Dynamics of pulsating swarmalators on a ring

We study a simple one-dimensional model of swarmalators, a generalization of phase oscillators that swarm around in space as well as synchronize internal oscillations in time. Previous studies of the model focused on Kuramoto-type couplings, where the phase interactions are governed by phase differences. Here we consider Winfree-type coupling, where the interactions are multiplicative, determined by the product of a phase response function $R(θ)$ and phase pulse function $P(θ)$. This more general interaction (from which the Kuramoto phase differences emerge after averaging) produces rich physics: six long-term modes of organization are found, which we characterize numerically and analytically.