Continuous versus Discontinuous Transitions in the $D$-Dimensional Generalized Kuramoto Model: Odd $D$ is Different

TL;DR

This paper extends the Kuramoto model to higher dimensions, revealing that in odd dimensions, the transition to collective coherence is discontinuous at zero coupling, unlike the continuous transition in 2D and even dimensions.

Contribution

It introduces a generalized $D$-dimensional Kuramoto model and uncovers the distinct discontinuous transition in odd dimensions, especially 3D, contrasting with known 2D behavior.

Findings

Discontinuous transition to coherence in odd $D$ at zero coupling.

Continuous transition in 2D and even $D$ at positive critical coupling.

Applicability to 3D swarming and flocking models.

Abstract

The Kuramoto model, originally proposed to model the dynamics of many interacting oscillators, has been used and generalized for a wide range of applications involving the collective behavior of large heterogeneous groups of dynamical units whose states are characterized by a scalar angle variable. One such application in which we are interested is the alignment of orientation vectors among members of a swarm. Despite being commonly used for this purpose, the Kuramoto model can only describe swarms in 2 dimensions, and hence the results obtained do not apply to the often relevant situation of swarms in 3 dimensions. Partly based on this motivation, as well as on relevance to the classical, mean-field, zero-temperature Heisenberg model with quenched site disorder, in this paper we study the Kuramoto model generalized to $D$ dimensions. We show that in the important case of 3 dimensions, as well as for any odd number of dimensions, the $D$-dimensional generalized Kuramoto model for heterogeneous units has dynamics that are remarkably different from the dynamics in 2 dimensions. In particular, for odd $D$ the transition to coherence occurs discontinuously as the inter-unit coupling constant $K$ is increased through zero, as opposed to the $D=2$ case (and, as we show, also the case of even $D$) for which the transition to coherence occurs continuously as $K$ increases through a positive critical value $K_c$. We also demonstrate the qualitative applicability of our results to related models constructed specifically to capture swarming and flocking dynamics in three dimensions.