Aperiodic mean-field evolutions in coupled map lattices

Losson, Jérôme; Vannitsem, Stéphane; Nicolis, Grégoire

doi:10.1103/physreve.57.4921

Cited by 4 publications

(4 citation statements)

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“…High-dimensional chaotic systems can give rise to remarkable collective phenomena: for instance, macroscopic (global) observables can display well-defined motions even when the microscopic elements, of which they are made up, behave chaotically and their number N is very large [47][48][49][50][51][52][53][54][55]. A particularly interesting case is the mean field behavior of globally coupled maps (GCMs) defined by the dynamics…”

Section: Macroscopic Chaosmentioning

confidence: 99%

“…where f is the map specifying the local dynamics, N is the number of microscopic elements and σ is the coupling strength. Collective behavior can be detected by looking at the mean field m(t ), upon varying the coupling σ and the map f (x), different types of behavior have been found [48][49][50][52][53][54][55], which can be classified as follows [53,54]. Standard chaos (a) corresponds to the natural expectation based on the central limit theorem.…”

Section: Macroscopic Chaosmentioning

confidence: 99%

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Finite size Lyapunov exponent: review on applications

Cencini

Vulpiani

2013

J. Phys. A: Math. Theor.

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In dynamical systems, the growth of infinitesimal perturbations is well characterized by the Lyapunov exponents. In many situations of interest, however, important phenomena involve finite amplitude perturbations, which are ruled by nonlinear dynamics out of tangent space, and thus cannot be captured by the standard Lyapunov exponents. We review the application of the finite size Lyapunov exponent (FSLE) for the characterization of noninfinitesimal perturbations in a variety of systems. In particular, we illustrate their usage in the context of predictability of systems with multiple spatiotemporal scales of geophysical relevance, in the characterization of nonlinear instabilities, and in some aspects of transport in fluid flows. We also discuss the application of the FSLE to more general aspects such as chaos-noise detection and coarse-grained descriptions of signals.

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Section: Macroscopic Chaosmentioning

confidence: 99%

Section: Macroscopic Chaosmentioning

confidence: 99%

Finite size Lyapunov exponent: review on applications

Cencini

Vulpiani

2013

J. Phys. A: Math. Theor.

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“…In our RM model, d departs from zero, due to the fluctuations of the FTLE caused by μt and s t , see Eq. (13). To ascertain whether this fluctuation persists in the thermodynamic limit, we computed d(N ) for several system sizes and fixed parameter values well inside region III (m = 2 and = 0.02).…”

Section: A Vanishing Diffusion Coefficientmentioning

confidence: 99%

“…An early striking discovery was the nonstationarity of the mean field in the infinite size limit [9][10][11][12]. Subsequently, several papers characterized the collective properties of chaos in turbulent GCMs [13][14][15][16]. Finally, Takeuchi et al [7] uncovered the delicate arrangement of the Lyapunov exponents underlying turbulent GCMs: The Lyapunov spectrum is apparently extensive, but "subextensive bands" persist for arbitrarily large system sizes at both ends of the Lyapunov spectrum.…”

Section: Introductionmentioning

confidence: 99%

Nonuniversal large-size asymptotics of the Lyapunov exponent in turbulent globally coupled maps

2021

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Globally coupled maps (GCMs) are prototypical examples of high-dimensional dynamical systems. Interestingly, GCMs formed by an ensemble of weakly coupled identical chaotic units generically exhibit a hyperchaotic 'turbulent' state. A decade ago, Takeuchi et al. [Phys. Rev. Lett. 107, 124101 (2011)] theorized that in turbulent GCMs the largest Lyapunov exponent (LE), λ(N ), depends logarithmically on the system size N : λ∞ − λ(N ) c/ ln N . We revisit the problem and analyze, by means of analytical and numerical techniques, turbulent GCMs with positive multipliers to show that there is a remarkable lack of universality, in conflict with the previous prediction. In fact, we find a power-law scaling λ∞ − λ(N ) c/N γ , where γ is a parameterdependent exponent in the range 0 < γ ≤ 1. However, for strongly dissimilar multipliers, the LE varies with N in a slower fashion, which is here numerically explored. Although our analysis is only valid for GCMs with positive multipliers, it suggests that a universal convergence law for the LE cannot be taken for granted in general GCMs.

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Spatio-Temporal Chaos and Vacuum Fluctuations of Quantized Fields

Beck¹

2002

Advanced Series in Nonlinear Dynamics

163

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PrefaceThis book is written for an interdisciplinary readership of graduate students and researchers interested in nonlinear dynamics, stochastic processes, statistical mechanics on the one hand and high energy physics, quantum field theory, string theory on the other. In fact, one of the goals that I had in mind when writing this book was to make particle physicists become interested in nonlinear dynamics, and nonlinear physicists become interested in particle physics. Why that? Didn't so far these two subjects evolve quite independently from each other? So what is this book about? Mathematically, the subject of the book are coupled map lattices exhibiting spatio-temporal chaotic behaviour. Physically, the subject is a topic that lies at the heart of elementary particle physics: There are about 25 free parameters in the standard model of electroweak and strong interactions, namely the coupling strengths of the three interactions, the fermion and boson masses, and various mass mixing angles. These parameters are not fixed at all by the standard model itself, they are just measured in experiments, and a natural question is why these free parameters take on the numerical values that we observe in nature and not some other values. It will turn out that the answer is closely related to certain distinguished types of coupled map lattices that we will consider in this book as suitable models of vacuum fluctuations. These dynamical systems, called 'chaotic strings' in the following, are observed to have minimum vacuum energy for the observed standard model parameters. They yield an extension of ordinary quantization schemes which can account for the free parameters.In this sense this book deals with both, nonlinear dynamics and high energy physics. So far only very few original papers have been published Vlll Preface on this very new subject. With the current book I hope to make these important new applications for coupled chaotic dynamical systems accessible to a broad readership. The book consists of 12 chapters. The first few chapters will mainly concentrate onto the theory of the relevant class of coupled map lattices, their use for second quantization purposes, and their physical interpretation in terms of vacuum fluctuations. In the later chapters concrete numerical results are presented and these are then related to standard model phenomenology. Sections marked with an asterisk can be omitted at a first reading, these sections deal with interesting side issues which, however, are not necessary for the logical development of the following chapters. In view of the fact that (unfortunately!) many readers may not have the time to read this book from the beginning to the end, I included a very detailed summary as a self-contained chapter 12. This summary contains the most important concepts and results of this book and is written in a self-consistent way, i.e. no knowledge of previous chapters is required.The research described in this book developed over a longer period of time at various places. I started to work o...

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Aperiodic mean-field evolutions in coupled map lattices

Cited by 4 publications

References 26 publications

Finite size Lyapunov exponent: review on applications

Finite size Lyapunov exponent: review on applications

Nonuniversal large-size asymptotics of the Lyapunov exponent in turbulent globally coupled maps

Spatio-Temporal Chaos and Vacuum Fluctuations of Quantized Fields

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