Rigorous numerics in floquet theory: Computing stable and unstable bundles of periodic orbits
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 F(t) is a canonical decomposition of the form F(t) = Q(t)...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2013 |
| País: | España |
| Institución: | Basque Center for Applied Mathematics (BCAM) |
| Repositorio: | BIRD. BCAM's Institutional Repository Data |
| OAI Identifier: | oai:bird.bcamath.org:20.500.11824/644 |
| Acceso en línea: | http://hdl.handle.net/20.500.11824/644 |
| Access Level: | acceso abierto |
| Palabra clave: | Contraction mapping theorem Floquet theory Fundamental matrix solutions Periodic orbits Rigorous numerics Tangent bundles |
| Sumario: | 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 F(t) is a canonical decomposition of the form F(t) = Q(t)eRt, 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. |
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