<<Graphics`Graphics3D`

Change of Basis

The action of our matrix under `nice' basis

Since we're working in our `nice' basis where the action is (relatively) simple, we can just write down the matrix.

stretch = 20/19 ; shrink = 99/100 ;

B = ({{stretch, 0, 0}, {0, shrink * Cos[π/20], -shrink * Sin[π/20]}, {0, shrink * Sin[π/20], shrink * Cos[π/20]}})

( {{20/19, 0, 0}, {0, 99/100 cos(π/20), -99/100 sin(π/20)}, {0, 99/100 sin(π/20), 99/100 cos(π/20)}} )

First we'll plot our initial point.

[Graphics:HTMLFiles/index_5.gif]

Let's see what happens to our system as time progresses.  We'll stay in the `nice' coordinate system.

futureSteps = 110 ;

[Graphics:HTMLFiles/futureTrajectory.gif]

If that was a little hard to follow, here's the trail of the system as time progresses.

[Graphics:HTMLFiles/index_120.gif]

The transformation in the standard coordinates

Our nice basis is given by the column vectors of the following matrix.

S = ( {{1, 1, 0}, {1, 0, 1}, {0, -1, 0}} )

( {{1, 1, 0}, {1, 0, 1}, {0, -1, 0}} )

A = S . B . Inverse[S]

Here's our new initial point.

[Graphics:HTMLFiles/index_126.gif]

-Graphics3D -

Here's the system moving through time.

futureSteps = 110 ;

[Graphics:HTMLFiles/futureTrajectoryStandard.gif]

Here's the trail of our system.

[Graphics:HTMLFiles/index_242.gif]

-Graphics3D -