Homotopy continuation for characteristic roots of delay differential equations using the Lambert W function

In this work, we develop a homotopy continuation method to find the characteristic roots of delay differential equations with multiple delays. We introduce a homotopy parameter into the characteristic equation in such a way that for =0 this equation contains only one exponential term (corresponding to the largest delay) and for =1 the original characteristic equation is recovered. For =0, all the characteristic roots can be expressed in terms of the Lambert W function. Pseudo-arclength continuation is then used to trace the roots as a function of . We demonstrate the method on several test cases. Cases where it may fail are also discussed.