to solve convection diffusion equation. • Stability, accuracy and numerical diffusion for NEM are analyzed for the first time. • Stability and numerical diffusion depend on the NEM expansion order and its parity. • NEMs have higher accuracy than both second order upwind and QUICK scheme. • NEMs with different expansion orders are integrated into a unified discrete form. - Abstract: The traditional finite difference method or finite volume method (FDM or FVM) is used for HTGR thermal-hydraulic calculation at present. However, both FDM and FVM require the fine mesh sizes to achieve the desired precision and thus result in a limited efficiency. Therefore, a more efficient and accurate numerical method needs to be developed. Nodal expansion method (NEM) can achieve high accuracy even on the coarse meshes in the reactor physics analysis so that the number of spatial meshes and computational cost can be largely decreased. Because of higher efficiency and accuracy, NEM can be innovatively applied to thermal-hydraulic calculation. In the paper, NEMs with different orders of basis functions are successfully developed and applied to multi-dimensional steady convection diffusion equation. Numerical results show that NEMs with three or higher order basis functions can track the reference solutions very well and are superior to second order upwind scheme and QUICK scheme. However, the false diffusion and unphysical oscillation behavior are discovered for NEMs. To explain the reasons for the above-mentioned behaviors, the stability, accuracy and numerical diffusion properties of NEM are analyzed by the Fourier analysis, and by comparing with exact solutions of difference and differential equation. The theoretical analysis results show that the accuracy of NEM increases with the expansion order. However, the stability and numerical diffusion properties depend not only on the order of basis functions but also on the parity of

Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions

KAUST Repository

Gerbi, Stéphane

2011-12-01

In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions, related to the KelvinVoigt damping. Global existence and asymptotic stability of solutions starting in a stable set are proved. Blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained. © 2011 Elsevier Ltd. All rights reserved.

Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions

KAUST Repository

Gerbi, Sté phane; Said-Houari, Belkacem

2011-01-01

In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions, related to the KelvinVoigt damping. Global existence and asymptotic stability of solutions starting in a stable set are proved. Blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained. © 2011 Elsevier Ltd. All rights reserved.

Stability of KAM tori for nonlinear Schrödinger equation

CERN Document Server

Cong, Hongzi; Yuan, Xiaoping

2016-01-01

The authors prove the long time stability of KAM tori (thus quasi-periodic solutions) for nonlinear Schrödinger equation \\sqrt{-1}\\, u_{t}=u_{xx}-M_{\\xi}u+\\varepsilon|u|^2u, subject to Dirichlet boundary conditions u(t,0)=u(t,\\pi)=0, where M_{\\xi} is a real Fourier multiplier. More precisely, they show that, for a typical Fourier multiplier M_{\\xi}, any solution with the initial datum in the \\delta-neighborhood of a KAM torus still stays in the 2\\delta-neighborhood of the KAM torus for a polynomial long time such as |t|\\leq \\delta^{-\\mathcal{M}} for any given \\mathcal M with 0\\leq \\mathcal{M}\\leq C(\\varepsilon), where C(\\varepsilon) is a constant depending on \\varepsilon and C(\\varepsilon)\\rightarrow\\infty as \\varepsilon\\rightarrow0.

Stability and square integrability of derivatives of solutions of nonlinear fourth order differential equations with delay

Directory of Open Access Journals (Sweden)

Erdal Korkmaz

2017-06-01

Full Text Available Abstract In this paper, we give sufficient conditions for the boundedness, uniform asymptotic stability and square integrability of the solutions to a certain fourth order non-autonomous differential equations with delay by using Lyapunov’s second method. The results obtained essentially improve, include and complement the results in the literature.

Existence, uniqueness, and stability of stochastic neutral functional differential equations of Sobolev-type

Energy Technology Data Exchange (ETDEWEB)

Yang, Xuetao; Zhu, Quanxin, E-mail: zqx22@126.com [School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, Jiangsu (China)

2015-12-15

In this paper, we are mainly concerned with a class of stochastic neutral functional differential equations of Sobolev-type with Poisson jumps. Under two different sets of conditions, we establish the existence of the mild solution by applying the Leray-Schauder alternative theory and the Sadakovskii’s fixed point theorem, respectively. Furthermore, we use the Bihari’s inequality to prove the Osgood type uniqueness. Also, the mean square exponential stability is investigated by applying the Gronwall inequality. Finally, two examples are given to illustrate the theory results.

Stability problems for linear hyperbolic systems

International Nuclear Information System (INIS)

Eckhoff, K.S.

1975-05-01

The stability properties for the trivial solution of a general linear hyperbolic system of partial differential equations of the first order are studied. It is shown that results may be obtained by studying the stability properties of certain systems of ordinary differential equations which can be constructed from the hyperbolic system (the so-called transport equations). In some cases the associated stability problem for the transport equations can in fact be shown to be equivalent to the stability problem for the hyperbolic system, but in general the transport equations will only give the necessary conditions for stability. (Auth.)

T-Stability of the Heun Method and Balanced Method for Solving Stochastic Differential Delay Equations

Directory of Open Access Journals (Sweden)

Xiaolin Zhu

2014-01-01

Full Text Available This paper studies the T-stability of the Heun method and balanced method for solving stochastic differential delay equations (SDDEs. Two T-stable conditions of the Heun method are obtained for two kinds of linear SDDEs. Moreover, two conditions under which the balanced method is T-stable are obtained for two kinds of linear SDDEs. Some numerical examples verify the theoretical results proposed.

Stability and oscillation of two coupled Duffing equations with time delay state feedback

International Nuclear Information System (INIS)

El-Bassiouny, A F

2006-01-01

This paper presents an analytical study of the simultaneous principal parametric resonances of two coupled Duffing equations with time delay state feedback. The concept of an equivalent damping related to the delay feedback is proposed and the appropriate choice of the feedback gains and the time delay is discussed from the viewpoint of vibration control. The method of multiple scales is used to determine a set of ordinary differential equations governing the modulation of the amplitudes and phases of the two modes. The first order approximation of the resonances are derived and the effect of time delay on the resonances is investigated. The fixed points correspond to a periodic motion for the starting system and we show the frequency-response curves. We analyse the effect of time delay and the other different parameters on these oscillations. The stability of the fixed points is examined by using the variational method. Numerical solutions are carried out and graphical representations of the results are presented and discussed. Increasing in the time delay τ given decreasing and increasing in the regions of definition and stability respectively and the first mode has decreased magnitudes. The multivalued solutions disappear when decreasing the coefficients of cubic nonlinearities of the second mode α 3 and the detuning parameter σ 2 respectively. Both modes shift to the left for increasing linear feedback gain v 1 and the coefficient of parametric excitation f 1 respectively

Stability of parallel flows

CERN Document Server

Betchov, R

2012-01-01

Stability of Parallel Flows provides information pertinent to hydrodynamical stability. This book explores the stability problems that occur in various fields, including electronics, mechanics, oceanography, administration, economics, as well as naval and aeronautical engineering. Organized into two parts encompassing 10 chapters, this book starts with an overview of the general equations of a two-dimensional incompressible flow. This text then explores the stability of a laminar boundary layer and presents the equation of the inviscid approximation. Other chapters present the general equation

Stability of plane wave solutions of the two-space-dimensional nonlinear Schroedinger equation

International Nuclear Information System (INIS)

Martin, D.U.; Yuen, H.C.; Saffman, P.G.

1980-01-01

The stability of plane, periodic solutions of the two-dimensional nonlinear Schroedinger equation to infinitesimal, two-dimensional perturbation has been calculated and verified numerically. For standing wave disturbances, instability is found for both odd and even modes; as the period of the unperturbed solution increases, the instability associated with the odd modes remains but that associated with the even mode disappears, which is consistent with the results of Zakharov and Rubenchik, Saffman and Yuen and Ablowitz and Segur on the stability of solitons. In addition, we have identified travelling wave instabilities for the even mode perturbations which are absent in the long-wave limit. Extrapolation to the case of an unperturbed solution with infinite period suggests that these instabilities may also be present for the soliton. In other words, the soliton is unstable to odd, standing-wave perturbations, and very likely also to even, travelling-wave perturbations. (orig.)

Orbital stability of standing waves of a class of fractional Schrödinger equations with Hartree-type nonlinearity

KAUST Repository

Cho, Yonggeun; Fall, Mouhamed M.; Hajaiej, Hichem; Markowich, Peter A.; Trabelsi, Saber

2016-01-01

This paper is devoted to the mathematical analysis of a class of nonlinear fractional Schrödinger equations with a general Hartree-type integrand. We show the well-posedness of the associated Cauchy problem and prove the existence and stability

Implications of a wavepacket formulation for the nonlinear parabolized stability equations to hypersonic boundary layers

Science.gov (United States)

Kuehl, Joseph

2016-11-01

The parabolized stability equations (PSE) have been developed as an efficient and powerful tool for studying the stability of advection-dominated laminar flows. In this work, a new "wavepacket" formulation of the PSE is presented. This method accounts for the influence of finite-bandwidth-frequency distributions on nonlinear stability calculations. The methodology is motivated by convolution integrals and is found to appropriately represent nonlinear energy transfer between primary modes and harmonics, in particular nonlinear feedback, via a "nonlinear coupling coefficient." It is found that traditional discrete mode formulations overestimate nonlinear feedback by approximately 70%. This results in smaller maximum disturbance amplitudes than those observed experimentally. The new formulation corrects this overestimation, accounts for the generation of side lobes responsible for spectral broadening and results in disturbance saturation amplitudes consistent with experiment. A Mach 6 flared-cone example is presented. Support from the AFOSR Young Investigator Program via Grant FA9550-15-1-0129 is gratefully acknowledges.

Motions in a Bose condensate: X. New results on stability of axisymmetric solitary waves of the Gross-Pitaevskii equation

OpenAIRE

Berloff, Natalia G.; Roberts, Paul H.

2004-01-01

The stability of the axisymmetric solitary waves of the Gross-Pitaevskii (GP) equation is investigated. The Implicitly Restarted Arnoldi Method for banded matrices with shift-invert was used to solve the linearised spectral stability problem. The rarefaction solitary waves on the upper branch of the Jones-Roberts dispersion curve are shown to be unstable to axisymmetric infinitesimal perturbations, whereas the solitary waves on the lower branch and all two-dimensional solitary waves are linea...

Transient Growth Analysis of Compressible Boundary Layers with Parabolized Stability Equations

Science.gov (United States)

Paredes, Pedro; Choudhari, Meelan M.; Li, Fei; Chang, Chau-Lyan

2016-01-01

The linear form of parabolized linear stability equations (PSE) is used in a variational approach to extend the previous body of results for the optimal, non-modal disturbance growth in boundary layer flows. This methodology includes the non-parallel effects associated with the spatial development of boundary layer flows. As noted in literature, the optimal initial disturbances correspond to steady counter-rotating stream-wise vortices, which subsequently lead to the formation of stream-wise-elongated structures, i.e., streaks, via a lift-up effect. The parameter space for optimal growth is extended to the hypersonic Mach number regime without any high enthalpy effects, and the effect of wall cooling is studied with particular emphasis on the role of the initial disturbance location and the value of the span-wise wavenumber that leads to the maximum energy growth up to a specified location. Unlike previous predictions that used a basic state obtained from a self-similar solution to the boundary layer equations, mean flow solutions based on the full Navier-Stokes (NS) equations are used in select cases to help account for the viscous-inviscid interaction near the leading edge of the plate and also for the weak shock wave emanating from that region. These differences in the base flow lead to an increasing reduction with Mach number in the magnitude of optimal growth relative to the predictions based on self-similar mean-flow approximation. Finally, the maximum optimal energy gain for the favorable pressure gradient boundary layer near a planar stagnation point is found to be substantially weaker than that in a zero pressure gradient Blasius boundary layer.

Pramana – Journal of Physics | Indian Academy of Sciences

Indian Academy of Sciences (India)

Kuramoto equation) has been considered. By using the Painlevé analysis and Lie group analysis methods, the Painlevé properties and symmetries have been studied. Some solutions of the reduced ODEs are obtained.

Exponential Stability of the Monotubular Heat Exchanger Equation with Time Delay in Boundary Observation

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Xue-Lian Jin

2017-01-01

Full Text Available The exponential stability of the monotubular heat exchanger equation with boundary observation possessing a time delay and inner control was investigated. Firstly, the close-loop system was translated into an abstract Cauchy problem in the suitable state space. A uniformly bounded C0-semigroup generated by the close-loop system, which implies that the unique solution of the system exists, was shown. Secondly, the spectrum configuration of the closed-loop system was analyzed and the eventual differentiability and the eventual compactness of the semigroup were shown by the resolvent estimates on some resolvent sets. This implies that the spectrum-determined growth assumption holds. Finally, a sufficient condition, which is related to the physical parameters in the system and is independent of the time delay, of the exponential stability of the closed-loop system was given.

Using the charge-stabilization technique in the double ionization potential equation-of-motion calculations with dianion references.

Science.gov (United States)

Kuś, Tomasz; Krylov, Anna I

2011-08-28

The charge-stabilization method is applied to double ionization potential equation-of-motion (EOM-DIP) calculations to stabilize unstable dianion reference functions. The auto-ionizing character of the dianionic reference states spoils the numeric performance of EOM-DIP limiting applications of this method. We demonstrate that reliable excitation energies can be computed by EOM-DIP using a stabilized resonance wave function instead of the lowest energy solution corresponding to the neutral + free electron(s) state of the system. The details of charge-stabilization procedure are discussed and illustrated by examples. The choice of optimal stabilizing Coulomb potential, which is strong enough to stabilize the dianion reference, yet, minimally perturbs the target states of the neutral, is the crux of the approach. Two algorithms of choosing optimal parameters of the stabilization potential are presented. One is based on the orbital energies, and another--on the basis set dependence of the total Hartree-Fock energy of the reference. Our benchmark calculations of the singlet-triplet energy gaps in several diradicals show a remarkable improvement of the EOM-DIP accuracy in problematic cases. Overall, the excitation energies in diradicals computed using the stabilized EOM-DIP are within 0.2 eV from the reference EOM spin-flip values. © 2011 American Institute of Physics

Numerical Solution of Heun Equation Via Linear Stochastic Differential Equation

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Hamidreza Rezazadeh

2014-05-01

Full Text Available In this paper, we intend to solve special kind of ordinary differential equations which is called Heun equations, by converting to a corresponding stochastic differential equation(S.D.E.. So, we construct a stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained solutions of these S.D.E.s compared with exact solution of corresponding differential equations.

Stability of Planar Rarefaction Wave to 3D Full Compressible Navier-Stokes Equations

Science.gov (United States)

Li, Lin-an; Wang, Teng; Wang, Yi

2018-05-01

We prove time-asymptotic stability toward the planar rarefaction wave for the three-dimensional full, compressible Navier-Stokes equations with the heat-conductivities in an infinite long flat nozzle domain {R × T^2} . Compared with one-dimensional case, the proof here is based on our new observations on the cancellations on the flux terms and viscous terms due to the underlying wave structures, which are crucial for overcoming the difficulties due to the wave propagation in the transverse directions x 2 and x 3 and its interactions with the planar rarefaction wave in x 1 direction.

Stability analysis for a delay differential equations model of a hydraulic turbine speed governor

Science.gov (United States)

Halanay, Andrei; Safta, Carmen A.; Dragoi, Constantin; Piraianu, Vlad F.

2017-01-01

The paper aims to study the dynamic behavior of a speed governor for a hydraulic turbine using a mathematical model. The nonlinear mathematical model proposed consists in a system of delay differential equations (DDE) to be compared with already established mathematical models of ordinary differential equations (ODE). A new kind of nonlinearity is introduced as a time delay. The delays can characterize different running conditions of the speed governor. For example, it is considered that spool displacement of hydraulic amplifier might be blocked due to oil impurities in the oil supply system and so the hydraulic amplifier has a time delay in comparison to the time control. Numerical simulations are presented in a comparative manner. A stability analysis of the hydraulic control system is performed, too. Conclusions of the dynamic behavior using the DDE model of a hydraulic turbine speed governor are useful in modeling and controlling hydropower plants.

Modeling boundary-layer transition in DNS and LES using Parabolized Stability Equations

Science.gov (United States)

Lozano-Duran, Adrian; Hack, M. J. Philipp; Moin, Parviz

2016-11-01

The modeling of the laminar region and the prediction of the point of transition remain key challenges in the numerical simulation of boundary layers. The issue is of particular relevance for wall-modeled large eddy simulations which require 10 to 100 times higher grid resolution in the thin laminar region than in the turbulent regime. Our study examines the potential of the nonlinear parabolized stability equations (PSE) to provide an accurate, yet computationally efficient treatment of the growth of disturbances in the pre-transitional flow regime. The PSE captures the nonlinear interactions that eventually induce breakdown to turbulence, and can as such identify the onset of transition without relying on empirical correlations. Since the local PSE solution at the point of transition is the solution of the Navier-Stokes equations, it provides a natural inflow condition for large eddy and direct simulations by avoiding unphysical transients. We show that in a classical H-type transition scenario, a combined PSE/DNS approach can reproduce the skin-friction distribution obtained in reference direct numerical simulations. The computational cost in the laminar region is reduced by several orders of magnitude. Funded by the Air Force Office of Scientific Research.

The role of driver nodes in managing epileptic seizures: Application of Kuramoto model.

Science.gov (United States)

Mohseni, Ali; Gharibzadeh, Shahriar; Bakouie, Fatemeh

2017-04-21

Synchronization is an important global phenomenon which could be found in a wide range of complex systems such as brain or electronic devices. However, in some circumstances the synchronized states are not desirable for the system and should be suppressed. For example, excessively synchronized activities in the brain network could be the root of neuronal disorders like epileptic seizures. According to the controllability theory of the complex networks, a minimum set of driver nodes has the ability to control the entire system. In this study, we examine the role of driver nodes in suppressing the excessive synchronization in a generalized Kuramoto model, which consists of two types of oscillators: contrarian and regular ones. We used two different structural topologies: Barabási-Albert scale-free (BASF) network and Caenorhabditis elegans (C.elegans) neuronal network. Our results show that contrarian driver nodes have the sufficient ability to break the synchronized level of the systems. In this case, the system coherency level is not fully suppressed that is avoiding dysfunctions of normal brain functions which require the neuronal synchronized activities. Moreover, in this case, the oscillators grouped in two distinct synchronized clusters that could be an indication of chaotic behavior of the system known as resting-state activity of the brain. Copyright © 2017 Elsevier Ltd. All rights reserved.

On parametric domain for asymptotic stability with probability one of zero solution of linear Ito stochastic differential equations

International Nuclear Information System (INIS)

Phan Thanh An; Phan Le Na; Ngo Quoc Chung

2004-05-01

We describe a practical implementation for finding parametric domain for asymptotic stability with probability one of zero solution of linear Ito stochastic differential equations based on Korenevskij and Mitropolskij's sufficient condition and our sufficient conditions. Numerical results show that all of these sufficient conditions are crucial in the implementation. (author)

Dromion-like structures and stability analysis in the variable coefficients complex Ginzburg–Landau equation

International Nuclear Information System (INIS)

Wong, Pring; Pang, Li-Hui; Huang, Long-Gang; Li, Yan-Qing; Lei, Ming; Liu, Wen-Jun

2015-01-01

The study of the complex Ginzburg–Landau equation, which can describe the fiber laser system, is of significance for ultra-fast laser. In this paper, dromion-like structures for the complex Ginzburg–Landau equation are considered due to their abundant nonlinear dynamics. Via the modified Hirota method and simplified assumption, the analytic dromion-like solution is obtained. The partial asymmetry of structure is particularly discussed, which arises from asymmetry of nonlinear and dispersion terms. Furthermore, the stability of dromion-like structures is analyzed. Oscillation structure emerges to exhibit strong interference when the dispersion loss is perturbed. Through the appropriate modulation of modified exponent parameter, the oscillation structure is transformed into two dromion-like structures. It indicates that the dromion-like structure is unstable, and the coherence intensity is affected by the modified exponent parameter. Results in this paper may be useful in accounting for some nonlinear phenomena in fiber laser systems, and understanding the essential role of modified Hirota method

Modeling boundary-layer transition in direct and large-eddy simulations using parabolized stability equations

Science.gov (United States)

Lozano-Durán, A.; Hack, M. J. P.; Moin, P.

2018-02-01

We examine the potential of the nonlinear parabolized stability equations (PSE) to provide an accurate yet computationally efficient treatment of the growth of disturbances in H-type transition to turbulence. The PSE capture the nonlinear interactions that eventually induce breakdown to turbulence and can as such identify the onset of transition without relying on empirical correlations. Since the local PSE solution at the onset of transition is a close approximation of the Navier-Stokes equations, it provides a natural inflow condition for direct numerical simulations (DNS) and large-eddy simulations (LES) by avoiding nonphysical transients. We show that a combined PSE-DNS approach, where the pretransitional region is modeled by the PSE, can reproduce the skin-friction distribution and downstream turbulent statistics from a DNS of the full domain. When the PSE are used in conjunction with wall-resolved and wall-modeled LES, the computational cost in both the laminar and turbulent regions is reduced by several orders of magnitude compared to DNS.

