Stochastically stable globally coupled maps with bistable thermodynamic limit

Details Stochastically stable globally coupled maps with bistable thermodynamic limit Stochastically stable globally coupled maps with bistable thermodynamic limit

2008-12-21

arxiv.org/abs/0812.4040v1

Communications in Mathematical Physics 292, 1 (2009) 237-270

Mathematics

Dynamical Systems

37 pages

Scientific paper

10.1007/s00220-009-0854-9

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.

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