TITLE:
On the Analytical and Numerical Approximation of Invariant Manifolds

Reprinted from "Les M\'ethodes Modernes de la Mec\'anique C\'eleste"
(Course given at Goutelas, France, 1989), D. Benest and C. Froeschl\'e
(eds.), pp. 285--329, Editions Fronti\`eres, Paris, 1990.

AUTHOR:
Carles Sim\'o

Departament de Matem\`atica Aplicada i An\`alisi,
Universitat de Barcelona, Gran Via, 585, 08007 Barcelona, Spain
<carles@maia.ub.es>

ABSTRACT:
The study of Dynamical Systems and, in particular, Celestial
Mechanics, requires a combination of analytical and numerical
methods. Most of the relevant objects in the phase space can be found
as solutions of equations, either in the phase space itself or in a
suitable functional space (which is approximated by a
finite-dimensional truncation in numerical computations).

In these lectures we consider, first, the continuation of solutions
(to any general problem posed by the objects we are looking for) when
they depend on some parameter. Then, the corresponding analysis of
bifurcations is presented when the differential of the function
determining the solutions has non-maximal rank.

After a quick review on fixed points and their stability and on
numerical integrators, the computation of Poincar\'e maps and their
differentials is presented. This is used for the computation of
periodic orbits, their stability and continuation. Some methods to
compute also quasi-periodic orbits are given.

As indicators of the behaviour of general orbits we stress on the
computation of Lyapunov exponents, warning about the correct
interpretation of what is really computed.

Concerning invariant manifolds, it is useful to have good local
analytic approximations. To this end some symbolic manipulation can be
required. This is simple close to fixed points. Near periodic orbits
or invariant tori it can pose more difficulties, but the general
principle is always the same: to ask for invariance. Having a local
approximation at hand we can globalize the manifolds numerically.
Finally, knowing how to compute invariant manifolds, the computation
of homoclinic and heteroclinic points, their tangencies, and the
variation with respect to parameters is shown to be a relatively
simple problem.

The formulations are presented in general, and several examples
illustrate a sample of topics.