TITLE:
Numerical integration of high-order variational equations of ODEs

AUTHORS:
Joan Gimeno^(1), Angel Jorba^(1), Marc Jorba-Cusco^(2),
Narcis Miguel^(3) and Maorong Zou^(4)

(1) Departament de Matematiques i Informatica
    Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain
(2) Universidad Internacional de la Rioja (UNIR),
    Av. de la Paz, 137, 26006 Logro\~{n}o, Spain
(3) PAL Robotics S.L., Carrer de Pujades, 77, 4-4, 08005 Barcelona, Spain
(4) Department of Mathematics, University of Texas at Austin,
    Austin, TX 78712, USA

E-mails: joan@maia.ub.es, angel@maia.ub.es, marc.jorba@unir.net,
         narcis.miguel@pal-robotics.com, mzou@math.utexas.edu

ABSTRACT:

This paper discusses the numerical integration of high-order
variational equations of ODEs. It is proved that, given a numerical
method (say, any Runge-Kutta or Taylor method), to use automatic
differentiation on this method (that is, using jet transport up to
order $p$ with a time step $h$ for the numerical integration) produces
exactly the same results as integrating the variational equations up
to of order $p$ with the same method and time step $h$ as before.
Finally, the paper discusses how to use jet transport to obtain power
expansions of Poincar\'e maps (either with spatial or temporal
Poincar\'e sections) and invariant manifolds. Some examples are
provided.