Asymptotic synchronous behavior of Kuramoto type models with frustrations

Seung-Yeal Ha; Yongduck Kim; Zhuchun Li

doi:10.3934/nhm.2014.9.33

Article Contents

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

This issue Previous Article Sparse stabilization of dynamical systems driven by attraction and avoidance forces Next Article Numerical network models and entropy principles for isothermal junction flow

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: August 31, 2013

Revised: January 31, 2014

Published: February 2014

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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.

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Mathematics Subject Classification: 92D25, 34C15, 76N10.

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