Determinação de integrais primeiras para sistemas dinâmicos em duas e três dimensões com um algoritmo linear

Abstract

This work proposes, initially, a broad approach with regard to the non-classificatory or generalist methods to deal with first and second order ordinary differential equations. More specifically, we will be interested in rational ODE’s (which are directly related to autonomous polynomial differential systems) and, to this end, our starting point consists of the study of theoretical structures introduced by the end of 19th century, in a reference to the works of S. Lie and G. Darboux. In this sense, we present some details contained in each of these theories in order to establish a mixed procedure called the S - function method, which was recently implemented by the UERJ research group working in the topics. It is widely known that such methods present complications when applied to some specific cases, and it will become evident throughout this reading. Thus, implementing and refining algorithms to deal with these type of equations, when integrable, has been the main motivation of the group. Therefore, the development at the very end of this work presents, as a proposal, a highly efficient method that allows us to determine the Darboux polynomials with a linear procedure (the eigenpolynomials of the vector field associated with these equations have fundamental importance for the construction of an integrating factor). In this aspect, we reinforce some ideas which are discussed in the preceding chapters (introducing a notation in conformity with the most recent developments of the group) and definitively establish the algorithms for our purposes, in such a way that, one of them applies to rational first order ODE’s presenting a Darbouxian general solution, while the other two are useful to deal with problems involving second order ODE’s, in subcases that arise as follows: rational second order ODE’s presenting one elementary first integral and one non-Liouvillian first integral; rational second order ODE’s with two Liouvillian first integrals, one of them being elementary. We conclude with a discussion on the performance of our method in computational terms, which we can surely assert, beforehand, represents a routine of singular efficiency (due to the linearity of the equations involved in the procedure) in terms of consumed memory and execution time. Additionally, we present an example as an application of our method to an equation of a non-linear oscillator (Duffing van-der Pol) which is of great interest for various applications.