Dynamics of phase oscillators under symmetry breaking and time delay couplings

We examine a nontrivial generalization of the coupled Kuramoto model with both symmetry-breaking interaction and time delay coupling. We show that this model exhibits three unique dynamical behaviors: incoherent state, stationary synchronized state, and standing wave state. In contrast to the symmetry-preserving coupling, time delay in the symmetry-breaking interaction results in three multistability regions. We elucidate that the latter two bistable regions emerge only in the presence of time delay in the symmetry-breaking coupling. The stability of the stationary synchronized state remains unaffected by the time delay interaction in the system. We deduce the low-dimensional equations for the complex order parameters using the Ott-Antonsen ansatz. We also deduce the Hopf, pitchfork, and saddle-node bifurcation curves from the evolution equations for the complex order parameters mediating the dynamical transitions. Simulation results of the original discrete set of equations for the generalized Kuramoto model agree well with the analytical bifurcation curves.