Time delay in the 1d swarmalator model

Abstract

We study the 1d swarmalator model augmented with time delayed coupling. Along with the familiar sync, async, and phase wave states, we find a family of unsteady states where the order parameters are time periodic, sometimes with clean oscillations, sometimes with irregular vacillations. The unsteady states are born in two ways: via a Hopf bifurcation from the phase wave, and a zero eigenvalue bifurcation from the async state. We find both of these boundary curves analytically. A surprising result is that stabilities of the async and sync states are independent of the delay {\tau}; they depend only on the coupling strength.