Universality of fractal dimension on time-independent Hamiltonian systems Juan F. Navarro * , M.C. Martínez Departamento de Matemática Aplicada, Universidad de Alicante, Carretera San Vicente del Raspeig, s/n – 03690, San Vicente del Raspeig, Alicante, Spain article info Keywords: Dynamical systems Chaotic motion Fractality Hénon–Heiles potential Escape abstract 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 symme- tries. 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 poten- tial 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. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction The aim of this paper is to investigate the evolution of the fractal dimension with respect to the energy in some two- dimensional time-independent Hamiltonian systems, which may be related to some situations in space physics. The standard method usually applied to describe and analyze these type of systems is to examine sets of initial conditions inside the potential well and observe which of the outcome possibilities correspond to each initial condition. A first impor- tant result obtained is the existence of fractal basin boundaries separating the possible modes of exit in the space of initial conditions [2], which is a suitable characterization of chaos. As a consequence, it is natural to measure the dimension of these boundaries, which will be obviously fractal. In order to carry out our study, we have selected the following three two-dimensional, time-independent Hamiltonians H 1  1 2 _ x 2 þ _ y 2 þ x 2 þ y 2    lx 2 y 2 ¼ h 1 ; H 2  1 2 _ x 2 þ _ y 2 þ x 2 þ y 2    lxy 2 ¼ h 2 ; H 3  1 2 _ x 2 þ _ y 2 þ x 2 þ y 2  2 3 y 3   þ lx 2 y ¼ h 3 ; ð1Þ all of them with polynomial potentials. These three Hamiltonians can be split up in the form H ¼ H 0 þ lH P , where H 0 is the integrable part and lH P is a perturbing correction. The first two systems correspond to two harmonic oscillators, coupled via 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.04.014 * Corresponding author. E-mail address: jf.navarro@ua.es (J.F. Navarro). Applied Mathematics and Computation 214 (2009) 462–467 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc