Solving Perturbed Dynamic Systems Using Schur Decomposition

Abstract

This work introduces a methodology for the generation of an approximate analytical solution to perturbed ordinary differential equations using Schur decomposition. This methodology is based on the use of operator theory to find a linear approximation to the ordinary differential equation in an expanded space of configuration. Once this linearization is performed, the Schur decomposition is used to transform the resultant differential equation into an upper triangular system that can be solved sequentially following the upper triangular structure. Based on these results, a perturbation technique is also proposed to study these problems. Finally, this work provides a set of algorithms to automate all the methodologies presented, with a special focus on polynomial differential equations and the use of Legendre polynomials to represent the dynamical system. Several examples of application are provided, including the Duffing oscillator, the Van der Pol oscillator, and the zonal harmonics problem around the moon.