A Semilinear Wave Equation with a Boundary Condition of Many-Point Type: Global Existence and Stability of Weak Solutions

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Giai Giang Vo

2015-01-01

Full Text Available This paper is devoted to the study of a wave equation with a boundary condition of many-point type. The existence of weak solutions is proved by using the Galerkin method. Also, the uniqueness and the stability of solutions are established.

Stability analysis of embedded solitons in the generalized third-order nonlinear Schroedinger equation

International Nuclear Information System (INIS)

Pelinovsky, Dmitry E.; Yang Jianke

2005-01-01

We study the generalized third-order nonlinear Schroedinger (NLS) equation which admits a one-parameter family of single-hump embedded solitons. Analyzing the spectrum of the linearization operator near the embedded soliton, we show that there exists a resonance pole in the left half-plane of the spectral parameter, which explains linear stability, rather than nonlinear semistability, of embedded solitons. Using exponentially weighted spaces, we approximate the resonance pole both analytically and numerically. We confirm in a near-integrable asymptotic limit that the resonance pole gives precisely the linear decay rate of parameters of the embedded soliton. Using conserved quantities, we qualitatively characterize the stable dynamics of embedded solitons

Functional equations with causal operators

CERN Document Server

Corduneanu, C

2003-01-01

Functional equations encompass most of the equations used in applied science and engineering: ordinary differential equations, integral equations of the Volterra type, equations with delayed argument, and integro-differential equations of the Volterra type. The basic theory of functional equations includes functional differential equations with causal operators. Functional Equations with Causal Operators explains the connection between equations with causal operators and the classical types of functional equations encountered by mathematicians and engineers. It details the fundamentals of linear equations and stability theory and provides several applications and examples.

Nonlinear Stability of ρ-Functional Equations in Latticetic Random Banach Lattice Spaces

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Mohammad Maleki V.

2018-02-01

Full Text Available In this paper, we prove the generalized nonlinear stability of the first and second of the following ρ -functional equations, G ( | a | Δ A * | b | Δ B * G ( | a | Δ A * * | b | − G ( | a | Δ B * * G ( | b | = ρ ( 2 G | a | Δ A * | b | 2 Δ B * G | a | Δ A * * | b | 2 − G ( | a | Δ B * * G ( | b | , and 2 G | a | Δ A * | b | 2 Δ B * G | a | Δ A * * | b | 2 − G ( | a | Δ B * * G ( | b | = ρ G ( | a | Δ A * | b | Δ B * G ( | a | Δ A * * | b | − G ( | a | Δ B * * G ( | b | in latticetic random Banach lattice spaces, where ρ is a fixed real or complex number with ρ ≠ 1 .

The concept of stability in numerical mathematics

CERN Document Server

Hackbusch, Wolfgang

2014-01-01

In this book, the author compares the meaning of stability in different subfields of numerical mathematics.  Concept of Stability in numerical mathematics opens by examining the stability of finite algorithms. A more precise definition of stability holds for quadrature and interpolation methods, which the following chapters focus on. The discussion then progresses to the numerical treatment of ordinary differential equations (ODEs). While one-step methods for ODEs are always stable, this is not the case for hyperbolic or parabolic differential equations, which are investigated next. The final chapters discuss stability for discretisations of elliptic differential equations and integral equations. In comparison among the subfields we discuss the practical importance of stability and the possible conflict between higher consistency order and stability.  

Stability Results, Almost Global Generalized Beltrami Fields and Applications to Vortex Structures in the Euler Equations

Science.gov (United States)

Enciso, Alberto; Poyato, David; Soler, Juan

2018-05-01

Strong Beltrami fields, that is, vector fields in three dimensions whose curl is the product of the field itself by a constant factor, have long played a key role in fluid mechanics and magnetohydrodynamics. In particular, they are the kind of stationary solutions of the Euler equations where one has been able to show the existence of vortex structures (vortex tubes and vortex lines) of arbitrarily complicated topology. On the contrary, there are very few results about the existence of generalized Beltrami fields, that is, divergence-free fields whose curl is the field times a non-constant function. In fact, generalized Beltrami fields (which are also stationary solutions to the Euler equations) have been recently shown to be rare, in the sense that for "most" proportionality factors there are no nontrivial Beltrami fields of high enough regularity (e.g., of class {C^{6,α}}), not even locally. Our objective in this work is to show that, nevertheless, there are "many" Beltrami fields with non-constant factor, even realizing arbitrarily complicated vortex structures. This fact is relevant in the study of turbulent configurations. The core results are an "almost global" stability theorem for strong Beltrami fields, which ensures that a global strong Beltrami field with suitable decay at infinity can be perturbed to get "many" Beltrami fields with non-constant factor of arbitrarily high regularity and defined in the exterior of an arbitrarily small ball, and a "local" stability theorem for generalized Beltrami fields, which is an analogous perturbative result which is valid for any kind of Beltrami field (not just with a constant factor) but only applies to small enough domains. The proof relies on an iterative scheme of Grad-Rubin type. For this purpose, we study the Neumann problem for the inhomogeneous Beltrami equation in exterior domains via a boundary integral equation method and we obtain Hölder estimates, a sharp decay at infinity and some compactness

Numerical resolution of Navier-Stokes equations coupled to the heat equation

International Nuclear Information System (INIS)

Zenouda, Jean-Claude

1970-08-01

The author proves a uniqueness theorem for the time dependent Navier-Stokes equations coupled with heat flow in the two-dimensional case. He studies stability and convergence of several finite - difference schemes to solve these equations. Numerical experiments are done in the case of a square domain. (author) [fr

Analyticity estimates for the Navier-Stokes equations

DEFF Research Database (Denmark)

Herbst, I.; Skibsted, Erik

We study spatial analyticity properties of solutions of the Navier-Stokes equation and obtain new growth rate estimates for the analyticity radius. We also study stability properties of strong global solutions of the Navier-Stokes equation with data in and prove a stability result...

Power-spectral-density relationship for retarded differential equations

Science.gov (United States)

Barker, L. K.

1974-01-01

The power spectral density (PSD) relationship between input and output of a set of linear differential-difference equations of the retarded type with real constant coefficients and delays is discussed. The form of the PSD relationship is identical with that applicable to unretarded equations. Since the PSD relationship is useful if and only if the system described by the equations is stable, the stability must be determined before applying the PSD relationship. Since it is sometimes difficult to determine the stability of retarded equations, such equations are often approximated by simpler forms. It is pointed out that some common approximations can lead to erroneous conclusions regarding the stability of a system and, therefore, to the possibility of obtaining PSD results which are not valid.

Global exponential stability of a class of retarded impulsive differential equations with applications

International Nuclear Information System (INIS)

Xia Yonghui; Wong, Patricia J.Y.

2009-01-01

This paper studies the dynamics of a class of retarded impulsive differential equations (IDE), which generalizes the delayed cellular neural networks (DCNN), delayed bidirectional associative memory (BAM) neural networks and some population growth models. Some sufficient criteria are obtained for the existence and global exponential stability of a unique equilibrium. When the impulsive jumps are absent, our results reduce to its corresponding results for the non-impulsive systems. The approaches are based on Banach's fixed point theorem, matrix theory and its spectral theory. Due to this method, our results generalize and improve many previous known results such as . Some examples are also included to illustrate the feasibility and effectiveness of the results obtained

Stability of mixing layers

Science.gov (United States)

Tam, Christopher; Krothapalli, A

1993-01-01

The research program for the first year of this project (see the original research proposal) consists of developing an explicit marching scheme for solving the parabolized stability equations (PSE). Performing mathematical analysis of the computational algorithm including numerical stability analysis and the determination of the proper boundary conditions needed at the boundary of the computation domain are implicit in the task. Before one can solve the parabolized stability equations for high-speed mixing layers, the mean flow must first be found. In the past, instability analysis of high-speed mixing layer has mostly been performed on mean flow profiles calculated by the boundary layer equations. In carrying out this project, it is believed that the boundary layer equations might not give an accurate enough nonparallel, nonlinear mean flow needed for parabolized stability analysis. A more accurate mean flow can, however, be found by solving the parabolized Navier-Stokes equations. The advantage of the parabolized Navier-Stokes equations is that its accuracy is consistent with the PSE method. Furthermore, the method of solution is similar. Hence, the major part of the effort of the work of this year has been devoted to the development of an explicit numerical marching scheme for the solution of the Parabolized Navier-Stokes equation as applied to the high-seed mixing layer problem.

Finite-time stability of discrete fractional delay systems: Gronwall inequality and stability criterion

Science.gov (United States)

Wu, Guo-Cheng; Baleanu, Dumitru; Zeng, Sheng-Da

2018-04-01

This study investigates finite-time stability of Caputo delta fractional difference equations. A generalized Gronwall inequality is given on a finite time domain. A finite-time stability criterion is proposed for fractional differential equations. Then the idea is extended to the discrete fractional case. A linear fractional difference equation with constant delays is considered and finite-time stable conditions are provided. One example is numerically illustrated to support the theoretical result.

Nonlinear Stability and Convergence of Two-Step Runge-Kutta Methods for Volterra Delay Integro-Differential Equations

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Haiyan Yuan

2013-01-01

Full Text Available This paper introduces the stability and convergence of two-step Runge-Kutta methods with compound quadrature formula for solving nonlinear Volterra delay integro-differential equations. First, the definitions of (k,l-algebraically stable and asymptotically stable are introduced; then the asymptotical stability of a (k,l-algebraically stable two-step Runge-Kutta method with 0

Differential equations

CERN Document Server

Barbu, Viorel

2016-01-01

This textbook is a comprehensive treatment of ordinary differential equations, concisely presenting basic and essential results in a rigorous manner. Including various examples from physics, mechanics, natural sciences, engineering and automatic theory, Differential Equations is a bridge between the abstract theory of differential equations and applied systems theory. Particular attention is given to the existence and uniqueness of the Cauchy problem, linear differential systems, stability theory and applications to first-order partial differential equations. Upper undergraduate students and researchers in applied mathematics and systems theory with a background in advanced calculus will find this book particularly useful. Supplementary topics are covered in an appendix enabling the book to be completely self-contained.

Computational partial differential equations using Matlab

CERN Document Server

Li, Jichun

2008-01-01

Brief Overview of Partial Differential Equations The parabolic equations The wave equations The elliptic equations Differential equations in broader areasA quick review of numerical methods for PDEsFinite Difference Methods for Parabolic Equations Introduction Theoretical issues: stability, consistence, and convergence 1-D parabolic equations2-D and 3-D parabolic equationsNumerical examples with MATLAB codesFinite Difference Methods for Hyperbolic Equations IntroductionSome basic difference schemes Dissipation and dispersion errors Extensions to conservation lawsThe second-order hyperbolic PDE

The impact of turbulent renewable energy production on power grid stability and quality

Science.gov (United States)

Schmietendorf, Katrin; Peinke, Joachim; Kamps, Oliver

2017-11-01

Feed-in fluctuations induced by renewables are one of the key challenges to the stability and quality of electrical power grids. In particular short-term fluctuations disturb the system on a time scale, on which load balancing does not operate yet and the system is intrinsically governed by self-organized synchronization. Wind and solar power are known to be strongly non-Gaussian with intermittent increment statistics in these time scales. We investigate the impact of short-term wind fluctuations on the basis of a Kuramoto-like power grid model considering stability in terms of desynchronization and frequency and voltage quality aspects. We present a procedure to generate realistic feed-in fluctuations with temporal correlations, Kolmogorov power spectrum and intermittent increments. By comparison to correlated Gaussian noise of the same spectrum and Gaussian white noise, we found out that while the correlations are essential to capture the likelihood of severe outages, the intermittent nature of wind power has significant consequences on power quality: intermittency is directly transferred into frequency and voltage fluctuations yielding a novel type of fluctuations, which is beyond engineering status of knowledge.

Aircraft nonlinear stability analysis and multidimensional stability region estimation under icing conditions

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Liang QU

2017-06-01

Full Text Available Icing is one of the crucial factors that could pose great threat to flight safety, and thus research on stability and stability region of aircraft safety under icing conditions is significant for control and flight. Nonlinear dynamical equations and models of aerodynamic coefficients of an aircraft are set up in this paper to study the stability and stability region of the aircraft under an icing condition. Firstly, the equilibrium points of the iced aircraft system are calculated and analyzed based on the theory of differential equation stability. Secondly, according to the correlation theory about equilibrium points and the stability region, this paper estimates the multidimensional stability region of the aircraft, based on which the stability regions before and after icing are compared. Finally, the results are confirmed by the time history analysis. The results can give a reference for stability analysis and envelope protection of the nonlinear system of an iced aircraft.

Influence of stochastic perturbations on the cluster explosive synchronization of second-order Kuramoto oscillators on networks.

Science.gov (United States)

Cao, Liang; Tian, Changhai; Wang, Zhenhua; Zhang, Xiyun; Liu, Zonghua

2018-02-01

Explosive synchronization in networked second-order Kuramoto oscillators has been well studied recently and it is revealed that the synchronization process is featured by cluster explosive synchronization. However, little attention has been paid to the influence of noise or perturbation. We here study this problem and discuss the influences of noise and perturbation. For the former, we interestingly find that noise has significant influence on the cluster explosive synchronization of those nodes with smaller degrees, i.e., their synchronization will change from the first-order to second-order transition and the critical points for both the forward and backward synchronization depend on the strength of noise. Especially, when the strength of noise is in an optimal range, a synchronization of the nodes with smaller degrees will be induced in the region of coupling strength where they do not display synchronization in the absence of noise. For the latter, we find that the effect of perturbation is similar to that of noise when its duration W is small. However, the perturbation will induce a change from cluster explosive synchronization to explosive synchronization when W is large. Furthermore, a brief theory is provided to explain the influence of perturbations on the critical points.

Iterative Splitting Methods for Differential Equations

CERN Document Server

Geiser, Juergen

2011-01-01

Iterative Splitting Methods for Differential Equations explains how to solve evolution equations via novel iterative-based splitting methods that efficiently use computational and memory resources. It focuses on systems of parabolic and hyperbolic equations, including convection-diffusion-reaction equations, heat equations, and wave equations. In the theoretical part of the book, the author discusses the main theorems and results of the stability and consistency analysis for ordinary differential equations. He then presents extensions of the iterative splitting methods to partial differential

On the stability, the periodic solutions and the resolution of certain types of non linear equations, and of non linearly coupled systems of these equations, appearing in betatronic oscillations

International Nuclear Information System (INIS)

Valat, J.

1960-12-01

Universal stability diagrams have been calculated and experimentally checked for Hill-Meissner type equations with square-wave coefficients. The study of these equations in the phase-plane has then made it possible to extend the periodic solution calculations to the case of non-linear differential equations with periodic square-wave coefficients. This theory has been checked experimentally. For non-linear coupled systems with constant coefficients, a search was first made for solutions giving an algebraic motion. The elliptical and Fuchs's functions solve such motions. The study of non-algebraic motions is more delicate, apart from the study of nonlinear Lissajous's motions. A functional analysis shows that it is possible however in certain cases to decouple the system and to find general solutions. For non-linear coupled systems with periodic square-wave coefficients it is then possible to calculate the conditions leading to periodic solutions, if the two non-linear associated systems with constant coefficients fall into one of the categories of the above paragraph. (author) [fr

Orbital stability of standing waves of a class of fractional Schrödinger equations with Hartree-type nonlinearity

KAUST Repository

Cho, Yonggeun

2016-05-04

This paper is devoted to the mathematical analysis of a class of nonlinear fractional Schrödinger equations with a general Hartree-type integrand. We show the well-posedness of the associated Cauchy problem and prove the existence and stability of standing waves under suitable assumptions on the nonlinearity. Our proofs rely on a contraction argument in mixed functional spaces and the concentration-compactness method. © 2015 World Scientific Publishing Company

Constitutive equations for discrete electromagnetic problems over polyhedral grids

International Nuclear Information System (INIS)

Codecasa, Lorenzo; Trevisan, Francesco

2007-01-01

In this paper a novel approach is proposed for constructing discrete counterparts of constitutive equations over polyhedral grids which ensure both consistency and stability of the algebraic equations discretizing an electromagnetic field problem. The idea is to construct discrete constitutive equations preserving the thermodynamic relations for constitutive equations. In this way, consistency and stability of the discrete equations are ensured. At the base, a purely geometric condition between the primal and the dual grids has to be satisfied for a given primal polyhedral grid, by properly choosing the dual grid. Numerical experiments demonstrate that the proposed discrete constitutive equations lead to accurate approximations of the electromagnetic field

STABILITY OF NONLINEAR NEUTRAL DIFFERENTIAL EQUATION VIA FIXED POINT

Institute of Scientific and Technical Information of China (English)

无

2012-01-01

In this paper,a nonlinear neutral differential equation is considered.By a fixed point theory,we give some conditions to ensure that the zero solution to the equation is asymptotically stable.Some existing results are improved and generalized.

Some Functional Equations Originating from Number Theory

Indian Academy of Sciences (India)

We will introduce new functional equations (3) and (4) which are strongly related to well-known formulae (1) and (2) of number theory, and investigate the solutions of the equations. Moreover, we will also study some stability problems of those equations.

Ordinary differential equations a graduate text

CERN Document Server

Bhamra, K S

2015-01-01

ORDINARY DIFFERENTIAL EQUATIONS: A Graduate Text presents a systematic and comprehensive introduction to ODEs for graduate and postgraduate students. The systematic organized text on differential inequalities, Gronwall's inequality, Nagumo's theorems, Osgood's criteria and applications of different equations of first order is dealt with in a greater depth. The book discusses qualitative and quantitative aspects of the Strum - Liouville problems, Green's function, integral equations, Laplace transform and is supported by a number of worked-out examples in each lesson to make the concepts clear. A lot of stress on stability theory is laid down, especially on Lyapunov and Poincare stability theory. A numerous figures in various lessons (in particular lessons dealing with stability theory) have been added to clarify the key concepts in DE theory. Nonlinear oscillation in conservative systems and Hamiltonian systems highlights basic nature of the systems considered. Perturbation techniques lesson deals in fairly d...

Exotic equilibria of Harary graphs and a new minimum degree lower bound for synchronization

KAUST Repository

Canale, Eduardo A.; Monzó n, Pablo

2015-01-01

© 2015 AIP Publishing LLC. This work is concerned with stability of equilibria in the homogeneous (equal frequencies) Kuramoto model of weakly coupled oscillators. In 2012 [R. Taylor, J. Phys. A: Math. Theor. 45, 1-15 (2012)], a sufficient condition

Stability and stabilization of nonlinear systems and Takagi-Sugeno's fuzzy models

Directory of Open Access Journals (Sweden)

Blanco Yann

2001-01-01

Full Text Available This paper outlines a methodology to study the stability of Takagi-Sugeno's (TS fuzzy models. The stability analysis of the TS model is performed using a quadratic Liapunov candidate function. This paper proposes a relaxation of Tanaka's stability condition: unlike related works, the equations to be solved are not Liapunov equations for each rule matrix, but a convex combination of them. The coefficients of this sums depend on the membership functions. This method is applied to the design of continuous controllers for the TS model. Three different control structures are investigated, among which the Parallel Distributed Compensation (PDC. An application to the inverted pendulum is proposed here.

A Stabilized Finite Element Method for Modified Poisson-Nernst-Planck Equations to Determine Ion Flow Through a Nanopore

Science.gov (United States)

Chaudhry, Jehanzeb Hameed; Comer, Jeffrey; Aksimentiev, Aleksei; Olson, Luke N.

2013-01-01

The conventional Poisson-Nernst-Planck equations do not account for the finite size of ions explicitly. This leads to solutions featuring unrealistically high ionic concentrations in the regions subject to external potentials, in particular, near highly charged surfaces. A modified form of the Poisson-Nernst-Planck equations accounts for steric effects and results in solutions with finite ion concentrations. Here, we evaluate numerical methods for solving the modified Poisson-Nernst-Planck equations by modeling electric field-driven transport of ions through a nanopore. We describe a novel, robust finite element solver that combines the applications of the Newton's method to the nonlinear Galerkin form of the equations, augmented with stabilization terms to appropriately handle the drift-diffusion processes. To make direct comparison with particle-based simulations possible, our method is specifically designed to produce solutions under periodic boundary conditions and to conserve the number of ions in the solution domain. We test our finite element solver on a set of challenging numerical experiments that include calculations of the ion distribution in a volume confined between two charged plates, calculations of the ionic current though a nanopore subject to an external electric field, and modeling the effect of a DNA molecule on the ion concentration and nanopore current. PMID:24363784

Algorithms For Integrating Nonlinear Differential Equations

Science.gov (United States)

Freed, A. D.; Walker, K. P.

1994-01-01

Improved algorithms developed for use in numerical integration of systems of nonhomogenous, nonlinear, first-order, ordinary differential equations. In comparison with integration algorithms, these algorithms offer greater stability and accuracy. Several asymptotically correct, thereby enabling retention of stability and accuracy when large increments of independent variable used. Accuracies attainable demonstrated by applying them to systems of nonlinear, first-order, differential equations that arise in study of viscoplastic behavior, spread of acquired immune-deficiency syndrome (AIDS) virus and predator/prey populations.

