Ring states in swarmalator systems

TL;DR

This paper investigates finite populations of swarmalators, revealing the existence and stability of ring and annular states, with explicit calculations of their properties, applicable to biological and physical systems exhibiting both swarming and synchronization.

Contribution

It extends previous continuum analyses by studying finite swarmalator populations, deriving criteria for ring state existence and stability, and explicitly calculating properties of annular distributions.

Findings

Ring states exist in finite swarmalator populations.

Criteria for stability of ring states are established.

Explicit formulas for annular distribution properties are provided.

Abstract

Synchronization is a universal phenomenon, seen in systems as diverse as superconducting Josephson junctions and discharging pacemaker cells. Here the elements have rhythmic state variables whose mutual influence promotes temporal order. A parallel form of order is seen in swarming systems, such as schools of fish or flocks of birds. Now the degrees of freedom are the individuals' positions, which get redistributed through interactions to form spatial structures. Systems capable of both swarming and synchronizing, dubbed swarmalators, have recently been proposed [O'Keeffe, Kevin P., and Steven H. Strogatz. "Swarmalators: Oscillators that sync and swarm." arXiv preprint arXiv:1701.05670 (2017)] and analyzed in the continuum limit. Here we extend this work by studying finite populations of swarmalators, whose phase similarity affects both their spatial attraction and repulsion. We find ring states, and compute criteria for their existence and stability. Larger populations can form annular distributions, whose density and inner and outer radii we calculate explicitly. These states may be observable in groups of Japanese tree frogs, magnetic colloids, and other systems with an interplay between swarming and synchronization.