Multifario - Computing Invariant Manifolds

This is a new application of multifario. A paper describing the algorithm in detail has been submitted to SIADS. A preprint is available here. I wanted to include it because it indicates that the representation of a manifold as a set of overlapping charts may be used in situations other than implicitly defined manifolds, as long as there is a way to project and compute tangent spaces. Also it makes nice pictures.

The Lorenz system is

At the parametervalues s=10, r=28, b=8/3 there are three stationary points. The origin has two stable and one unstable eigenvalue, and the two stationary points near (8.5,8.5,27) and (-8.5,-8.5,27) have one stable and two unstable eigenvalues.

I won't explain the algorithm, but it is based on "fattened" trajectories, which are the point, tangent space and curvature at each point on a trajectory. Here are two in the backward flow, away from the fixed point at the origin:

The two initial points differ only by 1e-7, which is of course one of the things that makes these problems difficult. Here are a couple of views of the surface covered by these fat trajectories:

The upper left image is for t<50. The upper right is for t<150, but a limit was put on the total number of fat trajectories, so not all the surface is filled in (and you can actually see the fat trajectories). The lower two are slices to show the interior structure.

Below we show the Lorenz attractor and how it fits together with the Lorenz manifold, which is the inset to the attractor.

Finally, here are several animations of the computation (QuickTime).

Starting well into the computation of the stable manifold of the origin
The stable manifold of the origin as the maximum time changes.
The unstable manifold of one of the fixedpoints at z=28. (This computation isn't working quite right yet - there are problems with the parallel sheets near the fixed point).

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