Smoothness of invariant manifolds and foliations for infinite dimensional random dynamical systems
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. ARTICLES .
September 2020 Vol. 63 No. 9: 1877–1912 https://doi.org/10.1007/s11425-019-1664-3
Smoothness of invariant manifolds and foliations for infinite dimensional random dynamical systems Dedicated to Professor Shantao Liao
Jun Shen1 , Kening Lu2,∗ & Weinian Zhang1 1Department 2Department
of Mathematics, Sichuan University, Chengdu 610064, China; of Mathematics, Brigham Young University, Provo, UT 84602, USA
Email: [email protected], [email protected], [email protected] Received December 4, 2019; accepted March 3, 2020; published online August 14, 2020
Abstract
In this paper, we investigate the smoothness of invariant manifolds and foliations for random dynam-
ical systems with nonuniform pseudo-hyperbolicity in Hilbert spaces. We discuss on the effect of temperedness and the spectral gaps in the nonuniform pseudo-hyperbolicity so as to prove the existence of invariant manifolds and invariant foliations, which preserve the C N,τ (ω) H¨ older smoothness of the random system in the space variable and the measurability of the random system in the sample point. Moreover, we also prove that the stable foliation is C N −1,τ (ω) in the base point. Keywords
random dynamical system, invariant manifold, invariant foliation, pseudo-hyperbolicity, measura-
bility MSC(2010)
60H15, 37D10, 57R30
Citation: Shen J, Lu K N, Zhang W N. Smoothness of invariant manifolds and foliations for infinite dimensional random dynamical systems. Sci China Math, 2020, 63: 1877–1912, https://doi.org/10.1007/s11425-0191664-3
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Introduction
Invariant manifolds and invariant foliations are important tools in reduction of high dimensional systems, bifurcation of homoclinic or heteroclinic loops and linearization of dynamical systems (see [11,22,33,44]). Extensive literatures (see, e.g., Kelley [28], Hale [25], Henry [26], Carr [9], Chow and Lu [13, 14], Bates and Jones [3], Chow et al. [12], Chow et al. [11], and Bates et al. [4–6]) are devoted to the theory of invariant manifolds including stable, unstable, center, center-stable, center-unstable manifolds and invariant foliations for deterministic dynamical systems of finite or infinite dimension. Among those works there are two common methods: one is Hadamard’s method (see [24]), which is based on Hadamard’s graph transforms, and the other is Lyapunov-Perron’s method (see [35, 39]), employing the variation of the constant formula and the exponential dichotomy. Those results all require the spectral gap condition, where the spectral gap is decided by distribution of Lyapunov exponents (see [38]) and its size guarantees the smoothness of invariant manifolds and invariant foliations (see [13]). Using these methods, Foias et al. [20], Mallet-Paret and Sell [36], Chow et al. [15], Fenichel [19] and Jones [27] studied inertial manifolds for dissipative systems, slow-fast manifolds for singular perturbations, and related topics. * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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Shen J
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