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This paper revisits the Taylor method for the numerical integration of initial value problems of Ordinary Differential Equations (ODEs). The main goal is to present a computer program that outputs a specific numerical integrator for a given set of ODEs. The generated code includes a function to compute the jet of derivatives of the solution up to a given order plus adaptive selection of order and step size at run time. The package provides support for several extended precision arithmetics, including user-defined types. The paper discusses the performance of the resulting integrator in some examples, showing that it is very competitive in many situations. This is especially true for integrations that require extended precision arithmetic. The main drawback is that the Taylor method is an explicit method, so it has all the limitations of these kind of schemes. For instance, it is not suitable for stiff systems.

This work focusses on quasiperiodic time-dependent perturbations of ordinary di erential equations near elliptic equilibrium points. This means studying _ x = ( A + "Q(t "))x + "g(t ") + h(x t ") where A is elliptic and h is O(x 2 ). It is shown that, under suitable hypothesis of analyticity, nonresonance and nondegeneracy with respect to ", t here exists a Cantorian set E such t hat for all " 2 E there exists a quasiperiodic solution such t hat it goes to zero when " does. This quasiperiodic solution has the same set of basic frequencies as the p e r t urbation. Moreover, the relative m easure of the set 0 " 0 ] n E in 0 " 0 ] is exponentially small in " 0 . T h e case g 0, h 0 (quasiperiodic Floquet theorem) is also considered.Finally, t he Hamiltonian case is studied. In this situation, most of the i n variant t o r i t hat are near the equilibrium p o i n t are not destroyed, but only slightly deformedand \ s h aken" in a quasiperiodic way. This quasiperiodic \shaking" has the s a m e basic frequenciesas the p e r t urbation. j rg:This kind of equations app e a r s i n m any problems. As an example, we can consider the equations of the motion near the Equilateral Libration points o f t he Earth-Moon system, including (quasiperiodic) perturbations coming f r o m t he noncircular motion This paper was written on October 28th, 1994. y Departament d e M a tem atica Aplicada I, ETSEIB, Universitat P olit ecnica de C a talunya, Diagonal 647,

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The purpose of this paper is to study the dynamics near a reducible lower dimensional invariant tori of a nite-dimensional autonomous Hamiltonian system withd egrees of freedom. We will focus in the case in which the torus has (some) elliptic directions. First, let us assume that the torus is totally elliptic. In this case, it is shown that the di usion time (the time to move a way from the torus) is exponentially big with the initial distance to the torus. The result is valid, in particular, when the torus is of maximal dimension and when it is of dimension 0 (elliptic point). In the maximal dimension case, our results coincide with previous ones. In the zero dimension case, our results improve the existing bounds in the literature. Let us assume now that the torus (of dimension r, 0 r <`) is partially elliptic (let us call m e to the number of these directions). In this case we show that, given a xed numberof elliptic directions (let us call m 1 m e to this number), there exist a Cantor family of invariant tori of dimension r+m 1 , that generalize the linear oscillations corresponding to these elliptic directions. Moreover, the Lebesgue measure of the complementary of this Cantor set (in the frequency space R r+m 1) is proven to beexponentially small with the distance to the initial torus. This is a sort of \Cantorian central manifold" theorem, in which the central manifold is completely lled up by i n variant tori and it is uniquely de ned. The proof of these results is based on the construction of suitable normal forms around the initial torus.