Quasi-potential Calculation and Minimum Action Method for Limit Cycle

TL;DR

This paper introduces a novel approach combining quasi-potential approximation and the minimum action method to efficiently compute noise-induced escape paths from stable limit cycles in dynamical systems.

Contribution

It develops a quadratic approximation of the quasi-potential near the limit cycle and integrates it with the minimum action method to improve numerical computation of escape paths.

Findings

Effective computation of escape paths from limit cycles.

Quadratic approximation via Riccati differential equation.

Improved numerical stability and accuracy.

Abstract

We study the noise-induced escape from a stable limit cycle of a non-gradient dynamical system driven by a small additive noise. The fact that the optimal transition path in this case is infinitely long imposes a severe numerical challenge to resolve it in the minimum action method. We first consider the landscape of the quasi-potential near the limit cycle, which characterizes the minimal cost of the noise to drive the system far away form the limit cycle. We derive and compute the quadratic approximation of this quasi-potential near the limit cy- cle in the form of a positive definite solution to a matrix-valued periodic Riccati differential equation on the limit cycle. We then combine this local approxima- tion in the neighbourhood of the limit cycle with the minimum action method applied outside of the neighbourhood. The neighbourhood size is selected to be compatible with the path discretization error. By several numerical examples, we show that this strategy effectively improve the minimum action method to compute the spiral optimal escape path from limit cycles in various systems.