Time delay in the 1d swarmalator model

TL;DR

This paper investigates the effects of time delay on the 1d swarmalator model, revealing new unsteady states and bifurcation phenomena, with stability depending solely on coupling strength.

Contribution

It introduces the analysis of time delayed coupling in the 1d swarmalator model and characterizes new unsteady states and bifurcation boundaries.

Findings

Unsteady states with periodic and irregular oscillations are identified.

Bifurcation boundaries are derived analytically for Hopf and zero eigenvalue bifurcations.

Stability of sync and async states is unaffected by delay, depending only on coupling strength.

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.