Wave Propagation in Nonlocally Coupled Oscillators with Noise

Unlike conservative oscillatory media, dissipative self-oscillatory media admit traveling waves without decay even in the presence of noise or other sources of randomness. This ability of wave transmission in random self-oscillatory media should be functionally relevant in a variety of living organisms for which randomness is unavoidable. For this type of systems, a critical strength of randomness is generally expected to exist such that below which the system is capable of sustaining undamped traveling waves. This critical point should be identical with the Hopf bifurcation point of an effective dynamical system obtained by properly renormalizing the effects of noise. It is reasonable to expect that a theory could be developed unambiguously on this issue in the particular case when each oscillator couples with sufficiently many oscillators, because a mean-field idea should be applicable then. In the present article, we carry out this program for nonlocally coupled phase oscillators with noise, and show how an effective dynamical equation can be derived, and how it is reduced to a small-amplitude equation near the bifurcation point. Our theory may be regarded as a natural extension of a previous theory 1) on the onset of collective oscillation for globally coupled phase oscillators with noise. Imagine an infinitely long array of nonlocally coupled oscillators which are distributed densely and subject to additive noise. By taking the continuum limit, the phases φ(x, t) of the oscillators are assumed to obey the following Langevin-type equation.