Collective dynamics of identical phase oscillators with high-order coupling

In this paper, we propose a framework to investigate the collective dynamics in ensembles of globally coupled phase oscillators when higher-order modes dominate the coupling. The spatiotemporal properties of the attractors in various regions of parameter space are analyzed. Furthermore, a detailed linear stability analysis proves that the stationary symmetric distribution is only neutrally stable in the marginal regime which stems from the generalized time-reversal symmetry. Moreover, the critical parameters of the transition among various regimes are determined analytically by both the Ott-Antonsen method and linear stability analysis, the transient dynamics are further revealed in terms of the characteristic curves method. Finally, for the more general initial condition the symmetric dynamics could be reduced to a rigorous three-dimensional manifold which shows that the neutrally stable chaos could also occur in this model for particular parameter. Our theoretical analysis and numerical results are consistent with each other, which can help us understand the dynamical properties in general system with higher-order harmonics couplings.