Random perturbations of hyperbolic dynamics

TL;DR

This paper studies how small random perturbations affect the stability and dynamics of hyperbolic systems, showing that the perturbed dynamics tends to stable fixed points despite large perturbations.

Contribution

It demonstrates that under certain conditions, random perturbations do not prevent the system from approaching stable fixed points, even with significant perturbation strength.

Findings

Perturbed dynamics converges to stable fixed points.

Stability persists despite large perturbations.

Results apply to high-dimensional hyperbolic systems.

Abstract

A sequence of large invertible matrices given by a small random perturbation around a fixed diagonal and positive matrix induces a random dynamics on a high-dimensional sphere. For a certain class of rotationally invariant random perturbations it is shown that the dynamics approaches the stable fixed points of the unperturbed matrix up to errors even if the strength of the perturbation is large compared to the relative increase of nearby diagonal entries of the unperturbed matrix specifying the local hyperbolicity.