Desynchronization of Random Dynamical System under Perturbation by an Intrinsic 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.