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More on Manifolds

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 \framebox{SELECT} 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 \framebox{Control-MIDDLE} 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 \framebox{SELECT} 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:

Lastly, make sure that the text item Period to Find: is still set to the value 5.

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 [*])


next up previous contents
Next: Saving and Loading Data Up: A Torus Diffeomorphism Previous: More on Fixed Points   Contents
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1998-11-02