Computing unstable manifolds of periodic orbits 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 autonomous delay differential equations (DDEs) with one fixed delay. Specifically, we grow the one-dimensional unstable manifold Wu(q) of an associated saddle fixed point q of a Poincaré map defined by a suitable Poincaré section Σ. Starting close to q along the linear approximation to Wu(q) 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 Wu(q)∩Σ of Wu(q) with Σ. 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 crisis bifurcation to chaos in a DDE model of a semiconductor laser with phase-conjugate feedback.