3D-fattened Thom map^*

For small positive epsilon, this map has an attracting invariant torus. The dynamics on the torus are determined by the saddle point at (0,0,0) - the stable and unstable manifolds of which have dense intersection on the torus. The continuation started at epsilon=0.0 and epsilon was increased. The normal hyperbolicity of the torus is lost around epsilon=0.469. In the pictures below, the torus appears to lose smoothness (a sharp ridge appears) for increasing epsilon. The second image provides some evidence for this loss of smoothness: small scale irregularities appear along the ridge.

Torus of the fattened Thom map, epsilon=0.469, discretized by 5000 third order elements.

Torus of the fattened Thom map, epsilon>0.469, discretized initially with 800 first order elements and adaptively refined.

3D-fattened sink map^**

This example helps illustrate that the algorithm converges to an invariant manifold independently of the dynamics on the manifold. For small positive epsilon, this map has an attracting torus. The dynamics on the torus are quite different than those for the fattened Thom map, however. For epsilon=0.0, the torus dynamics are determined by four fixed points (two saddles, a sink, and a source), and four invariant closed curves. In fact, the torus is the closure of the unstable manifold of the source. For small positive epsilon, almost all points on the torus are attracted to the sink. Hence, the torus may not be visualized by simple iteration. Normal hyperbolicity of the torus is lost for increasing epsilon at about epsilon=0.13, where two stable eigenvalues of the sink become complex conjugate. Past this epsilon value, the unstable manifold is computed, to epsilon=0.353, where the sink map fails to be a local diffeomorphism. The computed unstable manifold is not smooth at the sink. There is a plane of spiral dynamics orthogonal to the torus at the sink, parallel to the plane of the image below. In the computation below, one sees the torus being "wrapped up" into this spiral.

Torus of the fattened "sink" map, epsilon=0.353, discretized with 5000 first order elements.

The user-written program which uses the class library to perform this computation is provided as a class library usage example .

*H. M. Osinga,``Computing Invariant Manifolds,'' Ph.D. Thesis, University of Groningen, The Netherlands, 1996.

**H. W. Broer, A. Hagen and G. Vegter, Multiple purpose algorithms for invariant manifolds, Dynam. Contin. Discrete Implus. Systems B 10 (2003), 331--344.