Characteristic times in the standard map

TL;DR

This paper investigates and compares three characteristic times in the standard map, revealing empirical relations and the impact of stickiness on diffusion behavior, with implications for understanding chaotic dynamics.

Contribution

It introduces empirical relations for characteristic times in the standard map and analyzes how stickiness influences diffusion and recurrence properties.

Findings

Empirical relations between LCN, K, and A.

Stickiness time is much smaller than Poincare recurrence time.

Diffusion is anomalous inside islands and normal outside.

Abstract

We study and compare three characteristic times of the standard map, the Lyapunov time t_L, the Poincare recurrence time t_r and the stickiness (or escape) time t_{st}. The Lyapunov time is the inverse of the Lyapunov characteristic number LCN and in general is quite small. We find empirical relations for the LCN as a function of the nonlinearity parameter K and of the chaotic area A. We also find empirical relations for the Poincare recurrence time t_r as a function of the nonlinearity parameter K, of the chaotic area A and of the size of the box of initial conditions e. As a consequence we find relations between t_r and LCN. We compare the distributions of the stickiness time and the Poincare recurrence time. The stickiness time inside the sticky regions at the boundary of the islands of stability is orders of magnitude smaller than the Poincare recurrence time t_r and this affects the diffusion exponent mu, which converges always to the value mu=1. This is shown in an extreme stickiness case. The diffusion is anomalous (ballistic motion) inside the accelerator mode islands of stability with mu=2 but it is normal everywhere outside the islands with mu=1. In a particular case of extreme stickiness we find the hierarchy of islands around islands.