Synchronization and random attractors for reaction jump processes

TL;DR

This paper investigates how common noise induces a time-shifted synchronization in reaction jump processes, analyzing the conditions for synchronization and characterizing the structure of their random attractors.

Contribution

It introduces a framework for analyzing synchronization in reaction jump processes and characterizes the structure of their random attractors, including for birth-death processes.

Findings

Partial time-shifted synchronization occurs after a waiting time

Synchronization depends on initial states

Structure of the random attractor is characterized

Abstract

This work explores a synchronization-like phenomenon induced by common noise for continuous-time Markov jump processes given by chemical reaction networks. A corresponding random dynamical system is formulated in a two-step procedure, at first for the states of the embedded discrete-time Markov chain and then for the augmented Markov chain including also random jump times. We uncover a time-shifted synchronization in the sense that -- after some initial waiting time -- one trajectory exactly replicates another one with a certain time delay. Whether or not such a synchronization behaviour occurs depends on the combination of the initial states. We prove this partial time-shifted synchronization for the special setting of a birth-death process by analyzing the corresponding two-point motion of the embedded Markov chain and determine the structure of the associated random attractor. In this context, we also provide general results on existence and form of random attractors for discrete-time, discrete-space random dynamical systems.