Circular current induced by angular dynamics in swarmalator populations

TL;DR

The paper asks whether swarmalator populations can exhibit genuine collective currents driven by angular dynamics. It introduces a phase- and origin-dependent driving term that replaces the phase-dependent spatial coupling, analyzing both fixed-origin and dynamic-origin implementations. The results show a spiral outward current under a fixed origin when $| heta_s|<\pi/2$, and a bounded circular current around the center of mass when using a dynamic origin, with no-current states possible for $| heta_s|>\pi/2$; current emergence can be boosted by tuning initial phase distributions. This work provides a mechanism for controllable rotational flows in swarmalator-like systems and suggests future exploration of local-interaction variants and active current control.

Abstract

We propose a modified swarmalator model that generates collective rotational currents in phase synchronization. Our approach builds on the original swarmalator model [4], introducing a key modification: the phase-dependent terms in the spatial dynamics are replaced with a simpler driving term that depends on both the phase and a specified origin. We investigate the dynamics of this model through extensive numerical simulations. When the origin is fixed, spiral motions of synchronized and clustered swarmalators emerge from a finite fraction of random initial conditions, resulting in collective currents. To prevent the unrealistic divergence of these spirals, we introduce a dynamic origin, defined as the center of the swarmalators' positions. With this dynamic origin, the system evolves into rotating collective currents, where synchronized swarmalators form stable circular patterns. In both the fixed and dynamic origin cases, we also observe no-current states, in which synchronized swarmalators aggregate near the origin. Finally, we find that the formation of collective currents can be facilitated by tuning the phase variables either at initialization or during the system's evolution.

Figures (5)

• Figure 1: Direction of the driving motion induced by the $W$-term in Eq. \ref{['model']}. Here, $\tilde{\mathbf r}_i=(\tilde{x}_i,\tilde{y}_i)$ denotes the position of swarmalator $i$ in the rectangular coordinate system centered at the chosen origin $\tilde{O}$, i.e., $\tilde{\mathbf r}_i\equiv {\mathbf r}_i-\tilde{O}$. When $\tilde{O}=O$ for the fixed origin $O=(0,0)$, we have $\tilde{\mathbf r}_i= {\mathbf r}_i$. The direction vector $\hat{x}$ and $\hat{y}$ remain fixed regardless of the choice of $\tilde{O}$. The arrow originating from $\tilde{\mathbf r}_i$ indicates the direction of the driving by the $W$-term. As shown, the phase $\theta_i$ is measured counterclockwise from the radial direction of $\tilde{\mathbf r}_i$.
• Figure 2: (Color Online) Outward spiral, the numerical time-trajectory of the cluster of swarmalators by Eqs. \ref{['model']} and \ref{['model2']}. Initially, at time $t=0$, swarmalators are randomly distributed over $2\pi\times 2\pi$ square (see the left inset) with random phases whose values are represented in the color bar right. This initialization of positions and phases is used throughout this work unless otherwise specified. The data points representing the swarmalators' positions in the spatiotemporal space are depicted at each discrete time steps $t=0,100,200,..\,$. Phase-sync of $\theta_{\rm s}=1.329784..=\theta_i$ for all $i$ is reached between $t=200$ and $t=400$ (see the colors become same). After synchronization, the trajectory of outward spiral appears. The arrow is the direction of the outward motion. The seeming circle each data thereafter is the cluster of all the swarmalators in system. The middle inset is the zoom-in of such cluster, which reveals that is a disc formed by the swarmalators. The right inset is the projection of the spiral trajectory onto the $x$-$y$ plane. The solid curve is the theoretical prediction given by Eq. \ref{['rp']}. For the parameters, $A=0.01$, $W=0.04$, and $K=0.16$ are used, and the system size is $N=100$.
• Figure 3: (Color Online) The ratio $R$ of spiral emergence is evaluated across various combinations of $W$, $K$, and $N$. For each parameter set $(W,K,N)$, whereby $W=0.01,0.04,0.16,0.64$, $K=0.01,0.04,0.16,0.64.2.56$, and $N=50,100,200,400$, simulations are performed using 20 different initial configurations. The resulting values of $R$ are plotted in panels (a)-(d). The legend for $N$, shown in panel (d), applies to all panels.
• Figure 4: (Color Online) Snapshots of circular currents appearing for the model with position-center origin $O_{\rm PC}$ (see text). Each panel shows the distribution of swarmalators at a time long after transient period (the data points represent swarmalators). At the center of panel, the radius of circular pattern is written. The rotation direction is indicated with the arrow heads ($<<<$ or $>>>$). The pattern in each panel is composed of $N=100$ swarmalators. (a) shows the (single-)ring formed when $W=0.16$ and $K=1.28$. The double-ring in (b) appears when $W=0.02$ and $K=1.28$. (c) is the triple-ring at $W=0.01$ and $K=1.28$. The ring of large radius and of inhomogeneous spacing in (d) appears when $W=0.64$ and $K=2.56$.
• Figure 5: (Color Online) Surface of $R$ on $W$-$f$ plane obtained for $A=0.08$ and $N=400$ in (a); $R$ is the ratio of emergence of circular currents and $f$ is a phase-initialization parameter (see text for the detail of $f$). The curves both on surface and on bottom are the contours of $R$. For (a), we have scanned $W$-$f$ plane on the $0.2 \le W\le 0.8$ and $0.3 \le f\le 0.7$ region with $\Delta W=\Delta f=0.02$ step, i.e., $21\times31$ number of $W$ and $f$ pairs are examined note1. For each pair of $W$ and $f$, 200 random samples are tested. More contours of $R$ are on the $W$-$f$ panels in (b), of which colors represent the same values of $R$ listed at the upper-right corner of (a). For every panel, the $W$-$f$ plane is scanned in the same way done for (a). The used $A$ and $N$ is written in each panel.