# On the Analytical and Numerical Approximation of Invariant Manifolds

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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 Poincaré 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.

Reprinted from "Les Méthodes Modernes de la Mecénique Céleste" (Course given at Goutelas, France, 1989), D. Benest and C. Froeschlé (eds.), pp. 285--329, Editions Frontières, Paris, 1990.

 Author Institutional Affiliation Carles Simo Departament de Matemàtica Aplicada i Anàlisi Universitat de Barcelona (Spain) Author Email carles@maia.ub.es Tutorial Level Basic Tutorial Description Notes Contest Entry No