and John Toland : Analytic Theory of Global Bifurcation

University Press. All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher, except for reading and browsing via the World Wide Web. Users are not permitted to mount this file on any network servers. Introduction Consider a system of k scalar equations in the form F (λ, x) = 0 ∈ F k , (1.1) where x ∈ F n represents the state of a system and λ ∈ F m is a vector parameter which controls x. (Here F denotes the real or complex field.) A solution of (1.1) is a pair (λ, x) ∈ F m × F n and the goal is to say as much as possible qualitatively about the solution set. Since (1.1) is a finite-dimensional nonlinear equation it might seem unnecessarily restrictive or even pointless to distinguish between the λ and x variables. Why not instead write (λ, x) = Z ∈ F m+n and study the equation F (Z) = 0 where singularity theory is all that is needed? For example, when F : C m+n → C k is given by a power series expansion (that is, F is analytic), a solution Z 0 is called a bifurcation point if, in every neighbourhood of Z 0 , the solutions of F (Z) = 0 do not form a smooth manifold. Locally the solutions form an analytic variety, a finite union of analytic manifolds of possibly different dimensions. So the qualitative theory of F (Z) = 0 in complex finite dimensions is reasonably complete. However (i) in our applications λ is a parameter and the dependence on λ of the solution set is important; (ii) we are looking for a theory that gives the existence globally (i.e. not only in a neighbourhood of a point) of connected sets of solutions; (iii) we are particularly interested in the infinite-dimensional equation F (λ, x) = 0 (1.2) The set S λ normally depends on the choice of λ and usually varies continuously as λ varies. However, it sometimes happens that there is an abrupt change, a bifurca-tion, in the solution set, as λ passes through a particular point λ 0. For example, in Figure 1.1 the number of solutions changes from one to two as λ increases through λ 0. …