Collective dynamics of swarmalators with time-delayed interactions

In this work we studied the effects of time delay in interactions on collective dyanmics of a population of swarmalators - phase oscillators that synchronize in time while swarming through space. We found new collective behaviors induced by the delay - a pseudo-crystaline state in which swarmalators form a quasi-static cluster, and a boiling state in which swarmalators near the boundary of the cluster perform convective motions. In both cases, the route to these states takes place through a long oscillatory ``breathing'' transient. We developed a continuum theory that allowed reduction from 2N equations to just two. This theory captures the density profile at later stages of the breathing transient, as well as the frequency and decay rates of breathing oscillations. Swarmalators have recently been realized in robotics labs, and may also play a role as a useful model for embryonic development, and collective motion of organisms.