A two-dimensional swarmalator model with higher-order interactions

TL;DR

The paper investigates how higher-order phase interactions affect the collective dynamics of swarmalators constrained to move on a two-dimensional torus. It develops a tractable model with both pairwise and triadic phase couplings and analyzes identical and nonidentical populations using continuum and Ott–Antonsen methods, respectively, to derive stability and bifurcation conditions for multiple states. Key findings include the emergence of stationary states such as a spatially coherent state and a gas state that do not arise under purely pairwise interactions, as well as abrupt transitions and bistability among async, phase waves, and sync states driven by the higher-order term $K_2$. These results broaden the understanding of swarmalator dynamics under higher-order interactions and provide analytical tools for predicting state transitions in complex multi-agent systems. The work has potential implications for designing and interpreting coordinated behavior in artificial swarms and in natural systems where group interactions extend beyond pairwise coupling.

Abstract

We study a simple two-dimensional swarmalator model that incorporates higher-order phase interactions, uncovering a diverse range of collective states. The latter include spatially coherent and gas-like configurations, neither of which appear in models with only pairwise interactions. Additionally, we discover bistability between various states, a phenomenon that arises directly from the inclusion of higher-order interactions. By analyzing several of these emergent states analytically, both for identical and nonidentical populations of swarmalators, we gain deeper insights into their underlying mechanisms and stability conditions. Our findings broaden the understanding of swarmalator dynamics and open new avenues for exploring complex collective behaviors in systems governed by higher-order interactions.

Figures (11)

• Figure 1: Stationary states. Top row: snapshots of the states in $(x,y)$ plane with colors indicating the phase $\theta$, except panel (d) where the snapshot of the spatially coherent state is portrayed in $(x,\theta)$ plane. Middle row: variation of order parameters $S_{1}^{\pm}$ as a function of time, and bottom row: order parameters $T_{1}^{\pm}$ as a function of time. (a, f, k) Async state: $(J, K_1, K_2)=(1, -3, 1)$, (b, g, l) Thick phase wave state: $(J, K_1, K_2)=(-1, 2, 1)$, (c, h, m) Thin phase wave state: $(J, K_1, K_2)=(2, -1, 1)$, (d, i, n) Spatially coherent state: $(J, K_1, K_2)=(1, 2, -4)$, and (e, j, o) Sync state: $(J, K_1, K_2)=(1, 2, 1)$. States are obtained by integrating Eq. \ref{['model']} with $N=1000$ swarmalators for a total of $T=5000$ time units using an adaptive Julia Ode solver having relative tolerance of $10^{-8}$.
• Figure 1: (a) Phase transition scenario in the $K_1$-$K_2$ plane for nonidentical swarmalators. The spatial coupling strength $J$ is fixed here ($J=20$). We observe bistability between the collective states which is clearly visible for large $K_2$. The transitions are abrupt, and the solid and dashed lines correspond to the points of forward and backward transitions, respectively. We also notice that, when higher-order coupling is absent (i.e., $K_2=0$) or sufficiently small, the nature of the transition is smooth. (b) Dependence of the order parameters with respect to $K_1$ for Fixed $J=20$ and $K_2=12$. The solid curves represent the stable branches of the solution associated to the specific value of the order parameter, while the dotted curves denote unstable branches. The vertical dotted lines mark the critical values of $K_1$ where forward (right) and backward (left) transitions occur. The system exhibits three distinct bistable regions for this parameter set.
• Figure 1: Bistability between async and sync state for $(J,K_1,K_2)=(1,-2,3)$. The Upper and lower rows correspond to the snapshots and time evolution of the order parameters for the async and sync states, respectively.
• Figure 2: Nonstationary states. Top row: snapshots of the states in $(x,y)$ plane with colors indicating the phases. Middle row: order parameter $S_{1}^{\pm}$ as a function of time, and bottom row: order parameter $T_{1}^{\pm}$ as a function of time. (a, d, g) Gas state: $(J, K_1, K_2)=(1, -0.5, -4)$, (b, e, h) Thick phase wave: $(J, K_1, K_2)=(3, -1, -2)$, (c, f, i) Thin phase wave: $(J, K_1, K_2)=(4, -1, -1)$. The other parameters are $(N,T)=(1000,5000)$.
• Figure 2: Bistability between async and thin phase wave state for $(J,K_1,K_2)=(0.5,-1,0.8)$. The Upper and lower rows correspond to the snapshots and time evolution of the order parameters for the async and thin phase wave states, respectively.