Rigorous numerics in Floquet theory: computing stable and unstable bundles of periodic orbits

In this paper, a rigorous method to compute the Floquet normal form of fundamental matrix solutions of non-autonomous linear differential equations with periodic coefficients is introduced. The Floquet normal form of a fundamental matrix solution Φ(t) is a canonical decomposition of the form Φ(t) = Q(t)e^(Rt), where Q(t) is a real periodic matrix and R is a constant matrix. To compute rigorously the Floquet normal form, the idea is to use the regularity of Q(t) and to solve simultaneously for R and Q(t) with the contraction mapping theorem in a Banach space of rapidly decaying coefficients. The explicit knowledge of R and Q can then be used to construct, in a rigorous computer-assisted way, stable and unstable bundles of periodic orbits of vector fields.