Asymptotic synchronous behavior of Kuramoto type models with frustrations

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 ensembles of Kuramoto oscillators with frustration. 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. When two identical Kuramoto oscillator groups are mixed, we focus on the synchronous stage, which describes the relaxation of phase synchronization in each group from a segregation state. 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 [1] . 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. THE KURAMOTO MODEL WITH FRUSTRATIONS 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. More precisely, we present several sufficient conditions for the complete (frequency) synchronization in terms of the initial phase diameter, the (interaction) frustration, and the coupling strength. Synchronization is ubiquitous in various disciplines such as physics, biology, chemistry, and the social sciences [2]. However, rigorous mathematical treatments of synchronization were initiated only a few decades ago by two pioneers; namely, Winfree [3] and 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. Let θi = θi(t) be the phase of the ith Kuramoto oscillator. Further, let αji be the frustration between the jth and ith oscillators, which is assumed to be symmetric in i and j. In this situation, the dynamics of Kuramoto oscillators is governed by the following ODE system: