Anomalous diffusion in single and coupled standard maps with extensive chaotic phase spaces

TL;DR

This study explores how anomalous diffusion arises in standard maps due to accelerator modes and how coupling these maps can suppress such anomalous transport, leading to normal diffusion.

Contribution

It provides a systematic analysis of diffusion behaviors in single and coupled standard maps, highlighting how coupling configurations influence transport properties.

Findings

Accelerator modes induce anomalous diffusion in standard maps.

Coupling arrangements can suppress anomalous transport.

Certain configurations promote convergence to normal diffusion.

Abstract

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 in 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.