Communications in Mathematical Physics

, Volume 292, Issue 1, pp 237–270 | Cite as

Stochastically Stable Globally Coupled Maps with Bistable Thermodynamic Limit

  • Jean-Baptiste Bardet
  • Gerhard Keller
  • Roland Zweimüller
Article

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.

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Jean-Baptiste Bardet
    • 1
    • 2
  • Gerhard Keller
    • 3
  • Roland Zweimüller
    • 4
  1. 1.IRMARUniversité Rennes 1Rennes CedexFrance
  2. 2.LMRSUniversité de RouenSaint-Étienne-du-RouvrayFrance
  3. 3.Department MathematikUniversität Erlangen-NürnbergErlangenGermany
  4. 4.Fakultät für MathematikUniversität WienWienAustria

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