Sistemas dinâmicos em três dimensões com integrais primeiras Liouvillianas: abordagens concretas

Abstract

In this thesis we present a semi-algorithmic method to determine Liouvillian first integrals of polynomial dynamical systems in three dimensions. The procedure we present here is an extension of what we built in my master's thesis (where we were looking for elementary first integrals) and, initially, follows a similar line. However, in the case of non-elementary Liouvillian first integrals, the structure of the integrating factor changes and, thus, to determine it, it is necessary to determine exponential factors. Thus, in this work, we improve the method shown in (1) and develop two algorithms (semi): The first is an extension of the method shown in (2, 1) and requires the computation of Darboux polynomials and exponential factors of the vector field associated to the dynamical system; the second is based on a technique analogous to the one developed in (3) and is based on the determination of vector fields (in two dimensions) associated with the original 3D vector field. We also make two theoretical applications of the second algorithm (for two important cases): 1. Second order rational ordinary differential equations (2 rational ODEs) - We improve the efficiency of the algorithm developed in (3) and demonstrate that the determination of the associated vector fields constitutes a complete algorithm. 2. First order ordinary differential equations (1ODEs) with an elementary function - We extend the algorithm developed in (4) to include the resolution of 1ODEs with algebraic functions. The procedures we developed in this thesis were implemented in a computational package (LeapS3Dim) in Maple. The LeapS3Dim package has several commands that perform the intermediate calculations necessary to obtain the first integral, thus making it useful for research in physics and mathematics.