A Soluble Active Rotator Model Showing Phase Transitions via Mutual Entrainment

Some analytical results are obtained for a large population of limit-cycle oscillators modelled by a set of deterministic equations φ = ωi-N-1K ΣNj=1 sin (φi-φj+α) (i=1,2, …, N), where φi is the phase of the i-th oscillator and ωi'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 ωi. 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.