Averaging for random metastable systems

TL;DR

This paper studies how random perturbations affect metastable systems with multiple invariant states, showing that the invariant density can be approximated by averaging the original densities and analyzing the limiting structures.

Contribution

It introduces an explicit averaging method for invariant densities in randomly perturbed systems with multiple invariant sets and characterizes the limiting coherent structures as perturbations vanish.

Findings

Invariant density approximated by convex combination of original densities

Identification of the limit of the second Oseledets space as perturbation approaches zero

Application to random paired tent maps and generalizations to more invariant sets

Abstract

Random metastability occurs when an externally forced or noisy system possesses more than one state of apparent equilibrium. This work investigates a class of random dynamical systems, arising from perturbing a one-dimensional piecewise smooth expanding map of the interval with two invariant subintervals, each supporting a unique ergodic absolutely continuous invariant measure. Upon perturbation, this invariance is destroyed, allowing trajectories to randomly switch between subintervals. We show that the invariant density of the randomly perturbed system may be approximated by an explicit convex combination of the two initially invariant densities, obtained by averaging. Further, we also identify the limit of the second Oseledets space, or coherent structure, as the perturbation shrinks to zero. Our results are applied to random paired tent maps over ergodic, measure-preserving, and invertible driving systems. Finally, we provide generalisations to systems admitting more than two initially invariant sets.