Synchronization of phase oscillators with attractive and repulsive interconnections

We consider Kuramoto model of phase oscillators with attractive-repulsive interconnections, that is, part of interconnection gains can be negative. We generalize results on existence of phase locked solutions of input-output weighted sign-symmetric interconnections and present corresponding necessary and sufficient conditions for phase locking. We give an expression for synchronization frequency, which appears to be independent of signs appearing in the interconnection matrix. Local asymptotic stability of phase locked solutions is analysed and finally our results are illustrated through a demonstrative example.

[1]  F. Paganini,et al.  Global considerations on the Kuramoto model of sinusoidally coupled oscillators , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[2]  C. Miniussi,et al.  New insights into rhythmic brain activity from TMS–EEG studies , 2009, Trends in Cognitive Sciences.

[3]  Claudio Altafini,et al.  Consensus Problems on Networks With Antagonistic Interactions , 2013, IEEE Transactions on Automatic Control.

[4]  Florian Dörfler,et al.  Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators , 2009, Proceedings of the 2010 American Control Conference.

[5]  Hongkui Zeng,et al.  Differential tuning and population dynamics of excitatory and inhibitory neurons reflect differences in local intracortical connectivity , 2011, Nature Neuroscience.

[6]  Yoshiki Kuramoto,et al.  Cooperative Dynamics of Oscillator Community : A Study Based on Lattice of Rings , 1984 .

[7]  Oliver Mason,et al.  Global Phase-Locking in Finite Populations of Phase-Coupled Oscillators , 2007, SIAM J. Appl. Dyn. Syst..

[8]  D. Aeyels,et al.  Stability of phase locking in a ring of unidirectionally coupled oscillators , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[9]  S. Strogatz.  From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators , 2000 .

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

[11]  Jurgen Kurths,et al.  Synchronization in complex networks , 2008, 0805.2976.

[12]  Zoran Levnajić.  Emergent multistability and frustration in phase-repulsive networks of oscillators. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  M. Elowitz,et al.  A synthetic oscillatory network of transcriptional regulators , 2000, Nature.

[14]  Elena Panteley,et al.  Asymptotic phase synchronization of Kuramoto model with weighted non-symmetric interconnections: A case study , 2013, 52nd IEEE Conference on Decision and Control.

[15]  Richard Bellman,et al.  Introduction to Matrix Analysis , 1972 .

[16]  Hyunsuk Hong,et al.  Kuramoto model of coupled oscillators with positive and negative coupling parameters: an example of conformist and contrarian oscillators. , 2011, Physical review letters.

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

[18]  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..

[19]  Phil Husbands,et al.  Exploring the Kuramoto model of coupled oscillators in minimally cognitive evolutionary robotics tasks , 2010, IEEE Congress on Evolutionary Computation.

[20]  Peter Brown,et al.  Parkinsonian Beta Oscillations in the External Globus Pallidus and Their Relationship with Subthalamic Nucleus Activity , 2008, The Journal of Neuroscience.

[21]  Kaspar Anton Schindler,et al.  Synchronization and desynchronization in epilepsy: controversies and hypotheses , 2012, The Journal of physiology.

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

[23]  Hyunsuk Hong,et al.  Conformists and contrarians in a Kuramoto model with identical natural frequencies. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

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