Stochastic Resonance for a Model with Two Pathways

TL;DR

This paper investigates stochastic resonance in a model with two metastable states and pathways, analyzing invariant measures, escape times, and developing a new statistical test to detect resonance phenomena.

Contribution

It introduces a new statistical test, the conditional Kolmogorov-Smirnov test, for detecting stochastic resonance in complex systems with multiple pathways.

Findings

Escape time distribution reveals stochastic resonance.

Changing forcing direction introduces a resonance at double the frequency.

The conditional KS test reliably detects resonance even in sparse data.

Abstract

In this thesis we consider stochastic resonance for a diffusion with drift given by a potential, which has two metastable states and two pathways between them. Depending on the direction of the forcing the height of the two barriers, one for each path, will either oscillate alternating or in synchronisation. We consider a simplified model given by discrete and continuous time Markov Chains with two states. This was done for alternating and synchronised wells. The invariant measures are derived for both cases and shown to be constant for the synchronised case. A PDF for the escape time from an oscillatory potential is reviewed. Methods of detecting stochastic resonance are presented, which are linear response, signal-to-noise ratio, energy, out-of-phase measures, relative entropy and entropy. A new statistical test called the conditional Kolmogorov-Smirnov test is developed, which can be used to analyse stochastic resonance. An explicit two dimensional potential is introduced, the critical point structure derived and the dynamics, the invariant state and escape time studied numerically. The six measures are unable to detect the stochastic resonance in the case of synchronised saddles. The distribution of escape times however not only shows a clear sign of stochastic resonance, but changing the direction of the forcing from alternating to synchronised saddles an additional resonance at double the forcing frequency starts to appear. The conditional KS test reliably detects the stochastic resonance even for forcing quick enough and for data so sparse that the stochastic resonance is not obvious directly from the histogram of escape times.