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 Luck 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.
Highlights
The purpose of this paper is to study the dynamic interplay between distinct natural frequencies and phase shift in interactions among Kuramoto oscillators
Kuramoto and Sakaguchi [6] proposed a variant of the Kuramoto model in which the coupling function incorporated frustration so that richer dynamical phenomena would be observed than with no frustration
N i=1 sin(θj where Ωi is a natural frequency of the ith oscillator, K denotes the positive coupling strength, and N denotes the number of oscillators
Summary
The purpose of this paper is to study the dynamic interplay between distinct natural frequencies (intrinsic frustration) and phase shift in interactions (interaction frustration) among Kuramoto oscillators. We present several sufficient conditions for the complete (frequency) synchronization in terms of the initial phase diameter, the (interaction) frustration, and the coupling strength. Kuramoto [4,5], who introduced simple ODE models for limit-cycle oscillators. Kuramoto and Sakaguchi [6] proposed a variant of the Kuramoto model in which the coupling function incorporated frustration (phase shift) so that richer dynamical phenomena would be observed than with no frustration.
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