The dynamics of network coupled phase oscillators: an ensemble approach.

We consider the dynamics of many phase oscillators that interact through a coupling network. For a given network connectivity we further consider an ensemble of such systems where, for each ensemble member, the set of oscillator natural frequencies is independently and randomly chosen according to a given distribution function. We then seek a statistical description of the dynamics of this ensemble. Use of this approach allows us to apply the recently developed ansatz of Ott and Antonsen [Chaos 18, 037113 (2008)] to the marginal distribution of the ensemble of states at each node. This, in turn, results in a reduced set of ordinary differential equations determining these marginal distribution functions. The new set facilitates the analysis of network dynamics in several ways: (i) the time evolution of the reduced system of ensemble equations is much smoother, and thus numerical solutions can be obtained much faster by use of longer time steps; (ii) the new set of equations can be used as a basis for obtaining analytical results; and (iii) for a certain type of network, a reduction to a low dimensional description of the entire network dynamics is possible. We illustrate our approach with numerical experiments on a network version of the classical Kuramoto problem, first with a unimodal frequency distribution, and then with a bimodal distribution. In the latter case, the network dynamics is characterized by bifurcations and hysteresis involving a variety of steady and periodic attractors.