Mean-field coupling of identical expanding circle maps

TL;DR

This paper investigates globally coupled doubling maps, analyzing bifurcations, synchronization, and distribution evolution, extending understanding from finite to infinite site systems with new insights into their dynamical behavior.

Contribution

It reinterprets known bifurcation phenomena in coupled maps through synchronization and introduces a distribution-based perspective for infinite systems.

Findings

Identification of bifurcation values related to contraction and ergodicity loss

Interpretation of dynamical phenomena via synchronization

Observation of infinite system behaviors analogous to finite limit states

Abstract

Globally coupled doubling maps are studied in this paper. In this setting and for finitely many sites, two distinct bifurcation values of the coupling strength have been identified in the literature, corresponding to the emergence of contracting directions (\cite{koiller2010coupled}) and, specifically for $N=3$ sites, to the loss of ergodicity (\cite{fernandez2014breaking}). On the one hand, we reconsider these results and provide an interpretation of the observed dynamical phenomena in terms of the synchronization of the sites. On the other hand, we initiate a new point of view which focuses on the evolution of distributions and allows to incorporate the investigation of infinitely many sites. In particular we observe phenomena in the infinite system that is analogous to the limit states of the contracting regime of $N=3$ sites.