Once the periodic points of period 5 are found,
we can compute the stable and unstable manifolds
of these points. In the periodic orbit panel, first
click
on the Show choice box next
to the label Settings. This will reveal a
large collection of parameters which can be adjusted.
Under 1-D Manifold, change the value
of Stable Manifold Divisions per Step to 3.
This
determines the number of interpolating
points which we will use to ``fill in''
the space between two calculated points on the
unstable manifold.
Immediately below the Stable Manifold Divisions per Step
entry, edit the
Number of Steps text field to the value 300.
This
basically determines the length of the unstable manifold: the bigger
this value is, the longer the manifold will be.
Change the corresponding
Unstable Manifold entries similarly.
Then select the button item Add 1-D Manifold.
As we saw in the Lorenz system, on a color monitor
dstool displays the stable manifolds in red and
the unstable manifolds in blue. (Figure
)
An often discussed phenomenon in the study of nonlinear dynamics is the transverse intersection of the stable and unstable manifolds of a non-integrable system. These intersections imply the existence of infinitely many saddle points and the presence of Smale Horseshoes. The torus map has a homoclinic orbit which intersects itself transversally for a certain parameter range, so our goal in the next few paragraphs is to view this transverse intersection. As a word of warning, the computation involved is rather extensive, and may take a few minutes.
First we'll adjust the number of significant digits possible in panel items. Going back to the dstool command window, open the Settings menu button to select the entry Defaults.... On the window which pops up, change the Window Precision: entry from 6 to 12.
Now if it is not already open, use the Settings menu button
to open the Selected Point window. Select the input
field of the text item wx:
in this panel, and replace the old parameter value with
the value 0.59547797038.
Now clear the 2-D Image window
via
and then create the periodic
orbit panel (if it is not already created)
as is detailed in section 1.2.13.
First, in the Periodic Orbit panel, click
on
the Show choice box to the right of the Settings:
entry. We will edit the 1-D Manifold entries to produce
more complete stable and unstable manifolds.
As before, we want to modify how ``long'' the
manifolds are and how ``fine'' they are. Change
the input fields of the following text items on the
Periodic Orbit panel:
We are now ready to begin the computation of the
stable and unstable manifolds. First select
the button item Clear in the Periodic Orbit
panel. This action resets the text item # Found
to the value 0. Now press
Find in the
panel. If dstool does not find
two period orbits of period five, then continue to
select Find. Once the periodic points are found,
choose the button item Add 1-D Manifold. When the
manifolds have been computed, we want to blow up
a region about a saddle point.
Using the text
items Min: and Max: located in the
2-D Image window, change the x-range to
[0.23897, 0.23902] and change the y-range
to
[0.65355, 0.65358]. Now select Refresh
and the homoclinic tangle will appear.(Figure )