LOW-DIMENSIONAL COLLECTIVE CHAOS IN

STRONGLY- AND GLOBALLY-COUPLED NOISY MAPS

Hugues ChatéService de Physique de L'Etat Condensé, CEA-Saclay, France

Silvia De Monte Ecologie Théorique, Ecole Normale Supérieure, Paris, France

Francesco d'OvidioMétéorologie Dynamique, Ecole Normale Supérieure, Paris, France

Erik MosekildeDepartment of Physics, The Technical University of Denmark

OUTLINE

Phenomenology: identical and noisy globally-coupled maps

single-element versus mean-field

Motivation: a system globally-coupled biological oscillators

role of microscopic disorder? extract microscopic info from measurements?

Order parameter expansion: effective macroscopic dynamics

hierarchical structure and dimensionality of the macroscopic attractor

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METABOLIC OSCILLATIONS IN YEAST CELLS

S. Dano, P. G. Soerensen, F. Hynne, Nature 402 (1999)

●macroscopic measurements only (so far)

●role of microscopic disorder (internal/external)

●use noise to extract microscopic information from measurements?

POPULATIONS

MICROSCOPIC FEATURES

SINGLE-ELEMENT DYNAMICSDISORDER DISTRIBUTION

MACROSCOPIC OBSERVABLES

AVERAGES OVER THE POPULATIONORDER PARAMETERS

Additive noise: stochastic process from a given distribution

MODEL: GLOBALLY-COUPLED NOISY MAPS

Single-element dynamics(chaotic, excitable)

Coupling strength

Global coupling

Large body of work on noiseless case, mostly in weak-coupling limit

(clustering, existence of collective dynamics, dimensionality…)

Here consider noisy chaotic maps in strong-coupling regime and

work around the fully-synchronized deterministic limit

This regime first considered by Teramae and Kuramoto

« anomalous scaling »

SPIRIT OF OUR APPROACH

Beyond finite-size effects: bifurcation diagram of collective observable vs noise strength

N=50

N=100

Beyond finite-size effects: noise-induced bifurcations

N=1000

N=10000

N=1000000

NOISE-INDUCED MACROSCOPIC REGIMES

Macroscopic bifurcation diagram

N 106

K 0.4

Snapshot pdfs

Macroscopic bifurcation diagram suggests theexistence 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

ORDER PARAMETER EXPANSION: dynamics of mean-field

ORDER PARAMETERS

Noise term

Change of variables Series expansion

ORDER PARAMETER EXPANSION

Infinite population size

n-th order REDUCED SYSTEM: truncation to

n-dimensional map

slaved variables

ZEROTH-ORDER REDUCED SYSTEMtrivial result for K=1

Reduced system 2

Population of logistic maps

● Interaction between nonlinearities of the single element and noise features● Classification of noise distributions according to their macroscopic effect

ZEROTH-ORDER APPROXIMATIONquartic maps with different noise distributions

Gaussian noise

Uniform noise

X 1 a 2 bm4 a 6b 2 X2 bX4

x 1 ax2 bx4Single-element dynamics:

Reduced system

HIGHER-ORDER APPROXIMATIONSmacroscopic bifurcation diagram of logistic maps

Period 2

Period 4Chaos

second order

fourth order

x 1 ax2Single-element dynamics:

Reduced system to the second order

X 1 aX2 a 2

22 1 K

2a2 m4 3 4 4X2

2 3 22

HIGHER-ORDER APPROXIMATIONSfine structure of the macroscopic attractor

Folding of the first return map The order parameter expansion captures the hierarchical structure of the macroscopic attractor

Zeroth-order

Second order

Fourth order

Population

HIGHER-ORDER APPROXIMATIONSmacroscopic Lyapunov exponents

calculate Lyapunov exponents/vectors directly from evolution of distribution p(x)

1st vector

HIGHER-ORDER APPROXIMATIONSmacroscopic Lyapunov exponents

compare with exponents of reduced system(s)

Largest Lyapunov exponent

HIGHER-ORDER APPROXIMATIONSmacroscopic Lyapunov exponents

ANOMALOUS SCALING...

K=0.3

K=0.315

K=0.32

Rather well captured by expansion for K not too small...

Over finite range of K, or only at loss of synchronisation?If only at Kc, continuous or subcritical transition?

= normal scaling=

ANOMALOUS SCALING AND LYAPUNOV ANALYSIS

K=0.315

K=0.4

One positive exponent and infinitely-many ~1/σ in anomalous scaling regionNormal scaling: all negative exponents finite as σ=0

Preliminary results from direct pdf simulations...

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 microscopicsfrom global measurements

GLOBALLY-COUPLED MAPS WITH PARAMETER MISMATCH

COHERENT REGIMES

Different effects of noise and parameter mismatch are captured by the order parameter expansion

Period 2Chaos

● Dependence on the system size● Convergence for maximal coupling

EFFECT OF NOISE ON INDIVIDUAL AND MACROSCOPIC TRAJECTORIES

Weak noise Strong noise

Mean field pdfsingle-element pdf

Time series

Onset of macroscopic oscillations

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Population of excitable maps:

ZEROTH-ORDER APPROXIMATIONnon-polynomial maps

NOISE-INDUCED COHERENCE/COHERENCE RESONANCE

Breakdown of approximation at large

Reduced system for small and large K.