Remarks on the stability properties of the Kuramoto-Sakaguchi-Fokker-Planck equation with frustration

Abstract

We present stability properties of the Kuramoto–Sakaguchi–Fokker–Planck (in short, KS–FP) equation with frustration arising from the synchronization modeling of a large ensemble of weakly coupled oscillators under the effect of uniform frustration (phase-lag). Even in the absence of Fokker–Planck term (i.e., the diffusion term), the existence of frustration destroys a gradient flow structure of the original kinetic Kuramoto equation. Thus, useful machineries from the gradient flow theory cannot be used in the large-time behavior of the equation as it is. In this paper, we address two stability estimates for the KS–FP equation. First, we present a sufficient framework leading to the asymptotic stability of the incoherent state. Our stability framework has been formulated in terms of the probability density function of natural frequencies, coupling strength, diffusion coefficient, size of frustration and initial data. Second, we show that the KS–FP equation is structurally stable with respect to the size of frustration. More precisely, we show that the solution to the KS–FP equation tends to the corresponding solution to the KS–FP with zero frustration, as the size of frustration tends to zero under a suitable framework. © 2018, Springer International Publishing AG, part of Springer Nature.