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Extensive bounds on the topological entropy of repellers in piecewise expanding coupled map lattices

  • RICARDO COUTINHO (a1) (a2) and BASTIEN FERNANDEZ (a3)
Abstract
Abstract

Beyond the uncoupled regime, the rigorous description of the dynamics of (piecewise) expanding coupled map lattices remains largely incomplete. To address this issue, we study repellers of periodic chains of linearly coupled Lorenz-type maps which we analyze by means of symbolic dynamics. Whereas all symbolic codes are admissible for sufficiently small coupling intensity, when the interaction strength exceeds a chain length independent threshold, we prove that a large bunch of codes is pruned and an extensive decay follows suit for the topological entropy. This quantity, however, does not immediately drop off to 0. Instead, it is shown to be continuous at the threshold and to remain extensively bounded below by a positive number in a large part of the expanding regime. The analysis is firstly accomplished in a piecewise affine setting where all calculations are explicit and is then extended by continuation to coupled map lattices based on $C^1$-perturbations of the individual map.

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[1] V. Afraimovich and B. Fernandez . Topological properties of linearly coupled expanding map lattices. Nonlinearity 13 (2000), 973993.

[2] J.-B. Bardet and B. Fernandez . Extensive escape rate in lattices of weakly coupled expanding maps. Discrete Contin. Dyn. Syst. A 31 (2011), 669684.

[3] J.-B. Bardet and G. Keller . Phase transitions in a piecewise expanding coupled map lattice with linear nearest neighbour coupling. Nonlinearity 19 (2006), 21932210.

[5] L. Bunimovich and Y. Sinai . Space–time chaos in coupled map lattices. Nonlinearity 1 (1988), 491516.

[6] J.-R. Chazottes and B. Fernandez (eds) Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems (Lecture Notes in Physics, 671). Springer, Berlin, 2005.

[7] R. Coutinho , B. Fernandez and P. Guiraud . Symbolic dynamics of two coupled Lorenz maps: from uncoupled regime to synchronisation. Phys. D 237 (2008), 24442462.

[8] K. Falconer . Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. Wiley, Chichester, 2003.

[9] B. Fernandez . Global synchronisation in translation invariant coupled map lattices. Internat. J. Bifur. Chaos 18 (2008), 34553459.

[12] G. Gielis and R. MacKay . Coupled map lattices with phase transitions. Nonlinearity 13 (2000), 867888.

[13] J. Guckenheimer and P. Holmes . Nonlinear Oscillators, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York, 1983.

[14] M. Jiang and Y. Pesin . Equilibrium measures for coupled map lattices: existence, uniqueness and finite-dimensional approximations. Comm. Math. Phys. 193 (1998), 675711.

[15] J. Jost and M. Joy . Spectral properties and synchronization in coupled map lattices. Phys. Rev. E 65 (2001), 016201.

[16] K. Kaneko . Period-doubling of kink-antikink patterns, quasiperiodicity in antiferro-like structures and spatial intermittency in coupled logistic lattice. Progr. Theoret. Phys. 72 (1984), 480486.

[19] G. Keller , M. Künzle and T. Nowicki . Some phase transitions in coupled map lattices. Phys. D 59 (1992), 3951.

[20] G. Keller and C. Liverani . Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension. Comm. Math. Phys. 262 (2006), 3350.

[21] J. Koiller and L.-S. Young . Coupled map networks. Nonlinearity 23 (2010), 11211141.

[23] P. Lind , J. Corte-Real and J. Gallas . Modeling velocity in gradient flows with coupled-map lattices with advection. Phys. Rev. E 66 (2002), 016219.

[25] Y. Oono and S. Puri . Computationally efficient modeling of ordering of quenched phases. Phys. Rev. Lett. 58 (1987), 836839.

[28] R. Solé , J. Valls and J. Bascompte . Spiral waves, chaos and multiple attractors in lattice models of interacting populations. Phys. Lett. A 166 (1992), 123128.

[29] I. Waller and R. Kapral . Spatial and temporal structure in systems of coupled nonlinear oscillators. Phys. Rev. A 30 (1984), 20472055.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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