Continuation Methods for point clouds which represent smooth manifolds

Abstract

Point clouds which represent a smooth manifold arise in the study of nonlinear dynamics as observations of a system. They may represent attractors, or bifurcation diagrams, or simply the image of a smooth set of initial conditions under a flow. This talk will discuss what it means for a point cloud to represent a manifold, and how continuation methods can be used to compute these manifolds. Existing Manifold Learning techniques (LLE, ISOMap) are of this type, using one chart per point. This new approach allows for more complicated local approximations derived from many points, for example that are distributed near the manifold. Examples will be presented of a bifurcation diagram of an oscillator, and several invariant tori.