Exactly solvable phase oscillator models with synchronization dynamics

Populations of phase oscillators interacting globally through a general coupling function $f(x)$ have been considered. In the absence of precessing frequencies and for odd-coupling functions there exists a Lyapunov functional and the probability density evolves toward stable stationary states described by an equilibrium measure. We have then proposed a family of exactly solvable models with singular couplings which synchronize more easily as the coupling becomes less singular. The stationary solutions of the least singular coupling considered, $f(x)=$ sign$(x)$, have been found analytically in terms of elliptic functions. This last case is one of the few non trivial models for synchronization dynamics which can be analytically solved.