Hénon Map

Author

Eric W. Weisstein
January 28, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DynamicalSystems/HenonMap.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/HenonMap.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Henon Dissipative Map

α==1.4, β==0.3

Show[Graphics[{Red,
AbsolutePointSize[.1],
Point/@Drop[NestWhileList[{1-1.4#[[1]]^2+#[[2]],.3#[[1]]}&,{0,0},#.#<100.&,1,10^5],1000]
}],Frame->True,FrameLabel->TraditionalForm/@{x,y}]//Timing

α==0.2, β==1.01

With[{α=0.2,β=1.01},
Show[GraphicsArray[DensityPlot[
-Length[FixedPointList[Function[{1-α#[[1]]^2+#[[2]],β#[[1]]}],{x,y},450,SameTest->(#.#>450&)]],
Evaluate[{x,Sequence@@First[#]}],
Evaluate[{y,Sequence@@Last[#]}],
DisplayFunction->Identity,
AspectRatio->Automatic,
ColorFunction->Hue,
Mesh->False,
FrameLabel->TraditionalForm/@{(Subscript[x, n]),(Subscript[y, n])},
PlotPoints->1000
]&/@{{{-5,5},{-6,7}},{{0,3.5},{-4,-.01}}}
]]
]//Timing

α==0.2, β==0.9991

Show[GraphicsArray[{
With[{α=0.2,β=0.9991},
Graphics[{Red,
AbsolutePointSize[.2],
Point/@Drop[NestWhileList[{1-α#[[1]]^2+#[[2]],β#[[1]]}&,{0,0},#.#<100.&,1,10^5],500]
},
AspectRatio->Automatic,
Frame->True,
DisplayFunction->Identity,
PlotRange->{{-4.5,4.2},{-5,5}},
FrameLabel->TraditionalForm/@{x,y}]
],
With[{α=0.2,β=-0.9999},
Graphics[{Red,
AbsolutePointSize[.2],
Point/@Drop[NestWhileList[{1-α#[[1]]^2+#[[2]],β#[[1]]}&,{0,0},#.#<100.&,1,10^5],500]
},
AspectRatio->Automatic,
Frame->True,
DisplayFunction->Identity,
PlotRange->{{-0.3,1.2},{-1.3,0.3}},
FrameLabel->TraditionalForm/@{x,y}]
]
}]]//Timing

Henon quadratic area-preserving map