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

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 Kc. We also demonstrate the qualitative applicability of our results to related models constructed specifically to capture swarming and flocking dynamics in three dimensions.

Observing microscopic transitions from macroscopic bursts: Instability-mediated resetting in the incoherent regime of the D-dimensional generalized Kuramoto model

“…Recently, a generalization of these models has been introduced in which the state of the agent i is a D-dimensional unit vector, σ i (t), thus allowing for more degrees of freedom in the dynamics of the individual agents. In this generalized model the D-dimensional unit vector, σ i (t), is taken to evolve according to the real equation [12][13][14] ,…”

“…(2) with σ i , we obtain d|σ i | 2 /dt = 0, as required by our designation of σ i as a unit vector. In general, depending on the situation to be modeled, ρ(t) can be chosen in different ways 12,15 . In this paper we focus on the simplest interesting choice,…”

“…13 and 14. The generalization to heterogeneous rotation rates 12 (the situation to be considered in the present paper) makes Eq. (2) more similar to the original Kuramoto model and widens its range of applicability.…”

“…In what follows, as in Ref. 12, we assume that the rotation rate matrix W i is randomly generated for each i, by choosing each of its D(D − 1)/2 upper triangular matrix elements, w (i) pq (with p < q), independently from a zero-mean, Gaussian distribution function as described in Sec. II.…”

“…In Ref. 12 using Method (i), previously employed for the original Kuramoto problem, analysis giving the critical transition values for even D were obtained. These values are indicated by the vertical arrows in Fig.…”