Polynomial dispersion of trajectories in sticky dynamics

Abstract

Hamiltonian chaotic dynamics is, in general, not ergodic and the boundaries of the ergodic or quasiergodic area (stochastic sea, stochastic layers, stochastic webs, etc.) are sticky, i.e., trajectories can spend an arbitrarily long time in the vicinity of the boundaries with a nonexponentially small probability. Segments of trajectories imposed by the stickiness are called flights. The flights have polynomial dispersion that can lead to non-Gaussian statistics of displacements and to anomalous transport in phase space. In particular, the presence of flights influences the distribution of Poincaré recurrences. We use the distribution function of (l,t;epsilon, epsilon0) -separation of trajectories that at time instant t and trajectory length l are separated for the first time by epsilon<<1, being initially at a distance epsilon0 <<epsilon. The connection of this function, called the complexity function [Afraimovich and Zaslavsky, Chaos 13, 519 (2003)], with the distribution of Poincaré recurrences is established for three cases: (i) for the case of superdiffusion in standard and web maps for which the stickiness is defined by the boundaries of hierarchical sets of islands; (ii) for the case of the Sinai billiard with infinite horizon, where the stickiness is defined by zero-measure slits in the phase space; (iii) for the square billiard with a slit (bar-in-square billiard) where the Lyapunov exponent is zero and the stickiness is defined by the vicinity of the trajectory to the closest periodic trajectories obtained from the Diophantine approximation. Finally, the powerwise asymptotics of the Poincaré recurrences can be connected, in some cases, to the anomalous transport exponent.