Approximating Solutions of Linear Ordinary Differential Equations with Periodic Coefficients by Exact Picard Iterates

Linear ordinary differential equations (ODEs) with periodic coefficients appear in various interesting applications, such as determining the linear stability regions of systems of vertically driven multiple pendula. Sinha and Butcher [1, 2] have obtained very good approximations to the solutions of such equations by calculating approximate Picard iterates symbolically in the parameters on which the system depends. In this article we show an improvement to the method of Sinha and Butcher. We are able to calculate exact, rather then approximate, Picard iterates of high order. The key point in the programming is the necessity of introducing a user-defined function to carry out the integrations that appear in the definition of the Picard iterates. After introducing the concept of Picard iteration and explaining its fast implementation, we apply the method to determine the stability regions for linearized systems of vertically driven multiple pendula.