Parameterization of Invariant Manifolds for Periodic Orbits (II): A Posteriori Analysis and Computer Assisted Error Bounds

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 Castelli et al. (SIAM J Appl Dyn Syst 14(1):132–167, 2015) 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 preliminary 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 \(\mathbb {R}^3\) and a three dimensional manifold of a suspension bridge equation in \(\mathbb {R}^4\).