kitty3.nb

Sketching Arnold's cat

Sketches of Arnold ' s cat, or a cube, sphere, etc, complement t ... ts into the flow that aren ' t easily seem using just one representation or the other .

Definitions

To sketch the cat's image under the flow of the linear vector field Ax:

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[Graphics:HTMLFiles/kitty3_10.gif]

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RowBox[{RowBox[{RowBox[{ 3), , Use, , the, , (real parts of), , the, , eigenva ... pace, , span[(1, 1)], , of, , 3, , get, , scaled, , by, , 2.}], FontFamily -> Helvetica]}]

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[Graphics:HTMLFiles/kitty3_17.gif]

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[Graphics:HTMLFiles/kitty3_21.gif]

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`Normal form basis' is, e.g. {Re[{-i,1}],Im[{-i,1}]}={{0,1},{-1,0}}. The real part of the eigenvalues is positive, so there is an overall dilation.

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[Graphics:HTMLFiles/kitty3_32.gif]

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Some sample plots for repeated eigenvalues

An example with double eigenvalue 1

[Graphics:HTMLFiles/kitty3_39.gif]

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${{^t (1 + t), -^t t}, {^t t, -^t (-1 + t)}}$

Normal form for double eigenvalue 1 with a one dimensional eigenspace

[Graphics:HTMLFiles/kitty3_49.gif]

[Graphics:HTMLFiles/kitty3_52.gif]

${{^t, ^t t}, {0, ^t}}$

Created by Mathematica (November 14, 2003)