Summary: The DISC project deals with the numerical continuation of invariant manifolds using a method of discretizing global manifolds. It provides a geometrically natural algorithm which works in principal for a general class of manifolds, regardless of the restricted dynamics. Examples include limit sets, co-dimension 1 manifolds separating basins of attraction (separatrices), stable/unstable/center manifolds, nested hierarchies of attracting manifolds in dissipative systems and manifolds appearing in bifurcations. These manifolds are most often invisible to current numerical methods. The approach is based on the general principle of normal hyperbolicity, where the graph transform leads to the numerical algorithms. This gives a highly multiple purpose method. To implement the graph transform, a discrete tubular neighborhood and discrete sections of the associated vector bundle are constructed. From this, the discrete graph transform for the perturbed manifold and the discrete linear graph transform for the perturbed hyperbolic splitting follow.
Keywords: Invariant manifolds, normal hyperbolicity, chaotic dynamics, numerical continuation, bifurcation theory, computational geometry, graph transform.
Invariant manifolds give information about the global
structure of phase space. For example, a codimension 1 manifold may separate several basins of attraction. Invariant manifolds are
also used to simplify dynamical systems. The phase portrait near the manifold is trivial, so restricting the dynamical system to
the manifold effectively reduces the dimension of the system. In some cases invariant manifolds can give a complete qualitative
description of phase space. An example is a nested hierarchy of attracting manifolds in a dissipative system whose global
attractor is a fixed point. If the global attractor is more complicated, it may be contained in an attracting manifold which
contains the nontrivial dynamics. A simple example of the kind of thing we're after is a 3--torus attractor with phase lock
dynamics, where the aim is to visualize an unstable 2--torus separatrix inside this 3--torus. In the ambient phase space, both
the 3--torus attractor and the 2--torus, which is of saddle--type, may be computed using the method.
The key notion needed here is normal hyperbolicity of the invariant manifold. According to the Invariant Manifold Theorem, this guarantees the smooth persistence of the manifold under small perturbations of the system. Normal hyperbolicity generalizes the linearization method for hyperbolic fixed points and periodic orbits to higher dimensional manifolds. For an invariant submanifold V of a Riemannian manifold M, the normal hyperbolicity of V is exhibited by a splitting of TV M into invariant stable, unstable and center parts. The persistence property of normally hyperbolic manifolds enables us to develop insensitive numerical algorithms that compute the manifolds by numerical continuation. The two classical tools for proving the Invariant Manifold Theorem are the graph transform and the Lyapunov-Perron mapping. At each step of the continuation process, the graph transform is used to find the perturbed invariant manifold, chosen here due to its geometrical simplicity.
The graph transform G operates, at least locally, on functions f : E1 ---> E2 with graphs in phase space Rn = E1 + E2, where the splitting respects the dynamics. The globalization of this idea uses sections s : V---> N(V), where the vector bundle N(V) is associated with a tubular neighborhood of the submanifold V. A numerical algorithm requires a formulation where all the components have a finite representation. Thus, to implement the graph transform, discrete sections and a discrete tubular neighborhood of V are constructed. The base space of the vector bundle associated with the discrete tubular neighborhood is the polyhedron P of a simplicial complex supporting V. The tubular neighborhood is constructed by combining explicit local parametrizations of the Grassman manifold with simplex-wise affine interpolation. The discrete sections are locally polynomial and globally continuous. The discrete graph transform is built from these components, by analogy with the usual smooth case.