We study a homoclinic bifurcation in a general functional differential equation of mixed type. More precisely, we investigate the case when the asymptotic steady state of a homoclinic solution undergoes a Hopf bifurcation. Bifurcations of this kind are diffcult to analyse due to the lack of Fredholm properties. In particular, a straightforward application of a Lyapunov–Schmidt reduction is not possible.
As one of the main results we prove the existence of centre-stable and centre-unstable manifolds of steady states near homoclinic orbits. With their help, we can analyse the bifurcation scenario similar to the case for ordinary differential equations and can show the existence of solutions which bifurcate near the homoclinic orbit, are decaying in one direction and oscillatory in the other direction. These solutions can be visualized as an interaction of the homoclinic orbit and small periodic solutions that exist on account of the Hopf bifurcation, for exactly one asymptotic direction t→8 or t→−∞.