Analytic Theory of Global Bifurcation: An Introduction

by Boris Buffoni, John Toland

Overview

Rabinowitz's classical global bifurcation theory, which concerns the study in-the-large of parameter-dependent families of nonlinear equations, uses topological methods that address the problem of continuous parameter dependence of solutions by showing that there are connected sets of solutions of global extent. Even when the operators are infinitely differentiable in all the variables and parameters, connectedness here cannot in general be replaced by path-connectedness. However, in the context of real-analyticity there is an alternative theory of global bifurcation due to Dancer, which offers a much stronger notion of parameter dependence.


This book aims to develop from first principles Dancer's global bifurcation theory for one-parameter families of real-analytic operators in Banach spaces. It shows that there are globally defined continuous and locally real-analytic curves of solutions. In particular, in the real-analytic setting, local analysis can lead to global consequences—for example, as explained in detail here, those resulting from bifurcation from a simple eigenvalue. Included are accounts of analyticity and implicit function theorems in Banach spaces, classical results from the theory of finite-dimensional analytic varieties, and the links between these two and global existence theory.


Laying the foundations for more extensive studies of real-analyticity in infinite-dimensional problems and illustrating the theory with examples, Analytic Theory of Global Bifurcation is intended for graduate students and researchers in pure and applied analysis.

Product Details

ISBN-13:	9780691112985
Publisher:	Princeton University Press
Publication date:	02/02/2003
Series:	Princeton Series in Applied Mathematics , #9
Pages:	184
Product dimensions:	6.00(w) x 9.25(h) x (d)

About the Author

Boris Buffoni holds a Swiss National Science Foundation Professorship in Mathematics at the Swiss Federal Institute of Technology-Lausanne. John Toland is Professor of Mathematical Sciences at the University of Bath and a Senior Research Fellow of the UK's Engineering and Physical Sciences Research Council

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Analytic Theory of Global Bifurcation

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By Boris Buffoni John Toland

Princeton University Press

Copyright © 2003 Princeton University Press
All right reserved.
ISBN: 0-691-11298-3

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Chapter One

Introduction

Consider a system of k scalar equations in the form

F([lambda], x) = 0 [member of] [F.sup.k], (1.1)

where x [member of] [F.sup.n] represents the state of a system and [lambda] [member of] [F.sup.m] is a vector parameter which controls x. (Here F denotes the real or complex field.) A solution of (1.1) is a pair ([lambda], x) [member of] [F.sup.m] x [F.sup.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 [lambda] and x variables. Why not instead write ([lambda], x) = Z [member of] [F.sup.m+n] and study the equation F(Z) = 0 where singularity theory is all that is needed? For example, when F : [C.sup.m+n] [right arrow] [C.sup.k] is given by a power series expansion (that is, F is analytic), a solution [Z.sub.0] is called a bifurcation point if, in every neighbourhood of [Z.sub.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 [lambda] is a parameter and the dependence on [lambda] 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([lambda], x) = 0 (1.2)

when X and Y are real Banach spaces, F : R x X [arrow right] Y is real-analytic and

F([lambda], 0) [equivalent to] 0.

Let

[S.sub.[lambda]] = {x [member of] X : F([lambda], x) = 0}.

The set [S.sub.[lambda]] normally depends on the choice of [lambda] and usually varies continuously as [lambda] varies. However, it sometimes happens that there is an abrupt change, a bifurcation, in the solution set, as [lambda] passes through a particular point [[lambda].sub.0]. For example, in Figure 1.1 the number of solutions changes from one to two as [lambda] increases through [[lambda].sub.0]. For a general treatment of bifurcation theory, see [19].

At this stage it is useful to see an infinite-dimensional example in which the global solution set can be found explicitly.

1.1 EXAMPLE: BENDING AN ELASTIC ROD I

Consider an elastic rod of length L > 0 with one end fixed at the origin of the (x, y)-plane and with the other free to move on the x-axis under the influence of a force along the x-axis towards the origin. If we suppose that the length of the rod does not change (that it is incompressible) and if the force is big enough, then the rod will bend (see Figure 1.2).

We suppose that the rod always lies in the (x, y)-plane (there is no twisting out of the plane in the simple model which follows). To describe the rod's configuration let (x(s), y(s)) be the coordinates of a point at distance s (measured along the rod) from the end which is fixed at the origin. Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

the shape of the rod is given by the angle [empty set](s) between the tangent to the rod and the horizontal at the point (x(s), y(s)), s [member of] [0, L].

