Conférences

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Séminaire
Rigorous numerics in Floquet theory: computing stable and unstable bundles of periodic orbits
Roberto Castelli
Basque Center for Applied Mathematics - BCAM
The stability of a T-periodic orbit of a dynamical system is encoded in the spectral properties of the so-called monodromy matrix. Such matrix is defined as the solution at time T of the non-autonomous linear differential equations with periodic coefficients arising by linearizing  the dynamical system  around the periodic orbit. Up to now, the solution of such linear differential equation can not be analytically computed. In this talk, a new rigorous numerical method to compute fundamental matrix solutions of non-autonomous linear differential equations with periodic coefficients is introduced. Decomposing the fundamental matrix solutions P(t) by their Floquet normal forms, that is as product of real periodic and exponential matrices P(t)=Q(t)e^{Rt}, one solves simultaneously for R and for the Fourier coefficients of Q via a fixed point argument in a suitable Banach space of rapidly decaying coefficients. As an application, the method is used to  rigorously compute the monodromy matrix and the  stable and unstable bundles of periodic orbits.  An example is given in the context of the Lorenz equations.
Date: 2011-11-16 à 10:30
Endroit: PLT-2546