Synchronization in discrete-time, discrete-state Random Dynamical Systems

TL;DR

This paper characterizes the conditions under which discrete-time, discrete-state random dynamical systems synchronize, using ergodic theory and spectral analysis, with applications to biological networks.

Contribution

It provides a novel characterization of synchronization in discrete-state systems via Lyapunov exponents and spectral subspaces, extending understanding to finite and countable state spaces.

Findings

Synchronization occurs if and only if the Lyapunov exponent zero has simple multiplicity.

Partial synchronization is explained through partitioning the state set.

Applications to biological networks demonstrate practical relevance.

Abstract

We characterize synchronization phenomenon in discrete-time, discrete-state random dynamical systems, with random and probabilistic Boolean networks as particular examples. In terms of multiplicative ergodic properties of the induced linear cocycle, we show such a random dynamical system with finite state synchronizes if and only if the Lyapunov exponent $0$ has simple multiplicity. For the case of countable state space, characterization of synchronization is provided in terms of the spectral subspace corresponding to the Lyapunov exponent $-\infty$. In addition, for both cases of finite and countable state spaces, the mechanism of partial synchronization is described by partitioning the state set into synchronized subsets. Applications to biological networks are also discussed.