A Soluble Active Rotater Model Showing Phase Transitions via Mutual Entertainment

Some analytical results are obtained for a large population of limit-cycle oscillators modelled by a set of deterministic equations dot{φ} = Ω_i-N^-1K Sigma;^N_j=1 sin (φ_i-φ_j+α) (i=1,2, \cdots, N), where φ_i is the phase of the i-th oscillator and Ω_i's are parameters distributed randomly. The present work is a generalization of the previous one where the study was limited to the case of vanishing α and symmetric distribution of Ω_i. As in the previous case, a particular macroscopic solution of steady rotation is found, which branches off the trivial solution at some positive K. A computer simulation with N=1000 is carried out, which correctly reproduces our analytical results.