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Elements of applied bifurcation theory. 3rd ed. (English) Zbl 1082.37002
Applied Mathematical Sciences 112. New York, NY: Springer (ISBN 0-387-21906-4/hbk). xxii, 631 p. (2004).
In the third edition of this textbook [for a review of the first edition see: Elements of applied bifurcation theory, Applied Mathematical Sciences, 112, New York: Springer-Verlag (1995; Zbl 0829.58029)], the material again has been slightly extended while the main structure of the book was kept. The additional material is mostly connected to recent work of the author, for instance, a detailled analysis of the fold-flip bifurcation in two-dimensional mappings or a technique to determine simultanously the Taylor expansion of a center manifold and the normal form on the center manifold. This extension is in line with the author’s general approach to emphasize explicit and computational methods, in particular with the aid of symbolic computations. Most of the normal form transformations are therefore carried out explicitly and normal form coefficients are expressed in terms of the original parameters. This and the clear structure of the book allow applied scientists to use it as a reference book. Some of the more theoretical topics in dynamical systems theory such as center manifolds, rotation numbers or homoclinic tangencies are mentioned only briefly. While full proofs of all statements would certainly go beyond the scope of this book, it might be helpful to provide the interested reader with some additional references. Nevertheless, even where proofs are sketchy, some of the underlying ideas are explained very well. Almost ten years after its first publication, the third edition of Kuznetsov’s book on applied bifurcation theory is still very useful both as a textbook (preferrably in combination with a “classical” dynamical systems textbook) and as a reference work for researchers from the natural sciences, engineering or economics.

37-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory
37G05 Normal forms for dynamical systems
37G10 Bifurcations of singular points in dynamical systems
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
37G25 Bifurcations connected with nontransversal intersection in dynamical systems
37C29 Homoclinic and heteroclinic orbits for dynamical systems
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
34C23 Bifurcation theory for ordinary differential equations
65P30 Numerical bifurcation problems