Synchronized oscillations on a Kuramoto ring and their entrainment under periodic driving

Abstract: We consider a finite number of coupled oscillators as an adaptation of the Kuramoto model of populations of oscillators. The synchronized solutions are characterized by an integer $m$, the winding number, and a second integer $l$. Synchronized solutions of type ($m$, $l=0$) are all stable, and an explicit perturbative expression for these for large values of the coupling constant $K$ is presented. For low $K$, these solutions appear at certain specific values, each merging with a solution of type ($m$, $l=1$), both these solutions being stable for $K$ close to the relevant value. The ($m$, 0) solution continues to be stable for larger $K$, while the ($m$, 1) solution, on continuation in $K$ becomes unstable and then merges with a new type (with a different $m$ and l) of unstable solution. The ($m$,0) type solutions for large $K$ are in the nature of phase waves traveling round the ring. All the stable synchronized solutions are entrained by an external periodic driving, provided that the driving frequency is sufficiently close to the frequency of the synchronized population. A perturbative approach is outlined for the construction of the entrained solutions. For a given amplitude of driving, there is a certain maximum detuning between the two up to which the entrained solution persists. The question of stability of the entrained solution is addressed. The simplest situation involving three oscillators is investigated in details where the onset of synchronization is seen to occur through a tangent bifurcation at some critical value of $K$. Immediately before the appearance of the first synchronized solution, the system exhibits intermittent chaos with a typical scaling for the duration of the laminar phase. Under a periodic driving with an appropriately limited detuning, there occurs entrainment of the chaotic solution.