Characterizing nonstationary coherent states in globally coupled conformist and contrarian oscillators.

For decades, the description and characterization of nonstationary coherent states in coupled oscillators have not been available. We here consider the Kuramoto model consisting of conformist and contrarian oscillators. In the model, contrarians are chosen from a bimodal Lorentzian frequency distribution and flipped into conformists at random. A complete and systematic analytical treatment of the model is provided based on the Ott-Antonsen ansatz. In particular, we predict and analyze not only the stability of all stationary states (such as the incoherent, the π, and the traveling-wave states), but also that of the two nonstationary states: the Bellerophon and the oscillating-π state. The theoretical predictions are fully supported by extensive numerical simulations.

[1]  S. Boccaletti,et al.  Synchronization clusters emerge as the result of a global coupling among classical phase oscillators , 2019, New Journal of Physics.

[2]  Y. Zou,et al.  Clustering and Bellerophon state in Kuramoto model with second-order coupling. , 2019, Chaos.

[3]  S. Boccaletti,et al.  Origin of Bellerophon states in globally coupled phase oscillators , 2018, Physical Review E.

[4]  S. Boccaletti,et al.  Rhythmic synchronization and hybrid collective states of globally coupled oscillators , 2018, Scientific Reports.

[5]  Shuguang Guan,et al.  Coexistence of Quantized, Time Dependent, Clusters in Globally Coupled Oscillators. , 2016, Physical review letters.

[6]  Jie Zhou,et al.  Synchronization and Bellerophon states in conformist and contrarian oscillators , 2016, Scientific Reports.

[7]  Zonghua Liu,et al.  An efficient approach to suppress the negative role of contrarian oscillators in synchronization. , 2013, Chaos.

[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]  Jan R Engelbrecht,et al.  Structure of long-term average frequencies for Kuramoto oscillator systems. , 2012, Physical review letters.

[10]  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.

[11]  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.

[12]  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.

[13]  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.

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

[15]  Peter A Tass,et al.  Chimera states: the natural link between coherence and incoherence. , 2008, Physical review letters.

[16]  E. Ott,et al.  Synchronization in networks of networks: the onset of coherent collective behavior in systems of interacting populations of heterogeneous oscillators. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Zhilin Qu,et al.  Nonlinear dynamics of cardiac excitation-contraction coupling: an iterated map study. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  D. Zanette.  Synchronization and frustration in oscillator networks with attractive and repulsive interactions , 2005, cond-mat/0509063.

[19]  Juan M. López,et al.  Spontaneous emergence of contrarian-like behaviour in an opinion spreading model , 2005, physics/0508029.

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

[21]  S. Strogatz,et al.  Chimera states for coupled oscillators. , 2004, Physical review letters.

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

[23]  S. Galam.  Contrarian deterministic effects on opinion dynamics: “the hung elections scenario” , 2003, cond-mat/0307404.

[24]  Nancy Kopell,et al.  Synchronization in Networks of Excitatory and Inhibitory Neurons with Sparse, Random Connectivity , 2003, Neural Computation.

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

[26]  J. Acebrón,et al.  Breaking the symmetry in bimodal frequency distributions of globally coupled oscillators , 1997, patt-sol/9707003.

[27]  J. Crawford,et al.  Amplitude expansions for instabilities in populations of globally-coupled oscillators , 1993, patt-sol/9310005.

[28]  Renato Spigler,et al.  Nonlinear stability of incoherence and collective synchronization in a population of coupled oscillators , 1992 .

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

[30]  S. Strogatz,et al.  Stability of incoherence in a population of coupled oscillators , 1991 .

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

[32]  Di Yuan,et al.  Dynamics of the Kuramoto model in the presence of correlation between distributions of frequencies and coupling strengths. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

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