Networks and Heterogeneous Media (NHM)

Asymptotic synchronous behavior of Kuramoto type models with frustrations

Pages: 33 - 64, Volume 9, Issue 1, March 2014      doi:10.3934/nhm.2014.9.33

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Seung-Yeal Ha - Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747, South Korea (email)
Yongduck Kim - Department of Mathematical Sciences, Seoul National University, Seoul 151-747, South Korea (email)
Zhuchun Li - Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China (email)

Abstract: We present a quantitative asymptotic behavior of coupled Kuramoto oscillators with frustrations and give some sufficient conditions for the parameters and initial condition leading to phase or frequency synchronization. We consider three Kuramoto-type models with frustrations. First, we study a general case with nonidentical oscillators; i.e., the natural frequencies are distributed. Second, as a special case, we study an ensemble of two groups of identical oscillators. For these mixture of two identical Kuramoto oscillator groups, we study the relaxation dynamics from the mixed stage to the phase-locked states via the segregation stage. Finally, we consider a Kuramoto-type model that was recently derived from the Van der Pol equations for two coupled oscillator systems in the work of Lück and Pikovsky [27]. In this case, we provide a framework in which the phase synchronization of each group is attained. Moreover, the constant frustration causes the two groups to segregate from each other, although they have the same natural frequency. We also provide several numerical simulations to confirm our analytical results.

Keywords:  Kuramoto model, bicluster configurations, frustration, synchronization.
Mathematics Subject Classification:  92D25, 34C15, 76N10.

Received: September 2013;      Revised: February 2014;      Available Online: April 2014.