Hyperbolic normal forms and invariant manifolds: Astronomical applications

In recent years, the study of the dynamics induced by the invariant manifolds of unstable periodic orbits in nonlinear Hamiltonian dynamical systems has led to a number of applications in celestial mechanics and dynamical astronomy. Two applications of main current interest are i) space manifold dynamics, i.e. the use of the manifolds in space mission design, and, in a quite different context, ii) the study of spiral structure in galaxies. At present, most approaches to the computation of orbits associated with manifold dynamics (i.e. periodic or asymptotic orbits) rely either on the use of the so-called Poincar? - Lindstedt method, or on purely numerical methods. In the present article we briefly review an analytic method of computation of invariant manifolds, first introduced by Moser (1958), and developed in the canonical framework by Giorgilli (2001). We use a simple example to demonstrate how hyperbolic normal form computations can be performed, and we refer to the analytic continuation method of Ozorio de Almeida and co-workers, by which we can considerably extend the initial domain of convergence of Moser?s normal form.