Multilevel and Multi-index Monte Carlo methods for the McKean–Vlasov equation

KAUST Repository

Haji-Ali, Abdul-Lateef

2017-09-12

We address the approximation of functionals depending on a system of particles, described by stochastic differential equations (SDEs), in the mean-field limit when the number of particles approaches infinity. This problem is equivalent to estimating the weak solution of the limiting McKean–Vlasov SDE. To that end, our approach uses systems with finite numbers of particles and a time-stepping scheme. In this case, there are two discretization parameters: the number of time steps and the number of particles. Based on these two parameters, we consider different variants of the Monte Carlo and Multilevel Monte Carlo (MLMC) methods and show that, in the best case, the optimal work complexity of MLMC, to estimate the functional in one typical setting with an error tolerance of $$\\\\mathrm {TOL}$$TOL, is when using the partitioning estimator and the Milstein time-stepping scheme. We also consider a method that uses the recent Multi-index Monte Carlo method and show an improved work complexity in the same typical setting of . Our numerical experiments are carried out on the so-called Kuramoto model, a system of coupled oscillators.

Orbital stability of standing waves for a class of Schrödinger equations with unbounded potential

Directory of Open Access Journals (Sweden)

2006-01-01

Full Text Available This paper is concerned with the nonlinear Schrödinger equation with an unbounded potential i ϕ t = − Δ ϕ + V ( x ϕ − μ | ϕ | p − 1 ϕ − λ | ϕ | q − 1 ϕ , x ∈ ℝ N , t ≥ 0 , where 0$"> μ > 0 , 0,$"> λ > 0 , and 1 < p < q < 1 + 4 / N . The potential V ( x is bounded from below and satisfies V ( x → ∞ as | x | → ∞ . From variational calculus and a compactness lemma, the existence of standing waves and their orbital stability are obtained.

Generalized Stabilities of Euler-Lagrange-Jensen (a,b-Sextic Functional Equations in Quasi-β-Normed Spaces

Directory of Open Access Journals (Sweden)

John Michael Rassias

2017-07-01

Full Text Available The aim of this paper is to investigate generalized Ulam-Hyers stabilities of the following Euler-Lagrange-Jensen-$(a,b$-sextic functional equation $$ f(ax+by+f(bx+ay+(a-b^6\\left[f\\left(\\frac{ax-by}{a-b}\\right+f\\left(\\frac{bx-ay}{b-a}\\right\\right]\\\\ = 64(ab^2\\left(a^2+b^2\\right\\left[f\\left(\\frac{x+y}{2}\\right+f\\left(\\frac{x-y}{2}\\right\\right]\\\\ +2\\left(a^2-b^2\\right\\left(a^4-b^4\\right[f(x+f(y] $$ where $a\

Validation of three-dimensional incompressible spatial direct numerical simulation code: A comparison with linear stability and parabolic stability equation theories for boundary-layer transition on a flat plate

Science.gov (United States)

Joslin, Ronald D.; Streett, Craig L.; Chang, Chau-Lyan

1992-01-01

Spatially evolving instabilities in a boundary layer on a flat plate are computed by direct numerical simulation (DNS) of the incompressible Navier-Stokes equations. In a truncated physical domain, a nonstaggered mesh is used for the grid. A Chebyshev-collocation method is used normal to the wall; finite difference and compact difference methods are used in the streamwise direction; and a Fourier series is used in the spanwise direction. For time stepping, implicit Crank-Nicolson and explicit Runge-Kutta schemes are used to the time-splitting method. The influence-matrix technique is used to solve the pressure equation. At the outflow boundary, the buffer-domain technique is used to prevent convective wave reflection or upstream propagation of information from the boundary. Results of the DNS are compared with those from both linear stability theory (LST) and parabolized stability equation (PSE) theory. Computed disturbance amplitudes and phases are in very good agreement with those of LST (for small inflow disturbance amplitudes). A measure of the sensitivity of the inflow condition is demonstrated with both LST and PSE theory used to approximate inflows. Although the DNS numerics are very different than those of PSE theory, the results are in good agreement. A small discrepancy in the results that does occur is likely a result of the variation in PSE boundary condition treatment in the far field. Finally, a small-amplitude wave triad is forced at the inflow, and simulation results are compared with those of LST. Again, very good agreement is found between DNS and LST results for the 3-D simulations, the implication being that the disturbance amplitudes are sufficiently small that nonlinear interactions are negligible.

The existence and global exponential stability of a periodic solution of a class of delay differential equations

International Nuclear Information System (INIS)

Tang, X H; Zou, Xingfu

2009-01-01

By employing Schauder's fixed point theorem and a non-Liapunov method (matrix theory, inequality analysis), we obtain some new criteria that ensure existence and global exponential stability of a periodic solution to a class of functional differential equations. Applying these criteria to a cellular neural network with time delays (delayed cellular neural network, DCNN) under a periodic environment leads to some new results that improve and generalize many existing ones we know on this topic. These results are of great significance in designs and applications of globally stable periodic DCNNs

Element diameter free stability parameters for stabilized methods applied to fluids

International Nuclear Information System (INIS)

Franca, L.P.; Madureira, A.L.

1992-08-01

Stability parameters for stabilized methods in fluids are suggested. The computation of the largest eigenvalue of a generalized eigenvalue problem replaces controversial definitions of element diameters and inverse estimate constants, used heretofore to compute these stability parameters. The design is employed in the advective-diffusive model, incompressible Navier-Stokes equations and the Stokes problem. (author)

On two functional equations originating from number theory

Indian Academy of Sciences (India)

Reducing the functional equations introduced in Proc. Indian Acad. Sci. (Math. Sci.) 113(2) (2003) 91–98 and in Appl. Math. Lett. 21 (2008) 974–977 to equations in complex variables and quaternions, we find general solutions of the equations. We also obtain the stability of the equations.

Kuramoto model for infinite graphs with kernels

KAUST Repository

Canale, Eduardo; Tembine, Hamidou; Tempone, Raul; Zouraris, Georgios E.

2015-01-01

. We focus on circulant graphs which have enough symmetries to make the computations easier. We then focus on the asymptotic regime where an integro-partial differential equation is derived. Numerical analysis and convergence proofs of the Fokker

Plasma equilibrium and stability in stellarators

International Nuclear Information System (INIS)

Pustovitov, V.D.; Shafranov, V.D.

1987-01-01

A review of theoretical methods of investigating plasma equilibrium and stability in stellarators is given. Principles forming the basis of toroidal plasma equilibrium and its stabilization, and the main results of analytical theory and numerical calculations are presented. Configurations with spiral symmetry and usual stellarators with plane axis and spiral fields are considered in detail. Derivation of scalar two-dimensional equations, describing equilibrium in these systems is given. These equations were used to obtain one-dimensional equations for displacement and ellipticity of magnetic surfaces. The model of weak-elliptic displaced surfaces was used to consider the evolution of plasma equilibrium in stellarators after elevation of its pressure: change of profile of rotational transformation after change of plasma pressure, current generation during its fast heating and its successive damping due to finite plasma conductivity were described. The derivation of equations of small oscillations in the form, suitable for local disturbance investigation is presented. These equations were used to obtain Mercier criteria and ballon model equations. General sufficient conditions of plasma stability in systems with magnetic confinement were derived

MATLAB Stability and Control Toolbox Trim and Static Stability Module

Science.gov (United States)

Kenny, Sean P.; Crespo, Luis

2012-01-01

MATLAB Stability and Control Toolbox (MASCOT) utilizes geometric, aerodynamic, and inertial inputs to calculate air vehicle stability in a variety of critical flight conditions. The code is based on fundamental, non-linear equations of motion and is able to translate results into a qualitative, graphical scale useful to the non-expert. MASCOT was created to provide the conceptual aircraft designer accurate predictions of air vehicle stability and control characteristics. The code takes as input mass property data in the form of an inertia tensor, aerodynamic loading data, and propulsion (i.e. thrust) loading data. Using fundamental nonlinear equations of motion, MASCOT then calculates vehicle trim and static stability data for the desired flight condition(s). Available flight conditions include six horizontal and six landing rotation conditions with varying options for engine out, crosswind, and sideslip, plus three take-off rotation conditions. Results are displayed through a unique graphical interface developed to provide the non-stability and control expert conceptual design engineer a qualitative scale indicating whether the vehicle has acceptable, marginal, or unacceptable static stability characteristics. If desired, the user can also examine the detailed, quantitative results.

Nonlinear Fokker-Planck Equations Fundamentals and Applications

CERN Document Server

Frank, Till Daniel

2005-01-01

Providing an introduction to the theory of nonlinear Fokker-Planck equations, this book discusses fundamental properties of transient and stationary solutions, emphasizing the stability analysis of stationary solutions by means of self-consistency equations, linear stability analysis, and Lyapunov's direct method. Also treated are Langevin equations and correlation functions. Nonlinear Fokker-Planck Equations addresses various phenomena such as phase transitions, multistability of systems, synchronization, anomalous diffusion, cut-off solutions, travelling-wave solutions and the emergence of power law solutions. A nonlinear Fokker-Planck perspective to quantum statistics, generalized thermodynamics, and linear nonequilibrium thermodynamics is given. Theoretical concepts are illustrated where possible by simple examples. The book also reviews several applications in the fields of condensed matter physics, the physics of porous media and liquid crystals, accelerator physics, neurophysics, social sciences, popul...

Existence and Asymptotic Stability of Periodic Solutions of the Reaction-Diffusion Equations in the Case of a Rapid Reaction

Science.gov (United States)

Nefedov, N. N.; Nikulin, E. I.

2018-01-01

A singularly perturbed periodic in time problem for a parabolic reaction-diffusion equation in a two-dimensional domain is studied. The case of existence of an internal transition layer under the conditions of balanced and unbalanced rapid reaction is considered. An asymptotic expansion of a solution is constructed. To justify the asymptotic expansion thus constructed, the asymptotic method of differential inequalities is used. The Lyapunov asymptotic stability of a periodic solution is investigated.

State equations and stability of shock wave fronts in homogeneous and heterogeneous metallic medium

International Nuclear Information System (INIS)

Romain, Jean-Pierre

1977-01-01

This research thesis in physical sciences reports a theoretical and experimental study of some mechanical and thermodynamic aspects related to a shock wave propagation in homogeneous and heterogeneous metallic media: state equations, stability and instability of shock wave fronts. In the first part, the author reports the study of the Grueneisen coefficient for some metallic elements with known static and dynamic compression properties. The second part reports the experimental investigation of dynamic compressibility of some materials (lamellar Al-Cu compounds). The front shock wave propagation has been visualised, and experimental Hugoniot curves are compared with those deduced from a developed numeric model and other models. The bismuth Hugoniot curve is also determined, and the author compares the existence and nature of phase transitions obtained by static and dynamic compression

Thermoviscous Model Equations in Nonlinear Acoustics

DEFF Research Database (Denmark)

Rasmussen, Anders Rønne

Four nonlinear acoustical wave equations that apply to both perfect gasses and arbitrary fluids with a quadratic equation of state are studied. Shock and rarefaction wave solutions to the equations are studied. In order to assess the accuracy of the wave equations, their solutions are compared...... to solutions of the basic equations from which the wave equations are derived. A straightforward weakly nonlinear equation is the most accurate for shock modeling. A higher order wave equation is the most accurate for modeling of smooth disturbances. Investigations of the linear stability properties...... of solutions to the wave equations, reveal that the solutions may become unstable. Such instabilities are not found in the basic equations. Interacting shocks and standing shocks are investigated....

Nonlinear differential equations

Energy Technology Data Exchange (ETDEWEB)

Dresner, L.

1988-01-01

This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.

Nonlinear differential equations

International Nuclear Information System (INIS)

Dresner, L.

1988-01-01

This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics

Mode instability in one-dimensional anharmonic lattices: Variational equation approach

Science.gov (United States)

Yoshimura, K.

1999-03-01

The stability of normal mode oscillations has been studied in detail under the single-mode excitation condition for the Fermi-Pasta-Ulam-β lattice. Numerical experiments indicate that the mode stability depends strongly on k/N, where k is the wave number of the initially excited mode and N is the number of degrees of freedom in the system. It has been found that this feature does not change when N increases. We propose an average variational equation - approximate version of the variational equation - as a theoretical tool to facilitate a linear stability analysis. It is shown that this strong k/N dependence of the mode stability can be explained from the view point of the linear stability of the relevant orbits. We introduce a low-dimensional approximation of the average variational equation, which approximately describes the time evolution of variations in four normal mode amplitudes. The linear stability analysis based on this four-mode approximation demonstrates that the parametric instability mechanism plays a crucial role in the strong k/N dependence of the mode stability.

Dynamics of multi-frequency oscillator ensembles with resonant coupling

International Nuclear Information System (INIS)

Lueck, S.; Pikovsky, A.

2011-01-01

We study dynamics of populations of resonantly coupled oscillators having different frequencies. Starting from the coupled van der Pol equations we derive the Kuramoto-type phase model for the situation, where the natural frequencies of two interacting subpopulations are in relation 2:1. Depending on the parameter of coupling, ensembles can demonstrate fully synchronous clusters, partial synchrony (only one subpopulation synchronizes), or asynchrony in both subpopulations. Theoretical description of the dynamics based on the Watanabe-Strogatz approach is developed. -- Highlights: → Kuramoto model is generalized on the case of resonantly interacting oscillators having frequency ratio 2:1. → Regimes of full and partial synchrony, as well as non-synchronous ones are reported. → Analytical description is developed on the basis of the Watanabe-Strogatz approach.

Dynamics of multi-frequency oscillator ensembles with resonant coupling

Energy Technology Data Exchange (ETDEWEB)

Lueck, S. [Department of Physics and Astronomy, Potsdam University, Karl-Liebknecht-Str. 24-25, 14476 Potsdam (Germany); Pikovsky, A., E-mail: pikovsky@stat.physik.uni-potsdam.de [Department of Physics and Astronomy, Potsdam University, Karl-Liebknecht-Str. 24-25, 14476 Potsdam (Germany)

2011-07-11

We study dynamics of populations of resonantly coupled oscillators having different frequencies. Starting from the coupled van der Pol equations we derive the Kuramoto-type phase model for the situation, where the natural frequencies of two interacting subpopulations are in relation 2:1. Depending on the parameter of coupling, ensembles can demonstrate fully synchronous clusters, partial synchrony (only one subpopulation synchronizes), or asynchrony in both subpopulations. Theoretical description of the dynamics based on the Watanabe-Strogatz approach is developed. -- Highlights: → Kuramoto model is generalized on the case of resonantly interacting oscillators having frequency ratio 2:1. → Regimes of full and partial synchrony, as well as non-synchronous ones are reported. → Analytical description is developed on the basis of the Watanabe-Strogatz approach.

A numerical technique for enhanced efficiency and stability for the solution of the nuclear reactor equation

International Nuclear Information System (INIS)

Khotylev, V.A.; Hoogenboom, J.E.

1996-01-01

The paper presents new techniques for the solution of the nuclear reactor equation in diffusion approximation, that has enhanced efficiency and stability. The code system based on the new technique solves a number of steady-state and/or transient problems with coupled thermal hydraulics in one-, two-, or three dimensional geometry with reduced CPU time as compared to similar code systems of previous generations if well-posed neutronics problems are considered. Automated detection of ill-posed problem and selection of the appropriate numerical method makes the new code system capable of yielding a correct solution for wider range of problems without user intervention. (author)

A numerical technique for enhanced efficiency and stability for the solution of the nuclear reactor equation

Energy Technology Data Exchange (ETDEWEB)

Khotylev, V.A.; Hoogenboom, J.E. [Delft Univ. of Technology, Interfaculty Reactor Inst., Delft (Netherlands)

1996-07-01

The paper presents new techniques for the solution of the nuclear reactor equation in diffusion approximation, that has enhanced efficiency and stability. The code system based on the new technique solves a number of steady-state and/or transient problems with coupled thermal hydraulics in one-, two-, or three dimensional geometry with reduced CPU time as compared to similar code systems of previous generations if well-posed neutronics problems are considered. Automated detection of ill-posed problem and selection of the appropriate numerical method makes the new code system capable of yielding a correct solution for wider range of problems without user intervention. (author)

On the Stability of Three-Dimensional Boundary Layers. Part 1; Linear and Nonlinear Stability

Science.gov (United States)

Janke, Erik; Balakumar, Ponnampalam

1999-01-01

The primary stability of incompressible three-dimensional boundary layers is investigated using the Parabolized Stability Equations (PSE). We compute the evolution of stationary and traveling disturbances in the linear and nonlinear region prior to transition. As model problems, we choose Swept Hiemenz Flow and the DLR Transition Experiment. The primary stability results for Swept Hiemenz Flow agree very well with computations by Malik et al. For the DLR Experiment, the mean flow profiles are obtained by solving the boundary layer equations for the measured pressure distribution. Both linear and nonlinear results show very good agreement with the experiment.

Nonlinear physical systems spectral analysis, stability and bifurcations

CERN Document Server

Kirillov, Oleg N

2013-01-01

Bringing together 18 chapters written by leading experts in dynamical systems, operator theory, partial differential equations, and solid and fluid mechanics, this book presents state-of-the-art approaches to a wide spectrum of new and challenging stability problems.Nonlinear Physical Systems: Spectral Analysis, Stability and Bifurcations focuses on problems of spectral analysis, stability and bifurcations arising in the nonlinear partial differential equations of modern physics. Bifurcations and stability of solitary waves, geometrical optics stability analysis in hydro- and magnetohydrodynam

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...

Determination of source terms in a degenerate parabolic equation

International Nuclear Information System (INIS)

Cannarsa, P; Tort, J; Yamamoto, M

2010-01-01

In this paper, we prove Lipschitz stability results for inverse source problems relative to parabolic equations. We use the method introduced by Imanuvilov and Yamamoto in 1998 based on Carleman estimates. What is new here is that we study a class of one-dimensional degenerate parabolic equations. In our model, the diffusion coefficient vanishes at one extreme point of the domain. Instead of the classical Carleman estimates obtained by Fursikov and Imanuvilov for non degenerate equations, we use and extend some recent Carleman estimates for degenerate equations obtained by Cannarsa, Martinez and Vancostenoble. Finally, we obtain Lipschitz stability results in inverse source problems for our class of degenerate parabolic equations both in the case of a boundary observation and in the case of a locally distributed observation

Electronic representation of wave equation

Energy Technology Data Exchange (ETDEWEB)

Veigend, Petr; Kunovský, Jiří, E-mail: kunovsky@fit.vutbr.cz; Kocina, Filip; Nečasová, Gabriela; Valenta, Václav [University of Technology, Faculty of Information Technology, Božetěchova 2, 612 66 Brno (Czech Republic); Šátek, Václav [IT4Innovations, VŠB Technical University of Ostrava, 17. listopadu 15/2172, 708 33 Ostrava-Poruba (Czech Republic); University of Technology, Faculty of Information Technology, Božetěchova 2, 612 66 Brno (Czech Republic)

2016-06-08

The Taylor series method for solving differential equations represents a non-traditional way of a numerical solution. Even though this method is not much preferred in the literature, experimental calculations done at the Department of Intelligent Systems of the Faculty of Information Technology of TU Brno have verified that the accuracy and stability of the Taylor series method exceeds the currently used algorithms for numerically solving differential equations. This paper deals with solution of Telegraph equation using modelling of a series small pieces of the wire. Corresponding differential equations are solved by the Modern Taylor Series Method.

Numerical method for the eigenvalue problem and the singular equation by using the multi-grid method and application to ordinary differential equation

International Nuclear Information System (INIS)

Kanki, Takashi; Uyama, Tadao; Tokuda, Shinji.

1995-07-01

In the numerical method to compute the matching data which are necessary for resistive MHD stability analyses, it is required to solve the eigenvalue problem and the associated singular equation. An iterative method is developed to solve the eigenvalue problem and the singular equation. In this method, the eigenvalue problem is replaced with an equivalent nonlinear equation and a singular equation is derived from Newton's method for the nonlinear equation. The multi-grid method (MGM), a high speed iterative method, can be applied to this method. The convergence of the eigenvalue and the eigenvector, and the CPU time in this method are investigated for a model equation. It is confirmed from the numerical results that this method is effective for solving the eigenvalue problem and the singular equation with numerical stability and high accuracy. It is shown by improving the MGM that the CPU time for this method is 50 times shorter than that of the direct method. (author)

Partial differential equations and their applications

International Nuclear Information System (INIS)

Gauthier-Villars

1998-01-01

This book is dedicated to the French mathematician J.L.Lions. It represents a compilation of articles from about 80 authors. The topics treated are diverse but the more or less commune matter is the study of the characteristics of some partial differential equations. Stability, optimal approximation, numerical resolution, particular applications are among the subjects reviewed. An article deals with the MHD stability of fusion plasmas in tokamaks, another presents the scientific and technical challenges of nuclear energy in France. The latter that contains no equations can be considered as an enjoyable break in a sea of about 40 mathematical articles. (A.C.)

Investigation of the Stability via Shadowing Property

Directory of Open Access Journals (Sweden)

Koh Heejeong

2009-01-01

Full Text Available The shadowing property is to find an exact solution to an iterated map that remains close to an approximate solution. In this article, using shadowing property, we show the stability of the following equation in normed group: , where , and is a mapping. And we prove that the even mapping which satisfies the above equation is quadratic and also the Hyers-Ulam stability of the functional equation in Banach spaces.

