Collocation Methods for the Computation of Periodic Solutions of Delay Diierential Equations Collocation Methods for the Computation of Periodic Solutions of Delay Diierential Equations

In this paper we investigate collocation methods for the computation of periodic solutions of autonomous delay diierential equations (DDEs). Periodic solutions are found by solving a periodic two-point boundary value problem, which is an innnite-dimensional problem for DDEs, in contrast to the case of ordinary diierential equations. We investigate three collocation methods based on piecewise polyno-mials. 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 super-convergence at the mesh points can be lost or recovered depending on the DDE model under consideration and on the choice of colloca-tion discretisation. We end with a brief discussion of adaptive mesh selection.