Asymptotic synchronous behavior of Kuramoto type models with frustrations

Seung-Yeal Ha; Yongduck Kim; Zhuchun Li; Seung-Yeal Ha; Yongduck Kim; Zhuchun Li

doi:10.3934/nhm.2014.9.33

Networks and Heterogeneous Media

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

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Asymptotic synchronous behavior of Kuramoto type models with frustrations

1.
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747
2.
Department of Mathematical Sciences, Seoul National University, Seoul 151-747
3.
Department of Mathematics, Harbin Institute of Technology, Harbin 150001

Received: 01 September 2013 Revised: 01 February 2014
92D25, 34C15, 76N10.

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.
- Kuramoto model,
- bicluster configurations,
- frustration,
- synchronization
Citation: Seung-Yeal Ha, Yongduck Kim, Zhuchun Li. Asymptotic synchronous behavior of Kuramoto type models with frustrations[J]. Networks and Heterogeneous Media, 2014, 9(1): 33-64. doi: 10.3934/nhm.2014.9.33

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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.

References

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