Abstract: In this paper we develop mathematically rigorous computer assisted
techniques for studying high order Fourier-Taylor parameterizations of local stable/unstable
manifolds for hyperbolic periodic orbits of analytic vector fields. We exploit the numerical
methods developed in [1] in order to obtain a high order Fourier-Taylor series expansion of
the parameterization. The main result of the present work is an a-posteriori theorem which
provides mathematically rigorous error bounds. The hypotheses of the theorem are checked with
computer assistance. The argument relies on a sequence of prelimi- nary computer assisted proofs
where we validate the numerical approximation of the periodic orbit, its stable/unstable
normal bundles, and the jets of the manifold to some desired order M. We illustrate our
method by implementing validated computations for two dimensional manifolds in the Lorenz
equations in R3 and a three dimensional manifold of a suspension bridge equation in R4.
Authors:
Roberto Castelli, J.P. Lessard, and J.D. Mireles James
Preprint:
(to appear in the Journal of Dynamics and Differential Equations).
Computer Assisted Proof Codes:
(codes require the IntLab package)