Topological synchronisation or a simple attractor?

TL;DR

This paper investigates topological synchronization in coupled systems, demonstrating it as the emergence of an attractor with a multifractal structure supported by an absolutely continuous physical measure, supported by numerical and theoretical analysis.

Contribution

It provides a new explanation of topological synchronization as attractor formation and analyzes the multifractal nature of the associated physical measure.

Findings

Synchronization corresponds to attractor creation with increasing coupling.

The attractor supports an absolutely continuous physical measure.

The physical measure exhibits a multifractal structure.

Abstract

A few recent papers introduced the concept of topological synchronisation. We refer in particular to \cite{TS}, where the theory was illustrated by means of a skew product system, coupling two logistic maps. In this case, we show that the topological synchronisation could be easily explained as the birth of an attractor for increasing values of the coupling strength and the mutual convergence of two marginal empirical measures. Numerical computations based on a careful analysis of the Lyapunov exponents suggest that the attractor supports an absolutely continuous physical measure (acpm). We finally show that for some unimodal maps such acpm exhibit a multifractal structure.