Localized Stable Manifolds for Whiskered Tori in Coupled Map Lattices with Decaying Interaction

TL;DR

This paper establishes the existence of localized invariant manifolds for whiskered tori in coupled lattice systems with decaying interactions, extending previous results to non-symplectic systems and generalizing stable manifold concepts.

Contribution

It introduces invariant manifold theorems for whiskered tori in coupled map lattices with decay interactions, including non-resonant and localized manifolds, without requiring symplectic structure.

Findings

Invariant manifolds are localized near specific sites.

Results apply to both maps and flows, with proofs for maps implying flow results.

Theorems extend the concept of stable manifolds to non-symplectic, coupled lattice systems.

Abstract

In this paper we consider lattice systems coupled by local interactions. We prove invariant manifold theorems for whiskered tori (we recall that whiskered tori are quasi-periodic solutions with exponentially contracting and expanding directions in the linearized system). The invariant manifolds we construct generalize the usual (strong) (un) stable manifolds and allow us to consider also non-resonant manifolds. We show that if the whiskered tori are localized near a collection of specific sites, then so are the invariant manifolds. We recall that the existence of localized whiskered tori has recently been proven for symplectic maps and flows in Fontich et. al. (submitted), but our results do not need that the systems are symplectic. For simplicity we will present first the main results for maps, but we will show tha the result for maps imply the results for flows. It is also true that the results for flows can be proved directly following the same ideas.