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

Bardet, Jean-Baptiste; Keller, Gerhard

doi:10.1088/0951-7715/19/9/012

Cited by 15 publications

(49 citation statements)

References 26 publications

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“…Exceptions to this failure are repellers of weakly coupled chains of maps with Cantor repelling set [12,13] and specially designed CML for which the coupling operator preserves the uncoupled Markov partition [2,8,14,17,21,35,36,39]. Independently of grammatical issues, proofs of uniqueness of the physical measure in the weak coupling regime (analogue to the uniqueness of the high temperature phase) have been provided using perturbative approaches from the uncoupled limit [1,5,6,16,19,27,28].…”

Section: Introductionmentioning

confidence: 99%

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

Fernandez

2014

J Stat Phys

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To identify and to explain coupling-induced phase transitions in Coupled Map Lattices (CML) has been a lingering enigma for about two decades. In numerical simulations, this phenomenon has always been observed preceded by a lowering of the Lyapunov dimension, suggesting that the transition might require changes of linear stability. Yet, recent proofs of co-existence of several phases in specially designed models work in the expanding regime where all Lyapunov exponents remain positive.In this paper, we consider a family of CML composed by piecewise expanding individual map, global interaction and finite number N of sites, in the weak coupling regime where the CML is uniformly expanding. We show, mathematically for N = 3 and numerically for N 3, that a transition in the asymptotic dynamics occurs as the coupling strength increases. The transition breaks the (Milnor) attractor into several chaotic pieces of positive Lebesgue measure, with distinct empiric averages. It goes along with various symmetry breaking, quantified by means of magnetization-type characteristics.Despite that it only addresses finite-dimensional systems, to some extend, this result reconciles the previous ones as it shows that loss of ergodicity/symmetry breaking can occur in basic CML, independently of any decay in the Lyapunov dimension.August 28, 2013.

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Section: Introductionmentioning

confidence: 99%

“…(Besides, phase transitions have been established in models with prescribed symbolic grammar mentioned above, that mimic probabilistic cellular automata for which the phenomenon had been rigorously proved [2,14,17]. )…”

Section: Introductionmentioning

confidence: 99%

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

Fernandez

2014

J Stat Phys

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“…They are undoubtedly important in order to argue spatio-temporal chaos on the mathematically proper basis [7,16,18,19,21]. An examination of the behavior of infinite-size systems requires that we first take the limit N → ∞ and then p → ∞, at variance with usual statistical mechanics where the limit is taken over sizes of all dimensions simultaneously.…”

Section: Discussionmentioning

confidence: 99%

“…Theoretically, it can be defined as a qualitative change in the statistical behavior of typical orbits in a single mixing attractor which does not change topologically [16,17,18,19], by which we exclude bifurcations coming up even in finite-dimensional dynamical systems. For the definition of "qualitative change," the analogy with that in equilibrium phase transitions is used.…”

Section: Introductionmentioning

confidence: 99%

Role of unstable periodic orbits in phase transitions of coupled map lattices

Takeuchi

Sano

2007

Phys. Rev. E

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The thermodynamic formalism for dynamical systems with many degrees of freedom is extended to deal with time averages and fluctuations of some macroscopic quantity along typical orbits, and applied to coupled map lattices exhibiting phase transitions. Thereby, it turns out that a seed of phase transition is embedded as an anomalous distribution of unstable periodic orbits, which appears as a so-called q-phase transition in the spatio-temporal configuration space. This intimate relation between phase transitions and q-phase transitions leads to one natural way of defining transitions and their order in extended chaotic systems. Furthermore, a basis is obtained on which we can treat locally introduced control parameters as macroscopic "temperature" in some cases involved with phase transitions.

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“…From a theoretical viewpoint, chaotic CML are motivated by the endeavour to prove the occurrence of phase transitions in deterministic lattice dynamical systems [LJ98,MH93]. However, excepted in specially designed CML [GM00,BK06], phase transitions in CML have not been proved up to date. For a detailed discussion on this problem, we refer to [CF05].…”

Section: Introductionmentioning

confidence: 99%

Symbolic dynamics of two coupled Lorenz maps: From uncoupled regime to synchronisation

Coutinho

Fernandez

Guiraud

2008

Physica D: Nonlinear Phenomena

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The bounded dynamics of a system of two coupled piecewise affine and chaotic Lorenz maps is studied over the coupling range, from the uncoupled regime where the entropy is maximal, to the synchronized regime where the entropy is minimal. By formulating the problem in terms of symbolic dynamics, bounds on the set of orbit codes (or the set itself, depending on parameters) are determined which describe the way the dynamics is gradually affected as the coupling increases. Proofs rely on monotonicity properties of bounded orbit coordinates with respect to some partial ordering on the corresponding codes. The estimates are translated in terms of (bounds on the) entropy, which are monotonously decreasing with coupling and which are compared to the numerically computed entropy. A good agreement is found which indicates that these bounds capture the essential features of the transition from the uncoupled regime to synchronisation.

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

Cited by 15 publications

References 26 publications

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

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

Role of unstable periodic orbits in phase transitions of coupled map lattices

Symbolic dynamics of two coupled Lorenz maps: From uncoupled regime to synchronisation

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