Breaking of Ergodicity in Expanding Systems of Globally Coupled Piecewise Affine Circle Maps

Fernandez, Bastien

doi:10.1007/s10955-013-0903-9

Cited by 16 publications

(62 citation statements)

References 42 publications

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“…Coupled dynamical systems in general, and globally coupled maps in particular have been extensively studied in the literature. Below we mention the papers that directly motivated our work, more complete lists of references can be found for instance in [16], [9], [15] and [5]. Our main interest here is to understand how different types of asymptotic phenomena arise in the system depending on the coupling strength ε.…”

Section: Introductionmentioning

confidence: 99%

Synchronization versus stability of the invariant distribution for a class of globally coupled maps

et al. 2018

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We study a class of globally coupled maps in the continuum limit, where the individual maps are expanding maps of the circle. The circle maps in question are such that the uncoupled system admits a unique absolutely continuous invariant measure (acim), which is furthermore mixing. Interaction arises in the form of diffusive coupling, which involves a function that is discontinuous on the circle. We show that for sufficiently small coupling strength the coupled map system admits a unique absolutely continuous invariant distribution, which depends on the coupling strength ε. Furthermore, the invariant density exponentially attracts all initial distributions considered in our framework. We also show that the dependence of the invariant density on the coupling strength ε is Lipschitz continuous in the BV norm.When the coupling is sufficiently strong, the limit behavior of the system is more complex. We prove that a wide class of initial measures approach a point mass with support moving chaotically on the circle. This can be interpreted as synchronization in a chaotic state.

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Section: Introductionmentioning

confidence: 99%

Synchronization versus stability of the invariant distribution for a class of globally coupled maps

et al. 2018

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“…the Z 2 -symmetry generated by −Id T D is systematically broken. The analytic proofs of ergodicity breaking in [12,30,31] established the existence of so-called InAsUP (see Definition 1 below) for all larger than thresholds that are remarkably close to the D above. InAsUP were guessed using trajectory renderings as above.…”

Section: Empirical Results From Numerical Trajectoriesmentioning

confidence: 89%

“…t k=0 P k for some , so that the construction can12 Strict inequalities in this definition are chosen on purpose. Indeed, excluding polytope facets is a convenient way to exclude discontinuities from InAsUP construction.…”

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confidence: 99%

Computer-Assisted Proof of Loss of Ergodicity by Symmetry Breaking in Expanding Coupled Maps

Fernandez

2019

Ann. Henri Poincaré

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From a dynamical viewpoint, basic phase transitions of statistical mechanics can be regarded as a breaking of ergodicity. While many random models exhibiting such transitions at the thermodynamics limit exist, finite-dimensional examples with deterministic dynamics on a chaotic attractor are rare, if at all existent. Here, the dynamics of a family of N coupled expanding circle maps is investigated in a parameter regime where absolutely continuous invariant measures are known to exist. At first, empirical evidence is given of symmetry breaking of the ergodic components upon increase of the coupling strength, suggesting that breaking of ergodicity should occur for every integer N > 2. Then, a numerical algorithm is proposed which aims to rigorously construct asymmetric ergodic components of positive Lebesgue measure. Due to the explosive growth of the required computational resources, the algorithm successfully terminates for small values of N only. However, this approach shows that phase transitions should be provable for systems of arbitrary number of particles with erratic dynamics, in a purely deterministic setting, without any reference to random processes.

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“…To give a brief overview of the existing literature, we only cite papers connected closely to our work. For more complete lists, see the references in [11], [5], [10] and the collection [4]. Most results concern the case of coupling strength close to zero.…”

Section: Introductionmentioning

confidence: 99%

“…By standard results in the literature [10], the map has a unique mixing acim for ε values close to zero. Fernandez [5] indicated, by numerically calculating certain order parameters, that multiple acims emerge when the value of the coupling parameter is increased. In other words, ergodicity is broken at some critical value of ε.…”

Section: Introductionmentioning

confidence: 99%

Symmetry breaking in a globally coupled map of four sites

Sélley¹

2018

Discrete &Amp; Continuous Dynamical Systems - A

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A system of four globally coupled doubling maps is studied in this paper. It is known that such systems have a unique absolutely continuous invariant measure (acim) for weak interaction, but the case of stronger coupling is still unexplored. As in the case of three coupled sites [14], we prove the existence of a critical value of the coupling parameter at which multiple acims appear. Our proof has several new ingredients in comparison to the one presented in [14]. We strongly exploit the symmetries of the dynamics in the course of the argument. This simplifies the computations considerably, and gives us a precise description of the geometry and symmetry properties of the arising asymmetric invariant sets. Some new phenomena are observed which are not present in the case of three sites. In particular, the asymmetric invariant sets arise in areas of the phase space which are transient for weaker coupling and a nontrivial symmetric invariant set emerges, shaped by an underlying centrally symmetric Lorenz map. We state some conjectures on further invariant sets, indicating that unlike the case of three sites, ergodicity breaks down in many steps, and not all of them are accompanied by symmetry breaking.

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Breaking of Ergodicity in Expanding Systems of Globally Coupled Piecewise Affine Circle Maps

Cited by 16 publications

References 42 publications

Synchronization versus stability of the invariant distribution for a class of globally coupled maps

Synchronization versus stability of the invariant distribution for a class of globally coupled maps

Computer-Assisted Proof of Loss of Ergodicity by Symmetry Breaking in Expanding Coupled Maps

Symmetry breaking in a globally coupled map of four sites

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