AMS – American Mathematical Society Mathematical Reviews Featured Review of the articles:
Almost twenty years have now passed since the first rigorous proofs of the existence of Hopf bifurcations in functional-differential equations (FDEs). Since then, many authors have contributed to the computational issue of calculating the direction of bifurcation/stability of such bifurcations. Normal forms have played a fundamental role in this problem, dating from the work of S. N. Chow and J. Mallet-Paret [J. Differential Equations 26 (1977), no. 1, 112–159; MR0488151 (58 #7718)].

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 [Theory and applications of Hopf bifurcation, Cambridge Univ. Press, Cambridge, 1981; MR0603442 (82j:58089)], where the Poincaré normal form is used.) Frequently, the relationship between the coefficients in the normal form and the coefficients in the original FDE is not easily determined, obscuring the effects of model parameters on system behavior.

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 x'(t)=f(x(t),x(t-1)).

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 Delay differential equations and dynamical systems (Claremont, CA, 1990), 161–175, Lecture Notes in Math., 1475, Springer, Berlin, 1991; MR1132027 (92k:34092)].

Reviewed by Harlan W. Stech