The Hénon-Heiles System

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In 1964, Henon and Heiles introduced a model to study the motion of a star through a galaxy with a two-dimensional, non-integrable potential, the result of adding a perturbation to an integrable harmonic potential. This Demonstration shows a plot of the phase-space trajectory and the Poincaré map. This system is an often-cited example of chaotic dynamics.

The equations of motion are (OverDot[x])=(Subscript[p, x]), (OverDot[y])=(Subscript[p, y]), (OverDot[(Subscript[p, x])])=-x-2x y, and (OverDot[(Subscript[p, y])])=-y-(x^2) +(y^2), where x and y are the coordinates and (Subscript[p, x]) and (Subscript[p, y]) are the momenta. The energy is equal to ((Subscript[p, x])^2)/2+((Subscript[p, y])^2)/2+(x^2)/2+(x^2)y+(y^2)/2-(y^3)/3.

Initialization CodeResourceCreateNotebook

If you provide initialization code, include a SaveDefinitions->True option in the Manipulate.

ManipulateResourceCreateNotebook

Manipulate[
With[{sdata=
Reap@NDSolve[{{
(Subscript[y, 1])'[t]==Mod[(Subscript[y, 3])[t],2,-1],
(Subscript[y, 2])'[t]==Mod[(Subscript[y, 4])[t],2,-1],
(Subscript[y, 3])'[t]==Mod[-(Subscript[y, 1])[t] (1+2 (Subscript[y, 2])[t]),2,-1],
(Subscript[y, 4])'[t]==Mod[-(((Subscript[y, 1])[t])^2)+(-1+(Subscript[y, 2])[t]) (Subscript[y, 2])[t],2,-1]},
{(Subscript[y, 1])[0]==x0,(Subscript[y, 2])[0]==y0,(Subscript[y, 3])[0]==px,(Subscript[y, 4])[0]==py}},
{(Subscript[y, 1])[t],(Subscript[y, 3])[t],(Subscript[y, 2])[t],(Subscript[y, 4])[t]}, {t, tf},Method->{"EventLocator","Event"->(Subscript[y, 1])[t],"EventAction":> Sow[{(Subscript[y, 2])[t], (Subscript[y, 4])[t],(Subscript[y, 3])[t]}],"EventLocationMethod"->"LinearInterpolation",
"Method"->{"SymplecticPartitionedRungeKutta","DifferenceOrder"->4,"PositionVariables"->{(Subscript[y, 1])[t], (Subscript[y, 2])[t]}}},
StartingStepSize->0.25, MaxSteps->∞]},
Labeled[With[{pp=ParametricPlot3D[Evaluate[{(Subscript[y, 1])[t],(Subscript[y, 2])[t], (Subscript[y, 4])[t]}/.First@sdata],{t,0,tf},ViewPoint->{1,0,-6},Boxed->False,Axes->False,ImageSize->300,SphericalRegion-> True,ColorFunction->(ColorData["ThermometerColors"][#4]&)]},Row[{pp,Graphics[If[sdata[[2]]=!={},MapThread[{ColorData["ThermometerColors"][#1],Point[#2]}&,{(Last/@sdata[[-1,1]]),Most/@sdata[[-1,1]]}],{}],Axes->False,Frame->True,AspectRatio->1,ImageSize->300]}]],Text@Row[{"energy = ",(px^2)/2+(py^2)/2+(x0^2)/2+(x0^2)y0+(y0^2)/2-(y0^3)/3}]]],
{{tf,400,Style["t", Italic]},4,3500, Appearance-> "Labeled"},
{{x0,-.1,Subscript[Style["x", Italic],"0"]},-.1,.01,Appearance-> "Labeled"},
{{y0,-.2,Subscript[Style["y", Italic],"0"]},-.2,.65,Appearance-> "Labeled"},
{{px,0.257229,Subscript[Style["p", Italic],Style["x", Italic]]},-0.257229,.38,Appearance-> "Labeled"},
{{py,-.05,Subscript[Style["p", Italic],Style["y", Italic]]},-.05,.05,Appearance-> "Labeled"}]

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Source & Additional Information

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Enrique Zeleny

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Details, references or citation information

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References for demonstration.

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Hamiltonian

Poincaré Map

Phase Space Trajectory

Hénon-Heiles Equation

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Link to a related demonstration

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Hamiltonian

Poincaré Map

Phase Space Trajectory

Hénon–Heiles Equation

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Published DateResourceCreateNotebook

2012-04-17

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6

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