Synchronization of weakly perturbed Markov chain oscillators

TL;DR

This paper develops a theoretical framework using second order perturbation theory to analyze the synchronization and response of weakly perturbed Markov chain oscillators, applicable to complex stochastic systems.

Contribution

It introduces a method to derive mean frequency and phase diffusion expressions for discrete state oscillators with time-dependent transition rates, and proposes a global control strategy for optimizing their response.

Findings

Derived formulas for mean frequency and phase diffusion constant.

Presented a global control method for complex transition networks.

Applicable to systems with strong stochasticity and complex topologies.

Abstract

Rate processes are simple and analytically tractable models for many dynamical systems which switch stochastically between a discrete set of quasi stationary states but they may also approximate continuous processes by coarse grained, symbolic dynamics. In contrast to limit cycle oscillators which are weakly perturbed by noise, the stochasticity in such systems may be strong and more complicated system topologies than the circle can be considered. Here we employ second order, time dependent perturbation theory to derive expressions for the mean frequency and phase diffusion constant of discrete state oscillators coupled or driven through weakly time dependent transition rates. We also describe a method of global control to optimize the response of the mean frequency in complex transition networks.