Self-similar transport in incomplete chaos

Abstract

Particle chaotic dynamics along a stochastic web is studied for three-dimensional Hamiltonian flow with hexagonal symmetry in a plane. Two different classes of dynamical motion, obtained by different values of a control parameter, and corresponding to normal and anomalous diffusion, have been considered and compared. It is shown that the anomalous transport can be characterized by powerlike wings of the distribution function of displacement, flights which are similar to Lévy flights, approximate trappings of orbits near the boundary layer of islands, and anomalous behavior of the moments of a distribution function considered as a function of the number of the moment. The main result is related to the self-similar properties of different topological and dynamical characteristics of the particle motion. This self-similarity appears in the Weierstrass-like random-walk process that is responsible for the anomalous transport exponent in the mean-moment dependence on t. This exponent can be expressed as a ratio of fractal dimensions of space and time sets in the Weierstrass-like process. An explicit form for the expression of the anomalous transport exponent through the local topological properties of orbits has been given.