Revision to the humidity correction equation in the calculation formulae of the air refractive index based on a phase step interferometer with three frequency-stabilized lasers

International Nuclear Information System (INIS)

Chen, Qianghua; Zhang, Mengce; Liu, Shuaijie; He, Yongxi; Luo, Huifu; Luo, Jun; Lv, Weiwei

2016-01-01

At present the formulae proposed by G Boensch and E Potulski in 1998 (Boensch and Potulski 1998 Metrologia 35 133–9) are mostly used to calculate the air refractive index. However, the humidity correction equation in the formulae is derived by using the light source of a Cd lamp whose light frequency stability is poor and at a narrow temperature range, around 20 °C. So it is no longer suitable in present optical precision measurements. To solve this problem, we propose a refractive index measurement system based on phase step interferometer with three frequency stabilized lasers (532 nm, 633 nm, 780 nm), corrected coefficients of the humidity are measured and a corresponding revised humidity correction equation is acquired. Meanwhile, the application temperature range is extended from 14.6 °C to 25.0 °C. The experiment comparison results at the temperature of 22.2–23.2 °C show the accuracy by the presented equation is better than that of Boensch and Potulski. (paper)

Kinetic stability analyses in a bumpy cylinder

International Nuclear Information System (INIS)

Dominguez, R.R.; Berk, H.L.

1981-01-01

Recent interest in the ELMO Bumpy Torus (EBT) has prompted a number of stability analyses of both the hot electron rings and the toroidal plasma. Typically these works employ the local approximation, neglecting radial eigenmode structure and ballooning effects to perform the stability analysis. In the present work we develop a fully kinetic formalism for performing nonlocal stability analyses in a bumpy cylinder. We show that the Vlasov-Maxwell integral equations (with one ignorable coordinate) are self-adjoint and hence amenable to analysis using numerical techniques developed for self-adjoint systems of equations. The representation we obtain for the kernel of the Vlasov-Maxwell equations is a differential operator of arbitrarily high order. This form leads to a manifestly self-adjoint system of differential equations for long wavelength modes

On stability of the solutions of inverse problem for determining the right-hand side of a degenerate parabolic equation with two independent variables

Science.gov (United States)

Kamynin, V. L.; Bukharova, T. I.

2017-01-01

We prove the estimates of stability with respect to perturbations of input data for the solutions of inverse problems for degenerate parabolic equations with unbounded coefficients. An important feature of these estimates is that the constants in these estimates are written out explicitly by the input data of the problem.

The stability of locus equation slopes across stop consonant voicing/aspiration

Science.gov (United States)

Sussman, Harvey M.; Modarresi, Golnaz

2004-05-01

The consistency of locus equation slopes as phonetic descriptors of stop place in CV sequences across voiced and voiceless aspirated stops was explored in the speech of five male speakers of American English and two male speakers of Persian. Using traditional locus equation measurement sites for F2 onsets, voiceless labial and coronal stops had significantly lower locus equation slopes relative to their voiced counterparts, whereas velars failed to show voicing differences. When locus equations were derived using F2 onsets for voiced stops that were measured closer to the stop release burst, comparable to the protocol for measuring voiceless aspirated stops, no significant effects of voicing/aspiration on locus equation slopes were observed. This methodological factor, rather than an underlying phonetic-based explanation, provides a reasonable account for the observed flatter locus equation slopes of voiceless labial and coronal stops relative to voiced cognates reported in previous studies [Molis et al., J. Acoust. Soc. Am. 95, 2925 (1994); O. Engstrand and B. Lindblom, PHONUM 4, 101-104]. [Work supported by NIH.

Numerical instability of time-discretized one-point kinetic equations

International Nuclear Information System (INIS)

Hashimoto, Kengo; Ikeda, Hideaki; Takeda, Toshikazu

2000-01-01

The one-point kinetic equations with numerical errors induced by the explicit, implicit and Crank-Nicolson integration methods are derived. The zero-power transfer functions based on the present equations are demonstrated to investigate the numerical stability of the discretized systems. These demonstrations indicate unconditional stability for the implicit and Crank-Nicolson methods but present the possibility of numerical instability for the explicit method. An upper limit of time mesh spacing for the stability is formulated and several numerical calculations are made to confirm the validity of this formula

A Fixed Point Approach to the Stability of a General Mixed AQCQ-Functional Equation in Non-Archimedean Normed Spaces

Directory of Open Access Journals (Sweden)

Tian Zhou Xu

2010-01-01

Full Text Available Using the fixed point methods, we prove the generalized Hyers-Ulam stability of the general mixed additive-quadratic-cubic-quartic functional equation f(x+ky+f(x−ky=k2f(x+y+k2f(x−y+2(1−k2f(x+((k4−k2/12[f(2y+f(−2y−4f(y−4f(−y] for a fixed integer k with k≠0,±1 in non-Archimedean normed spaces.

Analysis of spectral methods for the homogeneous Boltzmann equation

KAUST Repository

Filbet, Francis; Mouhot, Clé ment

2011-01-01

The development of accurate and fast algorithms for the Boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis. Recently, several works were devoted to the derivation of spectrally accurate schemes for the Boltzmann equation, but very few of them were concerned with the stability analysis of the method. In particular there was no result of stability except when the method was modified in order to enforce the positivity preservation, which destroys the spectral accuracy. In this paper we propose a new method to study the stability of homogeneous Boltzmann equations perturbed by smoothed balanced operators which do not preserve positivity of the distribution. This method takes advantage of the "spreading" property of the collision, together with estimates on regularity and entropy production. As an application we prove stability and convergence of spectral methods for the Boltzmann equation, when the discretization parameter is large enough (with explicit bound). © 2010 American Mathematical Society.

Analysis of spectral methods for the homogeneous Boltzmann equation

KAUST Repository

Filbet, Francis

2011-04-01

The development of accurate and fast algorithms for the Boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis. Recently, several works were devoted to the derivation of spectrally accurate schemes for the Boltzmann equation, but very few of them were concerned with the stability analysis of the method. In particular there was no result of stability except when the method was modified in order to enforce the positivity preservation, which destroys the spectral accuracy. In this paper we propose a new method to study the stability of homogeneous Boltzmann equations perturbed by smoothed balanced operators which do not preserve positivity of the distribution. This method takes advantage of the "spreading" property of the collision, together with estimates on regularity and entropy production. As an application we prove stability and convergence of spectral methods for the Boltzmann equation, when the discretization parameter is large enough (with explicit bound). © 2010 American Mathematical Society.

Integrability and structural stability of solutions to the Ginzburg-Landau equation

Science.gov (United States)

Keefe, Laurence R.

1986-01-01

The integrability of the Ginzburg-Landau equation is studied to investigate if the existence of chaotic solutions found numerically could have been predicted a priori. The equation is shown not to possess the Painleveproperty, except for a special case of the coefficients that corresponds to the integrable, nonlinear Schroedinger (NLS) equation. Regarding the Ginzburg-Landau equation as a dissipative perturbation of the NLS, numerical experiments show all but one of a family of two-tori solutions, possessed by the NLS under particular conditions, to disappear under real perturbations to the NLS coefficients of O(10 to the -6th).

Mathieu's Equation and its Generalizations: Overview of Stability Charts and their Features

DEFF Research Database (Denmark)

Kovacic, Ivana; Rand, Richard H.; Sah, Si Mohamed

2018-01-01

This work is concerned with Mathieu's equation - a classical differential equation, which has the form of a linear second-order ordinary differential equation with Cosine-type periodic forcing of the stiffness coefficient, and its different generalizations/extensions. These extensions include...... and features, and how it differs from that of the classical Mathieu's equation....

Mathematical modeling of the dynamic stability of fluid conveying pipe based on integral equation formulations

International Nuclear Information System (INIS)

Elfelsoufi, Z.; Azrar, L.

2016-01-01

In this paper, a mathematical modeling of flutter and divergence analyses of fluid conveying pipes based on integral equation formulations is presented. Dynamic stability problems related to fluid pressure, velocity, tension, topography slope and viscoelastic supports and foundations are formulated. A methodological approach is presented and the required matrices, associated to the influencing fluid and pipe parameters, are explicitly given. Internal discretizations are used allowing to investigate the deformation, the bending moment, slope and shear force at internal points. Velocity–frequency, pressure-frequency and tension-frequency curves are analyzed for various fluid parameters and internal elastic supports. Critical values of divergence and flutter behaviors with respect to various fluid parameters are investigated. This model is general and allows the study of dynamic stability of tubes crossed by stationary and instationary fluid on various types of supports. Accurate predictions can be obtained and are of particular interest for a better performance and for an optimal safety of piping system installations. - Highlights: • Modeling the flutter and divergence of fluid conveying pipes based on RBF. • Dynamic analysis of a fluid conveying pipe with generalized boundary conditions. • Considered parameters fluid are the pressure, tension, slopes topography, velocity. • Internal support increase the critical velocity value. • This methodologies determine the fluid parameters effects.

The nonlinear heat equation with state–dependent parameters and its connection to the Burgers’ and the potential Burgers’ equation

DEFF Research Database (Denmark)

Backi, Christoph Josef; Bendtsen, Jan Dimon; Leth, John-Josef

2014-01-01

In this work the stability properties of a nonlinear partial differential equation (PDE) with state–dependent parameters is investigated. Among other things, the PDE describes freezing of foodstuff, and is closely related to the (Potential) Burgers’ Equation. We show that for certain forms of coe...

Controllability and Stabilization of Bilinear and Semilinear Partial Differential Equations

DEFF Research Database (Denmark)

Krishnaswamy, Vijayaraghavan

The topic of the thesis is the investigation of the question of controllability of weakly nonlinear partial differntial equations. The method is based on the Hilbert Uniqueness Method.......The topic of the thesis is the investigation of the question of controllability of weakly nonlinear partial differntial equations. The method is based on the Hilbert Uniqueness Method....

Differential equations methods and applications

CERN Document Server

Said-Houari, Belkacem

2015-01-01

This book presents a variety of techniques for solving ordinary differential equations analytically and features a wealth of examples. Focusing on the modeling of real-world phenomena, it begins with a basic introduction to differential equations, followed by linear and nonlinear first order equations and a detailed treatment of the second order linear equations. After presenting solution methods for the Laplace transform and power series, it lastly presents systems of equations and offers an introduction to the stability theory. To help readers practice the theory covered, two types of exercises are provided: those that illustrate the general theory, and others designed to expand on the text material. Detailed solutions to all the exercises are included. The book is excellently suited for use as a textbook for an undergraduate class (of all disciplines) in ordinary differential equations. .

Development of structural correlations and synchronization from adaptive rewiring in networks of Kuramoto oscillators

Science.gov (United States)

Papadopoulos, Lia; Kim, Jason Z.; Kurths, Jürgen; Bassett, Danielle S.

2017-07-01

Synchronization of non-identical oscillators coupled through complex networks is an important example of collective behavior, and it is interesting to ask how the structural organization of network interactions influences this process. Several studies have explored and uncovered optimal topologies for synchronization by making purposeful alterations to a network. On the other hand, the connectivity patterns of many natural systems are often not static, but are rather modulated over time according to their dynamics. However, this co-evolution and the extent to which the dynamics of the individual units can shape the organization of the network itself are less well understood. Here, we study initially randomly connected but locally adaptive networks of Kuramoto oscillators. In particular, the system employs a co-evolutionary rewiring strategy that depends only on the instantaneous, pairwise phase differences of neighboring oscillators, and that conserves the total number of edges, allowing the effects of local reorganization to be isolated. We find that a simple rule—which preserves connections between more out-of-phase oscillators while rewiring connections between more in-phase oscillators—can cause initially disordered networks to organize into more structured topologies that support enhanced synchronization dynamics. We examine how this process unfolds over time, finding a dependence on the intrinsic frequencies of the oscillators, the global coupling, and the network density, in terms of how the adaptive mechanism reorganizes the network and influences the dynamics. Importantly, for large enough coupling and after sufficient adaptation, the resulting networks exhibit interesting characteristics, including degree-frequency and frequency-neighbor frequency correlations. These properties have previously been associated with optimal synchronization or explosive transitions in which the networks were constructed using global information. On the contrary, by

Weakly nonparallel and curvature effects on stationary crossflow instability: Comparison of results from multiple-scales analysis and parabolized stability equations

Science.gov (United States)

Singer, Bart A.; Choudhari, Meelan; Li, Fei

1995-01-01

A multiple-scales approach is used to approximate the effects of nonparallelism and streamwise surface curvature on the growth of stationary crossflow vortices in incompressible, three-dimesional boundary layers. The results agree with results predicted by solving the parabolized stability equations in regions where the nonparallelism is sufficiently weak. As the nonparallelism increases, the agreement between the two approaches worsens. An attempt has been made to quantify the nonparallelism on flow stability in terms of a nondimensional number that describes the rate of change of the mean flow relative to the disturbance wavelength. We find that the above nondimensional number provides useful information about the adequacy of the multiple-scales approximation for different disturbances for a given flow geometry, but the number does not collapse data for different flow geometries onto a single curve.

A stabilized Runge–Kutta–Legendre method for explicit super-time-stepping of parabolic and mixed equations

International Nuclear Information System (INIS)

Meyer, Chad D.; Balsara, Dinshaw S.; Aslam, Tariq D.

2014-01-01

Parabolic partial differential equations appear in several physical problems, including problems that have a dominant hyperbolic part coupled to a sub-dominant parabolic component. Explicit methods for their solution are easy to implement but have very restrictive time step constraints. Implicit solution methods can be unconditionally stable but have the disadvantage of being computationally costly or difficult to implement. Super-time-stepping methods for treating parabolic terms in mixed type partial differential equations occupy an intermediate position. In such methods each superstep takes “s” explicit Runge–Kutta-like time-steps to advance the parabolic terms by a time-step that is s 2 times larger than a single explicit time-step. The expanded stability is usually obtained by mapping the short recursion relation of the explicit Runge–Kutta scheme to the recursion relation of some well-known, stable polynomial. Prior work has built temporally first- and second-order accurate super-time-stepping methods around the recursion relation associated with Chebyshev polynomials. Since their stability is based on the boundedness of the Chebyshev polynomials, these methods have been called RKC1 and RKC2. In this work we build temporally first- and second-order accurate super-time-stepping methods around the recursion relation associated with Legendre polynomials. We call these methods RKL1 and RKL2. The RKL1 method is first-order accurate in time; the RKL2 method is second-order accurate in time. We verify that the newly-designed RKL1 and RKL2 schemes have a very desirable monotonicity preserving property for one-dimensional problems – a solution that is monotone at the beginning of a time step retains that property at the end of that time step. It is shown that RKL1 and RKL2 methods are stable for all values of the diffusion coefficient up to the maximum value. We call this a convex monotonicity preserving property and show by examples that it is very useful

Non-linear wave equations:Mathematical techniques

International Nuclear Information System (INIS)

1978-01-01

An account of certain well-established mathematical methods, which prove useful to deal with non-linear partial differential equations is presented. Within the strict framework of Functional Analysis, it describes Semigroup Techniques in Banach Spaces as well as variational approaches towards critical points. Detailed proofs are given of the existence of local and global solutions of the Cauchy problem and of the stability of stationary solutions. The formal approach based upon invariance under Lie transformations deserves attention due to its wide range of applicability, even if the explicit solutions thus obtained do not allow for a deep analysis of the equations. A compre ensive introduction to the inverse scattering approach and to the solution concept for certain non-linear equations of physical interest are also presented. A detailed discussion is made about certain convergence and stability problems which arise in importance need not be emphasized. (author) [es

Convergence of method of lines approximations to partial differential equations

International Nuclear Information System (INIS)

Verwer, J.G.; Sanz-Serna, J.M.

1984-01-01

Many existing numerical schemes for evolutionary problems in partial differential equations (PDEs) can be viewed as method of lines (MOL) schemes. This paper treats the convergence of one-step MOL schemes. The main purpose is to set up a general framework for a convergence analysis applicable to nonlinear problems. The stability materials for this framework are taken from the field of nonlinear stiff ODEs. In this connection, important concepts are the logarithmic matrix norm and C-stability. A nonlinear parabolic equation and the cubic Schroedinger equation are used for illustrating the ideas. (Auth.)

Analytical Solution of Pantograph Equation with Incommensurate Delay

Science.gov (United States)

Patade, Jayvant; Bhalekar, Sachin

2017-08-01

Pantograph equation is a delay differential equation (DDE) arising in electrodynamics. This paper studies the pantograph equation with two delays. The existence, uniqueness, stability and convergence results for DDEs are presented. The series solution of the proposed equation is obtained by using Daftardar-Gejji and Jafari method and given in terms of a special function. This new special function has several properties and relations with other functions. Further, we generalize the proposed equation to fractional-order case and obtain its solution.

Attractors for equations of mathematical physics

CERN Document Server

Chepyzhov, Vladimir V

2001-01-01

One of the major problems in the study of evolution equations of mathematical physics is the investigation of the behavior of the solutions to these equations when time is large or tends to infinity. The related important questions concern the stability of solutions or the character of the instability if a solution is unstable. In the last few decades, considerable progress in this area has been achieved in the study of autonomous evolution partial differential equations. For a number of basic evolution equations of mathematical physics, it was shown that the long time behavior of their soluti

Motions in a Bose condensate: X. New results on the stability of axisymmetric solitary waves of the Gross-Pitaevskii equation

International Nuclear Information System (INIS)

Berloff, Natalia G; Roberts, Paul H

2004-01-01

The stability of the axisymmetric solitary waves of the Gross-Pitaevskii (GP) equation is investigated. The implicitly restarted Arnoldi method for banded matrices with shift-invert is used to solve the linearized spectral stability problem. The rarefaction solitary waves on the upper branch of the Jones-Roberts dispersion curve are shown to be unstable to axisymmetric infinitesimal perturbations, whereas the solitary waves on the lower branch and all two-dimensional solitary waves are linearly stable. The growth rates of the instabilities on the upper branch are so small that an arbitrarily specified initial perturbation of a rarefaction wave at first usually evolves towards the upper branch as it acoustically radiates away its excess energy. This is demonstrated through numerical integrations of the GP equation starting from an initial state consisting of an unstable rarefaction wave and random non-axisymmetric noise. The resulting solution evolves towards, and remains for a significant time in the vicinity of, an unperturbed unstable rarefaction wave. It is shown however that, ultimately (or for an initial state extremely close to the upper branch), the solution evolves onto the lower branch or is completely dissipated as sound. It is shown how density depletions in uniform and trapped condensates can generate rarefaction waves, and a simple method is suggested by which such waves can be created in the laboratory

Motions in a Bose condensate: X. New results on the stability of axisymmetric solitary waves of the Gross-Pitaevskii equation

Energy Technology Data Exchange (ETDEWEB)

Berloff, Natalia G [Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA (United Kingdom); Roberts, Paul H [Department of Mathematics, University of California, Los Angeles, CA, 90095 (United States)

2004-11-26

The stability of the axisymmetric solitary waves of the Gross-Pitaevskii (GP) equation is investigated. The implicitly restarted Arnoldi method for banded matrices with shift-invert is used to solve the linearized spectral stability problem. The rarefaction solitary waves on the upper branch of the Jones-Roberts dispersion curve are shown to be unstable to axisymmetric infinitesimal perturbations, whereas the solitary waves on the lower branch and all two-dimensional solitary waves are linearly stable. The growth rates of the instabilities on the upper branch are so small that an arbitrarily specified initial perturbation of a rarefaction wave at first usually evolves towards the upper branch as it acoustically radiates away its excess energy. This is demonstrated through numerical integrations of the GP equation starting from an initial state consisting of an unstable rarefaction wave and random non-axisymmetric noise. The resulting solution evolves towards, and remains for a significant time in the vicinity of, an unperturbed unstable rarefaction wave. It is shown however that, ultimately (or for an initial state extremely close to the upper branch), the solution evolves onto the lower branch or is completely dissipated as sound. It is shown how density depletions in uniform and trapped condensates can generate rarefaction waves, and a simple method is suggested by which such waves can be created in the laboratory.

Linear stability analysis of heated parallel channels

International Nuclear Information System (INIS)

Nourbakhsh, H.P.; Isbin, H.S.

1982-01-01

An analyis is presented of thermal hydraulic stability of flow in parallel channels covering the range from inlet subcooling to exit superheat. The model is based on a one-dimensional drift velocity formulation of the two phase flow conservation equations. The system of equations is linearized by assuming small disturbances about the steady state. The dynamic response of the system to an inlet flow perturbation is derived yielding the characteristic equation which predicts the onset of instabilities. A specific application is carried out for homogeneous and regional uniformly heated systems. The particular case of equal characteristic frequencies of two-phase and single phase vapor region is studied in detail. The D-partition method and the Mikhailov stability criterion are used for determining the marginal stability boundary. Stability predictions from the present analysis are compared with the experimental data from the solar test facility. 8 references

Introduction to linear systems of differential equations

CERN Document Server

Adrianova, L Ya

1995-01-01

The theory of linear systems of differential equations is one of the cornerstones of the whole theory of differential equations. At its root is the concept of the Lyapunov characteristic exponent. In this book, Adrianova presents introductory material and further detailed discussions of Lyapunov exponents. She also discusses the structure of the space of solutions of linear systems. Classes of linear systems examined are from the narrowest to widest: 1)�autonomous, 2)�periodic, 3)�reducible to autonomous, 4)�nearly reducible to autonomous, 5)�regular. In addition, Adrianova considers the following: stability of linear systems and the influence of perturbations of the coefficients on the stability the criteria of uniform stability and of uniform asymptotic stability in terms of properties of the solutions several estimates of the growth rate of solutions of a linear system in terms of its coefficients How perturbations of the coefficients change all the elements of the spectrum of the system is defin...

