Emergent behaviors of the swarmalator model for position-phase aggregation

We study emergent dynamics of the swarmalator model [K. P. O'Keeffe, H. Hong and S. H. Strogatz, Oscillators that sync and swarm, Nature Commun. 8 (2017) 1504] describing the dynamic interplay of aggregation and synchronization dynamics for interacting many-particle systems. For the particle aggregation, we employ the nonlinear aggregation system with singular attractive-repulsive couplings depending on the phase differences, while we use the Kuramoto-type model with the singular coupling strength depending on the spatial distances for the dynamics of phases. We show how collective behaviors emerge from the dynamic interplay between position aggregation and phase synchronization. We introduce some sufficient framework leading to the positive minimal relative distances between particles and its uniform upper bound. We also show the convergence of position under some sufficient conditions.