Local attractor continuation of non-autonomously perturbed systems

TL;DR

This paper demonstrates that local attractors in dynamical systems persist under small non-autonomous perturbations, maintaining their attracting properties and converging to the original attractor, with implications for systems with random noise.

Contribution

It extends Conley theory to show the persistence and upper semicontinuous convergence of local attractors under small non-autonomous perturbations, including random noise.

Findings

Local attractors remain attractors under small perturbations.

Perturbed attractors have positive invariant neighborhoods.

Convergence of perturbed attractors to original attractor is upper semicontinuous.

Abstract

Using Conley theory we show that local attractors remain (past) attractors under small non-autonomous perturbations. In particular, the attractors of the perturbed systems will have positive invariant neighborhoods and converge upper semicontinuously to the original attractor. The result is split into a finite-dimensional part (locally compact) and an infinite-dimensional part (not necessarily locally compact). The finite-dimensional part will be applicable to bounded random noise, i.e. continuous time random dynamical systems on a locally compact metric space which are uniformly close the unperturbed deterministic system. The "closeness" will be defined via a (simpler version of) convergence coming from singular perturbations theory.