Swarmalators with higher harmonic coupling: Clustering and vacillating

We study the dynamics of a swarmalator model with higher harmonic phase coupling. We analyze stability, bifurcation and structural properties of several novel attracting states, including the formation of spatial clusters with distinct phases, and single spatial clusters with a small number of distinct phases. We use mean-field (centroid) dynamics to analytically determine inter-cluster distance. We also find states with two large clusters along with a small number of swarmalators that are trapped between the two clusters and vacillate (waver) between them. In the case of a single vacillator we use a mean-field reduction to reduce the dynamics to two-dimensions, which enables a detailed bifurcation analysis. We show excellent agreement between our reduced two-dimensional model and the dynamics and bifurcations of the full swarmalator model.