Fractal structures in the Hénon-Heiles Hamiltonian

Abstract

2007) 061208) have detected the presence of fractal escape basins in Hénon-Heiles potentials in the unbounded range. Upon fixing the energy value, these basins are detected on the (x, y) and (y, ẏ) planes. In this paper, we explore the appearance of different kinds of fractal structures. We present an analysis of the fractal structures on the escape basins of the (x, y) and (y, E) planes (allowing the energy value E to change and studying the fat-fractal exponent); later, we present these structures on the KAM tori for low energy values, on small regular islands inside the chaotic sea close to the critical energy level on the (y, E)-plane, and most interestingly, on small regular regions inside the escape region. These small regions of bounded motion and regular behavior appear after the critical escape energy, when most of the orbits are escape orbits.

Effects of periodic forcing in chaotic scattering

“…The exit basins associated to phase space in the Hénon-Heiles system can become very complicated in becoming the Wada property [28]. In this section we show that a similar structure holds for the system with forcing in which, for that purpose, the value of the frequency plays a crucial role.…”

On the use of surfaces of section in the N‐body ring problem

“…In 1969, Hénon introduced the section , to study the restricted three‐body problem. This plane has been also used by authors like Zotos, in his numerical exploration of the four‐body problem, and Barrio et al, in the analysis of the fractal structures in the Hénon‐Heiles potential . In this new plane, the initial conditions of the orbits are taken in the x axis, with x = x 0 , y =0, and with the initial velocity parallel to the y axis ( , ).…”

“…On the other hand, the escape of a particle from a dynamical system is another subject that has attracted much interest in the last decades . When the energy of a particle is larger than the energy of escape, the curve of zero velocity of the system opens and many particles escape from the potential well.…”

“…[3][4][5] On the other hand, the escape of a particle from a dynamical system is another subject that has attracted much interest in the last decades. [6][7][8][9][10][11][12][13][14][15][16][17][18][19] When the energy of a particle is larger than the energy of escape, the curve of zero velocity of the system opens and many particles escape from the potential well. However, there are regions of orbits that never escape or escape after a very long time.…”

Orbit classification in the planar circular Pluto-Charon system

“…Hénon back in the late 60s (Hénon, 1969), in order to overcome these limitations introduced a new type of plane which can provide information about areas of bounded and unbounded motion using the section y =ẋ = 0,ẏ > 0 (see also Barrio et al (2008)). In other words, all the initial conditions of the orbits of the test particles are launched from the x-axis with x = x 0 , parallel to the y-axis (y = 0).…”