Dynamical behavior of hydrodynamic Lyapunov modes in coupled map lattices

In our previous study of hydrodynamic Lyapunov modes (HLMs) in coupled map lattices, we found that there are two classes of systems with different lambda-k dispersion relations. For coupled circle maps we found the quadratic dispersion relations lambda approximately k2 and lambda approximately k for coupled standard maps. Here, we carry out further numerical experiments to investigate the dynamic Lyapunov vector (LV) structure factor which can provide additional information on the Lyapunov vector dynamics. The dynamic LV structure factor of coupled circle maps is found to have a single peak at omega=0 and can be well approximated by a single Lorentzian curve. This implies that the hydrodynamic Lyapunov modes in coupled circle maps are nonpropagating and show only diffusive motion. In contrast, the dynamic LV structure factor of coupled standard maps possesses two visible sharp peaks located symmetrically at +/- omega. The spectrum can be well approximated by the superposition of three Lorentzian curves centered at omega=0 and +/-omegau, respectively. In addition, the omega-k dispersion relation takes the form omegau=cuk for k --> 2pi/L. These facts suggest that the hydrodynamic Lyapunov modes in coupled standard maps are propagating. The HLMs in the two classes of systems are shown to have different dynamical behavior besides their difference in spatial structure. Moreover, our simulations demonstrate that adding damping to coupled standard maps turns the propagating modes into diffusive ones alongside a change of the lambda-k dispersion relation from lambda approximately k to lambda approximately k2. In cases of weak damping, there is a crossover in the dynamic LV structure factors; i.e., the spectra with smaller k are akin to those of coupled circle maps while the spectra with larger k are similar to those of coupled standard maps.