Analysis of Discrete and Hybrid Stochastic Systems by Nonlinear Contraction Theory

TL;DR

This paper extends stochastic contraction theory to discrete and hybrid systems, demonstrating that their trajectories converge in mean square and applying this to synchronize noisy oscillators.

Contribution

It introduces new stochastic contraction theorems for discrete and hybrid systems, expanding the applicability of contraction analysis.

Findings

Mean square distance between trajectories is bounded after transients.

Synchronization of noisy nonlinear oscillators is achieved.

Theoretical extension from continuous to discrete/hybrid systems.

Abstract

We investigate the stability properties of discrete and hybrid stochastic nonlinear dynamical systems. More precisely, we extend the stochastic contraction theorems (which were formulated for continuous systems) to the case of discrete and hybrid resetting systems. In particular, we show that the mean square distance between any two trajectories of a discrete (or hybrid resetting) contracting stochastic system is upper-bounded by a constant after exponential transients. Using these results, we study the synchronization of noisy nonlinear oscillators coupled by discrete noisy interactions.