On the double sphere model of synchronization

Abstract

The Kuramoto model has many higher-dimensional generalizations with similar synchronization behaviour, such as phase oscillators on the n-sphere which under suitable conditions can synchronize in any dimension n, with applications to swarming and flocking phenomena as well as to consensus and opinion formation. We consider further generalizations in which trajectories lie on the double sphere Sn−1×Sm−1 for any dimension m,n with synchronization properties that depend on the underlying parameters, and reduce to previously studied models for specific m or n. In such systems the synchronization properties on one sphere can be controlled from the other sphere by parameters such as the frequencies of oscillation and the connectivity coefficients. We show in particular that for these models the conformal group SO(n,m) acts as a time-evolution matrix, for example in the special case of the Kuramoto model there is an SO(2,1) matrix at each node which maps the initial value to the final value at any time t. For uniform coupling and identical frequencies, the time evolution of the Kuramoto model is controlled over all nodes by a single SO(2,1) matrix, a property which follows from the Watanabe–Strogatz reduction. This generalizes to the double sphere model with uniform coupling and identical frequency matrices, which we show is partially integrable for any m,n, and the system again evolves by means of a single time-evolution matrix in SO(n,m) which acts over all nodes. We explicitly perform the partial integration by means of a unit map which generalizes linear fractional transformations, equivalently Möbius maps Sn−1→Sn−1 for the case m=1, which have been previously used to reduce the n-sphere equations. As with the Kuramoto model, there exist conserved cross ratios which restrict all solutions to lie in a low-dimensional manifold. We parametrize SO(n,m) and so derive reduced equations, independent of N in number, which exactly solve the double sphere equations for any N. These models, and their further extensions to unit sphere systems with fifth-order nonlinearities, furnish a wide range of exactly reducible synchronization models which can be used to investigate systems for very large N.