Stickiness in three-dimensional volume preserving mappings

We investigate the orbital diffusion and the stickiness effects in the phase space of a 3-dimensional volume preserving mapping. We first briefly review the main results about the stickiness effects in 2-dimensional mappings. Then we extend this study to the 3-dimensional case, studying for the first time the behavior of orbits wandering in the 3-dimensional phase space and analyzing the role played by the hyperbolic invariant sets during the diffusion process. Our numerical results show that an orbit initially close to a set of invariant tori stays for very long times around the hyperbolic invariant sets near the tori. Orbits starting from the vicinity of invariant tori or from hyperbolic invariant sets have the same diffusion rule. These results indicate that the hyperbolic invariant sets play an essential role in the stickiness effects. The volume of phase space surrounded by an invariant torus and its variation with respect to the perturbation parameter influences the stickiness effects as well as the development of the hyperbolic invariant sets. Our calculations show that this volume decreases exponentially with the perturbation parameter and that it shrinks down with the period very fast.