Continuous vs. Discontinuous Transitions in the D-Dimensional Generalized Kuramoto Model: Odd D is different

In 1975, Yoshiki Kuramoto introduced a simple model to describe the collective dynamics of a set of interacting oscillators. In the model, each oscillator has a natural frequency, and is coupled equally to all other oscillators. Assuming a fixed spread in oscillator frequencies, Kuramoto showed that in the limit of a large number of oscillators, the model exhibits a continuous phase transition from asynchronous to synchronous behaviour with increasing inter-oscillator coupling. Since then, the model and generalizations of it have been widely used in exploring the synchronization behavior in groups of biological cells, fireflies, superconducting Josephson junctions, or the movements of swarms or flocks of organisms.
Many studies have focussed on oscillators described by a single scalar variable, but for  some applications – the behaviour of flocks of birds or schools of fish, for example – higher-dimensional spaces play a role. In a recent paper, LML Fellow Michelle Girvan and colleagues explore a generalisation of the Kuramoto model to such higher dimensional settings, with each oscillator described by a D-dimensional unit vector. They show that the macroscopic dynamics of the model depends strongly on the dimensionality of the system, with systems of odd or even dimension falling into two qualitatively distinct classes. For odd-dimensional systems, including the important case of three dimensions, they find that a phase transition from an incoherent to partially coherent regime takes place through a discontinuous, non-hysteretic transition as the coupling strength K increases through 0. In contrast, even-dimensional systems, undergo continuous transitions into a coherent phase at a positive critical coupling strength.
The paper is available at

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