L2-stability of the Vlasov-Maxwell-Boltzmann system near global Maxwellians

International Nuclear Information System (INIS)

Ha, Seung-Yeal; Xiao, Qinghua; Xiong, Linjie; Zhao, Huijiang

2013-01-01

We present a L 2 -stability theory of the Vlasov-Maxwell-Boltzmann system for the two-species collisional plasma. We show that in a perturbative regime of a global Maxwellian, the L 2 -distance between two strong solutions can be controlled by that between initial data in a Lipschitz manner. Our stability result extends earlier results [Ha, S.-Y. and Xiao, Q.-H., “A revisiting to the L 2 -stability theory of the Boltzmann equation near global Maxwellians,” (submitted) and Ha, S.-Y., Yang, X.-F., and Yun, S.-B., “L 2 stability theory of the Boltzmann equation near a global Maxwellian,” Arch. Ration. Mech. Anal. 197, 657–688 (2010)] on the L 2 -stability of the Boltzmann equation to the Boltzmann equation coupled with self-consistent external forces. As a direct application of our stability result, we show that classical solutions in Duan et al. [“Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space,” Commun. Pure Appl. Math. 24, 1497–1546 (2011)] and Guo [“The Vlasov-Maxwell-Boltzmann system near Maxwellians,” Invent. Math. 153(3), 593–630 (2003)] satisfy a uniform L 2 -stability estimate. This is the first result on the L 2 -stability of the Boltzmann equation coupled with self-consistent field equations in three dimensions

An introduction to geometric theory of fully nonlinear parabolic equations

International Nuclear Information System (INIS)

Lunardi, A.

1991-01-01

We study a class of nonlinear evolution equations in general Banach space being an abstract version of fully nonlinear parabolic equations. In addition to results of existence, uniqueness and continuous dependence on the data, we give some qualitative results about stability of the stationary solutions, existence and stability of the periodic orbits. We apply such results to some parabolic problems arising from combustion theory. (author). 24 refs

Nonlinear stability of supersonic jets

Science.gov (United States)

Tiwari, S. N. (Principal Investigator); Bhat, T. R. S. (Principal Investigator)

1996-01-01

The stability calculations made for a shock-free supersonic jet using the model based on parabolized stability equations are presented. In this analysis the large scale structures, which play a dominant role in the mixing as well as the noise radiated, are modeled as instability waves. This model takes into consideration non-parallel flow effects and also nonlinear interaction of the instability waves. The stability calculations have been performed for different frequencies and mode numbers over a range of jet operating temperatures. Comparisons are made, where appropriate, with the solutions to Rayleigh's equation (linear, inviscid analysis with the assumption of parallel flow). The comparison of the solutions obtained using the two approaches show very good agreement.

A model for construction of efficiency and stability maps of hot working processes in polycrystalline metallic materials using the Garofalo's equation

International Nuclear Information System (INIS)

Rieiro, I.; Fernandez, A.; Martinez, A.; Casi, M.

1998-01-01

Has been developed a fast and easy method for to evaluate the efficiency of the process and some or possible stabilities situations in the hot working process for the polycrystalline metallic materials (p.m.m.), by the obtained dates in the Garofalo's equation resolution for the steady state creep and for wide ranges of the work variables, stress, strain rate and temperature. These method use the fitting parameters obtained for the equation mentioned and of their physical meaning. Has been developed the numerical treatment from our previously developed software for the analysis of creep and we can obtained the efficiency energetic maps for the creep and the more generally advises areas for the hot working. Further more has been obtained maps for parameters of great physical significance; f.e. the effective activation energy for different areas of the materials hot working, and in addition has been developed a method for obtained the values of {n P L} for the different ranges of power-law application, when has been obtained the value of { n G } in the Garofalo's equation. (Author) 13 refs

Existence and stability of periodic solutions of an impulsive differential equation and application to CD8 T-cell differentiation.

Science.gov (United States)

Girel, Simon; Crauste, Fabien

2018-06-01

Unequal partitioning of the molecular content at cell division has been shown to be a source of heterogeneity in a cell population. We propose to model this phenomenon with the help of a scalar, nonlinear impulsive differential equation (IDE). To study the effect of molecular partitioning at cell division on the effector/memory cell-fate decision in a CD8 T-cell lineage, we study an IDE describing the concentration of the protein Tbet in a CD8 T-cell, where impulses are associated to cell division. We discuss how the degree of asymmetry of molecular partitioning can affect the process of cell differentiation and the phenotypical heterogeneity of a cell population. We show that a moderate degree of asymmetry is necessary and sufficient to observe irreversible differentiation. We consider, in a second part, a general autonomous IDE with fixed times of impulse and a specific form of impulse function. We establish properties of the solutions of that equation, most of them obtained under the hypothesis that impulses occur periodically. In particular, we show how to investigate the existence of periodic solutions and their stability by studying the flow of an autonomous differential equation. Then we apply those properties to prove the results presented in the first part.

Pulse-coupled mixed-mode oscillators: Cluster states and extreme noise sensitivity

Science.gov (United States)

Karamchandani, Avinash J.; Graham, James N.; Riecke, Hermann

2018-04-01

Motivated by rhythms in the olfactory system of the brain, we investigate the synchronization of all-to-all pulse-coupled neuronal oscillators exhibiting various types of mixed-mode oscillations (MMOs) composed of sub-threshold oscillations (STOs) and action potentials ("spikes"). We focus particularly on the impact of the delay in the interaction. In the weak-coupling regime, we reduce the system to a Kuramoto-type equation with non-sinusoidal phase coupling and the associated Fokker-Planck equation. Its linear stability analysis identifies the appearance of various cluster states. Their type depends sensitively on the delay and the width of the pulses. Interestingly, long delays do not imply slow population rhythms, and the number of emerging clusters only loosely depends on the number of STOs. Direct simulations of the oscillator equations reveal that for quantitative agreement of the weak-coupling theory the coupling strength and the noise have to be extremely small. Even moderate noise leads to significant skipping of STO cycles, which can enhance the diffusion coefficient in the Fokker-Planck equation by two orders of magnitude. Introducing an effective diffusion coefficient extends the range of agreement significantly. Numerical simulations of the Fokker-Planck equation reveal bistability and solutions with oscillatory order parameters that result from nonlinear mode interactions. These are confirmed in simulations of the full spiking model.

Elements of magnetohydrodynamic stability theory

International Nuclear Information System (INIS)

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

Numerical Computation of Detonation Stability

KAUST Repository

Kabanov, Dmitry

2018-06-03

Detonation is a supersonic mode of combustion that is modeled by a system of conservation laws of compressible fluid mechanics coupled with the equations describing thermodynamic and chemical properties of the fluid. Mathematically, these governing equations admit steady-state travelling-wave solutions consisting of a leading shock wave followed by a reaction zone. However, such solutions are often unstable to perturbations and rarely observed in laboratory experiments. The goal of this work is to study the stability of travelling-wave solutions of detonation models by the following novel approach. We linearize the governing equations about a base travelling-wave solution and solve the resultant linearized problem using high-order numerical methods. The results of these computations are postprocessed using dynamic mode decomposition to extract growth rates and frequencies of the perturbations and predict stability of travelling-wave solutions to infinitesimal perturbations. We apply this approach to two models based on the reactive Euler equations for perfect gases. For the first model with a one-step reaction mechanism, we find agreement of our results with the results of normal-mode analysis. For the second model with a two-step mechanism, we find that both types of admissible travelling-wave solutions exhibit the same stability spectra. Then we investigate the Fickett’s detonation analogue coupled with a particular reaction-rate expression. In addition to the linear stability analysis of this model, we demonstrate that it exhibits rich nonlinear dynamics with multiple bifurcations and chaotic behavior.

Discrete nonlinear Schrodinger equations with arbitrarily high-order nonlinearities

DEFF Research Database (Denmark)

Khare, A.; Rasmussen, Kim Ø; Salerno, M.

2006-01-01

-Ladik equation. As a common property, these equations possess three kinds of exact analytical stationary solutions for which the Peierls-Nabarro barrier is zero. Several properties of these solutions, including stability, discrete breathers, and moving solutions, are investigated.......A class of discrete nonlinear Schrodinger equations with arbitrarily high-order nonlinearities is introduced. These equations are derived from the same Hamiltonian using different Poisson brackets and include as particular cases the saturable discrete nonlinear Schrodinger equation and the Ablowitz...

Generalized synchronization and coherent structures in spatially extended systems

International Nuclear Information System (INIS)

Basnarkov, Lasko; Duane, Gregory S.; Kocarev, Ljupco

2014-01-01

We study the synchronization of a coupled pair of one-dimensional Kuramoto–Sivashinsky systems, with equations augmented by a third-space-derivative term. With two different values of a system parameter, the two systems synchronize in the generalized sense. The phenomenon persists even in the extreme case when one of the equations is missing the extra term. Master–slave synchronization error is small, so the generalized synchronization relationship is useful for predicting the state of the master from that of the slave, or conversely, for controlling the slave. The spatial density of coupling points required to bring about generalized synchronization appears to be related to the wavelength of traveling wave solutions, and more generally to the width of coherent structures in the separate systems

Solution of a modified Lame equation with an integral term

International Nuclear Information System (INIS)

Hagelstein, P.L.

1978-01-01

We consider an equation which occurs in the stability analysis of a passively modelocked laser system in which the pulses overlap. The equation is related to a Lame equation and can be written su(x) =]d 2 /dx 2 -[(2-m)-6dn 2 (x,m)

Numerical simulation of fractional Cable equation of spiny neuronal dendrites

Directory of Open Access Journals (Sweden)

N.H. Sweilam

2014-03-01

Full Text Available In this article, numerical study for the fractional Cable equation which is fundamental equations for modeling neuronal dynamics is introduced by using weighted average of finite difference methods. The stability analysis of the proposed methods is given by a recently proposed procedure similar to the standard John von Neumann stability analysis. A simple and an accurate stability criterion valid for different discretization schemes of the fractional derivative and arbitrary weight factor is introduced and checked numerically. Numerical results, figures, and comparisons have been presented to confirm the theoretical results and efficiency of the proposed method.

Asymptotic behavior and stability of second order neutral delay differential equations

NARCIS (Netherlands)

Chen, G.L.; van Gaans, O.W.; Verduyn Lunel, Sjoerd

2014-01-01

We study the asymptotic behavior of a class of second order neutral delay differential equations by both a spectral projection method and an ordinary differential equation method approach. We discuss the relation of these two methods and illustrate some features using examples. Furthermore, a fixed

The Price Equation, Gradient Dynamics, and Continuous Trait Game Theory.

Science.gov (United States)

Lehtonen, Jussi

2018-01-01

A recent article convincingly nominated the Price equation as the fundamental theorem of evolution and used it as a foundation to derive several other theorems. A major section of evolutionary theory that was not addressed is that of game theory and gradient dynamics of continuous traits with frequency-dependent fitness. Deriving fundamental results in these fields under the unifying framework of the Price equation illuminates similarities and differences between approaches and allows a simple, unified view of game-theoretical and dynamic concepts. Using Taylor polynomials and the Price equation, I derive a dynamic measure of evolutionary change, a condition for singular points, the convergence stability criterion, and an alternative interpretation of evolutionary stability. Furthermore, by applying the Price equation to a multivariable Taylor polynomial, the direct fitness approach to kin selection emerges. Finally, I compare these results to the mean gradient equation of quantitative genetics and the canonical equation of adaptive dynamics.

A Quadratic Spring Equation

Science.gov (United States)

Fay, Temple H.

2010-01-01

Through numerical investigations, we study examples of the forced quadratic spring equation [image omitted]. By performing trial-and-error numerical experiments, we demonstrate the existence of stability boundaries in the phase plane indicating initial conditions yielding bounded solutions, investigate the resonance boundary in the [omega]…

Stochastic nonlinear beam equations

Czech Academy of Sciences Publication Activity Database

Brzezniak, Z.; Maslowski, Bohdan; Seidler, Jan

2005-01-01

Roč. 132, č. 1 (2005), s. 119-149 ISSN 0178-8051 R&D Projects: GA ČR(CZ) GA201/01/1197 Institutional research plan: CEZ:AV0Z10190503 Keywords : stochastic beam equation * stability Subject RIV: BA - General Mathematics Impact factor: 0.896, year: 2005

Two-level schemes for the advection equation

Science.gov (United States)

Vabishchevich, Petr N.

2018-06-01

The advection equation is the basis for mathematical models of continuum mechanics. In the approximate solution of nonstationary problems it is necessary to inherit main properties of the conservatism and monotonicity of the solution. In this paper, the advection equation is written in the symmetric form, where the advection operator is the half-sum of advection operators in conservative (divergent) and non-conservative (characteristic) forms. The advection operator is skew-symmetric. Standard finite element approximations in space are used. The standard explicit two-level scheme for the advection equation is absolutely unstable. New conditionally stable regularized schemes are constructed, on the basis of the general theory of stability (well-posedness) of operator-difference schemes, the stability conditions of the explicit Lax-Wendroff scheme are established. Unconditionally stable and conservative schemes are implicit schemes of the second (Crank-Nicolson scheme) and fourth order. The conditionally stable implicit Lax-Wendroff scheme is constructed. The accuracy of the investigated explicit and implicit two-level schemes for an approximate solution of the advection equation is illustrated by the numerical results of a model two-dimensional problem.

Polynomial asymptotic stability of damped stochastic differential equations

Directory of Open Access Journals (Sweden)

John Appleby

2004-08-01

Full Text Available The paper studies the polynomial convergence of solutions of a scalar nonlinear It\\^{o} stochastic differential equation\\[dX(t = -f(X(t\\,dt + \\sigma(t\\,dB(t\\] where it is known, {\\it a priori}, that $\\lim_{t\\rightarrow\\infty} X(t=0$, a.s. The intensity of the stochastic perturbation $\\sigma$ is a deterministic, continuous and square integrable function, which tends to zero more quickly than a polynomially decaying function. The function $f$ obeys $\\lim_{x\\rightarrow 0}\\mbox{sgn}(xf(x/|x|^\\beta = a$, for some $\\beta>1$, and $a>0$.We study two asymptotic regimes: when $\\sigma$ tends to zero sufficiently quickly the polynomial decay rate of solutions is the same as for the deterministic equation (when $\\sigma\\equiv0$. When $\\sigma$ decays more slowly, a weaker almost sure polynomial upper bound on the decay rate of solutions is established. Results which establish the necessity for $\\sigma$ to decay polynomially in order to guarantee the almost sure polynomial decay of solutions are also proven.

On the stability of solitary waves for classical scalar fields

International Nuclear Information System (INIS)

Blanchard, P.; Stubbe, J.; Vazquez, L.

1986-01-01

We study the stability for the bound states of lowest action of certain nonlinear Klein-Gordon and Schroedinger equations by applying the Shatah-Strauss formalism. We extend the range of application of this formalism by using a recent existence theorem for minimum action solutions to a large class of equations including logarithmic Klein-Gordon equation and logarithmic Schroedinger equation and scalar fields with fractional non-linearities. Furthermore we discuss the relation between different stability criteria considered in the literature. (orig.)

Magnetohydrodynamic stability of tokamak plasmas with poloidal mode coupling

International Nuclear Information System (INIS)

Shigueoka, H.; Sakanaka, P.H.

1987-01-01

The stability behavior with respect to internal modes is examined for a class of tokamak equilibria with non-circular cross sections. The surfaces of the constant poloidal magnetic flux ψ (R,Z) are obtained numerically by solving the Grad-Shafranov's equation with a specified shape for the outmost plasma surface. The equation of motion for ideal MHD stability is written in a ortogonal coordinate system (ψ, χ, φ). Th e stability analysis is performance numerically in a truncated set of coupled m (poloidal wave number) equations. The calculations involve no approximations, and so all parameters of the equilibrium solution can be arbitrarily varied. (author) [pt

On the behaviour of solutions of parabolic equations for large values of time

International Nuclear Information System (INIS)

Denisov, V N

2005-01-01

This paper is a survey of classical and new results on stabilization of solutions of the Cauchy problem and mixed problems for second-order linear parabolic equations. Proofs are given for some new results about exact sufficient conditions on the behaviour of lower-order coefficients of the parabolic equation; these conditions ensure stabilization of a solution of the Cauchy problem for the parabolic equation in the class of bounded or increasing initial functions

Spectral representations of neutron-star equations of state

International Nuclear Information System (INIS)

Lindblom, Lee

2010-01-01

Methods are developed for constructing spectral representations of cold (barotropic) neutron-star equations of state. These representations are faithful in the sense that every physical equation of state has a representation of this type and conversely every such representation satisfies the minimal thermodynamic stability criteria required of any physical equation of state. These spectral representations are also efficient, in the sense that only a few spectral coefficients are generally required to represent neutron-star equations of state quiet accurately. This accuracy and efficiency is illustrated by constructing spectral fits to a large collection of 'realistic' neutron-star equations of state.

Pseudodifferential Perturbations and Stabilization of Distributed Parameter Systems: Dirichlet Feedback Control Problems

DEFF Research Database (Denmark)

Pedersen, Michael

1991-01-01

The stabilization problems for parabolic and hyperbolic partial differential equations with Dirichlet boundary condition are considered. The systems are stabilized by a boundary feedback in(1) The operator equation,(2) The boundary condition,(3) Both the operator equation and the boundary condition...... turns out to be a shortcut to some of the stabilization results of Lasiecka and Triggiani in [J. Differential Equations, 47 (1983), pp. 245-272], [SIAM J. Control Optim., 21(1983), pp. 766-802], and [Appl. Math. Optim., 8(1981), pp. 1-37], and it illuminates to some extent how a change of boundary...

Functional differential equations with unbounded delay in extrapolation spaces

Directory of Open Access Journals (Sweden)

Mostafa Adimy

2014-08-01

Full Text Available We study the existence, regularity and stability of solutions for nonlinear partial neutral functional differential equations with unbounded delay and a Hille-Yosida operator on a Banach space X. We consider two nonlinear perturbations: the first one is a function taking its values in X and the second one is a function belonging to a space larger than X, an extrapolated space. We use the extrapolation techniques to prove the existence and regularity of solutions and we establish a linearization principle for the stability of the equilibria of our equation.

A heat equation for freezing processes with phase change

DEFF Research Database (Denmark)

Backi, Christoph Josef; Bendtsen, Jan Dimon; Leth, John-Josef

2016-01-01

In this work, the stability properties as well as possible applications of a partial differential equation (PDE) with state-dependent parameters are investigated. Among other things, the PDE describes freezing of foodstuff, and is closely related to the (potential) Burgers’ equation. We show that...

Novel Exponentially Fitted Two-Derivative Runge-Kutta Methods with Equation-Dependent Coefficients for First-Order Differential Equations

Directory of Open Access Journals (Sweden)

Yanping Yang

2016-01-01

Full Text Available The construction of exponentially fitted two-derivative Runge-Kutta (EFTDRK methods for the numerical solution of first-order differential equations is investigated. The revised EFTDRK methods proposed, with equation-dependent coefficients, take into consideration the errors produced in the internal stages to the update. The local truncation errors and stability of the new methods are analyzed. The numerical results are reported to show the accuracy of the new methods.

Stability analysis of a class of fractional delay differential equations

Indian Academy of Sciences (India)

Abstract. In this paper we analyse stability of nonlinear fractional order delay differential equa- tions of the form Dα y(t) = af (y(t − τ )) − by(t), where Dα is a Caputo fractional derivative of order 0 < α ≤ 1. We describe stability regions using critical curves. To explain the proposed theory, we discuss fractional order logistic ...

The Relationship between Nonconservative Schemes and Initial Values of Nonlinear Evolution Equations

Institute of Scientific and Technical Information of China (English)

林万涛

2004-01-01

For the nonconservative schemes of the nonlinear evolution equations, taking the one-dimensional shallow water wave equation as an example, the necessary conditions of computational stability are given.Based on numerical tests, the relationship between the nonlinear computational stability and the construction of difference schemes, as well as the form of initial values, is further discussed. It is proved through both theoretical analysis and numerical tests that if the construction of difference schemes is definite, the computational stability of nonconservative schemes is decided by the form of initial values.

Finite Larmor radius stabilization of ballooning modes in tokamaks

International Nuclear Information System (INIS)

Tsang, K.T.

1980-07-01

A ballooning mode equation that includes full finite Larmor radius effects has been derived from the Vlasov equation for a circular tokamak equilibrium. Numerical solution of this equation shows that finite Larmor radius effects are stabilizing

Dynamical stability in fluid-structure interaction

International Nuclear Information System (INIS)

Planchard, J.; Thomas, B.

1991-01-01

The aim of the paper is to investigate the dynamical stability of a group of elastic tubes placed in a cross-flow which obeys to the Navier-Stokes equations. The stability of this coupled system is deduced from the study of a quadratic eigenvalue problem arising in the linearized equations. The instability occurs when the real part of one of the eigenvalues becomes positive; the steady state is then replaced by a time-periodic state which is stable (Hopf bifurcation phenomenon). Some numerical methods for solving the quadratic eigenvalue problem are described [fr

Modulation equations for spatially periodic systems: derivation and solutions

NARCIS (Netherlands)

Schielen, R.; Doelman, A.

