9R3. Bifurcation Theory: An Introduction With Applications to PBEs (Applied Mathematical Sciences 156 Series). - Edited by H Kielhofer (Inst for Math, Univ of Augsburg, Universitatsstr 14, Raum 2011, Augsburg D-86135 Germany). Springer-Verlag, New York. 2004. 346 pp. ISBN 0-387-40401-5. \$69.95.

Reviewed by HW Haslach Jr (Dept of Mech Eng, Univ. of Maryland, College Park MD 20742-3035).

This unified exposition of the single-parameter bifurcation response of an operator on infinite dimensional space is organized into three sections: an abstract development of local and then global bifurcation and applications to elliptic and parabolic partial differential equations.

The local bifurcation theory, taking up about half the book, depends on the breakdown of the implicit function theorem and is based on the Lyapunov-Schmidt reduction for infinite dimensional spaces. The case that the Fre´chet derivative has a one-dimensional kernel includes the saddle-node and the various types of pitchfork bifurcations. A principle of exchange of stability is proved. A Hopf bifurcation theorem for periodic solutions is first proved for ordinary differential equations and then for retarded functional differential equations. A center manifold theorem is used for Hopf bifurcation of Hamiltonian, reversible, and conservative systems. A principle of exchange of stability for Hopf bifurcations is proved, and the stability of global continuation of solutions is examined. A principle of exchange of stability for period-doubling bifurcations is developed. The Newton polygon method is described for problems with a one-dimensional kernel in both the degenerate and nondegenerate cases. Floquet exponents are used to create a principle of exchange of stability for degenerate Hopf bifurcations. Some results are given for higher dimensional kernels of the Fre´chet derivative and multiparameter bifurcation.

The global bifurcation theory, subject of the second section, is based on the local index of the operator F, obtained from the Leray-Schauder degree which takes the place of the Brouwer degree of finite-dimensional problems. The crossing number, which is related to a change in Morse index, is defined from the index of the operator. Then if $DxF0,λ$ has an odd crossing number for a particular value of the bifurcation parameter λ, that value of the parameter gives a bifurcation point. A degree is defined for a class of Fredholm operators that includes some classes of elliptic operators and for potential operators, and global bifurcation theorems proved, often using Rabinowitz’s theorem.

The applications forming the third section begin with results on elliptic operators of second order acting on scalar functions. An example using the degenerate Newton polygon is developed. The particular case of the Cahn-Hilliard energy that describes a binary alloy is solved by taking the mass as the bifurcation parameter and applying a symmetry constraint to obtain a one-dimensional kernel. Then local bifurcations of the free nonlinear vibrations described by the nonlinear wave equation with the period as bifurcation parameter are examined in the one-dimensional case and some extensions are given to higher dimensions. Results are also obtained for the wave equation on the unit sphere. A Hopf bifurcation is analyzed for a nonlinear parabolic problem in which the principle of exchange of stability holds. Finally global bifurcation and continuation results are given for quasi-linear elliptic problems and are applied to the Euler-Lagrange equation for the Cahn-Hilliard energy problem.

This mathematics book requires substantial mathematical background. The book is directed to advanced readers rather than beginners so that some basic ideas are not defined. The author has separated the abstract description in the first two sections from the applications with the goal of clarity and allowing the results to be available for any application, but at the risk of making it harder to understand the significance of some of the ideas. In many applications to models of physical behavior, the hypotheses of the various theorems are quite difficult to verify. The book is very useful as a reference because it collects and organizes the bifurcation analysis of infinite-dimensional operators. It could also be used as a text in an advanced course on bifurcation theory with an emphasis on partial differential equations.