Elements of Applied Bifurcation Theory

• 1 Introduction to Dynamical Systems.- 2 Topological Equivalence, Bifurcations, and Structural Stability of Dynamical Systems.- 3 One-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems.- 4 One-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems.- 5 Bifurcations of Equilibria and Periodic Orbits in n-Dimensional Dynamical Systems.- 6 Bifurcations of Orbits Homoclinic and Heteroclinic to Hyperbolic Equilibria.- 7 Other One-Parameter Bifurcations in Continuous-Time Dynamical Systems.- 8 Two-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems.- 9 Two-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems.- 10 Numerical Analysis of Bifurcations.- A Basic Notions from Algebra, Analysis, and Geometry.- A.1 Algebra.- A.1.1 Matrices.- A.1.2 Vector spaces and linear transformations.- A.1.3 Eigenvectors and eigenvalues.- A.1.4 Invariant subspaces, generalized eigenvectors, and Jordan normal form.- A.1.5 FredholmAlternative Theorem.- A.1.6 Groups.- A.2 Analysis.- A.2.1 Implicit and Inverse Function Theorems.- A.2.2 Taylor expansion.- A.2.3 Metric, normed, and other spaces.- A.3 Geometry.- A.3.1 Sets.- A.3.2 Maps.- A.3.3 Manifolds.- References.

Review of earlier edition "I know of no other book that so clearly explains the basic phenomena of bifurcation theory." Math Reviews "The book is a fine addition to the dynamical systems literature. It is good to see, in our modern rush to quick publication, that we, as a mathematical community, still have time to bring together, and in such a readable and considered form, the important results on our subject." Bulletin of the AMS From the reviews of the third edition: "In the third edition of this textbook, the material again has been slightly extended while the main structure of the book was kept. ... the clear structure of the book allows applied scientists to use it as a reference book. ... Kuznetsov's book on applied bifurcation theory is still very useful both as a textbook and as a reference work for researchers from the natural sciences, engineering or economics." (Jorg Harterich, Zentralblatt MATH, Vol. 1082, 2006) "This book deals with the theory of dynamical systems relevant for applications. The material is presented in a systematic and very readable form. It covers recent developments in bifurcation theory, with special attention to efficient numerical implementations. The text aims at an audience of graduate and Ph.D. students in applied mathematics, and researchers in science and engineering, who use dynamical systems and bifurcation analysis as a tool. Each chapter contains useful examples and many illustrations." (Dirk Roose, Bulletin of the Belgian Mathematical Society, 2007)