Castelli, Roberto; Lessard, Jean-Philippe Rigorous numerics in Floquet theory: computing stable and unstable bundles of periodic orbits. (English) Zbl 1293.37033 SIAM J. Appl. Dyn. Syst. 12, No. 1, 204-245 (2013). Summary: In this paper, a rigorous method to compute Floquet normal forms of fundamental matrix solutions of nonautonomous linear differential equations with periodic coefficients is introduced. The Floquet normal form of a fundamental matrix solution \(\Phi(t)\) is a canonical decomposition of the form \(\Phi(t)=Q(t)e^{Rt}\), where \(Q(t)\) is a real periodic matrix and \(R\) is a constant matrix. To rigorously compute the Floquet normal form, the idea is to use the regularity of \(Q(t)\) and to simultaneously solve 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. The new proposed method does not require rigorous numerical integration of the ODE. Cited in 17 Documents MSC: 37M99 Approximation methods and numerical treatment of dynamical systems 37B55 Topological dynamics of nonautonomous systems 37C27 Periodic orbits of vector fields and flows 65G99 Error analysis and interval analysis 34D05 Asymptotic properties of solutions to ordinary differential equations Keywords:rigorous numerics; Floquet theory; fundamental matrix solutions; contraction mapping theorem; periodic orbits; tangent bundles Software:INTLAB PDFBibTeX XMLCite \textit{R. Castelli} and \textit{J.-P. Lessard}, SIAM J. Appl. Dyn. Syst. 12, No. 1, 204--245 (2013; Zbl 1293.37033) Full Text: DOI arXiv Link