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Analytic theory of global bifurcation. An introduction. (English) Zbl 1021.47044
Princeton Series in Applied Mathematics. Princeton, NJ: Princeton University Press. x, 169 p. (2003).
In this book, the authors consider the theory of generally nonlinear equations and the bifurcation of their solutions. In this connection, they treat both linear and nonlinear functional analysis. The book contains in its Part 1, “Linear and Nonlinear Functional Analysis”, the theory of linear functional analysis, calculus in Banach spaces and multilinear and analytic functions. Part 2 develops the theory of analytic varieties with analytic function and polynomials. Part 3 treats local and global bifurcation and is subdivided into Chapters and Sections as follows: 8. Local bifurcation theory: 8.1 A necessary condition, 8.2 Lyapunov-Schmidt reduction, 8.3 Crandall-Rabinowitz transversality, 8.4 Bifurcation from a simple eigenvalue, 8.5. Bending an elastic road I, 8.6 Bifurcation of periodic solutions, 8.7 Notes on sources. 9. Global bifurcation theory: 9.1 Global one-dimensional branches, 9.2 Global Analytic bifurcation in cones, 9.3 Bending an elastic rod II, 9.4 Notes on Sources.
In Part 4, the authors present applications of this theory to Stokes waves as steady periodic water waves and the global existence of Stokes waves. Parts 1 and 2 are on an advanced textbook level and Parts 3 and 4 are more geared to specialists in the field.

47J15 Abstract bifurcation theory involving nonlinear operators
58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
58C15 Implicit function theorems; global Newton methods on manifolds
76D07 Stokes and related (Oseen, etc.) flows
37Gxx Local and nonlocal bifurcation theory for dynamical systems
37-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory