Swarmalators with frequency-weighted interactions

TL;DR

This paper presents a solvable 1D swarmalator model on a ring where coupling is weighted by intrinsic frequencies $|\,\omega_i\,|$, leading to three distinct collective states: asynchronous, phase-wave, and a novel bi-strip mixed state with antipodal clusters split by frequency sign. A self-consistency framework in sum–difference coordinates yields precise thresholds for transitions, including an AS-to-PW onset at $J_{+,c}=2\pi$ and a PW-to-BM boundary defined by locked-condition integrals, with bistable windows and hysteresis confirmed by numerical simulations. The results extend the understanding of how heterogeneity in interaction rules shapes the interplay between swarm dynamics and synchronization, and show how frequency-weighted coupling can produce robust clustering even in unimodal frequency distributions. The findings offer a foundation for exploring higher-dimensional generalizations and alternative heterogeneities, with potential relevance to engineered swarms and chemotactic collective behavior.

Abstract

We investigate the role of frequency-weighted interactions in a solvable model of one-dimensional (1D) swarmalators confined to a ring, where both spatial and phase couplings are scaled by the heterogeneous natural frequencies of individual agents. Our analysis identifies three distinct collective states: the asynchronous state , the phase-wave state , and the bistrip mixed state characterized by antipodal clusters that are internally split into frequency-dependent sub-strips. We further establish that the onset of abrupt transitions are driven by heterogeneous coupling. Using a self-consistency analysis, we precisely determine the conditions for dynamical transitions among the identified states, thereby extending the theoretical understanding of swarmalator dynamics under heterogeneous interaction rules, which are in good agreement with the numerical simulation results.

Figures (7)

• Figure 1: Phase space portraits ($x$ vs $\theta$) of the three primary states observed for a fixed spatial coupling $J=12$. (a) The disordered AS at $K=-8$, (b) PW at $K=0.5$, and (c) BM at $K=10$.
• Figure 2: Subclusters of the BM state for $J=12$, $K=10$ in the $(x,\theta)$ plane, with colors indicating the sign of its intrinsic frequency $\omega$
• Figure 3: Sweep of the coupling parameter $K$ at fixed $J=12$, showing the order parameters $S_{\pm}$. The intermediate intervals $R_{I}$ and $R_{II}$ indicate bistable windows between AS–PW and PW–BM states. Solid curves represent forward sweeps ($S^{\mathrm{fw}}_{+}$, $S^{\mathrm{fw}}_{-}$), while dashed curves denote backward sweeps ($S^{\mathrm{bw}}_{+}$, $S^{\mathrm{bw}}_{-}$). The $\blacktriangle$ markers indicate the stable BM branch obtained from self-consistency analysis.
• Figure 4: $J$–$K$ parameter space illustrating the full set of observed states. The intermediate regions $R_{I}$ and $R_{II}$ denote bistable windows between AS–PW and PW–BM states, while $R_{III}$ represents a multistable regime in which all three states coexist.
• Figure 5: Numerical continuation of the PW branch in the $(S_+,L)$ plane. The nonzero PW solution emerges through a saddle–node bifurcation at $L_{c} \approx 7.2$(solid vertical line), corresponding to the backward PW$\to$AS boundary. Here, the circular markers represent stable solutions, whereas the red x markers are unstable solutions.