Swarming Lattice in Frustrated Vicsek-Kuramoto Systems

Abstract

We introduce a frustration parameter $α$ into the Vicsek-Kuramoto systems of self-propelled particles. While the system exhibits conventional synchronized states, such as global phase synchronization and swarming, for low frustration ($α< π/2$), beyond the critical point $α= π/2$, a Hopf-Turing bifurcation drives a transition to a resting hexagonal lattice, accompanied by spatiotemporal patterns such as vortex lattices and dual-cluster lattices with oscillatory unit-cell motions. Lattice dominance is governed by coupling strength and interaction radius, with a clear parametric boundary balancing pattern periodicity and particle dynamics. Our results demonstrate that purely orientational interactions are sufficient to form symmetric lattices, challenging the necessity of spatial forces and illuminating the mechanisms driving lattice formation in active matter systems.

Figures (4)

• Figure 1: (a)-(h) Spatially homogeneous sync (a), swarming sync (b), chimera (c), vortex lattice (d), dual-cluster lattice (f), dual-lane lattice (h) states, and their coexistence states (e, g), at different frustration $\alpha$. Arrow orientation/color indicates instantaneous phase $\theta_i$. (i) Phase diagram and order parameters versus $\alpha$. Blue/green regions (divided by $\alpha=\pi/2$) indicate sync and lattice states, respectively. Dashed horizontal line shows $p_{\mathrm{std}}$ when $\max p(\theta, t)=1/2$ (dual-cluster sync). After reaching a steady state, order parameters were computed by averaging over 500 simulation steps. Other parameters: $K=20$, $d_0=1.55$, $L=7, N=2000, v=3$.
• Figure 2: (a, b) Respiration dynamics of vortex (a, $\alpha=0.6\pi$) and double clustered (b, $\alpha=0.9\pi$) unit cell. In (b), dashed circle indicates vortex cell volume. In (k), red/blue arrows show phase-divided clusters; dashed circles show estimated trajectories from mean instantaneous frequencies $\langle \dot{\theta}_i\rangle$. (c) Respiration amplitude over time for $\alpha=0.6\pi$. Other parameters are same as in Fig. \ref{['fig:snapshotsAndPhaseDiagram']}.
• Figure 3: Phase diagrams in the $(K, d_0)$ parameter space for different frustration $\alpha$ ($L=7, N=2000$, $v=3$). Black dashed lines mark boundaries (Eq. \ref{['eq:criticalLineOfKD0']}) between dominant and recessive lattice states.
• Figure 4: (a) Computations of the $\lambda(k)$ with the largest real part as a function of the continuous wavenumber $k$ and $\alpha$ in the truncated basis $M=10$. Other parameters as in Fig. \ref{['fig:snapshotsAndPhaseDiagram']}. (b) Measured mean lattice constant $\langle a \rangle$ compared to theoretical prediction. Horizontal and vertical error bars represent standard deviations across $K$ and unit cells, respectively. The black dashed line indicates a 1:1 correspondence.