Synchronization by degenerate noise

TL;DR

This paper establishes criteria for weak synchronization in stochastic systems driven by degenerate or non-Gaussian noise, demonstrating a noise-induced bifurcation in the Lorenz 63 system.

Contribution

It provides necessary and sufficient conditions for weak synchronization without requiring global swift transitivity, addressing an open problem in the field.

Findings

Weak attractor reduces to a single point at low noise intensity

No weak synchronization occurs at high noise intensity

Bifurcation in synchronization behavior depending on noise strength

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.