A generalization of the [formula omitted]-function method applied to a Duffing–Van der Pol forced oscillator

Abstract

Listen

Translate article

In Duarte et al. (2016) and Avellar et al. (2019), we have developed a method (we call it S-function method) that is successful in treating certain classes of rational second order ordinary differential equations (rational 2ODEs) that are particularly ‘resistant’ to canonical Lie methods and to Darbouxian approaches (extensions of the Prelle–Singer method). In this present paper, we generalize the S-function method making it capable of dealing with a class of elementary 2ODEs. We also present a Maple implementation of the method in a computational package – S++ – that is designed to provide a set of tools to allow the user to analyze the intermediary steps of the generalized S-function method. Finally, we apply this method to a Duffing–Van der Pol forced oscillator, obtaining an entirely new class of first integrals. Program summaryProgram Title:S++ – Generalization of the S-function Method.Program Files doi:http://dx.doi.org/10.17632/wr636kbd9m.1Licensing provisions: CC BY NC 3.0Programming language: Maple 17.Nature of problem: Search for first integrals of elementary 2ODEs.Solution method: The method of solution is based on an algorithm described in this paper.Additional comments including restrictions and unusual features: If the 2ODE that is being analyzed presents a very high degree in (x,y,z) or in the elementary function, then the method may not work well. Our implementation can deal with 2ODEs presenting elementary functions and find first integrals in many cases where the 2ODE under study cannot be reduced by other powerful solvers. Besides that, the package presents some useful research commands.