Competition for synchronization in a phase oscillator system

A system of phase oscillators with a Central Oscillator (CO) and a set of n Peripheral Oscillators (POs) is considered. Feed-forward and feedback connections between the CO and POs are determined by two interaction functions which are assumed to be smooth, odd, and periodic. To describe the competition of POs for synchronization with the CO, we study the asymptotic stability of fixed points corresponding to in-phase synchronization of a group of k POs, while other POs are in anti-phase with the CO. It is shown that stability conditions can be formulated in terms of four parameters that describe the slopes of the interaction functions at zero and half-period points. Analytical description of stability in terms of the regions in 4-dimensional parameter space is given. Combining stability analysis with the detailed study of geometry of invariant manifolds, the bifurcations of fixed points are investigated. We show that various dynamical regimes such as multistability, heteroclinic orbits, and chaos are possible. Analytical stability conditions for global synchronization of POs with the CO are formulated for the systems with local connections between POs. It is shown that synchronization in a large system with local connections becomes unstable even under weak desynchronizing influence from the CO. The application of the results to modeling in neuroscience and, in particular, for modeling visual attention is discussed.