Many structures of interest in dynamical systems can be written as the solution of either an algebraic system or a two point boundary value problem, which depends on a small number of parameters. Geometrically these solutions are pieces of low dimensional manifolds embedded in either spaces of dimension 2 up to around 100 (fixed points), or spaces of dimension in the thousands (periodic motions, homoclinic and heteroclinic trajectories). The boundaries of these pieces are singular sets, at which several pieces meet.

Previous continuation methods have computed tilings of the manifold (a union of non-overlapping neighborhoods). The main algorithmic difficulty is how to maintain this representation when a new tile is added, i.e. to ensure that the new tile does not overlap the old ones. Allgower and Schmidt's simplicial continuation method does this by using sections of a

mesh on the embedding space. In Keller's pseudo-arclength continuation (for one dimensional manifolds) the tiles are intervals, and the overlap is eliminated simply by trimming one side of the interval. In dimensions 2 and higher attempts have been made to use surface meshs as tilings, but it has proven difficult to maintain the tiling - you can't just trim one mesh cell against another and get compatible cells. (Rheinboldt and Brodzik have developed an algorithm which avoids local, but not global, overlap.) Simplicial continuation is not well suited to manifolds embedded in high dimensional spaces, so we need to extend predictor-corrector methods to higher dimensions.

I will describe a new predictor-corrector continuation method for computing implicitly defined manifolds which uses a covering instead of a tiling. Instead of using a mesh it represents the manifold as the union of overlapping neighborhoods, each centered at a point on the manifold. To extend the manifold, a new point is chosen from the boundary of this union

and a neighborhood of it is added to the collection. The problem of overlapping neighborhoods is now moot, but is replaced by the seemingly difficult task of finding a point on the boundary of a set of overlapping neighborhoods. I will show that when the neighborhoods are spherical, or nearly spherical, the boundary can be expressed very simply in terms of a certain set of convex polyhedra. Furthermore, a boundary point can be found in terms of the vertices of these polyhedra . The resulting algorithm is very simple, and since the new point need only be near the boundary, surprisingly robust.