Chaotic dynamics in periodic potentials - Lazarotto, Matheus Jean (2023)
Spatial diffusion of particles in periodic potential models has provided a good framework for studying the role of chaos in global properties of classical systems. Here a square-symmetric potential, classically modeled from an optical lattice Hamiltonian system, was initially used to numerically study diffusion transitions under variation of the control parameters. Sudden transitions between normal and ballistic regimes were found and characterized by inspection of topological changes taking place in phase-space. Particular transitions, correlated with increases in global stability area, were seen to occur for energy levels where local maxima points become accessible, deviating trajectories approaching them. These instabilities promote a slowing down of the dynamics and an island myriad bifurcation phenomenon, along with the suppression of long flights within the lattice. On further investigating the island myriad, its structure was found to be intimately related to the translational and rotational symmetries of the lattice potential. With a high fractal pattern, the myriad of islands is concentrically organized in isochronous chains, formed either by orbits with limited range or high escape transport. As the local maxima points change with the control parameters, the bifurcation of each chain sequentially follows a separatrix reconnection, as in a local non-twist scenario. Due to the myriads dependence on the tiling symmetry property of the square lattice, its presence was conjectured and confirmed also for a hexagonal lattice, although found in attenuated form due to extra instabilities in the potential. Beyond that, the numerical techniques applied for analyses along this work are of wide use and can be adapted to generic conservative systems, allowing their study as their parameters change in an automated way.