Minisymposium: Calculs rigoureux en systèmes dynamiques
Lundi, 17 juin
Salle Des Plaines A
Rigorous continuation of solutions of PDEs
Gameiro, Marcio
University of Sao Paulo at San Carlos

We present a rigorous numerical method to compute global smooth manifolds of solutions of infinite dimensional nonlinear problems. We use a parameter continuation method on a finite dimensional projection to construct a simplicial approximation of the manifold. This simplicial approximation is then used to construct local charts and an atlas of the global manifold in the infinite dimensional space. The idea behind the construction of the smooth charts is to use the so-called radii polynomials to verify the hypotheses of the uniform contraction principle on each simplex. The construction of the manifold is then finalized by proving smoothness along the common lower dimensional faces of adjacent simplices. The method is applied to compute one- and two-dimensional bifurcation manifold of equilibria and time periodic orbits for PDEs.

A bifurcation diagram of steady states for a system of PDEs via rigorous numerics, Part I: Theoretical description of the method
Vanicat, Matthieu
ENS Cachan, France

In this talk, I will present a method using rigorous numerics to compute several smooth branches of steady states for a system of PDEs, in the case of three reaction-diffusion equations depending on a diffusion parameter. The problem of showing the existence of steady states is similar to the one of finding a Newton-like operator fixed point. The corresponding operator is constructed explicitly in a neighbourhood of a numerical approximated solution. The proof of existence and uniqueness lies on the uniform contraction principle. An effective way to check the hypotheses of this theorem is to construct the radii polynomials which control the truncation error of the numerical approximation, and to show that these radii polynomials are simultaneously negative. This check is done rigorously using interval arithmetic. The implementation issues and results will be discussed in details in a talk by Maxime Breden.

A bifurcation diagram of steady states for a system of PDEs via rigorous numerics, Part II: Implementation issues and results
Breden, Maxime
ENS Cachan, France

As it will be explained by Matthieu Vanicat in another talk, computing smooth branches of steady states for some reaction diffusion system of PDEs and rigorously proving the validity of those results can be done numerically by a finite number of verifications. However the computational cost of this method can be significant. Thus I will discuss how to adapt the different parameters of the algorithm such as the dimension of the Galerkin projection and the decay rate of the Fourier coefficients, as well as the way of constructing whole branches of bifurcations, in order to improve the speed of the proof. Finally, to illustrate the interest of this method I will show on one particular system the kind of results that we were able to get such as the existence of several non trivial co-existing steady states.

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.