Universality of fractal dimension on time-independent Hamiltonian systems

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 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.

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 important 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

$$\begin{array}{cc}\hfill & H_1\equiv \frac{1}{2}\left({\dot{x}}^2+{\dot{y}}^2+x^2+y^2\right)-\mu x^2{y}^2=h_1\text{,}\hfill \\ \hfill & H_2\equiv \frac{1}{2}\left({\dot{x}}^2+{\dot{y}}^2+x^2+y^2\right)-\mu {\mathit{xy}}^2=h_2\text{,}\hfill \\ \hfill & H_3\equiv \frac{1}{2}\left({\dot{x}}^2+{\dot{y}}^2+x^2+y^2-\frac{2}{3}{y}^3\right)+\mu x^{2}y=h_3\text{,}\hfill \end{array}$$

all of them with polynomial potentials. These three Hamiltonians can be split up in the form $H=H_0+\mu H_P$, where $H_0$ is the integrable part and $\mu H_P$ is a perturbing correction. The first two systems correspond to two harmonic oscillators, coupled via a quartic and cubic term, respectively, which is multiplied by a coupling parameter μ; and the third system is characterized by a modified Hénon–Heiles potential. The parameter μ is non-negative and it is not assumed to be small. These potentials may represent the central parts of deformed galaxies.

It has been established [1] that, for each of these systems and for a given value of the energy $h_i(i=1\text{,}2\text{,}3)$, no orbit can escape if the parameter μ is lower than a critical value ${\mu }_{\mathit{esc}}$ which depends on the energy. For $\mu ⩾{\mu }_{\mathit{esc}}$, the isopotential surfaces open up to infinity drawing forth windows through which orbits may leave the potential well and escape from it. For each of these windows, there is an unstable periodic orbit across the opening, bouncing back and forth between the two “walls” of the pass. In paper [3], the authors analyze the shape and size of the windows of escape for the system defined by the Hamiltonian $H_1$, showing that they consist of a “main window” and of a hierarchy of secondary windows which are composed by intricate spiral structures.

The three potentials associated with Hamiltonians $H_1$, $H_2$ and $H_3$ exhibit different symmetries: $H_1$ is invariant under the transformation $x\to -x$, $y\to -y$, so the potential well opens up at four places along the diagonals of the configuration space $(x=\pm y)$. $H_2$ is symmetric with respect to $y\to -y$, and only presents two windows through which orbits may escape. Finally, for $\mu =1$, $H_3$ reduces to a Hénon–Heiles potential, which presents a $2\pi /3$ rotation symmetry.

Numerical experiments show that, although the escape probability $p(\mu \text{,}t)$ behaves rather irregularly during the first several times in a fashion that reflects the specific choice of initial conditions, this probability tends, after these initial transients, towards a constant value which does not depend on the choice of the sampling square of initial conditions. Therefore, due to the different type of symmetries of the Hamiltonians selected, the paper postulates the universality of this property in dynamical systems.

In this paper, we also compare the fractal dimension of the boundaries in the initial-condition space for the potentials associated with $H_1$, $H_2$ and $H_3$, in order to discover if there is another universal property in dynamical systems concerning the dimension of the initial-condition phase space.