Time delay in the 1d swarmalator model

TL;DR

This paper investigates a 1d swarmalator model with time-delayed coupling to understand how finite communication lags affect collective behavior. By transforming to sum/difference coordinates $(\xi,\eta)$, it reduces the dynamics to a pair of delayed Kuramoto-like equations with three parameters $(J',K',\tau)$ and derives exact stability criteria for the asynchronous, phase-wave, and synchronous states. A key contribution is the prediction of delay-induced unsteady states via a Hopf bifurcation from the phase wave and a zero-eigenvalue bifurcation from the async state, with explicit expressions for the Hopf boundary and a delay-dependent threshold. The results show that async and sync stability are independent of the delay $\tau$, while phase-wave stability depends on delay, and the work introduces a distributional perturbation framework that may extend to higher dimensions and practical swarm-robotics applications.

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 τ; they depend only on the coupling strength.

Figures (6)

• Figure 1: (a)-(c) The static collective states of 1d swarmalator model in the $N \rightarrow \infty$ limit for $(J',K') = (1,-2), (1,-0.5), (1,2)$ and $\tau = 1$. Here $(dt,T,N) = (0.1,100,200)$ and initial condition were drawn uniformly at random. (d) Stability diagram in the $(J,K)$ plane for $\tau=0$, shown as baseline, to give an idea of where in parameter space the various states occur.
• Figure 2: Bifurcation diagram in the $(K,\tau)$ plane for $J = 1$.
• Figure 3: Bifurcation diagram in the $(J,K)$ plane for $\tau = 0, 0.1, 0.25, 1.5$. The unsteady region is plotted in blue and is unlabeled due to space constraints.
• Figure 4: Left column, time series of order parameters for different values of $\tau$. Right column, associated power spectra. Parameters were $(J',K') = (1,-0.5)$ and simulations were for $N= 10^5$ for $T=10^3$ time units with stepsize $dt=0.01$.
• Figure 5: Same as Figure \ref{['time-series']} except we vary $K$ and keep $\tau=1$ constant.