Introduction to global bifurcation theory (Q2779226)

This book comes in the form of an introduction to global bifurcation theory. It is based on a post graduation lecture given in the Ecole Polytechnique Fédérale of Lausanne. Even if the lecture was made in the framework of an engineering school, the matter is presented in a very abstract form. In particular, the reader does not find any information on the classical global bifurcation theory of models related to ordinary differential equations with initial conditions. Remind that, since 1935, such a theory was developed, and continues to be dealt, in its most advanced aspects by the Russian Andronov' School. It would be expected that an introduction first deals with simpler cases, precisely those of ordinary differential equations with initial conditions. NEWLINENEWLINENEWLINEAs introduction the first chapter gives two ``attempts of definition'' of the bifurcation notion, an example of bending of an elastic rod (described by a two-dimensional ordinary differential equation with boundary conditions), and a historical presentation of results on Stokes waves (differential equation with partial derivatives model). Chapter two is a presentation of the differential calculus in Banach spaces, providing the basic mathematical tools used in the book. The third chapter is devoted to the local theory of bifurcations. In this framework, the Lyapunov-Schmidt local reduction of problems with infinite dimension to finite dimensional ones is considered for equations in Banach spaces, as well as the local bifurcation theory from a simple eigenvalue. Chapter 4, the longest one, deals with mathematical tools related to analytical sets. The global bifurcation theory is the subject of Chapter 5. The last chapter is devoted to an application to Stokes waves. NEWLINENEWLINENEWLINEThis book presents an evident interest for pure mathematicians, or other abstractly inclined readers, who want an initiation to the bifurcation theory. From this point of view it is well presented. But this will not be the case for engineers, researchers in applied fields of nonlinear dynamics. They will find an information more adapted to their works in other books.