Mean-field and mean-ensemble frequencies of a system of coupled oscillators

We investigate interacting phase oscillators whose mean field is at a different frequency from the mean or mode of their natural frequencies. The associated asymmetries lead to a macroscopic traveling wave. We show that the mean-ensemble frequency of such systems differs from their entrainment frequency. In some scenarios these frequencies take values that, counterintuitively, lie beyond the limits of the natural frequencies. The results indicate that a clear distinction should be drawn between the two variables describing the macroscopic dynamics of cooperative systems. This has important implications for real systems where a nontrivial distribution of parameters is common.

[1]  Arkady Pikovsky,et al.  Dynamics of heterogeneous oscillator ensembles in terms of collective variables , 2011 .

[2]  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.

[3]  Yoshiki Kuramoto,et al.  In International Symposium on Mathematical Problems in Theoretical Physics , 1975 .

[4]  Chris Arney.  Sync: The Emerging Science of Spontaneous Order , 2007 .

[5]  R. Spigler,et al.  The Kuramoto model: A simple paradigm for synchronization phenomena , 2005 .

[6]  E. Ott,et al.  Exact results for the Kuramoto model with a bimodal frequency distribution. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  E. Montbrió,et al.  Shear diversity prevents collective synchronization. , 2011, Physical review letters.

[8]  A Stefanovska,et al.  Stationary and traveling wave states of the Kuramoto model with an arbitrary distribution of frequencies and coupling strengths. , 2012, Physical review letters.

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

[10]  Edward Ott,et al.  Theoretical mechanics: Crowd synchrony on the Millennium Bridge , 2005, Nature.

[11]  Hidetsugu Sakaguchi,et al.  Cooperative Phenomena in Coupled Oscillator Systems under External Fields , 1988 .

[12]  Aneta Stefanovska,et al.  Asymmetry-induced effects in coupled phase-oscillator ensembles: Routes to synchronization. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  A. Stefanovska,et al.  Kuramoto model with time-varying parameters. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  J. Acebrón,et al.  Asymptotic description of transients and synchronized states of globally coupled oscillators , 1998 .

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

[16]  J. Cowan,et al.  Excitatory and inhibitory interactions in localized populations of model neurons. , 1972, Biophysical journal.

[17]  D. Zanette,et al.  Synchronization and clustering of phase oscillators with heterogeneous coupling , 2007 .

[18]  Lasko Basnarkov,et al.  Kuramoto model with asymmetric distribution of natural frequencies. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[20]  Diego Pazó,et al.  Existence of hysteresis in the Kuramoto model with bimodal frequency distributions. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  E. Ott,et al.  Low dimensional behavior of large systems of globally coupled oscillators. , 2008, Chaos.

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

[23]  K. Wiesenfeld.  New results on frequency-locking dynamics of disordered Josephson arrays , 1996 .

[24]  Aneta Stefanovska,et al.  Neuronal synchrony during anesthesia: a thalamocortical model. , 2008, Biophysical journal.

[25]  J. A. Acebron,et al.  Breaking the symmetry in bimodal frequency distributions of globally coupled oscillators , 1998 .