Applications of Centre Manifold Theory (Applied Mathematical Sciences)

Applied Mathematical Sciences I Volume 35

Jack Carr

Applications of Centre Manifold Theory

Springer-Verlag New York Heidelberg

Berlin

Jack Carr Department of Mathematics Heriot-Watt University Riccarton, Currie Edinburgh EH14 4AS Scotland

Library of Congress Cataloging in Publication Data Carr, Jack. Applications of centre manifold theory.

(Applied mathematical sciences; v. 35) "Based on a series of lectures given in the Lefschetz Center for Dynamical Systems in the Division of Applied Mathematics at Brown University during the academic year 1978-79"-Pref. 1. Manifolds (Mathematics) 2. Bifurcation theory. I. Title. II. Series: Applied mathematical sciences (Springer-Verlag New York Inc.); v. 35. QA1.A647 vol. 35 [QA613] 510s [516'.07] 81-4431 AACR2

All rights reserved.

No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. The use of general descriptive names, trade names, trademarks, etc. in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

© 1981 by Springer-Verlag New York Inc. Printed in the United States of America 9 8 7 6 5 4 3 2

1

To my parents

PREFACE

These notes are based on a series of lectures given in the Lefschetz Center for Dynamical Systems in the Division of Applied Mathematics at Brown University during the academic year 1978-79.

The purpose of the lectures was to give an introduction to the applications of centre manifold theory to differential equations.

Most of the material is presented in an informal

fashion, by means of worked examples in the hope that this clarifies the use of centre manifold theory.

The main application of centre manifold theory given in these notes is to dynamic bifurcation theory.

Dynamic

bifurcation theory is concerned with topological changes in the nature of the solutions of differential equations as parameters are varied.

Such an example is the creation of periodic

orbits from an equilibrium point as a parameter crosses a critical value.

In certain circumstances, the application of

centre manifold theory reduces the dimension of the system under investigation.

In this respect the centre manifold

theory plays the same role for dynamic problems as the Liapunov-Schmitt procedure plays for the analysis of static solutions.

Our use of centre manifold theory in bifurcation

problems follows that of Ruelle and Takens [57] and of Marsden and McCracken [51].

In order to make these notes more widely accessible,

we give a full account of centre manifold theory for finite dimensional systems. voted to this.

Indeed, the first five chapters are de-

Once the finite dimensional case is under-

stood, the step up to infinite dimensional problems is

Throughout these notes we give the

essentially technical.

simplest such theory, for example our equations are autonomous.

Once the core of an idea has been understood in a

simple setting, generalizations to more complicated situations are much more readily understood.

In Chapter 1, we state the main results of centre manifold theory for finite dimensional systems and we illustrate In Chapter 2, we prove

their use by a few simple examples.

the theorems which were stated in Chapter 1, and Chapter 3 contains further examples.

In Section 2 of Chapter 3 we out-

line Hopf bifurcation theory for 2-dimensional systems.

In

Section 3 of Chapter 3 we apply this theory to a singular perturbation problem which arises in biology.

In Example 3 of

Chapter 6 we apply the same theory to a system of partial differential equations.

In Chapter 4 we study a dynamic bifurca-

tion problem in the plane with two parameters.

Some of the

results in this chapter are new and, in particular, they confirm a conjecture of Takens [64].

Chapter 4 can be read in-

dependently of the rest of the notes.

In Chapter 5, we apply

the theory of Chapter 4 to a 4-dimensional system.

In Chap-

ter 6, we extend the centre manifold theory given in Chapter 2 to a simple class of infinite dimensional problems.

Fin-

ally, we illustrate their use in partial differential equations by means of some simple examples.

I first became interested in centre manifold theory through reading Dan Henry's Lecture Notes [34]. these notes is enormous.

My debt to

I would like to thank Jack K. Hale,

Dan Henry and John Mallet-Paret for many valuable discussions during the gestation period of these notes.

This work was done with the financial support of the United States Army, Durham, under AROD DAAG 29-76-G0294.

Jack Carr December 1980

TABLE OF CONTENTS Page CHAPTER 1. 1.1.

INTRODUCTION TO CENTRE MANIFOLD THEORY

Introduction

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Bifurcation Theory Comments on the Literature

CHAPTER 2. 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7.

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PROOFS OF THEOREMS

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CHAPTER 3.