1996-01-01

We study a class of partial dierential equations in one spatial dimension, which can be seen as model equations for the analysis of pattern formation in physical systems dened on unbounded, weakly oscillating domains. We perform a linear and weakly nonlinear stability analysis for solutions that

Dirac equation in low dimensions: The factorization method

Energy Technology Data Exchange (ETDEWEB)

Sánchez-Monroy, J.A., E-mail: antosan@if.usp.br [Instituto de Física, Universidade de São Paulo, 05508-090, São Paulo, SP (Brazil); Quimbay, C.J., E-mail: cjquimbayh@unal.edu.co [Departamento de Física, Universidad Nacional de Colombia, Bogotá, D. C. (Colombia); CIF, Bogotá (Colombia)

2014-11-15

We present a general approach to solve the (1+1) and (2+1)-dimensional Dirac equations in the presence of static scalar, pseudoscalar and gauge potentials, for the case in which the potentials have the same functional form and thus the factorization method can be applied. We show that the presence of electric potentials in the Dirac equation leads to two Klein–Gordon equations including an energy-dependent potential. We then generalize the factorization method for the case of energy-dependent Hamiltonians. Additionally, the shape invariance is generalized for a specific class of energy-dependent Hamiltonians. We also present a condition for the absence of the Klein paradox (stability of the Dirac sea), showing how Dirac particles in low dimensions can be confined for a wide family of potentials. - Highlights: • The low-dimensional Dirac equation in the presence of static potentials is solved. • The factorization method is generalized for energy-dependent Hamiltonians. • The shape invariance is generalized for energy-dependent Hamiltonians. • The stability of the Dirac sea is related to the existence of supersymmetric partner Hamiltonians.

Analysis of an Nth-order nonlinear differential-delay equation

Science.gov (United States)

Vallée, Réal; Marriott, Christopher

1989-01-01

The problem of a nonlinear dynamical system with delay and an overall response time which is distributed among N individual components is analyzed. Such a system can generally be modeled by an Nth-order nonlinear differential delay equation. A linear-stability analysis as well as a numerical simulation of that equation are performed and a comparison is made with the experimental results. Finally, a parallel is established between the first-order differential equation with delay and the Nth-order differential equation without delay.

Stability of non-linear constitutive formulations for viscoelastic fluids

CERN Document Server

Siginer, Dennis A

2014-01-01

Stability of Non-linear Constitutive Formulations for Viscoelastic Fluids provides a complete and up-to-date view of the field of constitutive equations for flowing viscoelastic fluids, in particular on their non-linear behavior, the stability of these constitutive equations that is their predictive power, and the impact of these constitutive equations on the dynamics of viscoelastic fluid flow in tubes. This book gives an overall view of the theories and attendant methodologies developed independently of thermodynamic considerations as well as those set within a thermodynamic framework to derive non-linear rheological constitutive equations for viscoelastic fluids. Developments in formulating Maxwell-like constitutive differential equations as well as single integral constitutive formulations are discussed in the light of Hadamard and dissipative type of instabilities.

Stabilization of computational procedures for constrained dynamical systems

Science.gov (United States)

Park, K. C.; Chiou, J. C.

1988-01-01

A new stabilization method of treating constraints in multibody dynamical systems is presented. By tailoring a penalty form of the constraint equations, the method achieves stabilization without artificial damping and yields a companion matrix differential equation for the constraint forces; hence, the constraint forces are obtained by integrating the companion differential equation for the constraint forces in time. A principal feature of the method is that the errors committed in each constraint condition decay with its corresponding characteristic time scale associated with its constraint force. Numerical experiments indicate that the method yields a marked improvement over existing techniques.

On the wall-normal velocity of the compressible boundary-layer equations

Science.gov (United States)

Pruett, C. David

1991-01-01

Numerical methods for the compressible boundary-layer equations are facilitated by transformation from the physical (x,y) plane to a computational (xi,eta) plane in which the evolution of the flow is 'slow' in the time-like xi direction. The commonly used Levy-Lees transformation results in a computationally well-behaved problem for a wide class of non-similar boundary-layer flows, but it complicates interpretation of the solution in physical space. Specifically, the transformation is inherently nonlinear, and the physical wall-normal velocity is transformed out of the problem and is not readily recovered. In light of recent research which shows mean-flow non-parallelism to significantly influence the stability of high-speed compressible flows, the contribution of the wall-normal velocity in the analysis of stability should not be routinely neglected. Conventional methods extract the wall-normal velocity in physical space from the continuity equation, using finite-difference techniques and interpolation procedures. The present spectrally-accurate method extracts the wall-normal velocity directly from the transformation itself, without interpolation, leaving the continuity equation free as a check on the quality of the solution. The present method for recovering wall-normal velocity, when used in conjunction with a highly-accurate spectral collocation method for solving the compressible boundary-layer equations, results in a discrete solution which is extraordinarily smooth and accurate, and which satisfies the continuity equation nearly to machine precision. These qualities make the method well suited to the computation of the non-parallel mean flows needed by spatial direct numerical simulations (DNS) and parabolized stability equation (PSE) approaches to the analysis of stability.

Stabilization of the norm of the solution of a mixed problem in an unbounded domain for parabolic equations of orders 4 and 6

International Nuclear Information System (INIS)

Mukminov, F Kh; Bikkulov, I M

2004-01-01

The behaviour as t→∞ of the solution of a mixed problem for parabolic equations in an unbounded domain with two exits to infinity is studied. A certain class of domains is distinguished, in which an estimate characterizing the stabilization of solutions and determined by the geometry of the domain is established. This estimate is proved to be sharp in a certain sense for a broad class of domains with two exits to infinity.

Remarks on boiling water reactor stability analysis. Pt. 2. Stability monitoring

Energy Technology Data Exchange (ETDEWEB)

Lange, Carsten; Hennig, Dieter; Hurtado, Antonio [Technische Univ. Dresden (Germany). Chair of Hydrogen and Nuclear Energy; Schuster, Roland [Kernkraftwerk Brunsbuettel GmbH und Co. oHG, Brunsbuettel (Germany); Lukas, Bernard [EnBW Kernkraft GmbH, Philippsburg (Germany). Kernkraftwerk Philippsburg; Aguirre, Carlos [Kernkraftwerk Leibstadt AG, Aargau (Switzerland)

2012-12-15

In part 1 of this article we explained the partly relative complex solution manifold of the differential equations describing the stability behaviour of a BWR, in particular the coexistence of different types of solutions, such as the coexistence of unstable limit cycles and stable fixed points are of interest from the operational safety point of view. The part 2 is devoted to the surveillance of the stability behaviour. We summarize some stability monitoring methods and suggest to support stability tests by RAM-ROM analyses in order to reveal in advance the stability 'landscape' of the BWR in a parameter region high sensitive for appearing of linear unstable states. The analysis of an especial stability test, performed at NPP Leibstadt (KKL), makes it clear that the measurement results can only be interpreted by application of bifurcation analysis. (orig.)

Applied analysis and differential equations

CERN Document Server

Cârj, Ovidiu

2007-01-01

This volume contains refereed research articles written by experts in the field of applied analysis, differential equations and related topics. Well-known leading mathematicians worldwide and prominent young scientists cover a diverse range of topics, including the most exciting recent developments. A broad range of topics of recent interest are treated: existence, uniqueness, viability, asymptotic stability, viscosity solutions, controllability and numerical analysis for ODE, PDE and stochastic equations. The scope of the book is wide, ranging from pure mathematics to various applied fields such as classical mechanics, biomedicine, and population dynamics.

An equation of movement for supporting a drilling machine

Energy Technology Data Exchange (ETDEWEB)

Totev, Sl

1982-01-01

Support of a drilling machine is examined and an equation of movement is written. The equation has an invariant form and may be used for theoretical study of support in order to determine the forces and to study the stability and endurance of the elements as a whole.

A remark on Dickey's stabilizing chain

International Nuclear Information System (INIS)

Svinin, Andrei K.

2011-01-01

We observe that Dickey's stabilizing chain can be naturally included into two-dimensional chain of infinitely many copies of equations of KP hierarchy. -- Highlights: → In this study we consider Dickey's stabilizing chain. → We construct two-dimensional chain of dressing truncated operators. → We show that Dickey's stabilizing chain can be included into two-dimensional chain of KP hierarchies.

Variational characterization of generalized Jacobi equations

International Nuclear Information System (INIS)

Casciaro, B.

1995-09-01

A Lagrangian depending on derivatives of the fields up to a generic order is considered, together with a series development around a given section. The problem of extremality and stability of action for this system is then addressed. Higher-order variations in the Lagrangian, the Euler-Lagrange equation, the expansion of the action, the D-invariant decomposition of the Lagrangian, the Jacobi equation, and a unified description of the Euler-Lag range and Jacobi equations are discussed. As a conclusion of the work it is stated that the theory of second variations is worthy to be revisited and a comment on a recent paper by Taub is made. 10 refs

A stabilized finite element formulation for the solution of the Navier-Stokes equations in axisymmetric geometry

International Nuclear Information System (INIS)

Souza, Altivo Monteiro de

2008-12-01

The world energy consumption has been increasing strongly in recent years. Nuclear energy has been regarded as a suitable option to supply this growing energy demand in industrial scale. In view of the need of improving the understanding and capacity of analysis of nuclear power plants, modern simulation techniques for flow and heat transfer problems are gaining greater importance. A large number of problems found in nuclear reactor engineering can be dealt assuming axial symmetry. Thus, in this work a stabilized finite element formulation for the solution of the Navier-Stokes and energy equations for axisymmetric problems have been developed and tested. The formulation has been implemented in the NS S OLVER M PI 2 D A program developed at the Parallel Computation Laboratory of the Instituto de Engenharia Nuclear (LCP/IEN) and is now available either for safety analysis or design of nuclear systems. (author)

Force-free thin flux tubes: Basic equations and stability

International Nuclear Information System (INIS)

Zhugzhda, Y.D.

1996-01-01

The thin flux tube approximation is considered for a straight, symmetrical, force-free, rigidly rotating flux tube. The derived set of equations describes tube, body sausage, and Alfveacute charn wave modes and is valid for any values of Β. The linear waves and instabilities of force-free flux tubes are considered. The comparison of approximate and exact solutions for an untwisted, nonrotating flux tube is performed. It is shown that the approximate and exact dispersion equations coincides, except the 20% discrepancy of sausage frequencies. An effective cross section is proposed to introduce the removal of this discrepancy. It makes the derived approximation correct for the force-free thin flux tube dynamics, except the detailed structure of radial eigenfunction. The dispersion of Alfveacute charn torsional waves in a force-free tubes appears. The valve effect of one directional propagation of waves in rotating twisted tube is revealed. The current and rotational sausage instabilities of a force-free, thin flux tube are considered. copyright 1996 American Institute of Physics

Equilibrium and stability of a rotating plasma

International Nuclear Information System (INIS)

Janssen, P.A.E.M.

1979-01-01

The author considers the equilibrium and stability of a rotating plasma. The kinetic equations for ions and electrons supplemented with the Maxwell equations and the appropriate boundary conditions are used. Two different models for the rotating plasma are considered: the equilibrium of a 'fast' rotating plasma (Magneto Hydrodynamic ordering) and the stability of a slowly rotating, 'weakly' unstable plasma (Finite Larmor Radius ordering). A striking difference between these orderings is the fact that, regarding the stability of the plasma, for a F.L.R. plasma viscosity effects due to the finite Larmor radius are important, whereas in a M.H.D. plasma they are negligible (at least to the required order). (Auth.)

Numerical methods for hydrodynamic stability problems

International Nuclear Information System (INIS)

Fujimura, Kaoru

1985-11-01

Numerical methods for solving the Orr-Sommerfeld equation, which is the fundamental equation of the hydrodynamic stability theory for various shear flows, are reviewed and typical numerical results are presented. The methods of asymptotic solution, finite difference methods, initial value methods and expansions in orthogonal functions are compared. (author)

Structural stability and chaotic solutions of perturbed Benjamin-Ono equations

International Nuclear Information System (INIS)

Birnir, B.; Morrison, P.J.

1986-11-01

A method for proving chaos in partial differential equations is discussed and applied to the Benjamin-Ono equation subject to perturbations. The perturbations are of two types: one that corresponds to viscous dissipation, the so-called Burger's term, and one that involves the Hilbert transform and has been used to model Landau damping. The method proves chaos in the PDE by proving temporal chaos in its pole solutions. The spatial structure of the pole solutions remains intact, but their positions are chaotic in time. Melnikov's method is invoked to show this temporal chaos. It is discovered that the pole behavior is very sensitive to the Burger's perturbation, but is quite insensitive to the perturbation involving the Hilbert transform

Unconditionally stable integration of Maxwell's equations

NARCIS (Netherlands)

J.G. Verwer (Jan); M.A. Botchev

2008-01-01

htmlabstractNumerical integration of Maxwell''s equations is often based on explicit methods accepting a stability step size restriction. In literature evidence is given that there is also a need for unconditionally stable methods, as exemplified by the successful alternating direction

Stability under persistent perturbation by white noise

International Nuclear Information System (INIS)

Kalyakin, L

2014-01-01

Deterministic dynamical system which has an asymptotical stable equilibrium is considered under persistent perturbation by white noise. It is well known that if the perturbation does not vanish in the equilibrium position then there is not Lyapunov's stability. The trajectories of the perturbed system diverge from the equilibrium to arbitrarily large distances with probability 1 in finite time. New concept of stability on a large time interval is discussed. The length of interval agrees the reciprocal quantity of the perturbation parameter. The measure of stability is the expectation of the square distance from the trajectory till the equilibrium position. The method of parabolic equation is applied to both estimate the expectation and prove such stability. The main breakthrough is the barrier function derived for the parabolic equation. The barrier is constructed by using the Lyapunov function of the unperturbed system

Whitham modulation theory for the two-dimensional Benjamin-Ono equation.

Science.gov (United States)

Ablowitz, Mark; Biondini, Gino; Wang, Qiao

2017-09-01

Whitham modulation theory for the two-dimensional Benjamin-Ono (2DBO) equation is presented. A system of five quasilinear first-order partial differential equations is derived. The system describes modulations of the traveling wave solutions of the 2DBO equation. These equations are transformed to a singularity-free hydrodynamic-like system referred to here as the 2DBO-Whitham system. Exact reductions of this system are discussed, the formulation of initial value problems is considered, and the system is used to study the transverse stability of traveling wave solutions of the 2DBO equation.

Modeling and Simulation of Swirl Stabilized Turbulent Non-Premixed Flames

Science.gov (United States)

Badillo-Rios, Salvador; Karagozian, Ann

2017-11-01

Flame stabilization is an important design criterion for many combustion chambers, especially at lean conditions and/or high power output, where insufficient stabilization can result in dangerous oscillations and noisy or damaged combustors. At high flow rates, swirling flow can offer a suitable stabilization mechanism, although understanding the dynamics of swirl-stabilized turbulent flames remains a significant challenge. Utilizing the General Equation and Mesh Solver (GEMS) code, which solves the Navier-Stokes equations along with the energy equation and five species equations, 2D axisymmetric and full 3D parametric studies and simulations are performed to guide the design and development of an experimental swirl combustor configuration and to study the effects of swirl on statistically stationary combustion. Results show that as the momentum of air is directed into the inner air inlet rather than the outer inlet of the swirl combustor, the central recirculating region becomes stronger and more unsteady, improving mixing and burning efficiency in that region. A high temperature region is found to occur as a result of burning of the trapped fuel from the central toroidal vortex. The effects of other parameters on flowfield and flame-stabilization dynamics are explored. Supported by ERC, Inc. (PS150006) and AFOSR (Dr. Chiping Li).

Equations governing the liquid-film flow over a plane with heat flux and interfacial phase change

International Nuclear Information System (INIS)

Spindler, B.

1983-01-01

The purpose of the study is to find a system of equations which can be used to study the linear stability of a liquid film flow over a plane exhibiting wall heat flux and interfacial phase change. The flow of such a film is governed by four groups of equations: the equations for mass balance, momentum and energy in the liquid; equations for the balance in the steam; equations for the balance at the liquid-steam interface; and the boundary conditions. Two flow patterns are considered - flow with upstream film and film condensation. Stability is studied by perturbation methods

Equations governing the liquid-film flow over a plane with heat flux and interfacial phase change

Science.gov (United States)

Spindler, B.

1983-08-01

The purpose of the study is to find a system of equations which can be used to study the linear stability of a liquid film flow over a plane exhibiting wall heat flux and interfacial phase change. The flow of such a film is governed by four groups of equations: the equations for mass balance, momentum and energy in the liquid; equations for the balance in the steam; equations for the balance at the liquid-steam interface; and the boundary conditions. Two flow patterns are considered - flow with upstream film and film condensation. Stability is studied by perturbation methods.

Stability Criteria for Differential Equations with Variable Time Delays

Science.gov (United States)

Schley, D.; Shail, R.; Gourley, S. A.

2002-01-01

Time delays are an important aspect of mathematical modelling, but often result in highly complicated equations which are difficult to treat analytically. In this paper it is shown how careful application of certain undergraduate tools such as the Method of Steps and the Principle of the Argument can yield significant results. Certain delay…

Mathematical analysis of Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation

OpenAIRE

Rozanova-Pierrat , Anna

2006-01-01

We consider the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation (ut — uux — βuxx)x— γΔy u = 0 in Sobolev spaces of functions periodic in x and with mean value zero. The derivation of KZK from the nonlinear isentropic Navier Stokes equations and the approximation of their solutions (for viscous and non viscous cases), the results of the existence, uniqueness, stability and blow-up of solution of KZK equation are obtained, also a result of existence of a smooth solution of Navier-Stokes system i...

Poisson-Nernst-Planck equations with steric effects - non-convexity and multiple stationary solutions

Science.gov (United States)

Gavish, Nir

2018-04-01

We study the existence and stability of stationary solutions of Poisson-Nernst-Planck equations with steric effects (PNP-steric equations) with two counter-charged species. We show that within a range of parameters, steric effects give rise to multiple solutions of the corresponding stationary equation that are smooth. The PNP-steric equation, however, is found to be ill-posed at the parameter regime where multiple solutions arise. Following these findings, we introduce a novel PNP-Cahn-Hilliard model, show that it is well-posed and that it admits multiple stationary solutions that are smooth and stable. The various branches of stationary solutions and their stability are mapped utilizing bifurcation analysis and numerical continuation methods.

Unconditionally stable integration of Maxwell's equations

NARCIS (Netherlands)

Verwer, J.G.; Bochev, Mikhail A.

Numerical integration of Maxwell's equations is often based on explicit methods accepting a stability step size restriction. In literature evidence is given that there is also a need for unconditionally stable methods, as exemplified by the successful alternating direction implicit finite difference

Unconditionally stable integration of Maxwell's equations

NARCIS (Netherlands)

J.G. Verwer (Jan); M.A. Botchev

2009-01-01

textabstractNumerical integration of Maxwell’s equations is often based on explicit methods accepting a stability step size restriction. In literature evidence is given that there is also a need for unconditionally stable methods, as exemplified by the successful alternating direction implicit –

General particle transport equation. Final report

International Nuclear Information System (INIS)

Lafi, A.Y.; Reyes, J.N. Jr.

1994-12-01

The general objectives of this research are as follows: (1) To develop fundamental models for fluid particle coalescence and breakage rates for incorporation into statistically based (Population Balance Approach or Monte Carlo Approach) two-phase thermal hydraulics codes. (2) To develop fundamental models for flow structure transitions based on stability theory and fluid particle interaction rates. This report details the derivation of the mass, momentum and energy conservation equations for a distribution of spherical, chemically non-reacting fluid particles of variable size and velocity. To study the effects of fluid particle interactions on interfacial transfer and flow structure requires detailed particulate flow conservation equations. The equations are derived using a particle continuity equation analogous to Boltzmann's transport equation. When coupled with the appropriate closure equations, the conservation equations can be used to model nonequilibrium, two-phase, dispersed, fluid flow behavior. Unlike the Eulerian volume and time averaged conservation equations, the statistically averaged conservation equations contain additional terms that take into account the change due to fluid particle interfacial acceleration and fluid particle dynamics. Two types of particle dynamics are considered; coalescence and breakage. Therefore, the rate of change due to particle dynamics will consider the gain and loss involved in these processes and implement phenomenological models for fluid particle breakage and coalescence

Relativistic dissipative hydrodynamics and the nuclear equation of state

International Nuclear Information System (INIS)

Olson, T.S.; Hiscock, W.A.

