1. NOISE-INDUCED COLLECTIVE REGIMES IN POPULATIONS OF GLOBALLY COUPLED MAPS Silvia De Monte Dept. of Physics, The Technical University of Denmark Francesco d'Ovidio IMEDEA, Palma de Mallorca, Spain Erik Mosekilde DTU, Lyngby, Denmark Hugues Chaté CEA Saclay, Gif-Sur-Yvette, France

2. OVERVIEW * Globally coupled maps with microscopic disorder macroscopic dynamics of large populations of noisy maps * Order parameter expansion maps with additive noise logistic and excitable maps macroscopic effects of different sources of disorder noise and parameter mismatch

3. POPULATIONS OF GLOBALLY COUPLED SYSTEMS Physical, chemical and biological population commonly modelled as globally coupled dynamical systems: ● Yeast cells in a Continuous-flow Stirred Tank Reactor (S. Dan ø, P.G. S ø rensen, Nature 1999) ● Electrochemical oscillators (J. Hudson, Science 2002) ● Josephson junction arrays ● Pancreatic Beta-cells ● Heart cells ● Clapping audiences ●...

4. MICROSCOPIC FEATURES MACROSCOPIC OBSERVABLES averages over the population MEAN FIELD Single-element dynamics ORDER PARMETERS Populations of dynamical systems with microscopic disorder

5. GLOBALLY COUPLED MAPS Logistic map Coupling constant Global coupling SINGLE-ELEMENT DYNAMICS MICROSCOPIC VARIABILITY ● Stochastic process (noise) ● Parameter mismatch Distribution of moments variance

6. GLOBALLY COUPLED NOISY MAPS Control parameter PERFECT SYNCHRONISATION from the talk of Ljupco Kocarev

7. Macroscopic cycle Mean field pdf single-element pdf Time series 'Macroscopic chaos' GLOBALLY COUPLED NOISY MAPS Control parameter

8. Can the mean field evolution be explained in terms of a MACROSCOPIC DYNAMICAL SYSTEM? GLOBALLY COUPLED NOISY MAPS macroscopic bifurcation diagram

9. ● Order parameter expansion S. De Monte, F. d'Ovidio, H. Chaté, and E. Mosekilde, PRL (2004) cond-mat/0405251 How to deal with systems with many-degrees of freedom? ● Nonlinear Perron-Frobenius equation Pikovsky and Kurts, PRL (1994) Topaj, Kye and Pikovsky, PRL (2001) Shibata and Kaneko, PRL (1999) ● Perturbation of the synchronous regime for weak noise 'The anomalous fluctuations could be visible through the higher moments in the range of stronger coupling where no anomaly is visible through lower moments.' Teramae and Kuramoto, PRE (2001)

10. ORDER PARAMETER EXPANSION: mean field ORDER PARAMETERS Noise term Change of variablesSeries expansion

11. ORDER PARAMETER EXPANSION Infinite population size n-th order REDUCED SYSTEM: truncation to n-dimensional map slaved variables

12. Zeroth-order reduced system exact result for K=1 Reduced system 2 Population of logistic maps ● Interaction between nolinearities of the single element and noise features ● Classification of noise distributions according to their macroscopic effect

13. Zeroth-order reduced system: remark #1 different noise distributions Gaussian noise Uniform noise Reduced system Population of quartic maps:

14. Onset of macroscopic oscillations *  Population of excitable maps: Zeroth-order reduced system: remark #2 non-polynomial maps

15. Breakdown of the approximation of small  Reduced system for small  and large K. Noise-induced coherence

16. Period 2 Period 4 Chaos Higher-order reduced systems macroscopic bifurcation diagram Reduced system to the second order Reduced system to the fourth order Reduced system to the second order

17. Higher-order approximations fine structure of the macroscopic attractor Folding of the first return map Hierarchically structured macroscopic attractor Zeroth-order Second order Fourth order Population Second order Zeroth-order

18. Higher-order reduced systems dimensionality of the macroscopic dynamics 'macroscopic' Lyapunov exponents

19. GLOBALLY COUPLED MAPS WITH PARAMETER MISMATCH Period 2 Chaos COHERENT REGIMES

20. GLOBALLY COUPLED MAPS WITH PARAMETER MISMATCH COHERENT REGIMES Different effects of noise and parameter mismatch are captured by the order parameter expansion Period 2 Chaos ● Dependence on the system size ● Convergence for maximal coupling

21. CONCLUSIONS AND PERSPECTIVES Microscopic disorder 'unfolds' the synchronous dynamics of globally and strongly coupled maps. The order parameter expansion provides a description of the collective behaviour in terms of effective degrees of freedom and macroscopic- level parameters. Potential applications to: ● coupled electronic circuits (Chua) ● pulsating lasers ● yeast cells suspensions ● reactive media with strong mixing

22. The reduced systems reflect the different microscopic properties of the population Noise Parameter mismatch

23.  Standard deviation of the parameter distribution X 32 Roessler systems with time scale mismatch Population of globally strongly coupled chaotic systems with parameter mismatch The collective behaviour in the coherent regimes ( ) can be periodic or stationary The bifurcation diagram is reproduced by two order parameters, the mean field and the 'shape parameter'

24. Higher-order approximations macroscopic Lyapunov exponents Largest Lyapunov exponent

25. Finite-size effects N=10 N=500 N=10000