Attractive and repulsive interactions in the one-dimensional swarmalator model

Abstract

We study a population of swarmalators, mobile variants of phase oscillators, which run on a ring and have both attractive and repulsive interactions. This one-dimensional (1D) swarmalator model produces several of collective states: the standard sync and async states as well as a splaylike "polarized" state and several unsteady states such as active bands or swirling. The model's simplicity allows us to describe some of the states analytically. The model can be considered as a toy model for real-world swarmalators such as vinegar eels and sperm which swarm in quasi-1D geometries.

Figures (7)

• Figure 1: Stationary Collective States. Scatter plots of four stationary states in the $(x_1, x_2)$ plane, where $(x_1,x_2)=(\cos{x},\sin{x})$ and the swarmalators are colored in terms of their phases. Simulations were run with $N=500$ swarmalators for variable numbers of time units $T$ and step size $dt=0.1$. (a) Static sync state for $(J,K_n,K_p,p)=$$(1,-0.5,0.5,1)$ and $T=100$. (b) Polarized state for $(J,K_n,K_p,p)=(1,-0.5,0.5,0.8)$ and $T=1000$. (c) Static phase wave state for $(J,K_n,K_p,p)=$$(1,-0.5,0.5,0.2)$ and $T=100$. (d) Static async state for $(J,K_n,K_p,p)=$$(1,-3,0.5,0.1)$ and $T=100$.
• Figure 2: Scatter plots in $(x,\theta)$ space. Distributions in $(x, \theta)$ space corresponding to different states. Simulations were run with $N=500$ swarmalators for variable numbers of time units $T$ and step size $dt=0.1$. Swarmalators coupling with $K_p$ and $K_n$ are presented as blue dots and red dots respectively. (a) Static sync state for $(J,K_n,K_p,p)=$$(1,-0.5,0.5,1)$ and $T=100$. (b) Polarized state for $(J,K_n,K_p,p)=(1,-0.5,0.5,0.8)$ and $T=500$. (c) Static phase wave state for $(J,K_n,K_p,p)=$$(1,-0.5,0.5,0.2)$ and $T=100$. (d) Static async state for $(J,K_n,K_p,p)=$$(1,-3,0.5,0.1)$ and $T=100$.
• Figure 3: Order parameters and averaged velocity for different coupling distributions. Asymptotic behavior of the order parameters $S_{max}$ := $\max(S_{+},S_{-}$) (blue dots) and $S_{min}$ := $\min(S_{+},S_{-}$)(red dots) versus $p$ for other parameters $(J,K_p,N,T,dt)$ = $(1,0.5,500,1000,0.1)$. (a) It shows the transition from static async to unsteady state with $p$ varying from 0 to 1 when $K_n = -2$. (b) It shows the transition from phase wave to polarized state with $p$ varying from 0 to 1 when $K_n=-0.8$. (c) It shows the transitions from phase wave to unsteady state and then polarized state when $K_n=-0.25$. Each data point represents the average of last $10\%$ realizations.
• Figure 4: Phase Diagram in $(P,K_n)$ Plane with fixed $K_p$=0.5. Each state is indicated by a distinct color. The black curves and lines represent the theoretical predictions. Parameters in simulation we used are $(J,N,T,dt)=(1,5000,1000,0.1)$.
• Figure 5: Unsteady Collective States.