Collocation methods for the computation of periodic solutions of delay differential equations

In this paper we investigate collocation methods for the computation of periodic solutions of autonomous delay differential equations ( DDEs). Periodic solutions are found by solving a periodic two-point boundary value problem, which is an infinite-dimensional problem for DDEs, in contrast to the case of ordinary differential equations. We investigate three collocation methods based on piecewise polynomials. We discuss computational issues and show numerical orders of convergence using an extensive number of tests. We compare our numerical results with known theoretical convergence results for initial value problems for DDEs. In particular, we show how superconvergence at the mesh points can be lost or recovered depending on the DDE model under consideration and on the choice of collocation discretization. We end with a brief discussion of adaptive mesh selection.