Invariant Manifolds for Random Dynamical Systems on Banach Spaces Exhibiting Generalized Dichotomies
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Invariant Manifolds for Random Dynamical Systems on Banach Spaces Exhibiting Generalized Dichotomies António J. G. Bento1,2
· Helder Vilarinho1,2
Received: 16 July 2018 / Revised: 23 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We prove the existence of measurable invariant manifolds for small perturbations of linear Random Dynamical Systems evolving on a Banach space and admitting a general type of dichotomy, both for continuous and discrete time. Moreover, the asymptotic behavior in the invariant manifold is similar to the one of the linear Random Dynamical System. Keywords Invariant manifolds · Random dynamical systems · Dichotomies Mathematics Subject Classification 37L55 · 37D10 · 37H99
1 Introduction One of the main issues in Dynamical Systems is the study of properties and structures (geometric, topological, ergodic, ...) that are invariant over time, either in the deterministic or in the random evolutionary systems. The study of invariant manifolds for deterministic dynamical systems goes back to the works of Hadamard [10], Lyapunov [15] and Perron [17–19]. For an historical background see for example [4]. In the Random Dynamical Systems (RDS) framework there are several works covering (local and/or global) center, stable, unstable and inertial manifolds for a variety of state spaces, that goes from the Euclidean space to Hilbert spaces or separable Banach spaces, either generated by stochastic or by random differential equations. The list of works on this subject is already too extensive to be completely written down here. We refer for [1,12,13,16,21]. See also [3,8,14,20] and references therein. For invariant manifolds of RDS on infinite dimensional Banach space see [3,4,9,12].
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António J. G. Bento [email protected] Helder Vilarinho [email protected] http://www.mat.ubi.pt/helder
1
Departamento de Matemática, Universidade da Beira Interior, 6201-001 Covilhã, Portugal
2
Centro de Matemática e Aplicações, Universidade da Beira Interior, 6201-001 Covilhã, Portugal
123
Journal of Dynamics and Differential Equations
In this work we prove the existence of random global invariant manifolds for RDS evolving on a Banach space (not necessarily separable), both in the continuous and in the discrete time settings. The RDS considered are obtained by perturbing linear RDS that admit a generalized dichotomy. In the deterministic cases this kind of dichotomies were considered in [5,6] and generalize the common nonuniform exponential condition that often arises from the Multiplicative Ergodic Theorem. To the best of our knowledge this kind of dichotomies were not considered before in the RDS setting. Moreover, the perturbations considered in this work satisfy some natural conditions that guarantee not only the existence of invariant manifolds but also some control on the dynamics. We notice that the separability of the state space is not assumed, which in the continuous time case requires some special attention to measurability and integrability issues. Moreover, for our pur
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