1989-01-01

The theory of dissipative, relativistic fluids due to Israel and Stewart is used to constrain the form of the nuclear equation of state. In the Israel-Stewart theory, there are conditions on the equation of state and other thermodynamic properties (the ''second-order'' coefficients) of a fluid which, if satisfied, guarantee that equilibria are stable and that fluid perturbations propagate causally and obey hyperbolic equations. The second-order coefficients in the Israel-Stewart theory, which are relaxation times for the dissipative degrees of freedom and coupling constants between different forms of dissipation, are derived for a free, degenerate Fermi gas. It is shown rigorously that the free, degenerate Fermi gas is stable (and hence causal) at all temperatures in this theory. These values for the second-order coefficients are then used in the stability conditions to constrain various proposed expressions for the nuclear ground-state energy. The stability conditions are found to provide significantly more stringent constraints on the proposed equations of state than the usual simple restriction that the adiabatic sound speed be less than the speed of light

STABLE STATIONARY STATES OF NON-LOCAL INTERACTION EQUATIONS

KAUST Repository

FELLNER, KLEMENS

2010-12-01

In this paper, we are interested in the large-time behaviour of a solution to a non-local interaction equation, where a density of particles/individuals evolves subject to an interaction potential and an external potential. It is known that for regular interaction potentials, stable stationary states of these equations are generically finite sums of Dirac masses. For a finite sum of Dirac masses, we give (i) a condition to be a stationary state, (ii) two necessary conditions of linear stability w.r.t. shifts and reallocations of individual Dirac masses, and (iii) show that these linear stability conditions imply local non-linear stability. Finally, we show that for regular repulsive interaction potential Wε converging to a singular repulsive interaction potential W, the Dirac-type stationary states ρ̄ ε approximate weakly a unique stationary state ρ̄ ∈ L∞. We illustrate our results with numerical examples. © 2010 World Scientific Publishing Company.

Boundary Observability and Stabilization for Westervelt Type Wave Equations without Interior Damping

International Nuclear Information System (INIS)

Kaltenbacher, Barbara

2010-01-01

In this paper we show boundary observability and boundary stabilizability by linear feedbacks for a class of nonlinear wave equations including the undamped Westervelt model used in nonlinear acoustics. We prove local existence for undamped generalized Westervelt equations with homogeneous Dirichlet boundary conditions as well as global existence and exponential decay with absorbing type boundary conditions.

Ion Motion Stability in Asymmetric Surface Electrode Ion Traps

Science.gov (United States)

Shaikh, Fayaz; Ozakin, Arkadas

2010-03-01

Many recently developed designs of the surface electrode ion traps for quantum information processing have asymmetry built into their geometries. The asymmetry helps rotate the trap axes to angles with respect to electrode surface that facilitate laser cooling of ions but introduces a relative angle between the RF and DC fields and invalidates the classical stability analysis of the symmetric case for which the equations of motion are decoupled. For asymmetric case the classical motion of a single ion is given by a coupled, multi-dimensional version of Mathieu's equation. In this poster we discuss the stability diagram of asymmetric surface traps by performing an approximate multiple scale perturbation analysis of the coupled Mathieu equations, and validate the results with numerical simulations. After obtaining the stability diagram for the linear fields, we simulate the motion of an ion in a given asymmetric surface trap, utilizing a method-of-moments calculation of the electrode fields. We obtain the stability diagram and compare it with the ideal case to find the region of validity. Finally, we compare the results of our stability analysis to experiments conducted on a microfabricated asymmetric surface trap.

Ordinary differential equations introduction to the theory of ordinary differential equations in the real domain

CERN Document Server

Kurzweil, J

1986-01-01

The author, Professor Kurzweil, is one of the world's top experts in the area of ordinary differential equations - a fact fully reflected in this book. Unlike many classical texts which concentrate primarily on methods of integration of differential equations, this book pursues a modern approach: the topic is discussed in full generality which, at the same time, permits us to gain a deep insight into the theory and to develop a fruitful intuition. The basic framework of the theory is expanded by considering further important topics like stability, dependence of a solution on a parameter, Car

Application of stabilization techniques in the dynamic analysis of multibody systems

Directory of Open Access Journals (Sweden)

Hajžman M.

2007-11-01

Full Text Available This paper is intended to the discussion of possible methods for the solution of the motion equations of constrained multibody systems. They can be formulated in the form of differential-algebraic equations and their numerical solution brings the problems of constraint violation and numerical stability. Therefore special methods were proposed to handle these problems. Various approaches for the numerical solution of equations are briefly reviewed and the application of the Baumgarte’s stabilization method on testing examples is shown. The paper was motivated by the effort to find the suitable solution methods for the equations of motion in the form of differentialalgebraic equations using the MATLAB standard computational system.

A nonlinear wave equation in nonadiabatic flame propagation

International Nuclear Information System (INIS)

Booty, M.R.; Matalon, M.; Matkowsky, B.J.

1988-01-01

The authors derive a nonlinear wave equation from the diffusional thermal model of gaseous combustion to describe the evolution of a flame front. The equation arises as a long wave theory, for values of the volumeric heat loss in a neighborhood of the extinction point (beyond which planar uniformly propagating flames cease to exist), and for Lewis numbers near the critical value beyond which uniformly propagating planar flames lose stability via a degenerate Hopf bifurcation. Analysis of the equation suggests the possibility of a singularity developing in finite time

The Klein–Gordon–Zakharov equations with the positive fractional ...

Indian Academy of Sciences (India)

2016-11-03

Nov 3, 2016 ... Gan et al [12,13] studied the instability of standing waves for KGZ equations. Linear stability analysis for periodic travelling waves of KGZ equa- tions are performed in refs [14,15]. In refs [16–19], finite difference schemes are proposed for the initial- boundary problem of the KGZ equations. The rest of the ...

Heavy ion collisions with A = 1057: Aspects of nuclear stability and the nuclear equation of state in coalescing neutron-star binary systems

International Nuclear Information System (INIS)

Mathews, G.J.; Wilson, J.R.; Evans, C.R.; Detweiler, S.L.

1987-12-01

The dynamics of the final stages of the coalescence of two neturon stars (such as the binary pulsar PSR 1913+16) is an unsolved problem in astrophysics. Such systems are probably efficient generators of gravitational radiation, and may be significant contributors to heavy-element nucleosynthesis. The input physics for the study of such systems is similar to that required for the strudy of heavy-ion collision hydrodynamics; e.g., a finite temperature nuclear equation of state, properties of nuclei away from stability, etc. We discuss the development of a relativistic hydrodynamics code in three spatial dimensions for the purpose of studying such neutron-star systems. The properties of the mass-radius relation (determined by the nuclear equation of state) may lead to a proposed mechanism by which hot, highly neutronized matter is ejected from the coalescing stars. This material is photodisintegrated into a free (mostly) neutron gas which may subsequently experience rapid-neutron capture (r-process) nucleosynthesis. 15 refs., 4 figs

Generalized modification in the lattice Bhatnagar-Gross-Krook model for incompressible Navier-Stokes equations and convection-diffusion equations.

Science.gov (United States)

Yang, Xuguang; Shi, Baochang; Chai, Zhenhua

2014-07-01

In this paper, two modified lattice Boltzmann Bhatnagar-Gross-Krook (LBGK) models for incompressible Navier-Stokes equations and convection-diffusion equations are proposed via the addition of correction terms in the evolution equations. Utilizing this modification, the value of the dimensionless relaxation time in the LBGK model can be kept in a proper range, and thus the stability of the LBGK model can be improved. Although some gradient operators are included in the correction terms, they can be computed efficiently using local computational schemes such that the present LBGK models still retain the intrinsic parallelism characteristic of the lattice Boltzmann method. Numerical studies of the steady Poiseuille flow and unsteady Womersley flow show that the modified LBGK model has a second-order convergence rate in space, and the compressibility effect in the common LBGK model can be eliminated. In addition, to test the stability of the present models, we also performed some simulations of the natural convection in a square cavity, and we found that the results agree well with those reported in the previous work, even at a very high Rayleigh number (Ra = 10(12)).

Langevin equations with multiplicative noise: application to domain growth

International Nuclear Information System (INIS)

Sancho, J.M.; Hernandez-Machado, A.; Ramirez-Piscina, L.; Lacasta, A.M.

1993-01-01

Langevin equations of Ginzburg-Landau form with multiplicative noise, are proposed to study the effects of fluctuations in domain growth. These equations are derived from a coarse-grained methodology. The Cahn-Hilliard-Cook linear stability analysis predicts some effects in the transitory regime. We also derive numerical algorithms for the computer simulation of these equations. The numerical results corroborate the analytical productions of the linear analysis. We also present simulation results for spinodal decomposition at large times. (author). 28 refs, 2 figs

Strongly nonlinear theory of rapid solidification near absolute stability

Science.gov (United States)

Kowal, Katarzyna N.; Altieri, Anthony L.; Davis, Stephen H.

2017-10-01

We investigate the nonlinear evolution of the morphological deformation of a solid-liquid interface of a binary melt under rapid solidification conditions near two absolute stability limits. The first of these involves the complete stabilization of the system to cellular instabilities as a result of large enough surface energy. We derive nonlinear evolution equations in several limits in this scenario and investigate the effect of interfacial disequilibrium on the nonlinear deformations that arise. In contrast to the morphological stability problem in equilibrium, in which only cellular instabilities appear and only one absolute stability boundary exists, in disequilibrium the system is prone to oscillatory instabilities and a second absolute stability boundary involving attachment kinetics arises. Large enough attachment kinetics stabilize the oscillatory instabilities. We derive a nonlinear evolution equation to describe the nonlinear development of the solid-liquid interface near this oscillatory absolute stability limit. We find that strong asymmetries develop with time. For uniform oscillations, the evolution equation for the interface reduces to the simple form f''+(βf')2+f =0 , where β is the disequilibrium parameter. Lastly, we investigate a distinguished limit near both absolute stability limits in which the system is prone to both cellular and oscillatory instabilities and derive a nonlinear evolution equation that captures the nonlinear deformations in this limit. Common to all these scenarios is the emergence of larger asymmetries in the resulting shapes of the solid-liquid interface with greater departures from equilibrium and larger morphological numbers. The disturbances additionally sharpen near the oscillatory absolute stability boundary, where the interface becomes deep-rooted. The oscillations are time-periodic only for small-enough initial amplitudes and their frequency depends on a single combination of physical parameters, including the

Stability and Hopf bifurcation in a delayed competitive web sites model

International Nuclear Information System (INIS)

Xiao Min; Cao Jinde

2006-01-01

The delayed differential equations modeling competitive web sites, based on the Lotka-Volterra competition equations, are considered. Firstly, the linear stability is investigated. It is found that there is a stability switch for time delay, and Hopf bifurcation occurs when time delay crosses through a critical value. Then the direction and stability of the bifurcated periodic solutions are determined, using the normal form theory and the center manifold reduction. Finally, some numerical simulations are carried out to illustrate the results found

Finite Larmor radius effects on the stability properties of internal modes of a z-pinch

International Nuclear Information System (INIS)

Aakerstedt, H.O.

1987-01-01

From the Vlasov-fluid model a set of approximate stability equations describing the stability of a cylindrically symmetric z-pinch is derived. The equations are derived in the limit of small gyroradius and include first order kinetic effects such as finite ion Larmor radius effects and resonant ion effects. Neglecting the resonant ion terms, we explicitly solve this set of equations for a constant current density profile leading to a dispersion relation. FLR effects are shown for the case of m=1 internal mode to be stabilizing and for large wavenumbers k, using a trial function approach, absolute stabilization is found. (author)

Some functional equations originating from number theory

Indian Academy of Sciences (India)

R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22

Moreover, we will also study some stability problems of those equations. ... Wisconsin in which he discussed a number of important unsolved problems [18]. ... According to a well-known theorem in number theory, a positive integer of the form.

Functional Equations in Fuzzy Banach Spaces

Directory of Open Access Journals (Sweden)

M. Eshaghi Gordji

2012-01-01

generalized Hyers-Ulam stability of the following additive-quadratic functional equation f(x+ky+f(x−ky=f(x+y+f(x−y+(2(k+1/kf(ky−2(k+1f(y for fixed integers k with k≠0,±1 in fuzzy Banach spaces.

Correlation of NMR and refractometry to evaluate the stability constant on carbon tetrachloride

International Nuclear Information System (INIS)

Modarress, H.; Pouryazadanpanah, N.

2004-01-01

An equation has been suggested which correlate the NMR and refractometry results to evaluate the stability constant of electron donor complexes. Using this equation the stability constant of complexation between carbon tetrachloride and toluene in cyclohexane has been studied by refractometry and NMR spectroscopy

Integration of equations of parabolic type by the method of nets

CERN Document Server

Saul'Yev, V K; Stark, M; Ulam, S

1964-01-01

International Series of Monographs in Pure and Applied Mathematics, Volume 54: Integration of Equations of Parabolic Type by the Method of Nets deals with solving parabolic partial differential equations using the method of nets. The first part of this volume focuses on the construction of net equations, with emphasis on the stability and accuracy of the approximating net equations. The method of nets or method of finite differences (used to define the corresponding numerical method in ordinary differential equations) is one of many different approximate methods of integration of partial diff

Evans functions and bifurcations of nonlinear waves of some nonlinear reaction diffusion equations

Science.gov (United States)

Zhang, Linghai

2017-10-01

The main purposes of this paper are to accomplish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear system of reaction diffusion equations ut =uxx + α [ βH (u - θ) - u ] - w, wt = ε (u - γw) and to establish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear scalar reaction diffusion equation ut =uxx + α [ βH (u - θ) - u ], under different conditions on the model constants. To establish the bifurcation for the system, we will study the existence and instability of a standing pulse solution if 0 1; the existence and instability of two standing wave fronts if 2 (1 + αγ) θ = αβγ and 0 traveling wave front as well as the existence and instability of a standing pulse solution if 0 traveling wave front as well as the existence and instability of an upside down standing pulse solution if 0 traveling wave back of the nonlinear scalar reaction diffusion equation ut =uxx + α [ βH (u - θ) - u ] -w0, where w0 = α (β - 2 θ) > 0 is a positive constant, if 0 motivation to study the existence, stability, instability and bifurcations of the nonlinear waves is to study the existence and stability/instability of infinitely many fast/slow multiple traveling pulse solutions of the nonlinear system of reaction diffusion equations. The existence and stability of infinitely many fast multiple traveling pulse solutions are of great interests in mathematical neuroscience.

Whitham modulation theory for the Kadomtsev- Petviashvili equation

Science.gov (United States)

Ablowitz, Mark J.; Biondini, Gino; Wang, Qiao

2017-08-01

The genus-1 Kadomtsev-Petviashvili (KP)-Whitham system is derived for both variants of the KP equation; namely the KPI and KPII equations. The basic properties of the KP-Whitham system, including symmetries, exact reductions and its possible complete integrability, together with the appropriate generalization of the one-dimensional Riemann problem for the Korteweg-de Vries equation are discussed. Finally, the KP-Whitham system is used to study the linear stability properties of the genus-1 solutions of the KPI and KPII equations; it is shown that all genus-1 solutions of KPI are linearly unstable, while all genus-1 solutions of KPII are linearly stable within the context of Whitham theory.

Stability of squashed Kaluza-Klein black holes

International Nuclear Information System (INIS)

Kimura, Masashi; Ishihara, Hideki; Murata, Keiju; Soda, Jiro

2008-01-01

The stability of squashed Kaluza-Klein black holes is studied. The squashed Kaluza-Klein black hole looks like a five-dimensional black hole in the vicinity of horizon and looks like a four-dimensional Minkowski spacetime with a circle at infinity. In this sense, squashed Kaluza-Klein black holes can be regarded as black holes in the Kaluza-Klein spacetimes. Using the symmetry of squashed Kaluza-Klein black holes, SU(2)xU(1)≅U(2), we obtain master equations for a part of the metric perturbations relevant to the stability. The analysis based on the master equations gives strong evidence for the stability of squashed Kaluza-Klein black holes. Hence, the squashed Kaluza-Klein black holes deserve to be taken seriously as realistic black holes in the Kaluza-Klein spacetime.

Optimal boundary control and boundary stabilization of hyperbolic systems

CERN Document Server

Gugat, Martin

2015-01-01

This brief considers recent results on optimal control and stabilization of systems governed by hyperbolic partial differential equations, specifically those in which the control action takes place at the boundary.  The wave equation is used as a typical example of a linear system, through which the author explores initial boundary value problems, concepts of exact controllability, optimal exact control, and boundary stabilization.  Nonlinear systems are also covered, with the Korteweg-de Vries and Burgers Equations serving as standard examples.  To keep the presentation as accessible as possible, the author uses the case of a system with a state that is defined on a finite space interval, so that there are only two boundary points where the system can be controlled.  Graduate and post-graduate students as well as researchers in the field will find this to be an accessible introduction to problems of optimal control and stabilization.

Two remarks on the stability of ordniary differential equations

International Nuclear Information System (INIS)

Vidossich, G.

1979-01-01

In this note the definition of Liapunov functions is extended, obtaining a generalization of the recent result by Lakshmikantham and Leela on the perturbation of Liapunov functions, and two sufficient conditions for the solvability of Olech's problem on global stability are pointed out. (author)

Vector domain decomposition schemes for parabolic equations

Science.gov (United States)

Vabishchevich, P. N.

2017-09-01

A new class of domain decomposition schemes for finding approximate solutions of timedependent problems for partial differential equations is proposed and studied. A boundary value problem for a second-order parabolic equation is used as a model problem. The general approach to the construction of domain decomposition schemes is based on partition of unity. Specifically, a vector problem is set up for solving problems in individual subdomains. Stability conditions for vector regionally additive schemes of first- and second-order accuracy are obtained.

Reminimization of energy integral and stability limit for non-ideal MHD (magnetohydrodynamic) plasma

International Nuclear Information System (INIS)

Kondoh, Y.

1988-03-01

The stability condition of relaxed states is derived from the energy principle for the non-ideal MHD plasma. An Euler equation for the reminimization of energy integral is derived and shown to give the marginal stable, non-singular perturbations for the stability condition. An extended stability limit for the β = 0 relaxed states is derived from the stability condition, with use of the eigenvalue analysis for the Euler equation. By using the perturbation method, the extended stability limit is solved in the 1st order approximation to explain the deviation of the experimental stability limit from the idealized stability limit by Taylor. A procedure to get overall stability limit against both the non-singular and the singular perturbations is discussed. 25 refs

Consistent three-equation model for thin films

Science.gov (United States)

Richard, Gael; Gisclon, Marguerite; Ruyer-Quil, Christian; Vila, Jean-Paul

2017-11-01

Numerical simulations of thin films of newtonian fluids down an inclined plane use reduced models for computational cost reasons. These models are usually derived by averaging over the fluid depth the physical equations of fluid mechanics with an asymptotic method in the long-wave limit. Two-equation models are based on the mass conservation equation and either on the momentum balance equation or on the work-energy theorem. We show that there is no two-equation model that is both consistent and theoretically coherent and that a third variable and a three-equation model are required to solve all theoretical contradictions. The linear and nonlinear properties of two and three-equation models are tested on various practical problems. We present a new consistent three-equation model with a simple mathematical structure which allows an easy and reliable numerical resolution. The numerical calculations agree fairly well with experimental measurements or with direct numerical resolutions for neutral stability curves, speed of kinematic waves and of solitary waves and depth profiles of wavy films. The model can also predict the flow reversal at the first capillary trough ahead of the main wave hump.

Numerical approximations of difference functional equations and applications

Directory of Open Access Journals (Sweden)

Zdzisław Kamont

2005-01-01

Full Text Available We give a theorem on the error estimate of approximate solutions for difference functional equations of the Volterra type. We apply this general result in the investigation of the stability of difference schemes generated by nonlinear first order partial differential functional equations and by parabolic problems. We show that all known results on difference methods for initial or initial boundary value problems can be obtained as particular cases of this general and simple result. We assume that the right hand sides of equations satisfy nonlinear estimates of the Perron type with respect to functional variables.

Stability of BDF-ADI Discretizations

KAUST Repository

Felí cio dos Reis, Joã o Miguel

2017-01-01

We present new results on absolute stability for BDF-ADI (Backward differentiation formula Alternating Direction Implicit) methods applied to a linear advection and diffusion equations. Unconditional absolute stability of the BDF2-ADI method is proven for advection and diffusion separately, as well as to the BDF3-ADI method for the purely-diffusive case. Conditional absolute stability of the BDF4-ADI is also proven for the purely-diffusive case, and stability regions for BDF3-ADI and BDF4- ADI are given in terms of the PDE coefficients and numerical parameters. Lastly, numerical experiments are presented to support the theoretical results and conjectures. These experiments also suggest future work.

Stability of BDF-ADI Discretizations

KAUST Repository

Felício dos Reis, João Miguel

2017-08-17

We present new results on absolute stability for BDF-ADI (Backward differentiation formula Alternating Direction Implicit) methods applied to a linear advection and diffusion equations. Unconditional absolute stability of the BDF2-ADI method is proven for advection and diffusion separately, as well as to the BDF3-ADI method for the purely-diffusive case. Conditional absolute stability of the BDF4-ADI is also proven for the purely-diffusive case, and stability regions for BDF3-ADI and BDF4- ADI are given in terms of the PDE coefficients and numerical parameters. Lastly, numerical experiments are presented to support the theoretical results and conjectures. These experiments also suggest future work.

Solutions to the equations describing materials with competing quadratic and cubic nonlinearities

International Nuclear Information System (INIS)

Li-Na, Zhao; Ji, Lin; Zi-Shuang, Tong

2009-01-01

The Lie group theoretical method is used to study the equations describing materials with competing quadratic and cubic nonlinearities. The equations share some of the nice properties of soliton equations. From the elliptic functions expansion method, we obtain large families of analytical solutions, in special cases, we have the periodic, kink and solitary solutions of the equations. Furthermore, we investigate the stability of these solutions under the perturbation of amplitude noises by numerical simulation

Taylor's series method for solving the nonlinear point kinetics equations

International Nuclear Information System (INIS)

Nahla, Abdallah A.

