Synchronisation on non-compact manifolds

Abstract

The Kuramoto model is a widely studied model of phase synchronisation which continues to display novel and interesting emergent behaviours with its many extensions and generalisations. We formulate a non-compact version of the Kuramoto model by replacing the compact symmetry group SO(2) with the non-compact group SO(1, 1). The N equations are similar to the Kuramoto equations, except that the trigonometric terms are replaced with hyperbolic functions. Solution trajectories are generally unbounded, lying on the unit hyperbola, but synchronisation occurs for any positive coupling constants and arbitrary driving parameters. We show solutions can develop singularities for negative coupling constants. We also develop a vector model of synchronisation on non-compact manifolds. By choosing the group SO(1, 2) trajectories are confined to a one-sheeted hyperboloid. This model also displays unbounded trajectories but can synchronise for negative couplings and restricted initial conditions. Singularities can occur for positive couplings or if the initial conditions are too widely distributed. We describe a physical interpretation of the SO(1, 1) model as a system of interacting relativistic particles in 1 + 1 spacetime dimensions.