Bifurcation in Mean Phase Portraits for Stochastic Dynamical Systems with Multiplicative Gaussian Noise

We investigate the bifurcation phenomena for stochastic systems with multiplicative Gaussian noise, by examining qualitative changes in mean phase portraits. Starting from the Fokker–Planck equation for the probability density function of solution processes, we compute the mean orbits and mean equilibrium states. A change in the number or stability type, when a parameter varies, indicates a stochastic bifurcation. Specifically, we study stochastic bifurcation for three prototypical dynamical systems (i.e. saddle-node, transcritical, and pitchfork systems) under multiplicative Gaussian noise, and have found some interesting phenomena in contrast to the corresponding deterministic counterparts.

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A Stochastic Pitchfork Bifurcation in Most Probable Phase Portraits

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418500177 ◽

2018 ◽

Vol 28 (01) ◽

pp. 1850017 ◽

Cited By ~ 8

Author(s):

Hui Wang ◽

Xiaoli Chen ◽

Jinqiao Duan

Keyword(s):

Gaussian Noise ◽

Pitchfork Bifurcation ◽

Phase Portraits ◽

Lévy Noise ◽

Stochastic Bifurcation ◽

Bifurcation Phenomena ◽

Pitchfork Bifurcations ◽

Non Gaussian ◽

Levy Noise ◽

Qualitative Changes

We study stochastic bifurcation for a system under multiplicative stable Lévy noise (an important class of non-Gaussian noise), by examining the qualitative changes of equilibrium states with its most probable phase portraits. We have found some peculiar bifurcation phenomena in contrast to the deterministic counterpart: (i) When the non-Gaussianity parameter in Lévy noise varies, there is either one, two or no backward pitchfork type bifurcations; (ii) When a parameter in the vector field varies, there are two or three forward pitchfork bifurcations; (iii) The non-Gaussian Lévy noise clearly leads to fundamentally more complex bifurcation scenarios, since in the special case of Gaussian noise, there is only one pitchfork bifurcation which is reminiscent of the deterministic situation.

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Entropy and Lyapunov Exponents Relationships in Stochastic Dynamical Systems

Problems Involving Thermal Hydraulics, Liquid Sloshing, and Extreme Loads on Structures ◽

10.1115/pvp2003-1822 ◽

2003 ◽

Cited By ~ 1

Author(s):

F. Jedrzejewski

Keyword(s):

Dynamical Systems ◽

Lyapunov Exponents ◽

Random Media ◽

Planck Equation ◽

Seismic Wave Propagation ◽

Noise Strength ◽

Stochastic Dynamical Systems ◽

Complementary Approach ◽

Entropy Information ◽

Stability Of Dynamical Systems

Stochastic differential equations and classical techniques related to the Fokker-Planck equation are standard bases for the analysis of nonlinear systems perturbed by noise, such as seismic wave propagation in random media and response of structures to turbulent wind. In this paper, a complementary approach based on entropy production is proposed to analyse the stochastic stability of dynamical systems. For a large class of stochastic dynamical systems, it is shown that the entropy information production is equal to the negative sum of Lyapunov exponents as the noise strength tends to zero. This result is correlated to the topological entropy property, which is in some cases such as the hyperbolic case, equal the sum of Lyapunov exponents. Several examples are given to illustrate the proposed procedure.

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The Fokker-Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solutions

10.1142/2347 ◽

1994 ◽

Cited By ~ 50

Author(s):

C Soize

Keyword(s):

Steady State ◽

Dynamical Systems ◽

Planck Equation ◽

Stochastic Dynamical Systems ◽

Fokker Planck Equation ◽

Steady State Solutions ◽

Fokker Planck

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Numerical Solution of Fokker-Planck Equation for Nonlinear Stochastic Dynamical Systems

IUTAM Symposium on Nonlinear Stochastic Dynamics and Control - IUTAM Bookseries ◽

10.1007/978-94-007-0732-0_8 ◽

2011 ◽

pp. 77-86

Author(s):

S. Narayanan ◽

Pankaj Kumar

Keyword(s):

Dynamical Systems ◽

Numerical Solution ◽

Planck Equation ◽

Stochastic Dynamical Systems ◽

Fokker Planck Equation ◽

Fokker Planck

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Universal Feedback Controllers and Inverse Optimality for Nonlinear Stochastic Systems

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.4045153 ◽

2019 ◽

Vol 142 (2) ◽

Author(s):

Wassim M. Haddad ◽

Xu Jin

Keyword(s):

Dynamical Systems ◽

Feedback Control ◽

Stochastic Control ◽

Stochastic Systems ◽

Sufficient Conditions ◽

Optimal Feedback Control ◽

Control Law ◽

Stochastic Dynamical Systems ◽

Control Lyapunov Function ◽

Inverse Optimality

Abstract In this paper, we develop a constructive finite time stabilizing feedback control law for stochastic dynamical systems driven by Wiener processes based on the existence of a stochastic control Lyapunov function. In addition, we present necessary and sufficient conditions for continuity of such controllers. Moreover, using stochastic control Lyapunov functions, we construct a universal inverse optimal feedback control law for nonlinear stochastic dynamical systems that possess guaranteed gain and sector margins. An illustrative numerical example involving the control of thermoacoustic instabilities in combustion processes is presented to demonstrate the efficacy of the proposed framework.

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