Hyperbolic periodic orbits in nongradient systems and small-noise-induced metastable transitions

TL;DR

This paper investigates small-noise-induced transitions in nongradient systems, revealing that such transitions can involve hyperbolic periodic orbits instead of saddle points, and proposes a numerical method to identify these orbits.

Contribution

It introduces a numerical approach based on the String method to identify hyperbolic periodic orbits involved in metastable transitions in nongradient systems.

Findings

MLPs can cross hyperbolic periodic orbits instead of saddle points in nongradient systems.

In orthogonal-type systems, the crossing location determines the transition rate maximum.

Counter-example shows non-uniqueness of crossing points in general cases.

Abstract

Small noise can induce rare transitions between metastable states, which can be characterized by Maximum Likelihood Paths (MLPs). Nongradient systems contrast gradient systems in that MLP does not have to cross the separatrix at a saddle point, but instead possibly at a point on a hyperbolic periodic orbit. A numerical approach for identifying such unstable periodic orbits is proposed based on String method. In a special class of nongradient systems (`orthogonal-type'), there are provably local MLPs that cross such saddle point or hyperbolic periodic orbit, and the separatrix crossing location determines the associated local maximum of transition rate. In general cases, however, the separatrix crossing may not determine a unique local maximum of the rate, as we numerically observed a counter-example in a sheared 2D-space Allen-Cahn SPDE. It is a reasonable conjecture that there are always local MLPs associated with each attractor on the separatrix, such as saddle point or hyperbolic periodic orbit; our numerical experiments did not disprove so.