Synchronization of stochastic hybrid oscillators driven by a common switching environment

TL;DR

This paper studies how stochastic hybrid oscillators, influenced by a shared Markov jump process, can synchronize, analyzing the Lyapunov exponent to understand the rate of phase synchronization in the fast switching limit.

Contribution

It introduces a novel analysis of synchronization driven by a common Markov jump environment, highlighting differences from white noise models and providing more accurate Lyapunov exponents.

Findings

Oscillators synchronize under shared switching environment.

Lyapunov coefficient determines phase difference decay rate.

Numerical simulations confirm theoretical predictions.

Abstract

Many systems in biology, physics and chemistry can be modeled through ordinary differential equations, which are piecewise smooth, but switch between different states according to a Markov jump process. In the fast switching limit, the dynamics converges to a deterministic ODE. In this paper we suppose that this limit ODE supports a stable limit cycle. We demonstrate that a set of such oscillators can synchronize when they are uncoupled, but they share the same switching Markov jump process. The latter is taken to represent the effect of a common randomly switching environment. We determine the leading order of the Lyapunov coefficient governing the rate of decay of the phase difference in the fast switching limit. The analysis bears some similarities to the classical analysis of synchronization of stochastic oscillators subject to common white noise. However the discrete nature of the Markov jump process raises some difficulties: in fact we find that the Lyapunov coefficient from the quasi-steady-state approximation differs from the Lyapunov coefficient one obtains from a second order perturbation expansion in the waiting time between jumps. Finally, we demonstrate synchronization numerically in the radial isochron clock model and show that the latter Lyapinov exponent is more accurate.