Computing unstable manifolds in delay differential equations

We present the first algorithm for computing unstable manifolds of saddle-type periodic orbits with one unstable Floquet multiplier in systems of delay differential equations (DDEs). Specifically, we grow the one-dimensional unstable manifold Wu(q) of an associated saddle fixed point q of a Poincare map defined by a suitable Poincar\'e section. Starting close to q along the linear approximation given by the associated eigenfunction, our algorithm grows the manifold as a sequence of points, where the distance between points is governed by the curvature of the one-dimensional intersection curve of Wu(q) with the Poincare section. Our algorithm makes it possible to study global bifurcations in DDEs. We illustrate this with the break-up of an invariant torus and a subsequent boundary crisis to chaos in a DDE model of a semiconductor laser with phase-conjugate feedback. See also the mpeg movie of growing unstable manifolds.