Universality of fractal dimension on time-independent Hamiltonian systems

摘要

This paper summarizes a numerical study of the dependence of the fractal dimension on the energy of certain open Hamiltonian systems, which present different kind of symmetries. Owing to the presence of chaos in these systems, it is not possible to make predictions on the way and the time of escape of the orbits starting inside the potential well. This fact causes the appearance of fractal boundaries in the initial-condition phase space. In order to compute its dimension, we use a simple method based on the perturbed orbits’ behavior. The results show that the fractal dimension function depends on the structure of the potential well, contrary to other properties, such us the probability of escape, which has already been postulated as universal in earlier papers (see for instance [C. Siopis, H.E. Kandrup, G. Contopoulos, R. Dvorak, Universal properties of escape in dynamical systems, Celest. Mech. Dyn. Astr. 65 (57–68) (1997)]), from the study of Hamiltonians with different number of possible exits.