Bifurcations in phase oscillator networks with a central element

A system of phase oscillators with identical natural frequencies and the star-like architecture of connections is considered. Interaction functions are described by two terms of Fourier expansion. Bifurcation analysis of small systems containing 3 or 4 oscillators has been performed. The results are summarized in bifurcation diagrams that provide a full description of the boundaries between regions with different dynamics and the types of bifurcations that lead to the changes in the topology of phase space. The bifurcations include changes of fixed point stability and formation (destruction) of limit and heteroclinic cycles. For the system with 4 oscillators chaotic behaviour has been investigated. The results can be useful to control system dynamics through an appropriate choice and variation of parameter values. The generalization of the results to the systems with an arbitrary number of oscillators and application of the results in computational neuroscience are discussed.