  We summarize various cases where chaotic orbits can be described
analytically. First we consider the case of a magnetic bottle where we have
non-resonant and resonant ordered and chaotic orbits. In the sequence we
consider the hyperbolic Henon map, where chaos appears mainly around the
origin, which is an unstable periodic orbit. In this case the chaotic orbits
around the origin are represented by analytic series (Moser series). We find
the domain of convergence of these Moser series and of similar series around
other unstable periodic orbits. The asymptotic manifolds from the various
unstable periodic orbits intersect at homoclinic and heteroclinic orbits that
are given analytically. Then we consider some Hamiltonian systems and we find
their homoclinic orbits by using a new method of analytic prolongation. An
application of astronomical interest is the domain of convergence of the
analytical series that determine the spiral structure of barred-spiral
galaxies.
