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Rigorous numerics in Floquet theory: computing stable and unstable bundles of periodic orbits. (English) Zbl 1293.37033

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.

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

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