Uniqueness of the SRB Measure for Piecewise Expanding Weakly Coupled Map Lattices in Any Dimension

Keller, Gerhard; Liverani, Carlangelo

doi:10.1007/s00220-005-1474-7

Cited by 43 publications

(85 citation statements)

References 31 publications

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“…To the best of our knowledge the only known result is in dimension 2 for the uncoupled systems given by the direct product of two piece-wise expanding and smooth maps of the circle, see [5], and it is consistent with our results. Nevertheless a few preliminary considerations 16 seem to indicate that the compound distribution (6.36) still holds with p in (6.37) replaced by 1 − θ n in (5.35), and more generally with the EI given by formulas (2.7), (2.6), with the quantities q k,l given by the right hand side of (5.27) when the transfer operator is not available. In particular one should recover a pure Poisson distribution when the size n of the lattice tends to infinity.…”

Section: Distribution Of the Number Of Successive Visitsmentioning

confidence: 99%

Extreme value theory for synchronization of coupled map lattices

et al. 2018

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We show that the probability of the appearance of synchronization in chaotic coupled map lattices is related to the distribution of the maximum of a certain observable evaluated along almost all orbits. We show that such a distribution belongs to the family of extreme value laws, whose parameters, namely the extremal index, allow us to get a detailed description of the probability of synchronization. Theoretical results are supported by robust numerical computations that allow us to go beyond the theoretical framework provided and are potentially applicable to physically relevant systems.

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Section: Distribution Of the Number Of Successive Visitsmentioning

confidence: 99%

Extreme value theory for synchronization of coupled map lattices

et al. 2018

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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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“…In order to state our results we adopt the notation of [20]: Let Ω = I Λ and let M(Ω) be the set of signed Borel measures on Ω.…”

Section: Introductionmentioning

confidence: 99%

Phase transitions in a piecewise expanding coupled map lattice with linear nearest neighbour coupling

Bardet

Keller

2006

Nonlinearity

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Abstract. We construct a mixing continuous piecewise linear map on [−1, 1] with the property that a two-dimensional lattice made of these maps with a linear north and east nearest neighbour coupling admits a phase transition. We also provide a modification of this construction where the local map is an expanding analytic circle map. The basic strategy is borroughed from [10], namely we compare the dynamics of the CML to those of a probabilistic cellular automaton of Toom's type, see [24] for a detailed discussion.

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Uniqueness of the SRB Measure for Piecewise Expanding Weakly Coupled Map Lattices in Any Dimension

Abstract: Abstract. We prove the existence of a unique SRB measure for a wide range of multidimensional weakly coupled map lattices. These include piecewise expanding maps with diffusive coupling.

Cited by 43 publications

References 31 publications

Extreme value theory for synchronization of coupled map lattices

Extreme value theory for synchronization of coupled map lattices

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

Phase transitions in a piecewise expanding coupled map lattice with linear nearest neighbour coupling

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