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Computing Manifolds

Before beginning this section of the tutorial, Section 1.2.13 must be completed. If the three equilibria have been found and marked, then we may compute the stable and unstable manifolds of the hyperbolic saddle located at the origin. At the moment, when these are of dimension greater than 1, dstool computes one dimensional pieces of these tangent at the equilibrium to the eigen-directions of the linearization, for those eigen-directions with real eigen-values.

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 longer, more detailed stable and unstable manifolds. Select the input field to the panel text item Unstable Manifold Divisions per Step:, and change the default value of 1 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. Below this field, change the Number of Steps: default entry of 200 to 2500. This basically determines the length of the unstable manifold: the bigger this value is, the longer the manifold will be.

Above the entries for unstable manifolds are corresponding ones for stable manifolds. Edit the Stable Manifold Divisions per Step: text value to 10.

After setting the above values, select the button item Add 1-D Manifold in the Periodic Orbit panel. (Figure [*]) If you are on a color monitor, dstool draws the stable manifolds in red and the unstable manifold in blue.


next up previous contents
Next: Multiple Orbits Up: The Lorenz Equations Previous: Computing Equilibria   Contents
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1998-11-02