Let P denote the applied force. Then the Euler-Bernoulli theory [5, 6] of bending says that the curvature of the rod at a point is proportional to the moment created by the force. In other words,

-k[empty set]'(s) = Py(s),

where k, the constant of proportionality, is determined by the material properties of the rod, Py(s) is the moment of the applied force and -[empty set]'(s) is the curvature at the point (x(s), y(s)). It follows that if P = 0 then [empty set] must be constant, and that constant must be 0 (mod 2[pi]) since y(L) = 0. From now on we consider only the case P > 0. Since y'(s) = sin [empty set](s) and y(0) = y(L) = 0 this gives

[empty set]"(s) + [lambda] sin [empty set](s) = 0, s [member of] [0, L], [empty set]'(0) = [empty set]'(L) = 0, (1.3)

where [lambda] = P/k > 0. If [empty set] is a solution of (1.3), then so is 2k[pi] + [empty set], for any k [member of] Z. We therefore assume that [empty set](0) [member of] (-[pi], [pi]). (If [empty set](0) = [+ or -] [pi] then [empty set] is a constant.)

For all [lambda] > 0, ([lambda], [empty set]) = ([lambda], 0) is a solution of (1.3). This means that the mathematical model of bending admits a solution representing a straight rod, irrespective of how large the applied force might be. These solutions, [empty set] = 0, [lambda] > 0 arbitrary, comprise the family of trivial solutions. To be realistic the model must also have solution corresponding to a bent rod (such as depicted in Figures 1.2 and 1.3). Note that any solution of (1.3) must satisfy the identity

[empty set]'[(s).sup.2] + 4[lambda] [sin.sup.2](1/2[empty set](s)) = 4[lambda] [sin.sup.2](1/2[[empty set].sub.0]), s[member of] [0, L], (1.4)

where [[empty set].sub.0] = [empty set](0). This means that ([empty set](s), [empty set]'(s)), s [member of] [0, L], lies on a segment of the curve in ([empty set], [empty set]')-phase space (see Figure 1.4) given implicitly by

{([empty set], [empty set]') [member of] [R.sup.2] : [[empty set].sup.2] + 4[lambda] [sin.sup.2] 1/2[[empty set].sub.0] = 4[lambda] [sin.sup.2] 1/2[[empty set].sub.0]}[subset][R.sup.2].

We therefore see that there is a solution joining (-|[[empty set].sub.0]|, 0) to (|[[empty set].sub.0]|, 0) in the half-space {([empty set], [empty set]') [member of] [R.sub.2], [empty set]' [greater than or equal to] 0} and one joining (|[empty set]0|, 0) to (-|[[empty set].sub.0]|, 0) in the half-space {([empty set], [empty set]') [member of] [R.sup.2], [empty set]' [greater than or equal to] 0}, the corresponding value of L being given by the formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [theta] [member of] [-[pi]/2, [pi]/2] is given by sin([empty set]/2) = sin(|[[empty set].sub.0]|/2) sin [theta]. In fact there are other solutions of (1.4) which in Figure 1.4 go around the curve 1/2K times for any positive integer K. For such solutions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This integral increases in |[[empty set].sub.0]| and converges to +[infinity] as |[[empty set].sub.0]| [arrow right] [pi].

Since L is the given length of the rod, this relation for each K is an implicit relation between [[empty set].sub.0] = [empty set](0) and [lambda] when ([lambda], [empty set]) is a solution of (1.3). We can best describe the situation with the aid of a bifurcation diagram in which [lambda] is the horizontal axis, [[empty set].sub.0] is the vertical axis, and L is fixed, see Figure 1.5.

The different curves correspond to different values of K, and it is easily checked that the Kth curve intersects the horizontal axis at [(K[pi]/L).sub.2].

It is fortunate but unusual that (1.3) can be reduced to (1.4) and that L can be calculated in terms of elliptic integrals. Because of this, solutions to (1.3) of all amplitudes can be found more-or-less explicitly. This is not the case for slightly more complicated problems and almost never for partial differential equations (PDEs). General methods suitable for PDE applications are based on a study of (1.2). To put (1.3) in such a setting let

X = {[empty set] [member of] [C.sup.2][0, L] : [empty set]'(0) = [empty set]'(L) = 0}, Y = C[0, L] and F([lambda], [empty set]) = [empty set]" + [lambda] sin [empty set] [member of] Y,

for all ([lambda], [empty set]) [member of] R x X. Then F : R x X [arrow right] Y is smooth (Chapter 3) and real-analytic (Chapter 4). In Chapter 8 we show how equation (1.2) can be reduced locally to a finite-dimensional problem. This means that if ([[lambda].sub.0], [x.sub.0]) satisfies (1.2) then there is a neighbourhood U of ([[lambda].sub.0], [x.sub.0]) in R x X, a neighbourhood V of ([[lambda].sub.0], 0) [member of] R x [R.sub.N] and an equation

f([lambda], z) = 0 [member of] [R.sup.M], ([lambda], z) [member of] R x [R.sup.N], N, M [member of] N,

such that the solutions of the two equations are in one-to-one correspondence. The reduction to finite dimensions in §8.2 is called Lyapunov-Schmidt reduction and leads immediately to a local bifurcation theory based on the implicit function theorem. In particular, it yields a classical relation between a nonlinear problem and its linearization.

