Stochastically stable globally coupled maps with bistable thermodynamic limit

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.

37 pages

References in corpus (3)

• Remarks on the mean field dynamics of networks of chaotic elements
• Collective motion occurs inevitably in a class of populations of globally coupled chaotic elements
• Confined chaotic behavior in collective motion for populations of globally coupled chaotic elements

Cited by in corpus (5)

• Ergodic averaging with and without invariant measures
• Self consistent transfer operators. Invariant measures, convergence to equilibrium, linear response and control of the statistical properties
• Synchronization versus stability of the invariant distribution for a class of globally coupled maps
• Mean-field coupling of identical expanding circle maps
• Globally coupled Anosov diffeomorphisms: Statistical properties