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Introduction A Simple Example Existence of Centre Manifolds Reduction Principle. Approximation of the Centre Manifold Properties of Centre Manifolds Global Invariant Manifolds for Singular Perturbation Problems. Centre Manifold Theorems for Maps. .

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EXAMPLES

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3.1. 3.2. 3.3.

Rate of Decay Estimates in Critical Cases. Hopf Bifurcation Hopf Bifurcation in a Singular Perturbation .

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Bifurcation of Maps .

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CHAPTER 4.

4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. 4.8. 4.9.

CHAPTER S. 5.1. 5.2. 5.3. 5.4.

Problem.

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Motivation

Centre Manifolds

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1.2. 1.3. 1.4.

Examples

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BIFURCATIONS WITH TWO PARAMETERS IN TWO SPACE DIMENSIONS . . . . . . . . . . . .

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Introduction

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Preliminaries .

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64 64 77

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More Scalin Completion of the Phase Portraits. Remarks and Exercises. Quadratic Nonlinearities

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APPLICATION TO A PANEL FLUTTER PROBLEM Introduction Reduction to a Second Order Equation Calculation of Linear Terms. Calculation of the Nonlinear Terms .

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Page INFINITE DIMENSIONAL PROBLEMS.

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Introduction

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Semigroup Theory Centre Manifolds

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CHAPTER 6.

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Examples

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REFERENCES

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136

INDEX .

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CHAPTER 1

INTRODUCTION TO CENTRE MANIFOLD THEORY

1.1.

Introduction

In this chapter we state the main results of centre manifold theory for finite dimensional systems and give some simple examples to illustrate their application. 1.2.

Motivation To motivate the study of centre manifolds we first

look at a simple example. x = ax3,

where

a

is a constant.

Consider the system y = -y + y2,

(1.2.1)

Since the equations are uncoupled

we can easily show that the zero solution of (1.2.1) is asymptotically stable if and only if

a < 0.

x = ax 3 + x2y y = -y + y + xy - x3.

Suppose now that

(1.2.2)

Since the equations are coupled we cannot immediately decide if the zero solution of (1.2.2) is asymptotically stable, but we might suspect that it is if

a < 0.

The key to understand-

ing the relation of equation (1.2.2) to equation (1.2.1) is t

INTRODUCTION TO CENTRE MANIFOLD THEORY

1.

2

an abstraction of the idea of uncoupled equations. A curve, say

y = h(x), defined for

lx1

small, is

said to be an invariant manifold for the system of differential equations x = f(x,Y),

if the solution lies on the curve

(1.2.3)

Y = g(x,Y),

of (1.2.3) through

(x(t),y(t))

y = h(x)

for small

t, i.e., y(t)

manifold.

=

is an invariant

Thus, for equation (1.2.1), y - 0

h(x(t)).

(x0,h(x0))

Note that in deciding upon the stability of the

zero solution of (1.2.1), the only important equation is x = ax3, that is, we only need study a first order equation on a particular invariant manifold.

The theory that we develop tells us that equation (1.2.1') has an invariant manifold

h(x) = 0(x2)

as

y = h(x),

jxj

small, with

Furthermore, the asymptotic stability

x + 0.

of the zero solution of (1.2.2) can be proved by studying a first order equation. u = au

This equation is given by + u2h(u) = au3 + 0(u4),

(1.2.4)

and we see that the zero solution of (1.2.4) is asymptotically stable if

a < 0

and unstable if

a > 0.

This tells us that

the zero solution of (1.2.2) is asymptotically stable if and unstable if

a < 0

a >

0

as we expected.

We are also able to use this method to obtain estimates for the rate of decay of solutions of (1.2.2) in the case a < 0.

with u(t) y(t)

For example, if (x(0),y(0))

(x(t),y(t))

is a solution of (1.2.2)

small, we prove that there is a solution

of (1.2.4) such that - h(u(t))(1+o(1))

as

x(t)

t -

u(t)(1+o(1)),

1.3.

Centre Manifolds

1.3.

Centre Manifolds

3

We first recall the definition of an invariant manifold for the equation x - N(x)

where

x E1Rn.

A set

is said to be a local invari-

S d1Rn

ant manifold for (1.3.1) if for of (1.3.1) with T >

x(O) = x0

x0 E S, the solution

is in

If we can always choose

0.

(1.3.1)

for

S

Iti

x(t)

where

< T

then we say that

T =

S

is an invariant manifold.

