Exactly solvable phase oscillator models with synchronization dynamics

TL;DR

This paper introduces a family of exactly solvable phase oscillator models with singular couplings, providing analytical solutions for synchronization dynamics, especially for the case where the coupling function is the sign function.

Contribution

It presents new exactly solvable models with singular couplings, including an analytical solution for the sign coupling case, advancing understanding of synchronization.

Findings

Models with less singular coupling synchronize more easily.

Analytical solutions for stationary states using elliptic functions.

The sign coupling model is one of the few analytically solvable synchronization models.

Abstract

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.