Binary Mixtures of Locally Coupled Mobile Oscillators

Abstract

We study synchronization dynamics in binary mixtures of locally coupled Kuramoto oscillators which perform Brownian motion in a two-dimensional box. We introduce two models, where in model $\cal I$ there are two type of oscillators, say $\cal A$ and $\cal B$, and any two similar oscillators tend to synchronize their phases, while any two dissimilar ones tend to be out of phase. In model $\cal J$, in contrast, the oscillators in subpopulation $\cal A$ behave as in model $\cal I$, while the oscillators in subpopulation $\cal B$ tend to be out of phase with all the others. In the real space all the oscillators in both models interact via a soft-core repulsive potential. Both subpopulations of model $\cal I$ and subpopulation $\cal A$ of model $\cal J$, by their own, exhibit a phase coherent attractor in a certain region of model parameters. The approach to the attractor, after an initial transient regime, is exponential with some characteristic synchronization time scale $\tau$. Numerical analysis reveals that the attractors of the two subpopulations survive within model $\cal I$, regardless of the composition of the mixture $\phi$ and the strength of the cross-population negative coupling constant $H$, and that $\tau$ sensitively depends on $\phi$, $H$ and the packing fraction. In particular, the ability of the oscillators to move and exchange neighbours can significantly decrease $\tau$. In contrast, model $\cal J$ predicts suppression of the synchronized state in subpopulation $\cal A$ and emergence of the coherent attractor in the "contrarians" subpopulation $\cal B$ for strong and weak cross-population coupling, respectively.