Rigorous and accurate enclosure of invariant manifolds on surfaces

Knowledge about stable and unstable manifolds of hyperbolic fixed points of certain maps is desirable in many fields of research, both in pure mathematics as well as in applications, ranging from forced oscillations to celestial mechanics and space mission design. We present a technique to find highly accurate polynomial approximations of local invariant manifolds for sufficiently smooth planar maps and rigorously enclose them with sharp interval remainder bounds using Taylor model techniques. Iteratively, significant portions of the global manifold tangle can be enclosed with high accuracy. Numerical examples are provided.