Title: Taylor method for the integration of ODE

Author: Carles Sim\'o

Dept. Matem\`atica Aplicada i An\`alisi, Univ. de Barcelona,
Gran Via 585, 08007 Barcelona, Spain
E-mail: carles@maia.ub.es

Lectures given at the LTI07 Advanced School on Long Time Integrations

Barcelona, 070903 to 070907

Contents

The method and its motivation, 
Recurrent computations, automatic differentiation,
A first example: H\'enon-Heiles model,
Tolerance, order and step selection, 
Stability,
Complexity,
Dense output and Poincar\'e maps,
Possible difficulties I: the harmonic oscillator,
Roundoff propagation,
Possible difficulties II: the Kepler problem,
Possible difficulties III: the van der Pol canards,
Some applications,
Unfolding the Hopf-saddle-node singularity and stability domain,
Stability boundaries. The planar RTBP in the outer region. The 3D RTBP around
      $L_{4,5}$,
Motion of the giant planets for the age of the solar system,
Arnol'd diffusion: the classical example and a galactic model,
Additional comments.