Problemas de centro e isocronía. Linealización t-homogénea de campos vectoriales

Abstract

We study several problems related to the qualitative analysis of the differential equations. We analyze the isochronicity problem of a singular point of a plane differential equations systems, for the cases of a centre and of a focus. We characterize a weak isochronous focus of finite order from the Poincaré-Dulac normal form and by means of the existence of a commutator of the vector field whose linear part can be null. We give several applications and examples. We study the existence of commutators of two classes of polynomial plane systems: the polynomial systems which are known as degenerated infinity systems and the systems with constant angular speed (rigid systems). We provide several applications. We deal with a family of nilpotent plane systems. We prove that this family has a generalized Lyapunov function and we give the Taylor expansion at the origin of the Poincaré map of these systems. We study the integrability of the plane quasihomogeneous systems and its relation with the Kowalevskaya exponents of these systems. We give necessary and sufficient conditions so that a $n$-dimensional vector field has the same structure as a quasi-homogeneous vector field, which is not lineal, in general. We conclude giving several applications of the obtained results.