Remarks on the mean field dynamics of networks of chaotic elements

Abstract

Fluctuations of the mean field of a globally coupled dynamical systems are discussed. The origin of hidden coherence is related with the instability of the fixed point solution of the self-consistent Perron-Frobenius equation. Collective dynamics in globally coupled tent maps are re-examined, both with the help of direct simulation and the Perron-Frobenius equation. Collective chaos in a single band state, and bifurcation against initial conditions in a two-band state are clarified with the return maps of the mean-field, Lyapunov spectra, and also the newly introduced Lyapunov exponent for the Perron-Frobenius equation. Future problems on the collective dynamics are discussed.

15 pages; original has 15 figures which are not included here. Submitted to the special issue of Physica D as the proceedings for "Gran Finale"

References in corpus (3)

• Rayleigh-Bénard Convection; Patterns, Chaos, Spatiotemporal Chaos and Turbulence
• Peeling the Onion of Order and Chaos in a High-dimensional Hamiltonian System
• Synchronization effects in the dynamical behavior of elevators

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• Heterogeneity Induced Order in Globally Coupled Chaotic Systems
• Driven Maps and the Emergence of Ordered Collective Behavior in Globally Coupled Maps
• Dynamics of Coupling Functions in Globally Coupled Maps: Size, Periodicity and Stability of Clusters
• Periodicity Manifestations in the Turbulent Regime of Globally Coupled Map Lattice
• Stochastically stable globally coupled maps with bistable thermodynamic limit
• Synchronization versus stability of the invariant distribution for a class of globally coupled maps
• Mean-field coupling of identical expanding circle maps
• Clustering and synchronization with positive Lyapunov exponents
• Cluster dynamics in systems with constant mean field coupling
• Turbulence in Globally Coupled Maps
• Collective motions in globally coupled tent maps with stochastic updating
• Large deviations from the thermodynamic limit in globally coupled maps
• Low-dimensional chaos in populations of strongly-coupled noisy maps
• Using machine-learning modelling to understand macroscopic dynamics in a system of coupled maps
• Periodicity Manifestations in the Non-Locally Coupled Maps