Pattern formation in symplectic coupled map lattices

Abstract

We investigate the emergence of spatio-temporal patterns in a one-dimensional symplectic coupled map lattice (CML) composed of periodically kicked rotors with nonlocal interactions. The system retains the Hamiltonian structure, preserving the phase space volume. The system exhibits cluster states driven by the stickiness effect, where chaotic trajectories wanders around regular structures. By changing the coupling strength and interaction range, we demonstrate the formation of mosaic-like and zigzag patterns associated with transient or persistent clustering. These patterns are analysed using the Kuramoto order parameter, Lyapunov spectrum, and spatial correlation integrals. Our results reveal that pattern formation correlates with suppressed local chaos and small values of Lyapunov exponent, indicating weak chaotic dynamics. The correlation analysis confirms the presence of coherent structures at small scales, which disappear in the strongly chaotic regime. These findings demonstrate how stickiness can induce clustering, given rise to complex collective behaviour in Hamiltonian systems with nonlocal couplings.