A Newton-Picard collocation method for periodic solutions of delay differential equations

This paper presents a collocation method with an iterative linear system solver to compute periodic solutions of a system of autonomous delay differential equations (DDEs). We show that the linearized collocation system is equivalent to a discretization of the linearized periodic boundary value problem (BVP). This linear BVP is solved using the Newton-Picard single shooting method ([Int. J. Bifurcation Chaos, 7 (1997), pp. 2547– 2560]). The Newton-Picard method combines a direct method in the subspace of the weakly stable and unstable modes with an iterative solver in the orthogonal complement. As a side effect, we also obtain good estimates for the dominant Floquet multipliers. We have implemented the method in the DDE-BIFTOOL environment to test our algorithm.