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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
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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". Their original idea was to find a third
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922:, 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
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Aguirre, Jacobo; Vallejo, Juan C.; Sanjuán, Miguel A. F. (2001-11-27). "Wada basins and chaotic invariant sets in the Hénon-Heiles system".
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355:{\displaystyle H={\frac {1}{2}}(p_{x}^{2}+p_{y}^{2})+{\frac {1}{2}}(x^{2}+y^{2})+\lambda \left(x^{2}y-{\frac {y^{3}}{3}}\right).}
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Hénon, M.; Heiles, C. (1964). "The applicability of the third integral of motion: Some numerical experiments".
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179:{\displaystyle V(x,y)={\frac {1}{2}}(x^{2}+y^{2})+\lambda \left(x^{2}y-{\frac {y^{3}}{3}}\right).}
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Hénon, Michel (1983), "Numerical exploration of
Hamiltonian Systems", in Iooss, G. (ed.),
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et al. had shown that in the Hénon–Heiles system the exit basins have the Wada property.
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899:{\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (x,y)=\left\Psi (x,y).}
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The Hénon–Heiles system (HHS) is defined by the following four equations:
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The corresponding two-dimensional Schrödinger equation is given by
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Hénon–Heiles system shows rich dynamical behavior. Usually the
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In the classical chaos community, the value of the parameter
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is usually taken as unity. Since HHS is specified in
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http://mathworld.wolfram.com/Henon-HeilesEquation.html
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1010:(6). American Physical Society (APS): 066208.
470:{\displaystyle {\dot {p_{x}}}=-x-2\lambda xy,}
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980:Chaotic Behaviour of Deterministic Systems
20:Contour plot of the Hénon–Heiles potential
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982:, Elsevier Science Ltd, pp. 53–170,
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516:{\displaystyle {\dot {y}}=p_{y},}
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667:Quantum Hénon–Heiles Hamiltonian
652:{\displaystyle \mathbb {R} ^{2}}
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1004:Physical Review E
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1071:Chaotic maps
1025:10261/342147
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46:Introduction
30:Michel Hénon
23:
673:Hamiltonian
191:Hamiltonian
34:Carl Heiles
1060:Categories
989:044486542X
930:References
1034:1063-651X
951:: 73–79.
876:Ψ
843:−
822:λ
768:∇
747:ℏ
743:−
714:Ψ
705:∂
701:∂
696:ℏ
618:λ
579:−
563:λ
560:−
554:−
545:˙
492:˙
456:λ
450:−
444:−
435:˙
382:˙
325:−
304:λ
149:−
128:λ
28:in 1962,
26:Princeton
24:While at
953:Bibcode
1032:
986:
1030:ISSN
984:ISBN
50:The
32:and
1020:hdl
1012:doi
961:doi
1062::
1028:.
1018:.
1008:64
1006:.
959:.
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663:.
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955::
894:.
891:)
888:y
885:,
882:x
879:(
872:]
867:)
861:3
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840:y
835:2
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803:+
798:2
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790:(
785:2
782:1
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772:2
761:m
758:2
751:2
736:[
732:=
729:)
726:y
723:,
720:x
717:(
708:t
693:i
645:2
640:R
595:.
592:)
587:2
583:y
574:2
570:x
566:(
557:y
551:=
540:y
536:p
511:,
506:y
502:p
498:=
489:y
465:,
462:y
459:x
453:2
447:x
441:=
430:x
426:p
401:,
396:x
392:p
388:=
379:x
350:.
346:)
340:3
335:3
331:y
322:y
317:2
313:x
308:(
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272:(
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259:+
256:)
251:2
246:y
242:p
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233:2
228:x
224:p
220:(
215:2
212:1
207:=
204:H
174:.
170:)
164:3
159:3
155:y
146:y
141:2
137:x
132:(
125:+
122:)
117:2
113:y
109:+
104:2
100:x
96:(
91:2
88:1
83:=
80:)
77:y
74:,
71:x
68:(
65:V
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