Contribución a la teoría cualitativa de sistemas dinámicos

Abstract

Throughout this memory, we approach three fundamental problems in the qualitative theory of dynamical systems. In a generai sense, the objective of the theory of dynamical systems is to determine the structure of the solutions set of systems that model the evolution over time of particular phenomena. Firstly, we consider the problem of determining the simplest analytical expression in which an autonomous system can be transformed, by means of changes in the state and time variables, and thus to compute the set of invariants of a vector field. This concept is referred to by some authors as a unique normal form (or hypernormal form). We will show that these simplifications can be carried out using linear procedures and we provide a method to obtain greater simplifications in the normal form, thereby arriving to calculate the unique normal form. Another problem that we have approached in this memory is the problem of the orbital reversibility of a system. It consists, roughly, in determining whether there is a change of variables in the state variables and a time reparametrization such that the resulting system is reversible with respect to a linear involution. We adapt the ideas of the normal form theory under equivalence to characterize the orbital reversibility. If we focus in planar systems, the idea is to reduce the system to a pre-normal form and then analyze the reversibility with respect to the axis (axis-reversibility) modulo orbital equivalence, through the properties of the invariant curves of the vector field. These techniques are applied to calculate families of centers. We complete some families of nilpotent centers that had been partially characterized in the literature. Finally, we have addressed the problem of the analytical integrability of a planar vector field. This problem, in planar vector fields whose first quasi-homogeneous component is Hamiltonian and its Hamiltonian function has simple factors in its factorization in C[x,y], is completely solved in [1]. However, when the Hamiltonian function of the first component has multiple factors in C[x,y] is an open problem. In this case is where our problem is framed. We have considered a degenerate nilpotent system, for which we obtain an equivalent orbital normal form, which we subsequently transform into a system whose first quasi-homogeneous component is irreducible. We provide necessary conditions for the integrability of the system.