Computer Algebra as a Tool for Analyzing Nonlinear Systems

Abstract

We have studied a variety of weakly perturbed nonlinear dynamical systems using the method of normal forms, a reduction scheme, introduced by Poincare in the late nineteenth century. The method was formalized by Birkhoff who applied it extensively to Hamiltonian mechanics. By invoking a near-identity coordinate transformation, the method of normal forms converts the nonlinear differential equations into simplified equations of motion for the zerothorder approximation to the true solution [1,2,3]. These coordinate transformations, nonlinear functions of the zeroth-order approximations, are found by solving a sequence of linear equations which are determined by the spectrum of the operator associated with the linear, unperturbed motion. The algebra related to these calculations is intensive and well-suited to the symbolic and programming capabilities of Maple. Through the use of procedures written in Maple we have performed high-order normal form computations and have investigated the exploitation of an inherent nonuniqueness of the zeroth-order approximation [4,5]. We will show how this nonuniqueness can be utilized to obtain transformed equations of motion that can be tailored to the needs of the investigator. Coupled with computer algebra, the method of normal forms is a potentially powerful tool for the examination of nonlinear systems [6].KeywordsNormal FormComputer AlgebraDuffing EquationUnperturbed MotionAnalyze Nonlinear SystemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.