Minisymposium: Calculs rigoureux en systèmes dynamiques
Rigorous computation and Floquet theory. A method to enclose the tangent bundles of periodic orbits
Castelli, Roberto
Basque Center for Applied Mathematics

The first ingredient necessary to parametrize the invariant manifolds of periodic orbits is the tangent bundle, that is the  the tangent space of the invariant manifold at the orbit. The tangent directions, as like as the stability
parameters, result  by integrating for one period a non-autonomous system of differential equations with periodic coefficients of the form $$\dot y=A(t) y,\quad A(t)\in \mathbb R^{n\times n}, ~~ \tau \ {\rm periodic} ~~~~~~~~~~~ (1)$$ obtained by linearizing the vector field around the periodic orbit.

In this talk we combine the Floquet theory and rigorous numerics to compute the real Floquet normal form decomposition $\Phi(t)=Q(t)e^{Rt}$ of the fundamental matrix solution of (1). Taking advantage of the periodicity of the function $Q(t)$, the methods aims at computing the Fourier coefficients of $Q(t)$ and the constant matrix $R$ by solving an infinite dimensional algebraic problem in a proper Banach space. As an application the enclosure of the tangent bundles associated to periodic orbits of the Lorenz system and the Arneodo system will be shown.

Lundi, 17 juin, 16h30
Salle Des Plaines A