Exactly Solvable Phase Oscillator Models with Synchronization Dynamics

Populations of phase oscillators interacting globally through a general coupling function $f\left(x\right)$ have been considered. We analyze the conditions required to ensure the existence of a Lyapunov functional giving close expressions for it in terms of a generating function. We have also proposed a family of exactly solvable models with singular couplings showing that it is possible to map the synchronization phenomenon into other physical problems. In particular, the stationary solutions of the least singular coupling considered, $f\left(x\right)=\mathrm{sgn}\left(x\right)$, have been found analytically in terms of elliptic functions. This last case is one of the few nontrivial models for synchronization dynamics which can be analytically solved.