Coherent regimes of globally coupled dynamical systems.

This Letter presents a method by which the mean field dynamics of a population of dynamical systems with parameter diversity and global coupling can be described in terms of a few macroscopic degrees of freedom. The method applies to populations of any size and functional form in the region of coherence. It requires linear variation or a narrow distribution for the dispersed parameter. Although an approximation, the method allows us to quantitatively study the transitions among the collective regimes as bifurcations of the effective macroscopic degrees of freedom. To illustrate, the phenomenon of oscillator death and the route to full locking are examined for chaotic oscillators with time scale mismatch.

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

[2]  A. Winfree.  The geometry of biological time , 1991 .

[3]  Yoshiki Kuramoto,et al.  Rhythms and turbulence in populations of chemical oscillators , 1981 .

[4]  Y. Yamaguchi,et al.  Theory of self-synchronization in the presence of native frequency distribution and external noises , 1984 .

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

[6]  K. Bar-Eli,et al.  On the stability of coupled chemical oscillators , 1985 .

[7]  Hadley,et al.  Phase locking of Josephson-junction series arrays. , 1988, Physical review. B, Condensed matter.

[8]  G. B. Ermentrout,et al.  The uniqueness and stability of the rest state for strongly coupled oscillators , 1989 .

[9]  M. Shiino,et al.  Synchronization of infinitely many coupled limit-cycle type oscillators , 1989 .

[10]  S. Strogatz,et al.  Phase diagram for the collective behavior of limit-cycle oscillators. , 1990, Physical review letters.

[11]  G. Ermentrout.  Oscillator death in populations of “all to all” coupled nonlinear oscillators , 1990 .

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

[13]  Hiroaki Daido.  Order Function Theory of Macroscopic Phase-Locking in Globally and Weakly Coupled Limit-Cycle Oscillators , 1997 .

[14]  A. Sherman,et al.  Diffusively coupled bursters: Effects of cell heterogeneity , 1998 .

[15]  Alexander S. Mikhailov,et al.  Mutual Synchronization of Molecular Turnover Cycles in Allosteric Enzymes , 1998 .

[16]  Sune Danø,et al.  Sustained oscillations in living cells , 1999, Nature.

[17]  Kunihiko Kaneko,et al.  Noiseless Collective Motion out of Noisy Chaos , 1998, chao-dyn/9812007.

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

[19]  S. De Monte,et al.  Synchronization of glycolytic oscillations in a yeast cell population. , 2002, Faraday discussions.

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

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

[22]  Mutual synchronization of molecular turnover cycles in allosteric enzymes III. Intramolecular cooperativity , 2002 .

[23]  ERRATUM: Dynamics of order parameters for globally coupled oscillators , 2002 .

[24]  John L Hudson,et al.  Emerging Coherence in a Population of Chemical Oscillators , 2002, Science.