1.2 PRINCIPLE OF LINEARIZATION

Roughly speaking, the principle of linearization [39] derives from the feeling that when F([lambda], 0) = 0 for all [lambda] and solutions with ||x|| small are sought, the nonlinear problem F([lambda], x) = 0 might as well be replaced with the linear equation [[partial derivative].sub.x]F[([lambda], 0)]x = 0, where [[partial derivative].sub.x]F[([lambda], 0)] denotes the linearization of F with respect to x at x = 0. Since sin [empty set] = [empty set] + O(|[[empty set]|.sup.3]) as [empty set] [arrow right] 0, the linearization of the elastic-rod problem at ([[lambda].sub.0], 0) is

[empty set]"(s) + [[lambda].sub.0][empty set](s) = 0, s [member of] [0, L], [empty set]'(0) = [empty set]'(L) = 0, [[lambda].sub.0] > 0,

and this problem has non-trivial solutions if and only if

[lambda]0 = [(K[pi]/L).sub.2] with [empty set](s) = cos(K[pi]s/L), K [member of] N.

The question is, can any inference be drawn from this about the nonlinear problem (1.3)? The answer is that in quite general situations (including equation (1.3) as a special case) [[lambda].sub.0] is a bifurcation point on the line of trivial solutions of (1.2) only if the linearized problem [d.sub.x]F[([[lambda].sub.0], 0)]x = 0 has a non-trivial solution. The fact that this is also sufficient for bifurcation from the line of trivial solutions for (1.3) (but not in general) is a consequence of the theory of bifurcation from a simple eigenvalue, see §8.4 and §]8.5.

1.3 GLOBAL THEORY

It is clear from Figure 1.5 that there is more to the solution set of equation (1.3) than is predicted by local theory. Global features of the diagram are not a consequence of finite-dimensional reduction methods alone. We will see in Chapter 9 how local bifurcation theory, the implicit function theorem and some elementary results on real-analytic varieties can be used to piece together a global picture of the solution set of (1.2), without assumptions about the size of the solutions under consideration. Provided some general functional-analytic structure is present and F is real-analytic, the global continuum C of solutions which bifurcates from the trivial solutions at a simple eigenvalue contains a continuous curve R with the following properties.

R = {([LAMBDA](s), K(s)) : s [member of] [0,[infinity])} [subset] C is either unbounded or forms a closed loop in R x X.

For each [s.sup.*] [element of] (0,[infinity]) there exists [[rho].sub.*] : (-1, 1) [arrow right] R (a re-parameterization) which is continuous, injective, and

[[rho].sup.*](0) = [s.sup.*], t [arrow right] ([LAMBDA]([[rho].sup.*](t)), K([[rho].sup.*](t)), t [member of] (-1, 1), is analytic.

This does not imply that R is locally a smooth curve. (The map [sigma] : (-1, 1) [arrow right] [R.sup.2] given by [sigma](t) = ([t.sup.2], [t.sup.3]) is real-analytic and its image is a curve with a cusp.) Nor does it preclude the possibility of secondary bifurcation points on R. In particular, since ([LAMBDA], K) : [0,[infinity]) [right arrow] R x X is not required to be globally injective; self-intersection of R (as in a figure eight) is not ruled out.

Secondary bifurcation points on the bifurcating branch, if any, are isolated.

See Theorem 9.1.1 for a complete statement and §9.3 for an application to the elastic-rod problem. This result about real-analytic global bifurcation from a simple eigenvalue is a sharpened version of a theorem due to Dancer. His general results [24, 26] deal with bifurcation from eigenvalues of higher multiplicity and give the path-connectedness of solutions sets that are not essentially one-dimensional. Since his hypotheses are less restrictive, his conclusions are necessarily somewhat less precise. The topological theory of global bifurcation without analyticity assumptions was developed slightly earlier, first for nonlinear Sturm-Liouville problems (such as (1.3)) by Crandall & Rabinowitz [21], then for partial differential equations by Rabinowitz [50], and for problems with a positivity structure by Dancer [25] and Turner [64]. Their basic tool was an infinite-dimensional topological degree function and the outcome was the existence of a global connected (but not always path-connected) set of solutions. Although it is sometimes possible to argue from the implicit function theorem that the connected set given by topological methods is a smooth curve in R x X, this approach fails if there is a secondary bifurcation point on the bifurcating branch.

