CHAPTER SELECTED TOPICS IN BIFURCATION THEORY This chapter introduces some basic ideas in bifurcation theory and gives a few examples of current interest. This subject is very large and the chapter cannot pretend to cover anything but a sampling. For this reason two sections-7.2 and 7.S-are written in the nature of surveys; we hope these will alert the reader to some of the important current literature. Sections 7.1 and 7.4 introduce basic ideas in the static and dynamic theory, respectively, and Sections 7.3 and 7.6 select a nontrivial example from each theory to work in detail. These examples are biased towards the authors' interests, so readers may wish to substitute one of their own choosing from the literature. 7.1 BASIC IDEAS OF STATIC BIFURCATION THEORY This section presents a few sample results in static bifurcation theory. The reader should understand that there are many different points of view in this subject and that the results can be refined in several directions. Some of the books that the serious reader should consult after reading this section are Keller and Antman [1969], Sattinger [1973], Nirenberg [1974J, Berger [1977], looss and Joseph [1980], and Antman [1983J. We begin with a few introductory remarks. Consider a beam free to move in a plane, distorted from its natural state by the application of a load A, as shown in Figure 7.1.1. For small A, the beam slightly compresses, but after a critical load Ac is reached, it buckles into one of two possible states. The compressed state is still there, but it has become unstable; the stability has been transferred from the original trivial solution to the stable buckled solutions. The situation 427
SELECTED TOPICS IN BIFURCATION THEORY
Jun 24, 2023
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