The years that have passed since the publication of the first edition of this book proved that the basic principles used to select and present the material made sense. The idea was to write a simple text that could serve as a seri ous introduction to the subject. Of course, the meaning of "simplicity" varies from person to person and from country to country. The word "introduction" contains even more ambiguity. To start reading this book, only a moder ate knowledge of linear algebra and calculus is required. Other preliminaries, qualified as "elementary" in modern mathematics, are explicitly formulated in the book. These include the Fredholm Alternative for linear systems and the multidimensional Implicit Function Theorem. Using these very limited tools, a framewo:k of notions, results, and methods is gradually built that allows one to read (and possibly write) scientific papers on bifurcations of nonlinear dynamical systems. Among other things, progress in the sciences means that mathematical results and methods that once were new become standard and routinely used by the research and development community. Hopefully, this edition of the book will contribute to this process. The book's structure has been kept intact. Most of the changes introduced reflect recent theoretical and software developments in which the author was involved. Important changes in the third edition can be summarized as follows. A new section devoted to the fold-flip bifurcation for maps has appeared in Chapter 9.
This is a book on nonlinear dynamical systems and their bifurcations under parameter variation. It provides a reader with a solid basis in dynamical systems theory, as well as explicit procedures for application of general mathematical results to particular problems. Special attention is given to efficient numerical implementations of the developed techniques. Several examples from recent research papers are used as illustrations. The book is designed for advanced undergraduate or graduate students in applied mathematics, as well as for Ph.D. students and researchers in physics, biology, engineering, and economics who use dynamical systems as model tools in their studies. A moderate mathematical background is assumed, and, whenever possible, only elementary mathematical tools are used. This new edition preserves the structure of the previous editions, while updating the context to incorporate recent theoretical and software developments and modern techniques for bifurcation analysis. Reviews of earlier editions: "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 "It is both a toolkit and a primer" - UK Nonlinear News
