Cluster synchronization in networks of coupled nonidentical dynamical systems.

In this paper, we study cluster synchronization in networks of coupled nonidentical dynamical systems. The vertices in the same cluster have the same dynamics of uncoupled node system but the uncoupled node systems in different clusters are different. We present conditions guaranteeing cluster synchronization and investigate the relation between cluster synchronization and the unweighted graph topology. We indicate that two conditions play key roles for cluster synchronization: the common intercluster coupling condition and the intracluster communication. From the latter one, we interpret the two cluster synchronization schemes by whether the edges of communication paths lie in inter- or intracluster. By this way, we classify clusters according to whether the communications between pairs of vertices in the same cluster still hold if the set of edges inter- or intracluster edges is removed. Also, we propose adaptive feedback algorithms to adapting the weights of the underlying graph, which can synchronize any bi-directed networks satisfying the conditions of common intercluster coupling and intracluster communication. We also give several numerical examples to illustrate the theoretical results.

[1]  J. Rogers.  Chaos , 1876, Molecular Vibrations.

[2]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[3]  J. P. Lasalle.  Some Extensions of Liapunov's Second Method , 1960 .

[4]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[5]  H. Fujisaka,et al.  Stability Theory of Synchronized Motion in Coupled-Oscillator Systems , 1983 .

[6]  Scaling Behaviors of Characteristic Exponents near Chaotic Transition Points , 1984 .

[7]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

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

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

[10]  Carroll,et al.  Synchronous chaos in coupled oscillator systems. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  D. Signorini,et al.  Neural networks , 1995, The Lancet.

[12]  Leon O. Chua,et al.  Autonomous cellular neural networks: a unified paradigm for pattern formation and active wave propagation , 1995 .

[13]  L. Chua,et al.  Synchronization in an array of linearly coupled dynamical systems , 1995 .

[14]  Nikolai F. Rulkov,et al.  Images of synchronized chaos: Experiments with circuits. , 1996, Chaos.

[15]  I. Stewart,et al.  From attractor to chaotic saddle: a tale of transverse instability , 1996 .

[16]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Non-linear Systems , 1998, Autom..

[17]  Bernd Blasius,et al.  Complex dynamics and phase synchronization in spatially extended ecological systems , 1999, Nature.

[18]  Thomas de Quincey.  [C] , 2000, The Works of Thomas De Quincey, Vol. 1: Writings, 1799–1820.

[19]  Belykh,et al.  Hierarchy and stability of partially synchronous oscillations of diffusively coupled dynamical systems , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[20]  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.

[21]  L. Glass.  Synchronization and rhythmic processes in physiology , 2001, Nature.

[22]  Jürgen Kurths,et al.  Synchronization - A Universal Concept in Nonlinear Sciences , 2001, Cambridge Nonlinear Science Series.

[23]  Xiao Fan Wang,et al.  Synchronization in scale-free dynamical networks: robustness and fragility , 2001, cond-mat/0105014.

[24]  Jürgen Kurths,et al.  Synchronization: Phase locking and frequency entrainment , 2001 .

[25]  Kevin M. Passino,et al.  Biomimicry of bacterial foraging for distributed optimization and control , 2002 .

[26]  S. Boccaletti,et al.  Synchronization of chaotic systems , 2001 .

[27]  J. Jost,et al.  Spectral properties and synchronization in coupled map lattices. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  Bernd Blasius,et al.  Complex Synchronization Phenomena in Ecological Systems , 2002 .

[29]  G. Rangarajan,et al.  Stability of synchronized chaos in coupled dynamical systems , 2002, nlin/0201037.

[30]  R. E. Amritkar,et al.  Self-organized and driven phase synchronization in coupled maps. , 2002, Physical review letters.

[31]  Guanrong Chen,et al.  Complex networks: small-world, scale-free and beyond , 2003 .

[32]  M. Hasler,et al.  Persistent clusters in lattices of coupled nonidentical chaotic systems. , 2003, Chaos.

[33]  G. Rangarajan,et al.  General stability analysis of synchronized dynamics in coupled systems. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  M. Hasler,et al.  Connection Graph Stability Method for Synchronized Coupled Chaotic Systems , 2004 .

[35]  M. Hasler,et al.  Blinking model and synchronization in small-world networks with a time-varying coupling , 2004 .

[36]  J. Kurths,et al.  Synchronization of two interacting populations of oscillators. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  Charles M. Gray,et al.  Synchronous oscillations in neuronal systems: Mechanisms and functions , 1994, Journal of Computational Neuroscience.

[38]  Kao-Shing Hwang,et al.  Cooperative strategy based on adaptive Q-learning for robot soccer systems , 2004, IEEE Transactions on Fuzzy Systems.

[39]  Sarika Jalan,et al.  Synchronized clusters in coupled map networks. II. Stability analysis. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  Debin Huang,et al.  A Simple Adaptive-feedback Controller for Identical Chaos Synchronization , 2022 .

[41]  A. Schnitzler,et al.  Normal and pathological oscillatory communication in the brain , 2005, Nature Reviews Neuroscience.

[42]  C. Wu.  Synchronization in networks of nonlinear dynamical systems coupled via a directed graph , 2005 .

[43]  Jinde Cao,et al.  Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters. , 2005, Chaos.

[44]  S. Boccaletti,et al.  Synchronization is enhanced in weighted complex networks. , 2005, Physical review letters.

[45]  Sarika Jalan,et al.  Synchronized clusters in coupled map networks. I. Numerical studies. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[46]  Changsong Zhou,et al.  Universality in the synchronization of weighted random networks. , 2006, Physical review letters.

[47]  Gang Zhang,et al.  A new method to realize cluster synchronization in connected chaotic networks. , 2006, Chaos.

[48]  Kevin M. Passino,et al.  Stable task load balancing strategies for Cooperative control of networked autonomous air vehicles , 2006, IEEE Transactions on Control Systems Technology.

[49]  Tianping Chen,et al.  New approach to synchronization analysis of linearly coupled ordinary differential systems , 2006 .

[50]  Changsong Zhou,et al.  Dynamical weights and enhanced synchronization in adaptive complex networks. , 2006, Physical review letters.

[51]  M Chavez,et al.  Synchronization in dynamical networks: evolution along commutative graphs. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[52]  V. Latora,et al.  Complex networks: Structure and dynamics , 2006 .

[53]  Jinde Cao,et al.  Global synchronization in arrays of delayed neural networks with constant and delayed coupling , 2006 .

[54]  Junan Lu,et al.  Adaptive synchronization of an uncertain complex dynamical network , 2006, IEEE Transactions on Automatic Control.

[55]  Jean-Jacques E. Slotine,et al.  Stable concurrent synchronization in dynamic system networks , 2005, Neural Networks.

[56]  Wenlian Lu.  Adaptive dynamical networks via neighborhood information: synchronization and pinning control. , 2007, Chaos.

[57]  E. Ott,et al.  Network synchronization of groups. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[58]  Tianping Chen,et al.  Boundedness and synchronization of y-coupled Lorenz systems with or without controllers ☆ , 2008 .

[59]  Tianping Chen,et al.  Partial synchronization in linearly and symmetrically coupled ordinary differential systems , 2009 .

[60]  Wei Wu,et al.  Cluster Synchronization of Linearly Coupled Complex Networks Under Pinning Control , 2009, IEEE Transactions on Circuits and Systems I: Regular Papers.

[61]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.