Stochastically stable globally coupled maps with bistable thermodynamic limit

TL;DR

This paper analyzes globally coupled interval maps with bistable thermodynamic limits, demonstrating unique invariant densities in finite systems and bifurcation phenomena in the infinite limit, revealing complex stability behavior.

Contribution

It introduces a detailed analysis of coupled maps with bistability, showing invariant densities and bifurcation structures in the thermodynamic limit.

Findings

Finite systems have unique, positive, analytic invariant densities.

Exponential decay of correlations in finite systems.

Infinite system exhibits a supercritical pitchfork bifurcation.

Abstract

We study systems of globally coupled interval maps, where the identical individual maps have two expanding, fractional linear, onto branches, and where the coupling is introduced via a parameter - common to all individual maps - that depends in an analytic way on the mean field of the system. We show: 1) For the range of coupling parameters we consider, finite-size coupled systems always have a unique invariant probability density which is strictly positive and analytic, and all finite-size systems exhibit exponential decay of correlations. 2) For the same range of parameters, the self-consistent Perron-Frobenius operator which captures essential aspects of the corresponding infinite-size system (arising as the limit of the above when the system size tends to infinity), undergoes a supercritical pitchfork bifurcation from a unique stable equilibrium to the coexistence of two stable and one unstable equilibrium.