Swarming Lattice in Frustrated Vicsek–Kuramoto Systems

Abstract We introduce a frustration parameter α into the Vicsek-Kuramoto system 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 stationary hexagonal lattice, accompanied by spatiotemporal patterns such as vortex lattices and dual-cluster lattices with respiration-like motions of unit cells. Lattice dominance is governed by coupling strength and interaction radius, with a clear separatrix explained by 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.