A sufficient condition for bifurcation in random dynamical systems

Chen, Xiaopeng; Duan, Jinqiao; Fu, Xinchu

doi:10.1090/s0002-9939-09-10093-x

Cited by 7 publications

(7 citation statements)

References 14 publications

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“…The same assertion holds for A λ(t) , λ(t) > λ * + δ. Thus, the bifurcation in the deterministic system (1) will be reflected in (2) by the change in the topological features of the quasi-static, noisy attractors [5] in the interval λ ∈ (λ * − ∆, λ * + ∆). In what follows, we describe the topological tools that can be used to measure, in a robust way, the changes in the topology of these attractors in the neighborhood of a bifurcation.…”

Section: 2mentioning

confidence: 99%

“…In models with stochasticity, bifurcations are particularly difficult to define, since the change in regime will depend on the particular realization of the underlying random variable. Some current approaches [5] use topological methods to devise sufficient conditions for bifurcations in stochastic dynamical systems, by using all possible realizations of the random variable. From a practical point of view, when the data generated by a system is acquired from a single, or a very limited number of realizations, such an approach may not be suitable.…”

Section: Introductionmentioning

confidence: 99%

“…The stochastic system in(5) with a slowly varying parameter undergoes a bifurcation in which a stable equilibrium changes to unstable and then grows to a periodic orbit. We show t vs. x.…”

mentioning

confidence: 99%

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Automatic Recognition and Tagging of Topologically Different Regimes in Dynamical Systems

Berwald¹,

Gidea²,

Vejdemo-Johansson³

2014

DNC

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Abstract. Complex systems are commonly modeled using nonlinear dynamical systems. These models are often high-dimensional and chaotic. An important goal in studying physical systems through the lens of mathematical models is to determine when the system undergoes changes in qualitative behavior. A detailed description of the dynamics can be difficult or impossible to obtain for high-dimensional and chaotic systems. Therefore, a more sensible goal is to recognize and mark transitions of a system between qualitatively different regimes of behavior. In practice, one is interested in developing techniques for detection of such transitions from sparse observations, possibly contaminated by noise. In this paper we develop a framework to accurately tag different regimes of complex systems based on topological features. In particular, our framework works with a high degree of success in picking out a cyclically orbiting regime from a stationary equilibrium regime in high-dimensional stochastic dynamical systems.

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Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Automatic Recognition and Tagging of Topologically Different Regimes in Dynamical Systems

Berwald¹,

Gidea²,

Vejdemo-Johansson³

2014

DNC

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“…However, for the preceding convenient definition, we refer the reader to [3]. (2) Although it is not obvious but the random conjugaison is in fact an equivalence relation on the set of continuous cocycles driven by the same noise.…”

Section: Remarksmentioning

confidence: 99%

On gradient-like random dynamical systems

Hmissi

2012

ESAIM: Proc.

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Abstract. This paper deals with some characterizations of gradient-like continuous random dynamical systems (RDS). More precisely, we establish an equivalence with the existence of random continuous section or with the existence of continuous and strict Liapunov function. However and contrary to the deterministic case, parallelizable RDS appear as a particular case of gradient-like RDS. The obtained results are generalizations of well-known analogous theorems in the framework of deterministic dynamical systems.MSC (2000): Primary: 37H10. Secondary: 39B52, 37B25, 37B35 Keywords: Random dynamical system (RDS), Translation, Cocycle, Random homeomorphism, Gradientlike RDS , Random section, Liapunov function, Parallelizable RDS, Backward cocycle. Résumé. Cet article est consacréà quelques caractérisations des systèmes dynamiques aléatoires(SDA) du type gradient. Plus précisément, nous montrons que cette notion estéquivalenteà l'existence d'une section aléatoire continue ouà l'existence d'une fonction de Liapunov stricte et continue. Mais, contrairement aux systèmes dynamiques déterministes, les SDA parallélisables apparaissent comme un cas particulier des SDA du type gradient. Les résultats obtenus généralisent des théorèmes bien connus dans le cadre déterministe.

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“…Stochastic bifurcations have been observed in a wide range of nonlinear systems in physical science and engineering. The existing works on stochastic bifurcation mostly are for stochastic dynamical systems with Gaussian noise and focus on the qualitative changes in stationary probability densities [Namachchivaya , 1990] as solutions of steady Fokker-Planck equations, invariant measures (together with their supports and Lyapunov spectra) and random point attractors [Arnold , 2003], or Conley index [Chen et al, 2009].…”

Section: Introductionmentioning

confidence: 99%

A Stochastic Pitchfork Bifurcation in Most Probable Phase Portraits

Wang

Chen

Duan

2018

Int. J. Bifurcation Chaos

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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 in 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 none 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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A sufficient condition for bifurcation in random dynamical systems

Cited by 7 publications

References 14 publications

Automatic Recognition and Tagging of Topologically Different Regimes in Dynamical Systems

Automatic Recognition and Tagging of Topologically Different Regimes in Dynamical Systems

On gradient-like random dynamical systems

A Stochastic Pitchfork Bifurcation in Most Probable Phase Portraits

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