Invariant Manifold Movies:

"Pendula Coupled by Torsion" Example

The following movies illustrate the unstable manifold of a hyperbolic saddle point for the system of two simple pendula coupled by a torsional spring (with Hooke's constant epsilon = 0.1).

Note: this manifold is a 2D object living in a 4D phase space; what we show is just a 3D projection. All the seeming self-intersections are artifacts of this projection. Color is used to indicate the fourth coordinate. The black lines forming a square indicates the ``cell'' boundaries (where one of the pendula is upright). Two corners of the square and the center are the saddle points. Other black lines show homoclinic and heteroclinic trajectories lying on the unstable manifold.

Revealing the inner structure:

Given a vector b =(b1, b2), and a constant B, we display only those points on the computed manifold,
for which (psi1 * b1 + psi2 * b2) is bigger than or equal to B.
The following movies are made by holding the vector b constant and allowing B to vary.
View 1: ``Slice away'' orthogonally to b = (1,1).
View 2: ``Slice away'' orthogonally to b = (-1,1).

``Growing'' the manifold:

View 1: The ``growing'' manifold viewed along the vector (0,1,0).
View 2: The manifold viewed from the usual perspective as it ``grows'' up to a longer distance-along-trajectory sigma. The manifold is displayed semi-transparently, except for the ``outer rim'' (opaque) representing the most recently added simplexes.

Soon to be added:

a detailed explanation of this example.

Note on GIF/MPEG viewers:

If your Internet connection is slow, you will get a better "movie" by saving the file first and then viewing the local copy on your computer.
Some of the animated GIF-viewers (e.g., QuickTime) may get confused by the colormaps used in the above files. These files are best viewed in the MS InternetExplorer or Netscape.

Relevant Links:

Our paper on "A fast method for approximating invariant manifolds".

Invariant manifold movies for the "Lorenz System" example.

This is a joint project with John Guckenheimer.