No abstract

We investigate the behavior of the Generalized Alignment Index of order k (GALIk ) for regular orbits of multidimensional Hamiltonian systems. The GALIk is an efficient chaos indicator, which asymptotically attains positive values for regular motion when 2≤k ≤N, with N being the dimension (D) of the torus on which the motion occurs. By considering several regular orbits in the neighborhood of two typical simple, stable periodic orbits of the Fermi-Pasta-Ulam-Tsingou (FPUT) β model for various values of the system's degrees of freedom, we show that the asymptotic GALIk values decrease when the order k of the index increases and when the orbit's energy approaches the periodic orbit's destabilization energy where the stability island vanishes, while they increase when the considered regular orbit moves further away from the periodic one for a fixed energy. In addition, by performing extensive numerical simulations we show that the behavior of the index does not depend on the choice of the initial deviation vectors needed for its evaluation.

We investigate the long-term diffusion transport and chaos properties of single and coupled standard maps. We consider model parameters that are known to induce anomalous diffusion in the maps' phase spaces, as opposed to normal diffusion which is associated with Gaussian distribution properties of the kinematic variables. This type of transport originates by the presence of the so-called accelerator modes, i.e. non-chaotic initial conditions which exhibit ballistic transport, which also affect the dynamics in their vicinity. We first systematically study the dynamics of single standard maps, investigating the impact of different ensembles of initial conditions on their behavior and asymptotic diffusion rates, as well as on the respective time-scales needed to acquire these rates. We consider sets of initial conditions in chaotic regions enclosing accelerator modes, which are not bounded by invariant tori. These types of chaotic initial conditions typically lead to normal diffusion transport. We then setup different arrangements of coupled standard maps and investigate their global diffusion properties and chaotic dynamics. Although individual maps bear accelerator modes causing anomalous transport, the global diffusion behavior of the coupled system turns out to depend on the specific configuration of the imposed coupling. Estimating the average diffusion properties for ensembles of initial conditions, as well as measuring the strength of chaos through computations of appropriate indicators, we find conditions and systems' arrangements which systematically favor the suppression of anomalous transport and long-term convergence to normal diffusion rates.