On algorithms for estimating computable error bounds for approximate periodic solutions of an autonomous delay differential equation

Abstract

Machine tool chatter has been characterized as isolated periodic solutions or limit cycles of delay differential equations. Determining the amplitude and frequency of the limit cycle is sometimes crucial to understanding and controlling the stability of machining operations. In Gilsinn [Gilsinn DE. Computable error bounds for approximate periodic solutions of autonomous delay differential equations, Nonlinear Dyn 2007;50:73–92] a result was proven that says that, given an approximate periodic solution and frequency of an autonomous delay differential equation that satisfies a certain non-criticality condition, there is an exact periodic solution and frequency in a computable neighborhood of the approximate solution and frequency. The proof required the estimation of a number of parameters and the verification of three inequalities. In this paper the details of the algorithms will be given for estimating the parameters required to verify the inequalities and to compute the final approximation errors. An application will be given to a Van der Pol oscillator with delay in the non-linear terms.