Based on a lecture course, this text gives a rigorous introduction to nonlinear analysis, dynamical systems and bifurcation theory including catastrophe theory. Wherever appropriate it emphasizes a geometrical or coordinate-free approach allowing a clear focus on the essential mathematical structures. It brings out features common to different branches of the subject while giving ample references for more advanced or technical developments.

Suitable for advanced undergraduates, postgraduates and researchers, this self-contained textbook provides an introduction to the mathematics lying at the foundations of bifurcation theory. The theory is built up gradually, beginning with the well-developed approach to singularity theory through right-equivalence. The text proceeds with contact equivalence of map-germs and finally presents the path formulation of bifurcation theory. This formulation, developed partly by the author, is more general and more flexible than the original one dating from the 1980s. A series of appendices discuss standard background material, such as calculus of several variables, existence and uniqueness theorems for ODEs, and some basic material on rings and modules. Based on the author's own teaching experience, the book contains numerous examples and illustrations. The wealth of end-of-chapter problems develop and reinforce understanding of the key ideas and techniques: solutions to a selection are provided.

This textbook gives a contemporary account of singularity theory and its principal application, bifurcation theory.

Providing readers with a solid basis in dynamical systems theory, as well as explicit procedures for application of general mathematical results to particular problems, the focus here is on efficient numerical implementations of the developed techniques. The book is designed for advanced undergraduates or graduates 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 first while updating the context to incorporate recent theoretical developments, in particular new and improved numerical methods for bifurcation analysis.

Bifurcation theory and catastrophe theory are two well-known areas within the field of dynamical systems. Both are studies of smooth systems, focusing on properties that seem to be manifestly non-smooth. Bifurcation theory is concerned with the sudden changes that occur in a system when one or more parameters are varied. Examples of such are familiar to students of differential equations, from phase portraits. Understanding the bifurcations of the differential equations that describe real physical systems provides important information about the behavior of the systems. Catastrophe theory became quite famous during the 1970's, mostly because of the sensation caused by the usually less than rigorous applications of its principal ideas to "hot topics", such as the characterization of personalities and the difference between a "genius" and a "maniac". Catastrophe theory is accurately described as singularity theory and its (genuine) applications. The authors of this book, previously published as Volume 5 of the Encyclopaedia, have given a masterly exposition of these two theories, with penetrating insight.

This volume is the record and product of two International Symposia on the Appli cation of Catastrophe Theory and Topological Concepts in Physics, held in May and December 1978 at the Institute for Information Sciences, University of TUbingen. The May Symposium centered around the conferral of an honorary doctorate upon Professor Rene Thom, Paris, by the Faculty of Physics of the University of TUbingen in recognition of his discovery of universal structure principles and the new di mension he has added to scientific knowledge by his pioneering work on structural stability and morphogenesis. Owing to the broad scope and rapid development of the field, the May Sympos,ium was followed in December by a second one on the same sub jects. The symposia, attended by more than 50 scientists, brought together mathe maticians, physicists, chemists and biologists to exchange ideas about the recent faSCinating impact of topological concepts on the physical sciences, and also to introduce young scientists to the field. The contributions, covering a wide spectrum, are summarized in the subsequent Introduction. The primary support of the Symposia was provided by the "Vereinigung der Freunde der Univertat TUbingen" (Association of the Benefactors of the University). We are particularly indebted to Dr. H. Doerner for his personal engagement and efficient help with the projects, both in his capacity as Secretary of the Association and as Administrative Director of the University.