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The Kuramoto model, originally proposed to model the dynamics of manyinteracting oscillators, has been used and generalized for a wide range ofapplications involving the collective behavior of large heterogeneous groups ofdynamical units whose states are characterized by a scalar angle variable. Onesuch application in which we are interested is the alignment of orientationvectors among members of a swarm. Despite being commonly used for this purpose,the Kuramoto model can only describe swarms in 2 dimensions, and hence theresults obtained do not apply to the often relevant situation of swarms in 3dimensions. Partly based on this motivation, as well as on relevance to theclassical, mean-field, zero-temperature Heisenberg model with quenched sitedisorder, 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 forany odd number of dimensions, the $D$-dimensional generalized Kuramoto modelfor heterogeneous units has dynamics that are remarkably different from thedynamics in 2 dimensions. In particular, for odd $D$ the transition tocoherence occurs discontinuously as the inter-unit coupling constant $K$ isincreased through zero, as opposed to the $D=2$ case (and, as we show, also thecase of even $D$) for which the transition to coherence occurs continuously as$K$ increases through a positive critical value $K_c$. We also demonstrate thequalitative applicability of our results to related models constructedspecifically to capture swarming and flocking dynamics in three dimensions.

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