Normal forms for semilinear functional differential equations in Banach spaces and applications. II. (Q1864088)

The author constructs normal forms for center and other invariant manifolds at equilibria for semilinear functional-differential equations in Banach spaces. The construction is based on a formal adjoint theory for the linearized equation at an equilibrium and on the existence of center manifolds for perturbed inhomogeneous equations. These fundamental results were obtained by the author, \textit{W. Huang} and {J. Wu}, [SIAM J. Math. Anal. 34, No. 1, 173--203 (2002; Zbl 1085.34064)]. In this nice paper, an algorithm is presented to compute normal forms. This is very important in getting the finite-dimensional ordinary differential equation in terms of the original infinite-dimensional functional-differential equation in order to determine the flow on center manifolds. As applications, Bogdanov-Takens and Hopf bifurcations are obtained for a delayed differential equation with spatial diffusion. For part I see [Trans. Am. Math. Soc. 352, 2217--2238 (2000; Zbl 0955.35008)].