Summary: We discuss the numerical computation of homoclinic and heteroclinic orbits in delay differential equations (DDEs). Such connecting orbits are approximated using projection boundary conditions, which involve the stable and unstable manifolds of a steady state solution. The stable manifold of a steady state solution of a DDE is infinite-dimensional, a problem which we circumvent by reformulating the end conditions using a special bilinear form.
The resulting boundary value problem is solved using a collocation method. We demonstrate results, showing homoclinic orbits in a model for neural activity and travelling wave solutions to the delayed Hodgkin-Huxley equation. Our numerical tests indicate convergence behaviour that corresponds to known theoretical results for ordinary differential equations and periodic boundary value problems for DDEs.