Synchronization between interacting ensembles of globally coupled chaotic maps

When two groups of globally coupled, identical chaotic maps interact, full synchronization (coherence) and different kinds of partial synchronization (clustering) can arise. In this paper, we identify the regions in parameter space that lead to coherence and clustering. The region of stability for the coherent state is shown to be independent of the number of maps in the ensembles. The borders of stability for the coherence region are obtained in terms of the transverse Lyapunov exponents and the nonlinearity parameter of the individual map. It is shown that for values of the nonlinearity parameter that correspond to periodic windows, the coherent state is stable in the unit square of the coupling parameter plane. For the simplest non-trivial case of four maps, we analyse the stability properties of the two-cluster states in detail and show cluster stability diagrams for particular values of the nonlinearity parameter of the logistic map. In the general case of 2N maps, we determine the invariant cluster manifolds in phase space and identify the corresponding cluster states. For each cluster state, we study the character of the in-cluster dynamics, which is found to be regular or chaotic.

[1]  ERRATUM: Dynamics of order parameters for globally coupled oscillators , 2002 .

[2]  K. Kaneko.  Clustering, coding, switching, hierarchical ordering, and control in a network of chaotic elements , 1990 .

[3]  E. Mosekilde,et al.  Chaotic Synchronization: Applications to Living Systems , 2002 .

[4]  J. Røttingen,et al.  Ruled by waves? Intracellular and intercellular calcium signalling. , 2000, Acta physiologica Scandinavica.

[5]  V N Belykh,et al.  Cluster synchronization modes in an ensemble of coupled chaotic oscillators. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  A. Scott,et al.  Excitation wave propagation as a possible mechanism for signal transmission in pancreatic islets of Langerhans. , 2001, Biophysical journal.

[7]  N. Holstein-Rathlou.  Synchronization of proximal intratubular pressure oscillations: evidence for interaction between nephrons , 1987, Pflügers Archiv.

[8]  S. Strogatz.  Exploring complex networks , 2001, Nature.

[9]  S. De Monte,et al.  Coherent regimes of globally coupled dynamical systems. , 2002, Physical review letters.

[10]  Kikuchi,et al.  Periodicity manifestations in the turbulent regime of the globally coupled map lattice , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  Kunihiko Kaneko,et al.  Complex Systems: Chaos and Beyond: A Constructive Approach with Applications in Life Sciences , 2000 .

[12]  I. Stewart,et al.  Coupled nonlinear oscillators and the symmetries of animal gaits , 1993 .

[13]  P. A. Tass,et al.  Effective desynchronization with a stimulation technique based on soft phase resetting , 2002 .

[14]  Vladimir belykh,et al.  Invariant manifolds and cluster synchronization in a family of locally coupled map lattices , 2000 .

[15]  W. D. Melo,et al.  ONE-DIMENSIONAL DYNAMICS , 2013 .

[16]  Role of asymmetric clusters in desynchronization of coherent motion , 2002 .

[17]  S H Strogatz,et al.  Coupled oscillators and biological synchronization. , 1993, Scientific American.

[18]  M. Golubitsky,et al.  Models of central pattern generators for quadruped locomotion I. Primary gaits , 2001, Journal of mathematical biology.

[19]  M. Golubitsky,et al.  Symmetry in locomotor central pattern generators and animal gaits , 1999, Nature.

[20]  Ian Stewart,et al.  A modular network for legged locomotion , 1998 .

[21]  A. Pikovsky,et al.  Resolving clusters in chaotic ensembles of globally coupled identical oscillators. , 2001, Physical review letters.

[22]  J. Milnor.  On the concept of attractor , 1985 .

[23]  M. Golubitsky,et al.  Models of central pattern generators for quadruped locomotion II. Secondary gaits , 2001, Journal of mathematical biology.

[24]  E. Mosekilde,et al.  Synchronization phenomena in nephron-nephron interaction. , 2001, Chaos.

[25]  E Mosekilde,et al.  Loss of coherence in a system of globally coupled maps. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.