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Erschienen in:
Patterns of Synchrony in Neuronal Networks: The Role of Synaptic Inputs Kapitel lesen Erstes Kapitel lesen

2015 | OriginalPaper | Buchkapitel

Self-Organized Regularity in Long-Range Systems

verfasst von : Xavier Leoncini

Erschienen in: Nonlinear Dynamics New Directions

Verlag: Springer International Publishing

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Abstract

Dynamics of many-body long-range interacting systems are investigated. Using the XY-Hamiltonian mean-field model as a case study, we show that regular trajectories, associated with invariant tori of the single-particle dynamics emerge as the number of particles is increased. Moreover, the construction of stationary solutions as well as studies of the maximal Lyapunov exponent of the systems show the same trend towards integrability. This feature provides a dynamical interpretation of the emergence of long-lasting out-of-equilibrium regimes observed generically in long-range systems. Extensions beyond the mean-field system are considered and display similar features.

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Vorheriges Kapitel Homoclinic Ω-Explosion: Hyperbolicity Intervals and Their Bifurcation Boundaries
Nächstes Kapitel Reducing the Sequential Dynamics of Excitatory Neural Networks to Synaptic Cellular Automata
Fußnoten
1
We do not differentiate between the observable M and its statistical average \(\langle M\rangle\), for simplicity in notations, except when needed
 
2
Note that even if one accounts for the linear stability criteria, there are still large sets of distributions which shall generate quasi-stationary dynamics.
 
3
Actually due to numerical error we renormalize to \((N+\sqrt{N}/4)\tilde{\rho}\), which is in range with expected typical fluctuations.
 
4
A comparison with higher scheme has been made and gave identical results.
 
5
Note that \(1/\log(N)\) scalings are also good, and can be explained by the fact that the typical time scale in the separatrix can be estimated from the period of regular motion of the last regular trajectory \(T\sim\log(\delta E)\sim\log(\delta M)\sim\log(N)\).
 
6
This approximation can be rigorously justified via a detailed expansion see [36] for details.
 
7
Actually only the ones with \(\kappa<\kappa_c\).
 
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Metadaten
Titel
Self-Organized Regularity in Long-Range Systems
verfasst von
Xavier Leoncini
Copyright-Jahr
2015
Buch
Nonlinear Dynamics New Directions

Print ISBN: 978-3-319-09863-0

Electronic ISBN: 978-3-319-09864-7

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-3-319-09864-7_4

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