Encontrando simetrias de Lie não-locais algoritmicamente

Abstract

In this work we present a new approach to calculate symmetries of second-order rational ordinary differential equations (rational 2EDOs). This method can calculate Lie symmetries (point symmetries, dynamic symmetries and non-local symmetries) algorithmically. The procedure is based on an idea arising from the formal equivalence between the total derivative operator and the vector field associated with the 2EDO over its solutions (Cartan’s vector field). Basically, from the formal representation of a Lie symmetry it is possible to extract information that allows using this symmetry in a practical way (in the 2EDO integration process) even in cases where the formal operation cannot be performed, that is, in cases where the symmetry is non-local. Furthermore, when the 2EDO in question depends on parameters, the procedure allows an analysis that determines the regions of the parameter space in which the 2EDO presents a first integral (a conserved quantity) Liouvillian, that is, the method identifies the regions in which the integrable cases are located. We present an analysis of the performance of the developed algorithms and finally, we apply our method to a forced Duffing-van der Pol oscillator (which presents chaotic behavior for arbitrary parameter values) and find the region in the parameter space in which the system is integrable. Furthermore, we found a first integral that had not been determined until then.