A Perturbation Method for Delay Differential Equations

Abstract

In this thesis we construct a perturbation method for delay differential equations (DDEs) based on the method of multiple scales for ordinary differential equations (ODEs) and ordinary difference equations (O$\Delta$Es). The method works for nonlinear DDEs, which are linear DDEs in the unperturbed case. The validity of the method is proven under certain conditions, such as a Lipschitz condition on the perturbation, and we illustrate how the method can be applied by working out several examples. We consider a delayed version of Mathieu's equation, which is especially useful, because it can be used when one linearizes a nonlinear oscillator around a period soluction. We also consider a quadratic perturbation. For these examples we have to analyse the relationship between the solutions of the characteristic equation. There already exists a perturbation method for DDEs, for which one solves a corresponding ODE, and uses this solution as an approximation. This method is only applicable when the influence of the delay is small, and is not always accurate due to the different natures of DDEs and ODEs. We study an example for which this method can be used, and show when it fails to give an accurate approximation. We then show how to use our perturbation method for this example, to obtain an accurate approximation.