Random Point Attractors Versus Random Set Attractors

Crauel, Hans

doi:10.1017/s0024610700001915

Cited by 100 publications

(95 citation statements)

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“…Unfortunately, it is impossible to make the following contents less abstract. But to provide necessary information to the readers who are keen to get a deeper understanding of the global random attractor, we still include the most typical results whose proofs can be found in Crauel [26,27]. Note skipping the following results will not affect the understanding of the applications to be studied later.…”

Section: Some Properties Of the Random Attractormentioning

confidence: 99%

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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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Section: Some Properties Of the Random Attractormentioning

confidence: 99%

“…In order to characterize a random attractor for a universe B, the following result was established in Crauel [27]. Theorem 4.5.…”

Section: Some Properties Of the Random Attractormentioning

confidence: 99%

Applied Nonautonomous and Random Dynamical Systems

Caraballo¹,

Han²

2016

SpringerBriefs in Mathematics

114

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“…The resulting random compact set ω∈Ω A(ω) is called a random attractor; it is also called a strong attractor since the convergence of remote initial data to the attractor holds almost surely for the Hausdorff semi-metric of the phase space X [20,33,38,39]. Moreover, when {θ t } is ergodic, then knowing t∈R A(θ t ω) yields ω∈Ω A(ω), and vice-versa.…”

Section: Random Attractormentioning

confidence: 99%

Stochastic climate dynamics: Random attractors and time-dependent invariant measures

Chekroun

Simonnet

Ghil

2011

Physica D: Nonlinear Phenomena

241

274

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This article attempts a unification of the two approaches that have dominated theoretical climate dynamics since its inception in the 1960s: the nonlinear deterministic and the linear stochastic one. This unification, via the theory of random dynamical systems (RDS), allows one to consider the detailed geometric structure of the random attractors associated with nonlinear, stochastically perturbed systems. A high-resolution numerical study of two highly idealized models of fundamental interest for climate dynamics allows one to obtain a good approximation of their global random attractors, as well as of the time-dependent invariant measures supported by these attractors; the latter are shown to be random Sinai-Ruelle-Bowen (SRB) measures. The first of the two models is a stochastically forced version of the classical Lorenz model. The second one is a low-dimensional, nonlinear stochastic model of the El Niño-Southern Oscillation (ENSO). Keywords: Climate Dynamics, Dynamical Systems, El Niño, Random Dynamical Systems, Stochastic ForcingThe geometric [1] and ergodic [2] theory of dynamical systems represents a significant achievement of the last century. In the meantime, the foundations of the stochastic calculus also led to the birth of a rigorous theory of time-dependent random phenomena. Historically, theoretical developments in climate dynamics have been largely motivated by these two complementary approaches, based on the work of E. N. Lorenz [3] and that of K. Hasselmann [4], respectively.It now seems clear that these two approaches complement, rather than exclude each other. Incomplete knowledge of small-, subgrid-scale processes, as well as computational limitations will always require one to account for these processes in a stochastic way. As a result of sensitive dependence on initial data and on parameters, numerical weather forecasts [5] as well as climate projections [6] are both expressed these days in probabilistic terms. In addition to the intrinsic challenge of addressing the nonlinearity along with the stochasticity of climatic processes, it is thus more convenient -and becoming more and more necessaryto rely on a model's (or set of models') probability density function (PDF) rather than on its individual, pointwise simulations or predictions.We show in this paper that finer, highly relevant and still computable statistics exist for stochastic nonlinear systems, which provide meaningful physical information not described by the PDF alone. These statistics are supported by a random attractor that extends the concept of a strange attractor [3,7] and of its invariant measures [2] from deterministic to stochastic dynamics.The attractor of a deterministic dynamical system provides crucial geometric information about its asymptotic regime as t → ∞, while the Sinaï-Ruelle-Bowen (SRB) measure provides, when it exists, the * Corresponding author.

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“…While it is natural to make dynamical assumptions in terms of the global attractor, any invariant measure is in fact supported by the point attractor [5].…”

Section: Now Setmentioning

confidence: 99%

Invariant Measures for Dissipative Systems and Generalised Banach Limits

2011

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Inspired by a theory due to Foias and coworkers (see, for example, Foias et al. Navier-Stokes equations and turbulence, Cambridge University Press, Cambridge, 2001) and recent work of Wang (Disc Cont Dyn Sys 23:521-540, 2009), we show that the generalised Banach limit can be used to construct invariant measures for continuous dynamical systems on metric spaces that have compact attracting sets, taking limits evaluated along individual trajectories. We also show that if the space is a reflexive separable Banach space, or if the dynamical system has a compact absorbing set, then rather than taking limits evaluated along individual trajectories, we can take an ensemble of initial conditions: the generalised Banach limit can be used to construct an invariant measure based on an arbitrary initial probability measure, and any invariant measure can be obtained in this way. We thus propose an alternative to the classical Krylov-Bogoliubov construction, which we show is also applicable in this situation.

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Random Point Attractors Versus Random Set Attractors

Cited by 100 publications

References 0 publications

Applied Nonautonomous and Random Dynamical Systems

Applied Nonautonomous and Random Dynamical Systems

Stochastic climate dynamics: Random attractors and time-dependent invariant measures

Invariant Measures for Dissipative Systems and Generalised Banach Limits

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