Local bifurcation and symmetry. (English) Zbl 0539.58022

Research Notes in Mathematics, 75. Boston-London-Melbourne: Pitman Advanced Publishing Program. IV, 350 p. £12.95 (1982).
The study of stability and bifurcations can be traced back to Euler and the Bernoullis, and that of symmetry at least to Haüy; these studies were done in physical situations, not purely mathematical. This book does have examples on circular and cylindrical plates; its emphasis is on pure mathematics, based on the applications. It follows Hilbert’s dictum of concentrating on ”sharper aids and simpler methods”, to generalize, to subsume particular cases and methods into powerful theorems of wide application: the references average one per page and cover a broad range of applications although not all: Aris, Cohen, Cole, Joseph, Koiter and Rosenblat do not feature prominently.
The aim is to examine a set of solutions to an equation dependent on a parameter. The interesting points in such a set are those where uniqueness fails, the bifurcation points. The author does this in the standard way of bifurcation theory. The book covers the main theory for partial differential equations and the symmetry group SO(2), for single and multiple eigenvalues, and with the Lyapunov-Schmidt reduction method. The book is a good review of the author’s own work and that of others over the period to 1981/82.
Reviewer: J.J.Cross


37G99 Local and nonlocal bifurcation theory for dynamical systems
58-02 Research exposition (monographs, survey articles) pertaining to global analysis