Two parameter family images
Stable/Unstable Manifolds for the two parameter family
For the parameters alpha = -2.0 and beta = 1.0, our map has two saddle
fixed points. One occurs at (2.28..., .64... ), and all the stable
unstable manifolds shown below correspond to this fixed point.
- Image 1 :
This is the unstable manifold seen in a neighborhood of radius 8
about the origin.
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Image 2 :
Previous image together with the stable manifold.
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Image 2a :
Alternative colors, a little further out. It seems difficult to go
much further out than this--the computer runs out of ram when computing
points on the stable mfld and then slows down drastically as it starts
thrashing the hard drive.
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Image 3 :
Unstable manifold, but image centered at [1,0,0] (y-axis is the
line at infinity; hence the second point where everything pinches off
is the point [1,b,0], which appears above [1,0,0] in this picture.
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Image 4 :
Previous image together with the stable manifold.
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Image 5 :
We redraw image3.jpg, blowing up [1,0,0]. The y-axis is
the exceptional curve and line at infinity now disappears.
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Image 6 :
Previous image together with the stable manifold.
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Image 7 :
Unstable manifold near [0,1,0]. The y-axis is the line at infinity
here. The x-axis is the (true) y-axis. The point [1,b,0] is again in the
picture.
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Image 8 :
Previous image together with the stable manifold.
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Image 9 :
Unstable manifold near the blowup of [0,1,0]. The y-axis is now the
exceptional curve, and the line at infinity disappears.
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Image 10 :
Previous image together with the stable manifold.
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Image 11 :
Blowing up [0,1,0] a second time. The y-axis is the new exceptional
curve. Unstable manifold is shown.
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Image 12 :
Previous image together with the stable manifold.
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