The existence of generalized synchronization of chaotic systems in complex networks.

The paper studies the existence of generalized synchronization in complex networks, which consist of chaotic systems. When a part of modified nodes are chaotic, and the others have asymptotically stable equilibriums or orbital asymptotically stable periodic solutions, under certain conditions, the existence of generalized synchronization can be turned to the problem of contractive fixed point in the family of Lipschitz functions. In addition, theoretical proofs are proposed to the exponential attractive property of generalized synchronization manifold. Numerical simulations validate the theory.

[1]  E. Hille,et al.  Lectures on ordinary differential equations , 1968 .

[2]  Carroll,et al.  Synchronization in chaotic systems. , 1990, Physical review letters.

[3]  L. Tsimring,et al.  Generalized synchronization of chaos in directionally coupled chaotic systems. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[4]  Parlitz,et al.  Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems. , 1996, Physical review letters.

[5]  H. Abarbanel,et al.  Generalized synchronization of chaos: The auxiliary system approach. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[6]  J. Yorke,et al.  Differentiable generalized synchronization of chaos , 1997 .

[7]  Jiong Ruan,et al.  An improved method in synchronization of chaotic svstems , 1998 .

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

[9]  Jiong Ruan,et al.  Synchronization of chaotic systems by feedback , 1999 .

[10]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[11]  J. Stark.  Regularity of invariant graphs for forced systems , 1999, Ergodic Theory and Dynamical Systems.

[12]  V S Afraimovich,et al.  Multivalued mappings in generalized chaos synchronization. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  Valentin Afraimovich,et al.  Synchronization in directionally coupled systems: Some rigorous results , 2001 .

[14]  Steven J Schiff,et al.  Limits to the experimental detection of nonlinear synchrony. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  J. M. Gonzalez-Miranda,et al.  Generalized synchronization in directionally coupled systems with identical individual dynamics. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Jun-an Lu,et al.  Synchronization of a unified chaotic system and the application in secure communication , 2002 .

[17]  Atsushi Uchida,et al.  Generalized synchronization of chaos in identical systems with hidden degrees of freedom. , 2003, Physical review letters.

[18]  O. Calvo,et al.  Characterization of the anticipated synchronization regime in the coupled FitzHugh–Nagumo model for neurons , 2003 .

[19]  Evelyn Sander,et al.  The geometry of chaos synchronization. , 2003, Chaos.

[20]  Fortunato Tito Arecchi,et al.  Chaotic Neuron Dynamics, Synchronization, and Feature Binding , 2003, Summer School on Neural Networks.

[21]  N. Rulkov,et al.  Detectability of nondifferentiable generalized synchrony. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Atsushi Uchida,et al.  Generalized synchronization of spatiotemporal chaos in a liquid crystal spatial light modulator. , 2004, Physical review letters.

[23]  S. Bowong.  Stability analysis for the synchronization of chaotic systems with different order: application to secure communications , 2004 .

[24]  Alexey A Koronovskii,et al.  Generalized synchronization: a modified system approach. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Jinde Cao,et al.  Adaptive synchronization in tree-like dynamical networks , 2007 .

[26]  Jinde Cao,et al.  Synchronization-based approach for parameters identification in delayed chaotic neural networks , 2007 .

[27]  Jin Zhou,et al.  Synchronization in complex delayed dynamical networks with impulsive effects , 2007 .

[28]  Yao-Chen Hung,et al.  Paths to globally generalized synchronization in scale-free networks. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  Aihua Hu,et al.  Stochastic linear generalized synchronization of chaotic systems via robust control , 2008 .

[30]  Manfeng Hu,et al.  Adaptive feedback controller for projective synchronization , 2008 .

[31]  Hölder continuity of two types of generalized synchronization manifold. , 2008, Chaos.

[32]  César Cruz-Hernández,et al.  Communicating via synchronized time-delay Chua’s circuits , 2008 .

[33]  Licheng Jiao,et al.  Synchronization in dynamic networks with nonsymmetrical time-delay coupling based on linear feedback controllers , 2008 .

[34]  Aihua Hu,et al.  Hölder continuity of generalized synchronization of three bidirectionally coupled chaotic systems , 2009 .

[35]  Hölder continuity of three types of generalized synchronization manifolds of non-autonomous systems , 2009 .