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

We consider a finite number of coupled oscillators on a ring 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, with solutions of type (m,l=0) being all stable. Following a number of recent works (see below) we indicate how the various solutions emerge as the coupling strength K is varied, presenting a perturbative expression for these for large K. The low K scenario is also briefly outlined, where the onset of synchronization by a tangent bifurcation is explained. The simplest situation involving three oscillators is described, where more than one tangent bifurcations are involved. Immediately before the tangent bifurcation leading to synchronization, the system exhibits the phenomenon of frequency- (or phase) splitting where more than one (usually two) phase clusters are involved. All the synchronized solutions are seen to be 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. Under a periodic driving with an appropriately limited detuning, there occurs entrainment of the phase-split solutions as well.