  We study the Kuramoto-Sakaguchi (KS) model composed by any N identical phase
oscillators symmetrically coupled. Ranging from local (one-to-one, R = 1) to
global (all-to-all, R = N/2) couplings, we derive the general solution that
describes the network dynamics next 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 KS model through bifurcation
analysis. Moreover, we present numerical evolutions of the network showing the
great accordance with our analytical one. The exact knowledge of the behavior
close to equilibriums is a fundamental step to investigate phenomena about
synchronization in networks. As an example, at the end we discuss the dynamics
behind chimera states from the point of view of our results.
