Invariant Manifold Movies:

"Lorenz System" Example

The following movies illustrate the stable manifold of the origin in the Lorenz system for the canonical parameter values.

In all of the following the color indicates the distance-to-origin-along-trajectory (sigma).

The stable manifold rotating about the z-axis.:

View 1: Computed up to Sigma=120.
View 2: Computed up to Sigma=150.

``Growing'' the manifold:

Two different views (rotated) of the stable manifold of the origin. The manifold is displayed semi-transparently, except for the ``outer rim'' (opaque) representing the most recently added simplexes. The attractor is shown in black.
View 1
View 2

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".

A great page on invariant manifolds of the Lorenz system by Bernd Krauskopf & Hinke Osinga.

Invariant manifold movies for the "Two Pendula Coupled by a Torsional Spring" example.

This is a joint project with John Guckenheimer.