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


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]Afraimovich, V. and Fernandez, B.. Topological properties of linearly coupled expanding map lattices. Nonlinearity 13 (2000), 973993.
[2]Bardet, J.-B. and Fernandez, B.. Extensive escape rate in lattices of weakly coupled expanding maps. Discrete Contin. Dyn. Syst. A 31 (2011), 669684.
[3]Bardet, J.-B. and Keller, G.. Phase transitions in a piecewise expanding coupled map lattice with linear nearest neighbour coupling. Nonlinearity 19 (2006), 21932210.
[4]Bricmont, J. and Kupiainen, A.. Coupled analytic maps. Phys. D 8 (1995), 379396.
[5]Bunimovich, L. and Sinai, Y.. Space–time chaos in coupled map lattices. Nonlinearity 1 (1988), 491516.
[6]Chazottes, J.-R. and Fernandez, B. (eds) Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems (Lecture Notes in Physics, 671). Springer, Berlin, 2005.
[7]Coutinho, R., Fernandez, B. and Guiraud, P.. Symbolic dynamics of two coupled Lorenz maps: from uncoupled regime to synchronisation. Phys. D 237 (2008), 24442462.
[8]Falconer, K.. Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. Wiley, Chichester, 2003.
[9]Fernandez, B.. Global synchronisation in translation invariant coupled map lattices. Internat. J. Bifur. Chaos 18 (2008), 34553459.
[10]Fernandez, B. and Guiraud, P.. Route to chaotic synchronisation in coupled map lattices: rigorous results. Discrete Contin. Dyn. Syst. B 4 (2004), 435455.
[11]Fernandez, B. and Jiang, M.. Coupling two unimodal maps of simple kneading sequences. Ergod. Th. & Dynam. Sys. 24 (2004), 107125.
[12]Gielis, G. and MacKay, R.. Coupled map lattices with phase transitions. Nonlinearity 13 (2000), 867888.
[13]Guckenheimer, J. and Holmes, P.. Nonlinear Oscillators, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York, 1983.
[14]Jiang, M. and Pesin, Y.. Equilibrium measures for coupled map lattices: existence, uniqueness and finite-dimensional approximations. Comm. Math. Phys. 193 (1998), 675711.
[15]Jost, J. and Joy, M.. Spectral properties and synchronization in coupled map lattices. Phys. Rev. E 65 (2001), 016201.
[16]Kaneko, K.. Period-doubling of kink-antikink patterns, quasiperiodicity in antiferro-like structures and spatial intermittency in coupled logistic lattice. Progr. Theoret. Phys. 72 (1984), 480486.
[17]Kaneko, K. (ed.) Theory and Applications of Coupled Map Lattices. Wiley, Chichester, 1993.
[18]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.
[19]Keller, G., Künzle, M. and Nowicki, T.. Some phase transitions in coupled map lattices. Phys. D 59 (1992), 3951.
[20]Keller, G. and Liverani, C.. Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension. Comm. Math. Phys. 262 (2006), 3350.
[21]Koiller, J. and Young, L.-S.. Coupled map networks. Nonlinearity 23 (2010), 11211141.
[22]Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.
[23]Lind, P., Corte-Real, J. and Gallas, J.. Modeling velocity in gradient flows with coupled-map lattices with advection. Phys. Rev. E 66 (2002), 016219.
[24]MacKay, R.. Dynamics of networks: features which persist from the uncoupled limit. Stochastic and Spatial Structures of Dynamical Systems, Lunel (1996), 81–104.
[25]Oono, Y. and Puri, S.. Computationally efficient modeling of ordering of quenched phases. Phys. Rev. Lett. 58 (1987), 836839.
[26]Rand, D.. The topological classification of Lorenz attractors. Math. Proc. Cambridge Philos. Soc. 83 (1978), 451460.
[27]Robinson, C.. Dynamical Systems, 2nd edn. CRC Press, Boca Raton, FL, 1999.
[28]Solé, R., Valls, J. and Bascompte, J.. Spiral waves, chaos and multiple attractors in lattice models of interacting populations. Phys. Lett. A 166 (1992), 123128.
[29]Waller, I. and Kapral, R.. 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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