Dynamics of pulsating swarmalators on a ring

TL;DR

This work extends swarmalator theory by implementing Winfree-type pulsatile coupling on a one-dimensional ring, replacing the standard Kuramoto-style phase differences with $R(\theta_i)P(\theta_j)$ interactions. By choosing $P(\theta)=1+\cos\theta$ and $R(\theta)=-\sin\theta$ and transforming to $\xi_i=x_i+\theta_i$ and $\eta_i=x_i-\theta_i$, the authors derive tractable equations and a rich set of order parameters to analyze joint spatial-phase organization. Numerically exploring the coupling plane $(J,K)$ reveals six long-term states, including two new configurations—$x$-antiphase sync and an intermediate mixed state—not present under Kuramoto coupling, alongside stability thresholds for four states obtained from Jacobian and continuum analyses. These results broaden the dynamical repertoire of swarmalators and offer analytical handles on pulsatile interactions relevant to fireflies, neurons, and other coupled oscillatory systems. Future work could incorporate frequency heterogeneities and varied coupling parameters to assess robustness and to connect with experimental realizations.

Abstract

We study a simple one-dimensional model of swarmalators, a generalization of phase oscillators that swarm around in space as well as synchronize internal oscillations in time. Previous studies of the model focused on Kuramoto-type couplings, where the phase interactions are governed by phase differences. Here we consider Winfree-type coupling, where the interactions are multiplicative, determined by the product of a phase response function $R(θ)$ and phase pulse function $P(θ)$. This more general interaction (from which the Kuramoto phase differences emerge after averaging) produces rich physics: six long-term modes of organization are found, which we characterize numerically and analytically.

Figures (7)

• Figure 1: Schematic diagram of the regions for the states for identical swarmalators. (a) Async: $(J, K)=(-0.5,-0.5)$, (b) intermediate mixed state: $(J, K)=(-0.28,0.72)$, (c) $\theta$ antiphase sync: $(J, K)=(0.0,0.5)$, (d) static $\pi$: $(J, K)=(0.5,0.5)$, (e) static phase wave: $(J, K)=(0.5,0.0)$, and (f) $x$ antiphase: $(J, K)=(0.5,-0.5)$. We run the simulations using Heun's method for $N=1000$ number of swarmalators for $T=5000$ time units with step-size $dt=0.1$. Initial positions and phases of the swarmalators are randomly drawn from $[-\pi, \pi]$. All the states are achieved after discarding the first $90\%$ data.
• Figure 2: phase space diagram of the order parameters in the $(J, K)$ plane demonstrating the regions of the collective states. (a) $S_+$, (b) $S_-$, (c) $R_{2x}$, and (d)$R_{2\theta}$. We simultaneously vary $J$ and $K$ over the interval $[-1.0, 1.0]$. Simulation parameters are $(dt, T, N)= (0.1, 5000, 10^3)$.[We will provide a detailed explanation of Figure (d) in Section \ref{['stability']}].
• Figure 3: (a) Scatter plot of the $x$ (antiphase) sync state $[(J,K)=(0.5,-0.5)]$ in $(x,\theta)$ plane, (b) & (c) denote the distribution of $x$ and $\theta$ in the $x$ (antiphase) sync state. (d) Scatter plot of the intermediate mixed state $[(J,K)=(-0.28,0.76)]$ in $(x,\theta)$ plane, (e) & (f) depict the corresponding distribution of $x$ and $\theta$. The simulation parameters are $(dt,T,N)=(0.1,5000,10^4)$.
• Figure 4: The black dashed line represents the prediction of Eq. \ref{['tt1']}; For numerical validation, we simulate the system for $N=2,3,4,5,20,50,100$ number of particles (as indicated by increasing the size of the red dots), which closely align with our analytical results. Simulation parameters $(dt,T,J,K)=(0.1,5\times10^4,0.5,-0.5)$. After wiping out the transients, we save the last $10\%$ data by taking $100$ realizations.
• Figure 5: A log-log plot of the time evolution of mean velocity $\langle v \rangle$ of the swarmalators at different values of $N$ for intermediate mixed state. We set $(J,K)=(-0.28,0.76)$.