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While at
Princeton in 1962,
Michel HĂ©non and
Carl Heiles worked on the non-linear motion of a star around a galactic center with the motion restricted to a plane. In 1964 they published an article titled "The applicability of the third integral of motion: Some numerical experiments".
[1] Their original idea was to find a third
integral of motion in a galactic dynamics. For that purpose they took a simplified two-dimensional nonlinear rotational symmetric potential and found that the third integral existed only for a limited number of initial conditions.
In the modern perspective the initial conditions that do not have the third integral of motion are called chaotic orbits.
Introduction
The HĂ©nonâHeiles potential can be expressed as
[2]
The HĂ©nonâHeiles
Hamiltonian can be written as
The HĂ©nonâHeiles system (HHS) is defined by the following four equations:
In the classical chaos community, the value of the parameter is usually taken as unity.
Since HHS is specified in , we need a Hamiltonian with 2 degrees of freedom to model it.
It can be solved for some cases using
Painlevé analysis.
Quantum HĂ©nonâHeiles Hamiltonian
In the quantum case the HĂ©nonâHeiles
Hamiltonian can be written as a two-dimensional
Schrödinger equation.
The corresponding two-dimensional Schrödinger equation is given by
Wada property of the exit basins
HĂ©nonâHeiles system shows rich dynamical behavior. Usually the
Wada property cannot be seen in the
Hamiltonian system, but HĂ©nonâHeiles exit basin shows an interesting Wada property. It can be seen that when the energy is greater than the critical energy, the HĂ©nonâHeiles system has three exit basins. In 2001
M. A. F. SanjuĂĄn et al.
[3] had shown that in the HĂ©nonâHeiles system the exit basins have the Wada property.
References
-
^ HĂ©non, M.; Heiles, C. (1964). "The applicability of the third integral of motion: Some numerical experiments". The Astronomical Journal. 69: 73â79.
Bibcode:
1964AJ.....69...73H.
doi:
10.1086/109234.
-
^ HĂ©non, Michel (1983), "Numerical exploration of Hamiltonian Systems", in Iooss, G. (ed.), Chaotic Behaviour of Deterministic Systems, Elsevier Science Ltd, pp. 53â170,
ISBN
044486542X
-
^ Aguirre, Jacobo; Vallejo, Juan C.; SanjuĂĄn, Miguel A. F. (2001-11-27). "Wada basins and chaotic invariant sets in the HĂ©non-Heiles system". Physical Review E. 64 (6). American Physical Society (APS): 066208.
doi:
10.1103/physreve.64.066208.
hdl:
10261/342147.
ISSN
1063-651X.
External links