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


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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© 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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