Synchronisation in Invertible Random Dynamical Systems on the Circle

TL;DR

This paper investigates the geometric properties and conditions for stable synchronization in random dynamical systems on the circle, focusing on crack points, invariant measures, and the effects of additive noise.

Contribution

It provides a geometric framework for understanding synchronization phenomena and characterizes stable synchronization in stochastic differential equations through subperiodicity.

Findings

Identification of crack points and invariant measures associated with synchronization.

Link between arc compressibility and stable synchronization.

Characterization of stable synchronization via subperiodicity in additive-noise SDEs.

Abstract

In this paper, we study geometric features of orientation-preserving random dynamical systems on the circle driven by memoryless noise that exhibit stable synchronisation: we consider crack points, invariant measures, and the link between synchronisation and compressibility of arcs; we also characterise stable synchronisation in additive-noise stochastic differential equations on the circle, in terms of "subperiodicity" of the vector field.