On dynamics of four globally phase-coupled oscillators with close frequencies

An array of four globally phase-coupled oscillators with slightly different eigenfrequencies is considered. In the case of equal frequencies the system is reduced to an integrable one with almost all phase trajectories being closed. In the case of different but close to each other eigenfrequencies the system is treated with the use of the averaging method. It is shown that probabilistic phenomena take place in the system: the phase flow is divided quasi-randomly between various regions of the phase space when passing through separatrices of the unperturbed problem. Formulas describing probabilities of the phase point transition to different regions are obtained.