LYAPUNOV FUNCTION FOR THE KURAMOTO MODEL OF NONLINEARLY COUPLED OSCILLATORS

A Lyapunov function for the phase-locked state of the Kuramoto model o f nonlinearly coupled oscillators is presented. It is also valid for f inite-range interactions and allows the introduction of thermodynamic formalism such as ground states and universality classes. For the Kura moto model, a minimum of the Lyapunov function corresponds to a ground state of a system with frustration: the interaction between the oscil lators, XY spins, is ferromagnetic, whereas the random frequencies ind uce random fields which try to break the ferromagnetic order, i.e., gl obal phase locking. The ensuing arguments imply asymptotic stability o f the phase-locked state (up to degeneracy) and hold for any probabili ty distribution of the frequencies. Special attention is given to disc rete distribution functions. We argue that in this case a perfect lock ing on each of the sublattices which correspond to the frequencies res ults, but that a partial locking of some but not all sublattices is no t to be expected. The order parameter of the phase-locked state is sho wn to have a strictly positive lower bound (r greater-than-or-equal-to 1/2), so that a continuous transition to a nonlocked state with vanis hing order parameter is to be excluded.