Qualitative Analysis of Dynamical Problems Using a Combination of Symbolic Algebra Coupled with Poincaré-Lindstedt Method

Abstract

Dynamical problems are governed by initial value problems (IVP), often nonlinear, that must be solved to understand the dynamical features of a problem. Numerical methods are very efficient and provide approximations with the precision required, if one is interested in solving a specific problem. Unfortunately, numerical approximations do not provide the insight necessary to understand how a solution depends on the parameters of the problem. Sometimes, perturbation methods help in the sense that they can provide an analytical approximation that shows how the parameters influence the solutions. However to solve a problem for large time intervals require high order approximations that are cumbersome to derive. This paper uses a symbolic method to derive approximation of an IVP using Poincaré-Lindstedt method. The resulting linear problems are combined to have several orders of approximations. The approximations are compared to understand their domain of validity. As a reference to estimate the quality of an approximation, a Runge-Kutta method is used for a specific value of the parameters, of course. To show the main features of the methodology, it is applied to a non-damped Duffing equation, the simplest nonlinear problem used in Mechanics. It is computed analytical approximations of displacement, velocity, and frequency, for any order of approximation and initial conditions desired by the user. To quantify how the order of approximation affect the results, the obtained analytical approximations are evaluated for different combinations of parameters values. The results show that the number of terms has a great influence in the accuracy of the approximation, specially when the term that controls the nonlinearities grows.