Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1894. / Vita.

Inaug.-Dis.--Göttingen. / Vita.

Dans le cas générique Yu. S. Ilyashenko a donné une solution pour le problème tangentielle du centre et le probème de la monodromie. Néanmoins, on ne connaît pas la solution pour tous les cas non-génériques. Dans cette thèse on étudie une famille des équations hamiltoniennes non-génériques dont l'hamiltonien est un produit de polynômes réels irréductibles de dégre supérieur ou égal à 1. On étudie cette famille dans le but d'avoir un modèle d'équation hamiltonienne qui nous permette de comprendre d'autres cas non-génériques. Cette famille ne satisfait pas necessairement les conditions de généricité de transversalité à l'infini et n'a pas nécessairement tous les points singuliers aux niveaux distincts. Nous considerons quelques conditions géomètriques sur les hamiltoniens qu'on appelle bon partage du plan proyective réel et bonne multiplicité à l'infini. Ces conditions nous servent pour calculer l'orbite par monodromie des cycles évanescents. On résout le problème de la monodromie pour deux sous-familles dans cette famille d'hamiltoniennes. Une d'elles satisfait que tous les points critiques de type centre sont à des niveux critiques distincts, et l'autre satisfait que l'hamiltonien est invariant par la réflexion par rapport à l'axe des y. En utilisant la solution du problème de la monodromie on résout aussi le problème tangentiel du centre pour ces familles. / In the generic case Yu. S. Ilyashenko gave a solution of the tangential center problem and the monodromy problem. However, a solution for all non-generic cases is not known. In this thesis we study a family of non-generic Hamiltonians, whose Hamiltonian is a product of real polynomials of degree equal or bigger than 1. We study this family with the idea that a good understanding of this Hamiltonian model could help us to understand other non-generic cases later. In this family the genericity assumption of transversality at infinity fails and the coincidence of the critical values for different critical points is allowed. We consider some geometric conditions on the Hamiltonians of this family that we call good divide of the real projective plane and good multiplicity at infinity. These conditions help us to compute the orbit under monodromy of vanishing cycles. We give a solution of the monodromy problem of two sub-families in this family. One of them satisfying that all the center critical points are at different critical levels, and the other satisfying that the Hamiltonian is invariant under the reflection with respect to the y-axis. Using the solution of the monodromy problem we also provide a solution of the tangential center problem for those families.

This PhD thesis consists of a summary and seven papers, where various applications of auto-validated computations are studied. In the first paper we describe a rigorous method to determine unknown parameters in a system of ordinary differential equations from measured data with known bounds on the noise of the measurements. Papers II, III, IV, and V are concerned with Abelian integrals. In Paper II, we construct an auto-validated algorithm to compute Abelian integrals. In Paper III we investigate, via an example, how one can use this algorithm to determine the possible configurations of limit cycles that can bifurcate from a given Hamiltonian vector field. In Paper IV we construct an example of a perturbation of degree five of a Hamiltonian vector field of degree five, with 27 limit cycles, and in Paper V we construct an example of a perturbation of degree seven of a Hamiltonian vector field of degree seven, with 53 limit cycles. These are new lower bounds for the maximum number of limit cycles that can bifurcate from a Hamiltonian vector field for those degrees. In Papers VI, and VII, we study a certain kind of normal form for real hyperbolic saddles, which is numerically robust. In Paper VI we describe an algorithm how to automatically compute these normal forms in the planar case. In Paper VII we use the properties of the normal form to compute local invariant manifolds in a neighbourhood of the saddle.