Cooperative Phenomena in Networks of Oscillators With Non-Identical Interactions and Dynamics

The incipience of synchrony in a diverse population of phase oscillators with non-identical interactions is an intriguing phenomenon. We study frequency synchronization of such oscillators composing networks with arbitrary topology in the context of the Kuramoto model and we show that its synchronization manifold is exponentially stable when the coupling has certain properties. Several example systems with periodic linear, cubic and sinusoidal coupling functions were examined, some including frustration and external fields. The numerical results confirmed the analytic findings and revealed some other interesting occurrences, like phase clustering in a synchronized network of strongly coupled oscillators. We also analyze the effects of the topology by considering random weighted networks.

[1]  Mark W. Spong,et al.  On Exponential Synchronization of Kuramoto Oscillators , 2009, IEEE Transactions on Automatic Control.

[2]  Oliver Mason,et al.  On Computing the Critical Coupling Coefficient for the Kuramoto Model on a Complete Bipartite Graph , 2009, SIAM J. Appl. Dyn. Syst..

[3]  A. Winfree.  Biological rhythms and the behavior of populations of coupled oscillators. , 1967, Journal of theoretical biology.

[4]  István Z Kiss,et al.  Predicting mutual entrainment of oscillators with experiment-based phase models. , 2005, Physical review letters.

[5]  H. J. Herrmann,et al.  How to suppress undesired synchronization , 2012, Scientific Reports.

[6]  Y. Kuramoto,et al.  A Soluble Active Rotater Model Showing Phase Transitions via Mutual Entertainment , 1986 .

[7]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[8]  D. Cumin,et al.  Generalising the Kuramoto Model for the study of Neuronal Synchronisation in the Brain , 2007 .

[9]  Nathaniel N. Urban,et al.  Predicting synchronized neural assemblies from experimentally estimated phase-resetting curves , 2006, Neurocomputing.

[10]  Wiesenfeld,et al.  Synchronization transitions in a disordered Josephson series array. , 1996, Physical review letters.

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

[12]  A. Jadbabaie,et al.  On the stability of the Kuramoto model of coupled nonlinear oscillators , 2005, Proceedings of the 2004 American Control Conference.

[13]  H. Nijmeijer,et al.  Cooperative oscillatory behavior of mutually coupled dynamical systems , 2001 .

[14]  G. Ermentrout.  Stable periodic solutions to discrete and continuum arrays of weakly coupled nonlinear oscillators , 1992 .

[15]  H. Daido,et al.  Order Function and Macroscopic Mutual Entrainment in Uniformly Coupled Limit-Cycle Oscillators , 1992 .

[16]  I. Gutman,et al.  Generalized inverse of the Laplacian matrix and some applications , 2004 .

[17]  J. Buck,et al.  Synchronous fireflies. , 1976, Scientific American.

[18]  Edward Ott,et al.  Theoretical mechanics: crowd synchrony on the Millennium Bridge. , 2005 .

[19]  Alessio Franci,et al.  Phase-locking between Kuramoto oscillators: Robustness to time-varying natural frequencies , 2010, 49th IEEE Conference on Decision and Control (CDC).

[20]  Richard M. Murray,et al.  Consensus problems in networks of agents with switching topology and time-delays , 2004, IEEE Transactions on Automatic Control.

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

[22]  Ljupco Kocarev,et al.  A unifying definition of synchronization for dynamical systems. , 1998, Chaos.

[23]  Martin W. McCall,et al.  Numerical simulation of a large number of coupled lasers , 1993 .

[24]  Yoshiki Kuramoto,et al.  Chemical Oscillations, Waves, and Turbulence , 1984, Springer Series in Synergetics.

[25]  Louis M. Pecora,et al.  Synchronizing chaotic circuits , 1991 .

[26]  Seung-Yeal Ha,et al.  Flocking and synchronization of particle models , 2010 .

[27]  Charles S. Peskin,et al.  Mathematical aspects of heart physiology , 1975 .

[28]  P. Rowlinson.  ALGEBRAIC GRAPH THEORY (Graduate Texts in Mathematics 207) By CHRIS GODSIL and GORDON ROYLE: 439 pp., £30.50, ISBN 0-387-95220-9 (Springer, New York, 2001). , 2002 .

[29]  Daido,et al.  Quasientrainment and slow relaxation in a population of oscillators with random and frustrated interactions. , 1992, Physical review letters.

[30]  Daido,et al.  Multibranch Entrainment and Scaling in Large Populations of Coupled Oscillators. , 1996, Physical review letters.

[31]  Reza Olfati-Saber,et al.  Consensus and Cooperation in Networked Multi-Agent Systems , 2007, Proceedings of the IEEE.

[32]  Lee DeVille,et al.  Fully synchronous solutions and the synchronization phase transition for the finite-N Kuramoto model. , 2011, Chaos.

[33]  Edward Ott,et al.  Cluster synchrony in systems of coupled phase oscillators with higher-order coupling. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  Maurizio Porfiri,et al.  Synchronization in Random Weighted Directed Networks , 2008, IEEE Transactions on Circuits and Systems I: Regular Papers.

[35]  Dirk Aeyels,et al.  Existence of Partial Entrainment and Stability of Phase Locking Behavior of Coupled Oscillators , 2004 .

[36]  J. L. Hemmen,et al.  Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators , 1993 .

[37]  R. C. Compton,et al.  Quasi-optical power combining using mutually synchronized oscillator arrays , 1991 .

[38]  S. Strogatz,et al.  The spectrum of the locked state for the Kuramoto model of coupled oscillators , 2005 .

[39]  B. Bollobás.  The evolution of random graphs , 1984 .

[40]  T. J. Walker,et al.  Acoustic Synchrony: Two Mechanisms in the Snowy Tree Cricket , 1969, Science.

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

[42]  Y. Kuramoto,et al.  Phase transitions in active rotator systems , 1986 .

[43]  Juan P. Torres,et al.  The Kuramoto model: A simple paradigm for synchronization phenomena , 2005 .

[44]  Richard M. Murray,et al.  Consensus Protocols for Undirected Networks of Dynamic Agents with Communication Time-Delays , 2003 .

[45]  Lin Huang,et al.  Consensus of Multiagent Systems and Synchronization of Complex Networks: A Unified Viewpoint , 2016, IEEE Transactions on Circuits and Systems I: Regular Papers.

[46]  Florian Dörfler,et al.  On the Critical Coupling for Kuramoto Oscillators , 2010, SIAM J. Appl. Dyn. Syst..

[47]  Seung-Yeal Ha,et al.  On the complete synchronization of the Kuramoto phase model , 2010 .