Nonautonomous and Random Attractors

Crauel, Hans; Kloeden, Peter E.

doi:10.1365/s13291-015-0115-0

Cited by 59 publications

(61 citation statements)

References 52 publications

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“…By using Proposition 5.5 again, we can then deduce that with positive probability, there are arbitrarily small finite-time Lyapunov exponents for any initial conditions. Note that we require that random attractor is measurable with respect to F ⊗B(R d ), in contrast to a weaker statement normally used in the literature (see also [11,Remark 4]).…”

Section: Bifurcation For Small Shearmentioning

confidence: 99%

Hopf bifurcation with additive noise

et al. 2018

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We consider the dynamics of a two-dimensional ordinary differential equation exhibiting a Hopf bifurcation subject to additive white noise and identify three dynamical phases: (I) a random attractor with uniform synchronisation of trajectories, (II) a random attractor with nonuniform synchronisation of trajectories and (III) a random attractor without synchronisation of trajectories. The random attractors in phases (I) and (II) are random equilibrium points with negative Lyapunov exponents while in phase (III) there is a so-called random strange attractor with positive Lyapunov exponent.We analyse the occurrence of the different dynamical phases as a function of the linear stability of the origin (deterministic Hopf bifurcation parameter) and shear (ampitude-phase coupling parameter). We show that small shear implies synchronisation and obtain that synchronisation cannot be uniform in the absence of linear stability at the origin or in the presence of sufficiently strong shear. We provide numerical results in support of a conjecture that irrespective of the linear stability of the origin, there is a critical strength of the shear at which the system dynamics loses synchronisation and enters phase (III).

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Section: Bifurcation For Small Shearmentioning

confidence: 99%

Hopf bifurcation with additive noise

et al. 2018

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“…Only the basic contents needed for the applications later this chapter are presented here. For more details the reader is referred to the recent survey by Crauel and Kloeden [31].…”

Section: Formulation Of Random Dynamical System and Random Attractormentioning

confidence: 99%

Applied Nonautonomous and Random Dynamical Systems

Caraballo¹,

Han²

2016

SpringerBriefs in Mathematics

114

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PrefaceDuring the past two decades, the theory of nonautonomous dynamical systems and random dynamical systems has made substantial progress in studying the long term dynamics of open systems subject to time-dependent or random forcing. However, most of the existing pertinent literatures are fairly technical and almost impenetrable for a general audience, except the most dedicated specialists. Moreover, the concepts and methods of nonautonomous and random dynamical systems, though well-established, are extremely nontrivial to apply to real-world problems. The aim of this work is to provide an accessible and broad introduction to the theory of nonautonomous and random dynamical systems, with an emphasis on applications of the theory to problems arising in the applied sciences and engineering.The book starts with basic concepts in the theory of autonomous dynamical systems which are easier to understand, and used as the motivation for the study of non-autonomous and random dynamical systems. Then the framework of nonautonomous dynamical systems is set up, including various approaches to analyze the long time behavior of non-autonomous problems. The major emphasis is given to the novel theory of pullback attractors, as it can be regarded as a natural extension of the autonomous theory and allows a larger variety of time-dependence forcing than other alternatives such as skew-product flows or cocycles. In the end the theory of random dynamical systems and random attractors is introduced and shown to fairly informative to the study of long term behavior of stochastic systems with random forcing.Each set of theory is illustrated by being applied to three different models, the chemostat model, the SIR epidemic model, and the Lorenz-84 model, in their autonomous, nonautonomous, and stochastic formulations, respectively. The techniques and methods adopted can be applied to the study of the long term behavior of a wide range of applications arising in applied sciences and engineering.

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“…Fix t ą 0. Since K B attracts B in probability, we have lim sÑ8 d`ϕpt`s, ϑ´sωqBpϑ´sωq, K B pϑ t ωq˘" 0 in probability (9) and lim sÑ8 d`ϕps, ϑ´sωqBpϑ´sωq, K B pωq˘" 0 in probability.…”

Section: Existence Of a Minimal Pullback Attractor For Bmentioning

confidence: 99%

“…For any set C Ă EˆΩ, we define C :" tpx, ωq : x P Cpωqu Remark 4. Note that it does not suffice to define random sets by demanding ω Þ Ñ d`x, Cpωq˘to be measurable for every x P E. In this case the associated tpx, ωq : x P Cpωqu need not be an element of E b F , see [9,Remark 4].…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Minimal random attractors

Crauel

Scheutzow

2018

Journal of Differential Equations

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It is well-known that random attractors of a random dynamical system are generally not unique. We show that for general pullback attractors and weak attractors, there is always a minimal (in the sense of smallest) random attractor which attracts a given family of (possibly random) sets. We provide an example which shows that this property need not hold for forward attractors. We point out that our concept of a random attractor is very general: The family of sets which are attracted is allowed to be completely arbitrary.

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Nonautonomous and Random Attractors

Cited by 59 publications

References 52 publications

Hopf bifurcation with additive noise

Hopf bifurcation with additive noise

Applied Nonautonomous and Random Dynamical Systems

Minimal random attractors

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