Lyapunov exponents, periodic orbits, and horseshoes for semiflows on Hilbert spaces

Abstract

Context and motivation.This work can be seen as a small step in a program to build an ergodic theory for infinite dimensional dynamical systems, a theory the domain of applicability of which will include systems defined by evolutionary PDEs. To reduce the scope, we focus on the ergodic theory of chaotic systems, on nonuniform hyperbolic theory, to be even more specific. In finite dimensions, a basic nonuniform hyperbolic theory already exists (see e.g. [8], [9], [11], [10], [2] and [4]). This body of results taken together provides a fairly good foundation for understanding chaotic phenomena on a qualitative, theoretical level. While an infinite dimensional theory is likely to be richer and more complex, there is no reason to reinvent all material from scratch. It is thus logical to start by determining which parts of finite dimensional hyperbolic theory can be extended to infinite dimensions. Our paper is an early step (though not the first step) in this effort. With an eye toward applications to systems defined by PDEs, emphasis will be given to continuous-time systems or semiflows. Furthermore, it is natural to first consider settings compatible with dissipative parabolic PDEs, for these systems have a finite dimensional flavor (see e.g. [14], [13], and [1]). We mention some previously known results for infinite dimensional systems that form the backdrop to the present work: On the infinitesimal level, i.e. on the level of Lyapunov exponents, generalizations of Oseledets' Multiplicative Ergodic Theorem [8] to operators of Hilbert and Banach spaces have been known for some time ([12], [7], [15] and [5]). Taking nonlinearity into consideration, local results, referring to results that pertain to behavior along one orbit at a time, such as the existence of local stable and unstable manifolds, have also been proved (see e.g.[12] and [5]). This paper is among the first (see also [15]) to discuss a result of a more "global" nature. Summary of results.In this paper, we consider a specific set of results from finite dimensional hyperbolic theory and show that they can be extended (with suitable modifications) to semiflows on Hilbert spaces. The finite dimensional results in question are due to A. Katok [2]. They assert, roughly speaking, the following: Let f be a C 2 diffeomorphism of a compact Riemannian manifold, and let μ be an f -invariant Borel probability measure. Assume that (f, μ) has nonzero Lyapunov exponents and positive metric entropy. Then horseshoes are present, and that implies, among other things, an abundance of hyperbolic periodic points.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 638 ZENG LIAN AND LAI-SANG YOUNG Katok's results were proved for diffeomorphisms of compact manifolds. As an intermediate step, we extended these results to mappings of Hilbert spaces [6]. Here we go one step further, proving analogous results for semiflows on Hilbert spaces satisfying conditions consistent with those in the program outlined above. The nozero-ex...

Lyapunov Exponents, Periodic Orbits and Horseshoes for Mappings of Hilbert Spaces

“…The time-independent case is treated in a forthcoming paper [8], which builds upon the present work and proves results analogous to Theorems A-D for semiflows on Hilbert spaces under the assumption that the system has at most one zero Lyapunov exponent.…”

“…[4,16,17]. With regard to our dynamical assumption of nonzero Lyapunov exponents, this is what causes us to consider systems that are periodically forced: PDEs with time-independent coefficients give rise to semiflows with zero Lyapunov exponents (see our forthcoming paper [8]), but there is no such constraint for time-T maps of periodically forced systems with forcing period T . Finally, periodic orbits of f = f T (the existence of which is asserted in Theorems A-D) correspond to periodic solutions of the original continuous-time system.…”

“…For given ε > 0, there exists Δ > 0 such that if x, y ∈ Γ l0 with |x − y| ≤ Δ, then for any z ∈ H and τ = u, c, s, |L y π τ y z| − |L x π τ x z| ≤ ε|z|. (8) Proof. Consider first the case τ = s, and let |z| = 1.…”

Entropy, Chaos and Weak Horseshoe for Infinite Dimensional Random Dynamical Systems

“…In [41], they extended Katok's results to C 2 differentiable maps with a nonuniformly hyperbolic compact invariant set supported by an invariant measure in a separable Hilbert space. In their second paper [42], Lian and Young went much further and studied a C 2 semiflow in a Hilbert space and proved that if it has a nonuniformly hyperbolic compact invariant set supported by an invariant measure, then the positive entropy implies the existence of horseshoes. In this case, the semiflow may have one simple zero Lyapunov exponent, which implies that the associated time-one map restricted to this invariant set is partially hyperbolic with one-dimensional center direction.…”

Observing Lyapunov Exponents of Infinite-Dimensional Dynamical Systems

“…Results in this direction include the existence of Sinai-Ruelle-Bowen (SRB) measures for periodically-kicked supercritical Hopf bifurcations in a concrete PDE context [16] and the existence of horseshoes in a general context [13, 14 ].…”