Dynamical Systems V

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 publ ished as Volume 5 of the Encyclopaedia, have given a masterly exposition of these two theories, with penetrating insight. §Bifurcation theory and catastrophe theory are two highlights in the study of dynamical systems. Another highlight is celestial mechanics, as found in EMS-3. The current volume builds on the basic material given in EMS 1 and can be considered as a continuation of it. The volume is a survey, intended for readers with some background in dynamical systems, thus aiming at a higher level than Arnol'd's famous little book, "Catastrophe Theory".