Normal forms for random diffeomorphisms

Arnold, Ludwig; Xu, Ke

doi:10.1007/bf01053806

Cited by 13 publications

(5 citation statements)

References 17 publications

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“…We follow Arnold and Xu [40]. Let cp be a coo RDS in JRd with time 1I' = Z, In contrast to the deterministic case, the assumption of a fixed point at 0 is without much loss of generality for an RDS with an invariant measure, see Lemma 7.2.1.…”

Section: The Random Cohomological Equationmentioning

confidence: 99%

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Random Dynamical Systems

Arnold¹

1998

Springer Monographs in Mathematics

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2,438

3,522

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Section: The Random Cohomological Equationmentioning

confidence: 99%

“…2.9 and 19.2]).Arnold and Xu[40] have worked out a Hilbert space set-up for equations(8.2.4) and(8.2.5). 0 E S(M(A)n)· In our adapted random basis ( M(A)~, M(A)~ and M(A)~ are the unstable, stable and neutral block, respectively.…”

mentioning

confidence: 99%

Random Dynamical Systems

Arnold¹

1998

Springer Monographs in Mathematics

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2,438

3,522

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“…The trajectory-based approach is often adopted under the framework of random dynamical systems, i.e., skew-product flows with ergodic measurepreserving base flows. By assuming vanishing noise at a reference equilibrium, noise perturbations of essential dynamics of a dynamical system are studied under the random dynamical system framework with respect to problems such as noise perturbations of invariant manifolds ( [5,16,17,44]), normal forms ( [3,4,6,35]), and stochastic bifurcations (see [3] and references therein). For a system of ordinary differential equations subject to white noise perturbations vanishing at a reference equilibrium, we refer the reader to [23] for some study of stochastic stability of the equilibrium (see also [37] for similar studies in infinite dimension).…”

Section: Introductionmentioning

confidence: 99%

Concentration and limit behaviors of stationary measures

Huang

Liu

et al. 2018

Physica D: Nonlinear Phenomena

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Abstract. In this paper, we study limit behaviors of stationary measures of the FokkerPlanck equations associated with a system of ordinary differential equations perturbed by a class of multiplicative including additive white noises. As the noises are vanishing, various results on the invariance and concentration of the limit measures are obtained. In particular, we show that if the noise perturbed systems admit a uniform Lyapunov function, then the stationary measures form a relatively sequentially compact set whose weak * -limits are invariant measures of the unperturbed system concentrated on its global attractor. In the case that the global attractor contains a strong local attractor, we further show that there exists a family of admissible multiplicative noises with respect to which all limit measures are actually concentrated on the local attractor; and on the contrary, in the presence of a strong local repeller in the global attractor, there exists a family of admissible multiplicative noises with respect to which no limit measure can be concentrated on the local repeller. Moreover, we show that if there is a strongly repelling equilibrium in the global attractor, then limit measures with respect to typical families of multiplicative noises are always concentrated away from the equilibrium. As applications of these results, an example of stochastic Hopf bifurcation is provided. Our study is closely related to the problem of noise stability of compact invariant sets and invariant measures of the unperturbed system.

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“…Normal form theory without any smallness assumption was developed for random dieomorphisms by Arnold and Xu in [8], and for random dierential equations in [9,10].…”

Section: Introductionmentioning

confidence: 99%

Normal forms for stochastic differential equations

Arnold¹,

Imkeller²

1998

Probab Theory Relat Fields

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We address the following problem from the intersection of dynamical systems and stochastic analysis: Two SDE dx t m j0 f j x t d j t and dx t m j0 g j x t d j t in R d with smooth coecients satisfying f j 0 g j 0 0 are said to be smoothly equivalent if there is a smooth random dieomorphism (coordinate transformation) hx with hxY 0 0 and hhxY 0 id which conjugates the corresponding local¯ows,where h t xs xt s À xt is the (ergodic) shift on the canonical Wiener space. The normal form problem for SDE consists in ®nding the``simplest possible'' member in the equivalence class of a given SDE, in particular in giving conditions under which it can be linearized (g j x hf j 0x).We develop a mathematically rigorous normal form theory for SDE which justi®es the engineering and physics literature on that problem. It is based on the multiplicative ergodic theorem and uses a uniform (with respect to a spatial parameter) Stratonovich calculus which allows the handling of non-adapted initial values and coecients in the stochastic version of the cohomological equation. Our Probab. Theory Relat. Fields 110, 559±588 (1998) Key words and phrases: Stochastic dierential equation ± Stochastic normal form ± Stochastic cohomological equation ± Anticipative calculus ± Random dynamical system ± Multiplicative ergodic theory ± Ane stochastic dierential equation ± Noisy Dung-van der Pol oscillator. main result (Theorem 3.2) is that an SDE is (formally) equivalent to its linearization if the latter is nonresonant.As a by-product, we prove a general theorem on the existence of a stationary solution of an anticipative ane SDE.The study of the Dung-van der Pol oscillator with small noise concludes the paper.

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Normal forms for random diffeomorphisms

Cited by 13 publications

References 17 publications

Random Dynamical Systems

Random Dynamical Systems

Concentration and limit behaviors of stationary measures

Normal forms for stochastic differential equations

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