Rotational Symmetry-Breaking effects in the Kuramoto model

We study the bifurcations and phase diagram for a network of identical Kuramoto oscillators with a coupling that explicitly breaks the rotational symmetry of the equations. Applying the Watanabe-Strogatz ansatz, the original N-dimensional dynamics of the network collapses to a system of dimension two. Our analytical exploration uncovers bifurcation mechanisms, including transcritical, saddle-node, heteroclinic, Hopf, and Bogdanov-Takens bifurcations, that dictate transitions between collective states. Numerical validation of the full system confirms emergent phenomena such as oscillation death, global synchronization, and multicluster dynamics. By integrating reduced-model bifurcation theory with largescale simulations, we map phase diagrams that link parameter regimes to distinct dynamical phases. This work offers insights into multistability and pattern formation in coupled oscillator systems. Notably, the multicluster obtained exhibits behavior closely resembling the frequency-synchronized clusters identified in Hodgkin-Huxley neuron models.