Metastability of Morse-Smale dynamical systems perturbed by heavy-tailed L\'evy type noise

TL;DR

This paper studies how finite-dimensional Morse-Smale dynamical systems with multiple attractors behave under small heavy-tailed Lévy noise, revealing metastable switching between attractors on an ε-dependent timescale.

Contribution

The authors extend previous exit time results to domains with characteristic boundaries and demonstrate metastability and Markov chain convergence under heavy-tailed Lévy perturbations.

Findings

Metastable behavior with ε-dependent switching times

Convergence to a Markov chain between attractors

Application to Duffing and chemical systems

Abstract

We consider a general class of finite dimensional deterministic dynamical systems with finitely many local attractors $K^i$ each of which supports a unique ergodic probability measure $P^i$, which includes in particular the class of Morse-Smale systems in any finite dimension. The dynamical system is perturbed by a multiplicative non-Gaussian heavy-tailed L\'evy type noise of small amplitude $\varepsilon>0$. Specifically we consider perturbations leading to a It\^o, Stratonovich and canonical (Marcus) stochastic differential equation. The respective asymptotic first exit time and location problem from each of the domains of attractions $D^i$ in case of inward pointing vector fields in the limit of $\varepsilon \to 0$ was solved by the authors in [J. Stoch. An. Appl. 32(1), 163-190, 2014]. We extend these results to domains with characteristic boundaries and show that the perturbed system exhibits a metastable behavior in the sense that there exits a unique $\varepsilon$-dependent time scale on which the random system converges to a continuous time Markov chain switching between the invariant measures $P^i$. As examples we consider $\alpha$-stable perturbations of the Duffing equation and a chemical system exhibiting a birhythmic behavior.