Analysis of dynamical systems via normal forms

Résumé

We study several problems related to the qualitative analysis of the systems of differential equations. We characterize the centers of quasi-homogeneous planar vector fields up to four degree, studying in each case, theirs integrability and reversibility and we make a study about the structural stability for planar quasi-homogeneous vector fields, more concretely, we characterize the quasi-homogeneous vector fields that are structurally stable. We describe the normal forai theory for vector fields and we deal with a case of Takens-Bogdanov singularity with a symmetry. We present a new decomposition which provides us a great simplification in the calculation of the normal form of vector fields whose first quasi-homogeneous component is Hamiltonian and we get a reduced noimal form, up to infinite order, of some families of degenerated vector fields. We study the existence of an inverse integrating factor and give necessary and sufficient conditions for the existence of a formal or algebraic inverse integrating factor. We apply the results obtained, for studying several families of polynomial vectors fields. Finally, we extend the normal form theory of planar vector fields to tridimensional vector fields and we give a reduced normal form, up to infinite order, for a particular case of them. We conclude with the calculation of a case of the Hopf-zero singularity and a case of the triple-zero singularity.