Desynchronization of Random Dynamical System under Perturbation by an Intrinsic Noise

TL;DR

This paper studies how intrinsic noise affects synchronization in random dynamical systems, revealing that in-sync periods follow a geometric distribution due to combined intrinsic and extrinsic noise effects.

Contribution

It introduces the concept of intrinsic noise as a perturbation in RDS, leading to the analysis of synchronization dynamics and the distribution of in-sync times.

Findings

In-sync times follow a geometric distribution asymptotically.

Intrinsic noise causes stochastic desynchronization between individuals.

Synchronization periods are influenced by both intrinsic and extrinsic noise.

Abstract

In the theory of random dynamical systems (RDS), individuals with different initial states follow a same law of motion that is stochastically changing with time | called extrinsic noise. In the present work, intrin- sic noises for each individual are considered as a perturbation to an RDS. This gives rise to random Markov systems (RMS) in which the law of mo- tion is still stochastically changing with time, but individuals also exhibit statistically independent variations, with each transition having a small probability not to follow the law. As a consequence, two individuals in an RMS system go through stochastically distributed periods of synchro- nization and desynchronization, driven by extrinsic and intrinsic noises respectively. We show that in-sync time, e.g., escaping from a random attractor, has a symptotic geometric distribution.