Hénon-Heiles System

Contour plot of the Hénon-Heiles potential

While at Princeton in 1962, Michel Hénon and Carl Heiles worked on the non-linear motion of a star around a galactic center where the motion is restricted to a plane. In 1964 they have published an article ^[1] titled 'The applicability of the third integral of motion: Some numerical experiments'. Their original idea was to find a third integral of motion in a galactic dynamics. For that purpose they have taken a simplified two-dimensional nonlinear axi-symmetric potential and found that the third integral exist only for a limited number of initial conditions. In the modern perspective these initial conditions which doesn't have the third integral motion are called chaotic orbits.

Introduction

The Hénon-Heiles Potential can be expressed as^[2]

$V(x,y) = \frac{1}{2}(x^2+y^2)+\lambda (x^2y - \frac{y^3}{3})$

The Hénon-Heiles Hamiltonian can be written as

$H=\frac1 2 (p_{x}^2+p_{y}^2)+\frac{1}{2}(x^2+y^2)+\lambda (x^2y - \frac{y^3}{3})$

The Hénon-Heiles System (HHS) is defined by the following four equations:

$\dot{x}=p_{x}$

$\dot{p_{x}}=-x-2\lambda xy$

$\dot{y}=p_{y}$

$\dot{p_{y}}=-y-\lambda(x^2-y^2)$

In the classical chaos community, the value of the parameter $\lambda$ is usually taken as unity. Since HHS is specified in $\R^2$, we need a Hamiltonian of degrees of freedom two to model it. It can be solved for some cases using Painlevé Analysis.

Quantum Henon-Heiles Hamiltonian

In the quantum case the Henon-Heiles Hamiltonian can be written as a two-dimensional Schrödinger equation.

The corresponding two-dimensional Schrödinger equation is given by

$i\hbar\frac{\partial}{\partial t} \Psi(x,y) = \left [ \frac{-\hbar^2}{2m}\nabla^2 + \frac1 2 (x^2+y^2 )+\lambda (x^2 y-\frac 1 3 y^3) \right ] \Psi(x,y)$.

Wada Property of the Exit Basins

Hénon-Heiles system shows rich dynamical behavior. Usually the wada property cannot be seen in the Hamiltonian system but Hénon-Heiles exit basin shows an interesting wada property. It can be seen that when the energy is greater than the critical energy, the Hénon-Heiles system has three exit basins. In 2001 M. A. F. Sanjuán et al. ^[3] had shown that in the Henon-Heiles system the exit basins have the wada property.

References

1. Hénon, M.; Heiles, C. (1964). "The applicability of the third integral of motion: Some numerical experiments". The Astronomical Journal 69: 73–79. Bibcode:1964AJ.....69...73H. doi:10.1086/109234.
2. Hénon, Michel (1983), "Numerical exploration of Hamiltonian Systems", in Iooss, G., Chaotic Behaviour of Deterministic Systems, Elsevier Science Ltd, pp. 53–170, ISBN 044486542X
3. Wada basins and chaotic invariant sets in the Henon-Heiles system, Phys. Rev. E 64, 066208 (2001)

External links

• http://mathworld.wolfram.com/Henon-HeilesEquation.html