Stability in the Kuramoto-Sakaguchi model for finite networks of identical oscillators

We study the Kuramoto-Sakaguchi model composed by N identical phase oscillators symmetrically coupled. Ranging from local (one-to-one, R=1 couplings, we derive a general solution that describes the network dynamics close to an equilibrium. Therewith, we build stability diagrams according to N and R bringing to the light a rich scenery of attractors, repellers, saddles, and non-hyperbolic equilibriums. Our result also uncovers the obscure repulsive regime of the model through bifurcation analysis. Numerical simulations show great accordance with our analytical studies. The exact knowledge of the behavior close to equilibriums may be a fundamental step to investigate phenomena about synchronization in networks. As an example, in the end, we discuss the dynamics behind chimera states from our results.