Bifurcation of the collective oscillatory state in phase oscillators with heterogeneity coupling

Abstract

We study the dynamical transition to the oscillatory state in a system of phase oscillators by considering the coupling heterogeneity. The bifurcation mechanism of the oscillatory state in the neighborhood of the critical point is clarified on the basis of two-time-scale analysis, and a theory which is equivalent to Crawford’s amplitude expansion based on the central manifold reduction. In contrast to the steady-state bifurcation, the $$\mathscr {O}(2)$$ symmetry, as well as the coupling scheme, denotes the properties of the eigenspectrum for the instability of the incoherence state, thereby determining the type for the emergence of the oscillatory state. We report that the bifurcation direction together with the stability of the oscillatory state can be changed by decreasing the noise strength or increasing the natural frequency heterogeneity. Theoretical predictions are consistent with numerical evidence, and our work provides another way for elucidating the dynamical phase transition in complex systems.