What is important is that in the analytic case a one-dimensional branch can be followed unambiguously through a secondary bifurcation point. In fact a one-dimensional branch is uniquely determined globally by its behaviour in an open set and can be parameterized globally, even when it intersects manifolds of solutions of different dimensions (see §7.5).

1.4 LAYOUT

We begin in Chapter 2 with a review, without proofs, of the linear functional analysis needed for nonlinear theory. Chapter 3 introduces the main results from nonlinear analysis, including the inverse and implicit function theorems for functions of limited differentiability in Banach spaces. Chapter 4 covers similar ground for analytic operators and operator equations in Banach spaces. In Chapters 5, 6 and 7 we consider finite-dimensional analyticity with particular regard to analyticity over the field R. We prove the classical theorems of Weierstrass on the reduction of an analytic equation to a canonical form which involves a polynomial equation for one variable in which the coefficients are analytic functions of the other variables.

Chapter 8 deals with the finite-dimensional reduction of infinite dimensional problems. When the infinite-dimensional problem involves analytic operators, so does the finite-dimensional reduction and the mapping from solutions of the latter to solutions of the former is also analytic. This chapter is the link between the theory of finite-dimensional analytic varieties and infinite-dimensional problems in Banach spaces.

Continues...

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Table of Contents

Preface ix
Chapter 1. Introduction 1
1.1 Example: Bending an Elastic Rod I 2
1.2 Principle of Linearization 5
1.3 Global Theory 6
1.4 Layout 7
PART 1. LINEAR AND NONLINEAR FUNCTIONAL ANALYSIS 9
Chapter 2. Linear Functional Analysis 11
2.1 Preliminaries and Notation 11
2.2 Subspaces 13
2.3 Dual Spaces 14
2.4 Linear Operators 15
2.5 Neumann Series 16
2.6 Projections and Subspaces 17
2.7 Compact and Fredholm Operators 18
2.8 Notes on Sources 20
Chapter 3. Calculus in Banach Spaces 21
3.1 Fréchet Differentiation 21
3.2 Higher Derivatives 27
3.3 Taylor's Theorem 31
3.4 Gradient Operators 32
3.5 Inverse and Implicit Function Theorems 35
3.6 Perturbation of a Simple Eigenvalue 38
3.7 Notes on Sources 40
Chapter 4. Multilinear and Analytic Operators 41
4.1 Bounded Multilinear Operators 41
4.2 Faà deBruno Formula 44
4.3 Analytic Operators 45
4.4 Analytic Functions of Two Variables 52
4.5 Analytic Inverse and Implicit Function Theorems 53
4.6 Notes on Sources 57
PART 2. ANALYTIC VARIETIES 59
Chapter 5. Analytic Functions on Fⁿ 61
5.1 Preliminaries 61
5.2 Weierstrass Division Theorem 64
5.3 Weierstrass Preparation Theorem 65
5.4 Riemann Extension Theorem 66
5.5 Notes on Sources 69
Chapter 6. Polynomials 70
6.1 Constant Coefficients 70
6.2 Variable Coefficients 74
6.3 Notes on Sources 77
Chapter 7. Analytic Varieties 78
7.1 F -Analytic Varieties 78
7.2 Weierstrass Analytic Varieties 81
7.3 Analytic Germs and Subspaces 86
7.4 Germs of C -analytic Varieties 88
7.5 One-dimensional Branches 95
7.6 Notes on Sources 99
PART 3. BIFURCATION THEORY 101
Chapter 8. Local Bifurcation Theory 103
8.1 A Necessary Condition 103
8.2 Lyapunov-Schmidt Reduction 104
8.3 Crandall-Rabinowitz Transversality 105
8.4 Bifurcation from a Simple Eigenvalue 109
8.5 Bending an Elastic Rod II 111
8.6 Bifurcation of Periodic Solutions 112
8.7 Notes on Sources 113
Chapter 9. Global Bifurcation Theory 114
9.1 Global One-Dimensional Branches 114
9.2 Global Analytic Bifurcation in Cones 120
9.3 Bending an Elastic Rod III 121
9.4 Notes on Sources 124
PART 4. STOKES WAVES 125
Chapter 10. Steady Periodic Water Waves 127
10.1 Euler Equations 127
10.2 One-dimensional Formulation 131
10.3 Main Equation 137
10.4 A Priori Bounds and Nekrasov's Equation 140
10.5 Weak Solutions Are Classical 146
10.6 Notes on Sources 151
Chapter 11. Global Existence of Stokes Waves 152
11.1 Local Bifurcation Theory 152
11.2 Global Bifurcation from = 1 154
11.3 Gradients, Morse Index and Bifurcation 157
11.4 Notes on Sources 159
Bibliography 161
Index 167