Invariant Gragh and Random Bony Attractors

TL;DR

This paper investigates the geometric structure and stability of random attractors in skew product dynamical systems with deterministic noise, revealing conditions for invariant graphs and bony attractors with non-uniform contraction.

Contribution

It identifies an open set of skew products with negative fiber Lyapunov exponents that admit invariant attractors as either continuous or bony graphs, advancing understanding of their stability and structure.

Findings

Existence of an open set of skew products with specific attractor types.

Attractors have negative fiber Lyapunov exponents and non-uniform contraction.

Invariant measures supported on these attractors vary continuously under perturbations.

Abstract

In this paper, we deal with random attractors for dynamical systems forced by a deterministic noise. These kind of systems are modeled as skew products where the dynamics of the forcing process are described by the base transformation. Here, we consider skew products over the Bernoulli shift with the unit interval fiber. We study the geometric structure of maximal attractors, the orbit stability and stability of mixing of these skew products under random perturbations of the fiber maps. We show that there exists an open set $\mathcal{U}$ in the space of such skew products so that any skew product belonging to this set admits an attractor which is either a continuous invariant graph or a bony graph attractor. These skew products have negative fiber Lyapunov exponents and their fiber maps are non-uniformly contracting, hence the non-uniform contraction rates are measured by Lyapnnov exponents. Furthermore, each skew product of $\mathcal{U}$ admits an invariant ergodic measure whose support is contained in that attractor. Additionally, we show that the invariant measure for the perturbed system is continuous in the Hutchinson metric.