The persistence of synchronization under $\alpha$-stable noise

TL;DR

This paper investigates how synchronization in nonlinear coupled systems persists despite $eta$-stable noise, introducing a new method linking synchronized and slow-fast systems, and demonstrating convergence and persistence of synchronization under averaging principles.

Contribution

It presents a novel technique connecting synchronized systems with slow-fast systems and proves the persistence of synchronization under $eta$-stable noise using averaging principles.

Findings

Convergence of the slow component to the mild solution of the averaging equation.

Persistence of synchronization under stationary random solutions.

A new method relating synchronized and slow-fast systems.

Abstract

This work is about the synchronization of nonlinear coupled dynamical systems driven by $\alpha$-stable noise. Firstly, we provide a novel technique to construct the relationship between synchronized system and slow-fast system. Secondly, we show that the slow component of original systems converges to the mild solution of the averaging equation under $L^{p}(1<p<\alpha)$ sense. Finally, using the results of averaging principle for stochastic dynamical system with two-time scales, we show that the synchronization effect is persisted provided equilibria are replaced by stationary random solutions.