Random Dynamical Systems

Arnold, Ludwig

doi:10.1007/978-3-662-12878-7

Cited by 2,434 publications

(3,671 citation statements)

References 166 publications

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“…Similarly to the autonomous case [Arn98], the noise of a nonautonomous random dynamical system is modelled by a base flow θ. Let (Ω, F, P) be a probability space with a σ-algebra F and a probability measure P, and let T be a time set (given by either R or Z).…”

Section: Nonautonomous Random Dynamical Systemsmentioning

confidence: 99%

A random dynamical systems perspective on stochastic resonance

et al. 2017

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Abstract. We study stochastic resonance in an over-damped approximation of the stochastic Duffing oscillator from a random dynamical systems point of view. We analyse this problem in the general framework of random dynamical systems with a nonautonomous forcing. We prove the existence of a unique global attracting random periodic orbit and a stationary periodic measure. We use the stationary periodic measure to define an indicator for the stochastic resonance.

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Section: Nonautonomous Random Dynamical Systemsmentioning

confidence: 99%

A random dynamical systems perspective on stochastic resonance

et al. 2017

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“…In this section we first present some concepts (from [1]) related to general random dynamical systems (RDSs) and random attractors that we require in the sequel. Our situation is, in fact, somewhat simpler, but to facilitate the reader's access to the literature we give more general definitions here.…”

Section: Preliminaries On Random Dynamical Systemsmentioning

confidence: 99%

“…This set-up establishes a time-dependent family θ that tracks the noise, and (Ω , F , P, θ ) is called a metric dynamical system [1].…”

Section: Preliminaries On Random Dynamical Systemsmentioning

confidence: 99%

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A Random Model for Immune Response to Virus in Fluctuating Environments

Asai

Caraballo

Han

et al. 2016

Advances in Dynamical Systems and Control

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In this work we study a model for virus dynamics with a random immune response and a random production rate of susceptible cells from cell proliferation. In traditional models for virus dynamics, the rate at which the viruses are cleared by the immune system is constant, and the rate at which susceptible cells are provided is constant or a function depending on the population of all cells. However, the human body in general is never stationary, and thus these rates can barely be constant. Here we assume that the human body is a random environment and models the rates by random processes, which result in a system of random differential equations. We then analyze the long term behavior of the random system, in particular the existence and geometric structure of the random attractor, by using the theory of random dynamical systems. Numerical simulations are provided to illustrate the theoretical result.

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“…Therefore, there are many works dealing with this topic and there exist a large amount of stability criteria for deterministic and stochastic systems. Among these criteria, the characteristic Lyapunov exponent is a powerful tool because it is important for explaining the chaos of the systems under consideration (see [1,2,10], etc.). We remark that studying the Lyapunov exponent of a function means comparing its growth rate with the growth rate of the exponential one.…”

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confidence: 99%

On an extension of the Lyapunov criterion of stability for quasi-linear systems via integral inequalities methods

Du¹

2005

Theor. Probability and Math. Statist.

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Abstract. In this article, we concern ourselves with a new concept for comparing the stability degree of two dynamical systems. By using the integral inequality method, we give a criterion which allows us to compare the growth rate of two Itô quasi-linear differential equations. It can be viewed as an extension of the Lyapunov criterion to the stochastic case. IntroductionIt is known that studying the asymptotic behavior of dynamical systems is important both in theory and in application. Therefore, there are many works dealing with this topic and there exist a large amount of stability criteria for deterministic and stochastic systems. Among these criteria, the characteristic Lyapunov exponent is a powerful tool because it is important for explaining the chaos of the systems under consideration (see [1,2,10], etc.). We remark that studying the Lyapunov exponent of a function means comparing its growth rate with the growth rate of the exponential one. However, the class of exponential functions is rather simple and it does not contain much information on the behavior of the function considered. By requirement of technical problems, we have sometimes to replace this class by a larger one, say C, and compare this function with elements of C in order to know its behavior, especially at t = ∞. To realize this we introduce a concept of comparing the behavior of trajectories of solutions of two differential equations as follows: a system is said to be better than a given one in the class C in view of stability (we will say that this system is more stable than the given one) if, whenever all trajectories of the given system starting from a small neighborhood of the origin 0 belong to the corridor {(t, x): t ≥ 0, |x| ≤ q t } generated by a positive function q t ∈ C, then all trajectories of the second system starting from a suitable neighborhood of 0 must have the same property.Naturally, this raises a question: when is one system more stable than another? Of course, it is a difficult problem even in the deterministic case. In this article, we deal with a criterion for comparing two quasi-linear systems. This problem is encountered when we investigate a nonlinear stochastic system via its first linear approximation.The article is organized as follows. Section 2 gives a concept of comparing two differential stochastic equations. In Section 3, we are concerned with the regularity of a 2000 Mathematics Subject Classification. Primary 60H10; Secondary 34F05, 93E15.

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Random Dynamical Systems

Cited by 2,434 publications

References 166 publications

A random dynamical systems perspective on stochastic resonance

A random dynamical systems perspective on stochastic resonance

A Random Model for Immune Response to Virus in Fluctuating Environments

On an extension of the Lyapunov criterion of stability for quasi-linear systems via integral inequalities methods

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