Normal forms for semilinear functional differential equations in Banach spaces and applications. Part II

A normal form theory for functional differential equations in Banach spaces of retarded type is addressed. The theory is based on a formal adjoint the ory for the linearized equation at an equilibrium and on the existence of c enter manifolds for perturbed inhomogeneous equations, established in the f irst part of this work under weaker hypotheses than those that usually appe ar in the literature. Based on these results, an algorithm to compute norma l forms on finite dimensional invariant manifolds of the origin is presente d. Such normal forms are important in obtaining the ordinary differential e quation giving the flow on center manifolds explicitly in terms of the orig inal functional differential equation. Applications to Bogdanov-Takens and Hopf bifurcations are presented.