Synchronization by degenerate noise
Xianming Liu, Xu Sun
TL;DR
The paper develops criteria for synchronization by noise under degenerate or non-Gaussian white noise within the random dynamical systems framework, replacing global swift transitivity with local conditions. It proves equivalence results linking local asymptotic stability and transitivity to synchronization, applicable even when noise is degenerate. The stochastic Lorenz 63 system is used to demonstrate a noise-intensity–driven bifurcation between synchronization and non-synchronization. Through concrete examples involving Poisson noise and degenerate-noise dynamics, the work broadens the understanding of synchronization by noise beyond nondegenerate Gaussian settings and provides practical criteria for analyzing such systems.
Abstract
In this paper, we derive several criteria for (weak) synchronization by noise without the global swift transitivity property. Our sufficient conditions for (weak) synchronization are necessary and can be applied to scenarios involving degenerate or non-Gaussian noise. These results partially answer the open question posed by Flandoli et al. (Probab Theory Relat Fields 168:511-556, 2017). As an application, we prove that the weak attractor for stochastic Lorenz 63 systems driven by degenerate noise consists of a single random point provided the noise intensity is small, and there is no weak synchronization if the noise intensity is large. This indicates that a bifurcation occurs in relation to the intensity of the noise.