2011-01-01

Highlights: → Taylor's series method for nonlinear point kinetics equations is applied. → The general order of derivatives are derived for this system. → Stability of Taylor's series method is studied. → Taylor's series method is A-stable for negative reactivity. → Taylor's series method is an accurate computational technique. - Abstract: Taylor's series method for solving the point reactor kinetics equations with multi-group of delayed neutrons in the presence of Newtonian temperature feedback reactivity is applied and programmed by FORTRAN. This system is the couples of the stiff nonlinear ordinary differential equations. This numerical method is based on the different order derivatives of the neutron density, the precursor concentrations of i-group of delayed neutrons and the reactivity. The r th order of derivatives are derived. The stability of Taylor's series method is discussed. Three sets of applications: step, ramp and temperature feedback reactivities are computed. Taylor's series method is an accurate computational technique and stable for negative step, negative ramp and temperature feedback reactivities. This method is useful than the traditional methods for solving the nonlinear point kinetics equations.

Thermodynamic analysis of transition pressure of δ-stabilized binary plutonium alloys

International Nuclear Information System (INIS)

Wang Qinghui

1992-01-01

The transformation of δ-stabilized binary plutonium alloys to α-Pu was studies by thermodynamic analysis. A transition pressure-composition equation which can characterize the high pressure transformation from δ to α was derived. Values calculated by the equation and values measured by experiments of published references have the same tendency. the following facts can be explained properly by this equation. (1)The transformation pressure increases linearly with the amount of an alloying element. (2) The slope of the plot of transformation pressure versus composition of δ-Pu alloys is inversely proportional to the minimum amount of solute required to retain δ-phase at room temperature and pressure. (3) Curves showing the relationship between transformation pressure and composition of various δ-stabilized binary alloys interact at the same point of zero solute (transformation pressure axis). In addition, some transformation pressures from δ to α of δ-stabilized alloys are predicted by using the modified theoretical equation

On the stability of an evaporating liquid surface

International Nuclear Information System (INIS)

Krahl, R; Bänsch, E

2012-01-01

The stability of the interface between a volatile liquid and a gaseous phase has been studied in this paper. We consider the case when the liquid volume is not a film and thus the thin-film approximation might not be valid. A linear stability analysis leads to the Orr–Sommerfeld equation for the stream function and a second-order differential equation for the temperature. This system is solved semi-analytically. A parameter study shows that surface tension is stabilizing, while viscosity is destabilizing the liquid surface. The capillary number is identified as the most significant factor. The analytical results were compared with the growth of an initial perturbation for the full system by direct numerical simulations, and excellent agreement was observed. (paper)

Unconditionally stable integration of Maxwell’s equations

NARCIS (Netherlands)

Verwer, J.G.; Botchev, M.A.

2009-01-01

Numerical integration of Maxwell’s equations is often based on explicit methods accepting a stability step size restriction. In literature evidence is given that there is also a need for unconditionally stable methods, as exemplified by the successful alternating direction implicit - finite

Constructing Hopf bifurcation lines for the stability of nonlinear systems with two time delays

Science.gov (United States)

Nguimdo, Romain Modeste

2018-03-01

Although the plethora real-life systems modeled by nonlinear systems with two independent time delays, the algebraic expressions for determining the stability of their fixed points remain the Achilles' heel. Typically, the approach for studying the stability of delay systems consists in finding the bifurcation lines separating the stable and unstable parameter regions. This work deals with the parametric construction of algebraic expressions and their use for the determination of the stability boundaries of fixed points in nonlinear systems with two independent time delays. In particular, we concentrate on the cases for which the stability of the fixed points can be ascertained from a characteristic equation corresponding to that of scalar two-delay differential equations, one-component dual-delay feedback, or nonscalar differential equations with two delays for which the characteristic equation for the stability analysis can be reduced to that of a scalar case. Then, we apply our obtained algebraic expressions to identify either the parameter regions of stable microwaves generated by dual-delay optoelectronic oscillators or the regions of amplitude death in identical coupled oscillators.

Nonlinear stability of Gardner breathers

Science.gov (United States)

Alejo, Miguel A.

2018-01-01

We show that breather solutions of the Gardner equation, a natural generalization of the KdV and mKdV equations, are H2 (R) stable. Through a variational approach, we characterize Gardner breathers as minimizers of a new Lyapunov functional and we study the associated spectral problem, through (i) the analysis of the spectrum of explicit linear systems (spectral stability), and (ii) controlling degenerated directions by using low regularity conservation laws.

Dynamics of partial differential equations

CERN Document Server

Wayne, C Eugene

2015-01-01

This book contains two review articles on the dynamics of partial differential equations that deal with closely related topics but can be read independently. Wayne reviews recent results on the global dynamics of the two-dimensional Navier-Stokes equations. This system exhibits stable vortex solutions: the topic of Wayne's contribution is how solutions that start from arbitrary initial conditions evolve towards stable vortices. Weinstein considers the dynamics of localized states in nonlinear Schrodinger and Gross-Pitaevskii equations that describe many optical and quantum systems. In this contribution, Weinstein reviews recent bifurcations results of solitary waves, their linear and nonlinear stability properties, and results about radiation damping where waves lose energy through radiation.   The articles, written independently, are combined into one volume to showcase the tools of dynamical systems theory at work in explaining qualitative phenomena associated with two classes of partial differential equ...

Differential equations, mechanics, and computation

CERN Document Server

Palais, Richard S

2009-01-01

This book provides a conceptual introduction to the theory of ordinary differential equations, concentrating on the initial value problem for equations of evolution and with applications to the calculus of variations and classical mechanics, along with a discussion of chaos theory and ecological models. It has a unified and visual introduction to the theory of numerical methods and a novel approach to the analysis of errors and stability of various numerical solution algorithms based on carefully chosen model problems. While the book would be suitable as a textbook for an undergraduate or elementary graduate course in ordinary differential equations, the authors have designed the text also to be useful for motivated students wishing to learn the material on their own or desiring to supplement an ODE textbook being used in a course they are taking with a text offering a more conceptual approach to the subject.

Asymptotic behavior of the plasma equation

International Nuclear Information System (INIS)

Kwong, Y.C.

1984-01-01

This paper is concerned with the plasma equation on a bounded smooth domain the N-dimensional Euclidean Space, with non-negative initial data and a homogenous Dirichlet boundary condition. It is known that there exists a finite extinction time T such that the solution decays to zero at T. Berryman and Holland investigated the stability of the profile of the solution as t is approaching T. However, they obtained their results at the expense of some very strong regularity assumptions. By invoking both the nonlinear semi-group theory and a standard regularizing scheme for the equation, the same results are proved without those assumptions by measuring the rate of decay of the solution and estimates are obtained on the time derivative as t is approaching T. As motivated by the regularity assumptions, both the interior and boundary regularities of the solution are studied. Finally, the nonlinearity of the plasma equation is perturbed and the same aspects for the perturbed equation are studied

A textbook on ordinary differential equations

CERN Document Server

Ahmad, Shair

2015-01-01

This book offers readers a primer on the theory and applications of Ordinary Differential Equations. The style used is simple, yet thorough and rigorous. Each chapter ends with a broad set of exercises that range from the routine to the more challenging and thought-provoking. Solutions to selected exercises can be found at the end of the book. The book contains many interesting examples on topics such as electric circuits, the pendulum equation, the logistic equation, the Lotka-Volterra system, the Laplace Transform, etc., which introduce students to a number of interesting aspects of the theory and applications. The work is mainly intended for students of Mathematics, Physics, Engineering, Computer Science and other areas of the natural and social sciences that use ordinary differential equations, and who have a firm grasp of Calculus and a minimal understanding of the basic concepts used in Linear Algebra. It also studies a few more advanced topics, such as Stability Theory and Boundary Value Problems, whic...

A textbook on ordinary differential equations

CERN Document Server

Ahmad, Shair

2014-01-01

The book is a primer of the theory of Ordinary Differential Equations. Each chapter is completed by a broad set of exercises; the reader will also find a set of solutions of selected exercises. The book contains many interesting examples as well (like the equations for the electric circuits, the pendium equation, the logistic equation, the Lotka-Volterra system, and many other) which introduce the reader to some interesting aspects of the theory and its applications. The work is mainly addressed to students of Mathematics, Physics, Engineering, Statistics, Computer Sciences, with  knowledge of Calculus and Linear Algebra, and contains more advanced topics for further developments, such as Laplace transform; Stability theory and existence of solutions to Boundary Value problems. The authors are preparing a complete solutions manual, containing solutions to all the exercises published in the book. The manual will be available Summer 2014. Instructors who wish to adopt the book may request the manual by writing...

Long-Term Dynamics of Autonomous Fractional Differential Equations

Science.gov (United States)

Liu, Tao; Xu, Wei; Xu, Yong; Han, Qun

This paper aims to investigate long-term dynamic behaviors of autonomous fractional differential equations with effective numerical method. The long-term dynamic behaviors predict where systems are heading after long-term evolution. We make some modification and transplant cell mapping methods to autonomous fractional differential equations. The mapping time duration of cell mapping is enlarged to deal with the long memory effect. Three illustrative examples, i.e. fractional Lotka-Volterra equation, fractional van der Pol oscillator and fractional Duffing equation, are studied with our revised generalized cell mapping method. We obtain long-term dynamics, such as attractors, basins of attraction, and saddles. Compared with some existing stability and numerical results, the validity of our method is verified. Furthermore, we find that the fractional order has its effect on the long-term dynamics of autonomous fractional differential equations.

On the stability, the periodic solutions and the resolution of certain types of non linear equations, and of non linearly coupled systems of these equations, appearing in betatronic oscillations; Sur la stabilite, les solutions periodiques et la resolution de certaines categories d'equations et systemes d'equations differentielles couplees non lineaires apparaissant dans les oscillations betatroniques

Energy Technology Data Exchange (ETDEWEB)

Valat, J [Commissariat a l' Energie Atomique, Saclay (France). Centre d' Etudes Nucleaires

1960-12-15

Universal stability diagrams have been calculated and experimentally checked for Hill-Meissner type equations with square-wave coefficients. The study of these equations in the phase-plane has then made it possible to extend the periodic solution calculations to the case of non-linear differential equations with periodic square-wave coefficients. This theory has been checked experimentally. For non-linear coupled systems with constant coefficients, a search was first made for solutions giving an algebraic motion. The elliptical and Fuchs's functions solve such motions. The study of non-algebraic motions is more delicate, apart from the study of nonlinear Lissajous's motions. A functional analysis shows that it is possible however in certain cases to decouple the system and to find general solutions. For non-linear coupled systems with periodic square-wave coefficients it is then possible to calculate the conditions leading to periodic solutions, if the two non-linear associated systems with constant coefficients fall into one of the categories of the above paragraph. (author) [French] Pour les equations du genre de Hill-Meissner a coefficients creneles, on a calcule des diagrammes universels de stabilite et ceux-ci ont ete verifies experimentalement. L'etude de ces equations dans le plan de phase a permis ensuite d'etendre le calcul des solutions periodiques au cas des equations differentielles non lineaires a coefficients periodiques creneles. Cette theorie a ete verifiee experimentalement. Pour Jes systemes couples non lineaires a coefficients constants, on a d'abord cherche les solutions menant a des mouvements algebriques. Les fonctions elliptiques et fuchsiennes uniformisent de tels mouvements. L'etude de mouvements non algebriques est plus delicate, a part l'etude des mouvements de Lissajous non lineaires. Une analyse fonctionnelle montre qu'il est toutefois possible dans certains cas de decoupler le systeme et de trouver des solutions generales. Pour les

Lattice Wigner equation

Science.gov (United States)

Solórzano, S.; Mendoza, M.; Succi, S.; Herrmann, H. J.

2018-01-01

We present a numerical scheme to solve the Wigner equation, based on a lattice discretization of momentum space. The moments of the Wigner function are recovered exactly, up to the desired order given by the number of discrete momenta retained in the discretization, which also determines the accuracy of the method. The Wigner equation is equipped with an additional collision operator, designed in such a way as to ensure numerical stability without affecting the evolution of the relevant moments of the Wigner function. The lattice Wigner scheme is validated for the case of quantum harmonic and anharmonic potentials, showing good agreement with theoretical results. It is further applied to the study of the transport properties of one- and two-dimensional open quantum systems with potential barriers. Finally, the computational viability of the scheme for the case of three-dimensional open systems is also illustrated.

The numerical dynamic for highly nonlinear partial differential equations

Science.gov (United States)

Lafon, A.; Yee, H. C.

1992-01-01

Problems associated with the numerical computation of highly nonlinear equations in computational fluid dynamics are set forth and analyzed in terms of the potential ranges of spurious behaviors. A reaction-convection equation with a nonlinear source term is employed to evaluate the effects related to spatial and temporal discretizations. The discretization of the source term is described according to several methods, and the various techniques are shown to have a significant effect on the stability of the spurious solutions. Traditional linearized stability analyses cannot provide the level of confidence required for accurate fluid dynamics computations, and the incorporation of nonlinear analysis is proposed. Nonlinear analysis based on nonlinear dynamical systems complements the conventional linear approach and is valuable in the analysis of hypersonic aerodynamics and combustion phenomena.

CIME course on Control of Partial Differential Equations

CERN Document Server

Alabau-Boussouira, Fatiha; Glass, Olivier; Le Rousseau, Jérôme; Zuazua, Enrique

2012-01-01

The term “control theory” refers to the body of results - theoretical, numerical and algorithmic - which have been developed to influence the evolution of the state of a given system in order to meet a prescribed performance criterion. Systems of interest to control theory may be of very different natures. This monograph is concerned with models that can be described by partial differential equations of evolution. It contains five major contributions and is connected to the CIME Course on Control of Partial Differential Equations that took place in Cetraro (CS, Italy), July 19 - 23, 2010.  Specifically, it covers the stabilization of evolution equations, control of the Liouville equation, control in fluid mechanics, control and numerics for the wave equation, and Carleman estimates for elliptic and parabolic equations with application to control. We are confident this work will provide an authoritative reference work for all scientists who are interested in this field, representing at the same time a fri...

Structure and stability of warm cores in neutron stars

Energy Technology Data Exchange (ETDEWEB)

Ibanez Cabanell, J M [Departamento de Mecanica y Astronomia, Facultad de Matematicas, Burjasot-Valencia (Spain)

1981-12-01

Relativistic equations of structure are solved using Lamb's equations of state for warm neutron degenerate matter. The stability of isothermal cores in neutron stars is discussed and also the possible compatibility of the results obtained with experimental evidence is shown.

Workshop on Numerical Methods for Ordinary Differential Equations

CERN Document Server

Gear, Charles; Russo, Elvira

1989-01-01

Developments in numerical initial value ode methods were the focal topic of the meeting at L'Aquila which explord the connections between the classical background and new research areas such as differental-algebraic equations, delay integral and integro-differential equations, stability properties, continuous extensions (interpolants for Runge-Kutta methods and their applications, effective stepsize control, parallel algorithms for small- and large-scale parallel architectures). The resulting proceedings address many of these topics in both research and survey papers.

Stability of Rotor Systems: A Complex Modelling Approach

DEFF Research Database (Denmark)

Kliem, Wolfhard; Pommer, Christian; Stoustrup, Jakob

1996-01-01

A large class of rotor systems can be modelled by a complex matrix differential equation of secondorder. The angular velocity of the rotor plays the role of a parameter. We apply the Lyapunov matrix equation in a complex setting and prove two new stability results which are compared...

Runge-Kutta and Hermite Collocation for a biological invasion problem modeled by a generalized Fisher equation

International Nuclear Information System (INIS)

Athanasakis, I E; Papadopoulou, E P; Saridakis, Y G

2014-01-01

Fisher's equation has been widely used to model the biological invasion of single-species communities in homogeneous one dimensional habitats. In this study we develop high order numerical methods to accurately capture the spatiotemporal dynamics of the generalized Fisher equation, a nonlinear reaction-diffusion equation characterized by density dependent non-linear diffusion. Working towards this direction we consider strong stability preserving Runge-Kutta (RK) temporal discretization schemes coupled with the Hermite cubic Collocation (HC) spatial discretization method. We investigate their convergence and stability properties to reveal efficient HC-RK pairs for the numerical treatment of the generalized Fisher equation. The Hadamard product is used to characterize the collocation discretized non linear equation terms as a first step for the treatment of generalized systems of relevant equations. Numerical experimentation is included to demonstrate the performance of the methods

Simple Numerical Schemes for the Korteweg-deVries Equation

International Nuclear Information System (INIS)

McKinstrie, C. J.; Kozlov, M.V.

2000-01-01

Two numerical schemes, which simulate the propagation of dispersive non-linear waves, are described. The first is a split-step Fourier scheme for the Korteweg-de Vries (KdV) equation. The second is a finite-difference scheme for the modified KdV equation. The stability and accuracy of both schemes are discussed. These simple schemes can be used to study a wide variety of physical processes that involve dispersive nonlinear waves

Simple Numerical Schemes for the Korteweg-deVries Equation

Energy Technology Data Exchange (ETDEWEB)

C. J. McKinstrie; M. V. Kozlov

2000-12-01

Two numerical schemes, which simulate the propagation of dispersive non-linear waves, are described. The first is a split-step Fourier scheme for the Korteweg-de Vries (KdV) equation. The second is a finite-difference scheme for the modified KdV equation. The stability and accuracy of both schemes are discussed. These simple schemes can be used to study a wide variety of physical processes that involve dispersive nonlinear waves.

Application of Arma Technique For Operation Stability of RSG-Gas

International Nuclear Information System (INIS)

Djudjuratisbela, Udju

2000-01-01

Application Of Arma Technique For Operation Stability Of RSG-Gas. Application of Fast Fourier Transport (FFT) method in the noise experiments data had been conducted to reactor kinetic parameter determination of RSG-Gas. Reactor stability that has closed relation to operation safety has not been measured yet. Noise analysis method and ARMA (Auto Regressive Moving Average) technique that has capability to identify mathematical model of the noise experimental data can be used for determination of kinetic/dynamic characteristic equation and its roots. From the roots of reactor characteristic equation, magnitude of natural frequency (fn), damping ratio (xi), damping frequency (fd), decay ratio (delta) and then reactor stability can be calculated

Numerical Solutions for Convection-Diffusion Equation through Non-Polynomial Spline

Directory of Open Access Journals (Sweden)

Ravi Kanth A.S.V.

2016-01-01

Full Text Available In this paper, numerical solutions for convection-diffusion equation via non-polynomial splines are studied. We purpose an implicit method based on non-polynomial spline functions for solving the convection-diffusion equation. The method is proven to be unconditionally stable by using Von Neumann technique. Numerical results are illustrated to demonstrate the efficiency and stability of the purposed method.

Steady state and linear stability analysis of a supercritical water natural circulation loop

International Nuclear Information System (INIS)

Sharma, Manish; Pilkhwal, D.S.; Vijayan, P.K.; Saha, D.; Sinha, R.K.

2010-01-01

Supercritical water (SCW) has excellent heat transfer characteristics as a coolant for nuclear reactors. Besides it results in high thermal efficiency of the plant. However, the flow can experience instabilities in supercritical water reactors, as the density change is very large for the supercritical fluids. A computer code SUCLIN using supercritical water properties has been developed to carry out the steady state and linear stability analysis of a SCW natural circulation loop. The conservation equations of mass, momentum and energy have been linearized by imposing small perturbation in flow rate, enthalpy, pressure and specific volume. The equations have been solved analytically to generate the characteristic equation. The roots of the equation determine the stability of the system. The code has been qualitatively assessed with published results and has been extensively used for studying the effect of diameter, height, heater inlet temperature, pressure and local loss coefficients on steady state and stability behavior of a Supercritical Water Natural Circulation Loop (SCWNCL). The present paper describes the linear stability analysis model and the results obtained in detail.

Solutions Stability of Initial Boundary Problem, Modeling of Dynamics of Some Discrete Continuum Mechanical System

Directory of Open Access Journals (Sweden)

D. A. Eliseev

2015-01-01

Full Text Available The solution stability of an initial boundary problem for a linear hybrid system of differential equations, which models the rotation of a rigid body with two elastic rods located in the same plane is studied in the paper. To an axis passing through the mass center of the rigid body perpendicularly to the rods location plane is applied the stabilizing moment proportional to the angle of the system rotation, derivative of the angle, integral of the angle. The external moment provides a feedback. A method of studying the behavior of solutions of the initial boundary problem is proposed. This method allows to exclude from the hybrid system of differential equations partial differential equations, which describe the dynamics of distributed elements of a mechanical system. It allows us to build one equation for an angle of the system rotation. Its characteristic equation defines the stability of solutions of all the system. In the space of feedback-coefficients the areas that provide the asymptotic stability of solutions of the initial boundary problem are built up.

Hyperbolic partial differential equations populations, reactors, tides and waves theory and applications

CERN Document Server

Witten, Matthew

1983-01-01

Hyperbolic Partial Differential Equations, Volume 1: Population, Reactors, Tides and Waves: Theory and Applications covers three general areas of hyperbolic partial differential equation applications. These areas include problems related to the McKendrick/Von Foerster population equations, other hyperbolic form equations, and the numerical solution.This text is composed of 15 chapters and begins with surveys of age specific population interactions, populations models of diffusion, nonlinear age dependent population growth with harvesting, local and global stability for the nonlinear renewal eq

MARG2D code. 1. Eigenvalue problem for two dimensional Newcomb equation