Mean First Passage Time in Periodic Attractors

TL;DR

This paper analyzes the mean first passage time between periodic attractors in a stochastic system, deriving an analytical expression and confirming it with simulations, highlighting the influence of attractor size, noise, and potential differences.

Contribution

It introduces an analytical formula for mean first passage time in systems with multiple periodic attractors, linking it to attractor size, noise, and potential differences, validated by simulations.

Findings

Derived an explicit formula for transition times between attractors.

Confirmed analytical results with numerical simulations.

Identified key factors influencing transition times in stochastic systems.

Abstract

The properties of the mean first passage time in a system characterized by multiple periodic attractors are studied. Using a transformation from a high dimensional space to 1D, the problem is reduced to a stochastic process along the path from the fixed point attractor to a saddle point located between two neighboring attractors. It is found that the time to switch between attractors depends on the effective size of the attractors, $\tau$, the noise, $\epsilon$, and the potential difference between the attractor and an adjacent saddle point as: $~T = {c \over \tau} \exp({\tau \over \epsilon} \Delta {\cal{U}})~$; the ratio between the sizes of the two attractors affects $\Delta {\cal{U}}$. The result is obtained analytically for small $\tau$ and confirmed by numerical simulations. Possible implications that may arise from the model and results are discussed.