LOW-DIMENSIONAL COLLECTIVE CHAOS IN STRONGLY- AND GLOBALLY-COUPLED NOISY MAPS

1. LOW-DIMENSIONAL COLLECTIVE CHAOS IN STRONGLY- AND GLOBALLY-COUPLED NOISY MAPS Hugues Chaté Service de Physique de L'Etat Condensé, CEA-Saclay, France Silvia De Monte Ecologie Théorique, Ecole Normale Supérieure, Paris, France Francesco d'Ovidio Météorologie Dynamique, Ecole Normale Supérieure, Paris, France Erik Mosekilde Department of Physics, The Technical University of Denmark

2. OUTLINE * Motivation: a system globally-coupled biological oscillators role of microscopic disorder? extract microscopic info from measurements? * Phenomenology: identical and noisy globally-coupled maps single-element versus mean-field * Order parameter expansion: effective macroscopic dynamics hierarchical structure and dimensionality of the macroscopic attractor

3. S. Dano, P. G. Soerensen, F. Hynne, Nature 402 (1999) METABOLIC OSCILLATIONS IN YEAST CELLS • macroscopic measurements only (so far) • role of microscopic disorder (internal/external) • use noise to extract microscopic information from measurements?

4. POPULATIONS MACROSCOPIC OBSERVABLES AVERAGES OVER THE POPULATION ORDER PARAMETERS MICROSCOPIC FEATURES SINGLE-ELEMENT DYNAMICS DISORDER DISTRIBUTION

5. MODEL: GLOBALLY-COUPLED NOISY MAPS Single-element dynamics (chaotic, excitable) Additive noise: stochastic process from a given distribution Global coupling Coupling strength

6. SPIRIT OF OUR APPROACH • Large body of work on noiseless case, mostly in weak-coupling limit • (clustering, existence of collective dynamics, dimensionality…) • Here considernoisy chaotic maps in strong-coupling regime and • work around the fully-synchronized deterministic limit • This regime first considered by Teramae and Kuramoto • « anomalous scaling »

7. Beyond finite-size effects: bifurcation diagram of collective observable vs noise strength N=50

8. Beyond finite-size effects: noise-induced bifurcations N=100

9. N=1000

10. N=10000

11. N=1000000

12. Snapshot pdfs NOISE-INDUCED MACROSCOPIC REGIMES Macroscopic bifurcation diagram

13. Macroscopic bifurcation diagram suggests the existence of an effective, low-dimensional dynamical system acting on macroscopic observables such as X This observation is at the heart of our approach, building hierarchically such an effective description

14. Change of variables Series expansion ORDER PARAMETER EXPANSION: dynamics of mean-field Noise term ORDER PARAMETERS

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

16. Reduced system Population of logistic maps 2 ZEROTH-ORDER REDUCED SYSTEM trivial result for K=1 • Interaction between nonlinearities of the single element and noise features • Classification of noise distributions according to their macroscopic effect

17. Uniform noise Single-element dynamics: Gaussian noise ZEROTH-ORDER APPROXIMATION quartic maps with different noise distributions Reduced system

18. Single-element dynamics: Chaos Period 4 second order Period 2 fourth order HIGHER-ORDER APPROXIMATIONS macroscopic bifurcation diagram of logistic maps Reduced system to the second order

19. Fourth order Population Second order Zeroth-order The order parameter expansion captures the hierarchical structure of the macroscopic attractor HIGHER-ORDER APPROXIMATIONS fine structure of the macroscopic attractor Folding of the first return map

20. HIGHER-ORDER APPROXIMATIONSmacroscopic Lyapunov exponents • calculate Lyapunov exponents/vectors directly from evolution of distribution p(x) 1st vector

21. Largest Lyapunov exponent HIGHER-ORDER APPROXIMATIONSmacroscopic Lyapunov exponents • compare with exponents of reduced system(s)

22. HIGHER-ORDER APPROXIMATIONS macroscopic Lyapunov exponents

23. ANOMALOUS SCALING... Over finite range of K, or only at loss of synchronisation? If only at Kc, continuous or subcritical transition? K=0.3 K=0.315 = normal scaling= K=0.32 Rather well captured by expansion for K not too small...

24. ANOMALOUS SCALING AND LYAPUNOV ANALYSIS Preliminary results from direct pdf simulations... K=0.315 K=0.4 One positive exponent and infinitely-many ~1/σ in anomalous scaling region Normal scaling: all negative exponents finite as σ=0

25. CONCLUSIONS AND PERSPECTIVES Microscopic disorder 'unfolds' the synchronous dynamics of globally and strongly coupled maps. Expansion provides a quantitatively-accurate hierarchical description of the collective dynamics in terms of macroscopic degrees of freedom and parameters. Anomalous scaling: • existence over finite range of parameters? • only at breakdown of synchronisation? • universality of transition? Applications: use effect of noise to learn about microscopics from global measurements

26. 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

27. Mean field pdf single-element pdf EFFECT OF NOISE ON INDIVIDUAL AND MACROSCOPIC TRAJECTORIES Time series Weak noise Strong noise

28. * s Onset of macroscopic oscillations ZEROTH-ORDER APPROXIMATION non-polynomial maps Population of excitable maps:

29. NOISE-INDUCED COHERENCE/COHERENCE RESONANCE Reduced system for small sand large K. Breakdown of approximation at large s