Center Manifold Theory for Functional Differential Equations of Mixed Type

We study the behaviour of solutions to nonlinear autonomous functional differential equations of mixed type in the neighbourhood of an equilibrium. We show that all solutions that remain sufficiently close to an equilibrium can be captured on a finite dimensional invariant center manifold, that inherits the smoothness of the nonlinearity. In addition, we provide a Hopf bifurcation theorem for such equations. We illustrate the application range of our results by discussing an economic life-cycle model that gives rise to functional differential equations of mixed type.