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Symbolic dynamics of two coupled Lorenz maps: From uncoupled regime to synchronisation |
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| Authors: | R Coutinho B Fernandez P Guiraud |
| Institution: | a Group of Mathematical Physics of the University of Lisbon, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal b Department of Mathematics, Instituto Superior Técnico (TU Lisbon), Av. Rovisco Pais, 1049-001 Lisboa, Portugal c Centre de Physique Théorique, CNRS &; Universités d’Aix-Marseille I et II, et de Toulon, Campus de Luminy Case 907, 13288 Marseille CEDEX 09, France d Centro de Modelamiento Matemático (CNRS UMI 2807), Departamento de Ingeniería Matemática, Av. Blanco Encalada 2120, Universidad de Chile, Santiago, Chile e Departamento de Estadística-CIMFAV, Gran Bretaña 1091, Universidad de Valparaíso, Valparaíso, Chile |
| Abstract: | 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 monotonically 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. |
| Keywords: | Coupled map lattices Chaotic dynamics Symbolic dynamics Topological entropy Synchronisation |
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