**MR1355889 (97a:34186a)**

Faria, Teresa; Magalhães, Luis T. Normal forms for retarded functional-differential equations and applications to Bogdanov-Takens singularity.

*J. Differential Equations***122**(1995), no. 2, 201–224.

**MR1355888 (97a:34186b)**

Faria, Teresa; Magalhães, Luis T. Normal forms for retarded functional-differential equations with parameters and applications to Hopf bifurcation.

*J. Differential Equations***122**(1995), no. 2, 181–200.

A variety of alternate approaches have appeared in the literature, often requiring the approximation of the center manifold before evaluating the normal form of the ordinary differential equation on the manifold. (See, for example, the book by B. D. Hassard, N. D. Kazarinoff and Y. H. Wan [

In the papers under review, the authors consider the computation of normal forms for ordinary differential equations describing the flow of FDEs in finite-dimensional invariant spaces associated with finitely many characteristic values for the linearized equation. The method, which does not involve center manifold approximation, is applied to the classic Hopf bifurcation problem, providing a useful comparison to other known methods. New ground is broken in its application to the Bogdanov-Takens singularity of delay-difference equations, including a detailed analysis of the scalar equation

The phase space chosen in this investigation is that of the Chow and Mallet-Paret paper, cited above. However, the methods developed here require a more thorough examination of the direct sum decomposition of the phase space. This is obtained by extension of the classical bilinear form for retarded FDEs and the use of an associated adjoint theory. These results are of independent interest, and perhaps as important as the normal form calculations to which they are applied. Although not discussed here, the algorithms of these papers raise the prospect that symbolic manipulation software can be applied to general FDE normal form calculations, as has been previously demonstrated in the case of Hopf bifurcations [J. M. Franke and H. W. Stech, in

Reviewed by Harlan W. Stech

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