Consider the system

Ax + f(x,y) By + g(x,y) where

x E1Rn, y E1Rm

A

and

and

such that all the eigenvalues of while all the eigenvalues of The functions

and

f

g

(1.3.2)

A

have zero real parts

have negative real parts.

B

are

are constant matrices

B

C2

f'(0,0) = 0, g(0,0) = 0, g'(0,0) = 0.

Jacobian matrix of If

and

f

f(0,0) - 0,

with

(Here, f'

f.) g

are identically zero then (1.3.2) has

two obvious invariant manifolds, namely

The invariant manifold

x =

0

x -

0

and

y = 0.

is called the stable manifold,

since if we restrict initial data to

The invariant manifold

tend to zero.

is the

x =

y =

0, all solutions 0

is called the

centre manifold.

In general, if (1.3.2) and if

h(0)

-

h

y - h(x)

is an invariant manifold for

is smooth, then it is called a centre manifold

0, h'(0) -

0.

We use the term centre manifold in

place of local centre manifold if the meaning is clear.

4

INTRODUCTION TO CENTRE MANIFOLD THEORY

1.

If

and

f

of (1.3.2) tend

are identically zero, then all solutions

g

exponentially fast, as

to solutions

t

of

x - Ax.

(1.3.3)

That is, the equation on the centre manifold determines the asymptotic behavior of solutions of the full equation modulo exponentially decaying terms.

We now give the analogue of

these results when

are non-zero.

f

and

g

These results

are proved in Chapter 2.

There exists a centre manifold for (1.3.2),

Theorem 1. y - h(x),

lxj

< 6, where

h

is

C2.

The flow on the centre manifold is governed by the n-dimensional system u = Au + f(u,h(u))

(1.3.4)

which generalizes the corresponding problem (1.3.3) for the linear case.

The next theorem tells us that (1.3.4) contains

all the necessary information needed to determine the asymptotic behavior of small solutions of (1.3.2). Theorem 2.

(a)

Suppose that the zero solution of (1.3.4) is

stable (asymptotically stable) (unstable).

Then the zero solu-

tion of (1.3.2) is stable (asymptotically stable) (unstable). (b)

stable.

Let

(x(O),y(O)) u(t)

Suppose that the zero solution of (1.3.4) is (x(t),y(t))

be a solution of (1.3.2) with

sufficiently small.

of (1.3.4) such that as

Then there exists a solution

t i m,

x(t) - u(t) + O(e- Yt) 11.3.5) y(t)

- h(u(t)) + O(e

yt)

Examples

1.4.

where

y >

5

is a constant.

0

If we substitute

y(t) = h(x(t))

into the second equa-

tion in (1.3.2) we obtain h'(x)[Ax + f(x,h(x))] = Bh(x) + g(x,h(x)). Equation (1.3.6) together with the conditions h'(0) = 0

(1.3.6)

h(0) = 0,

is the system to be solved for the centre manifold.

This is impossible, in general, since it is equivalent to The next result, however, shows that, in prin-

solving (1.3.2).

ciple, the centre manifold can be approximated to any degree of accuracy.

For functions

$:]Rn +]Rm

which are

CI

in a neigh-

borhood of the origin define

_ O'(x)[Ax + f(x,h(x))] - B$(x) - g(x,h(x))

(MO) (X)

Note that by (1.3.6), (Mh)(x) = 0. Theorem 3.

origin in

Let ]Rn

into

Suppose that as as

1.4.

x - 0,

jh(x)

with

]Rm

x _ 0, -

C1

be a

$

mapping of a neighborhood of the $(0) = 0

(M$)(x) - O(jxjq)

and

$'(0) =

where

0.

q > 1.

Then

$(x)l = O(Ixl q)

Examples

We now consider a few simple examples to illustrate the use of the above results. Example 1.

Consider the system x - xy + ax3 + by2x Y - -Y + cx2 + dx2Y.

INTRODUCTION TO CENTRE MANIFOLD THEORY

1.

6

By Theorem 1, equation (1.4.1) has a centre manifold To approximate

we set

h

(M0)(x) = m'(x)[xm(x) + ax3 + bxm2(x)l + m(x) If

'(x)

=

0(x2)

if

m(x)

=

cx2,

cx2 + 0(x4).

y = h(x).

(MO) (x) = 4(x)

then

cx2

Hence,

cx2 + 0(x4).

-

dx2m(x)

-

0(x4), so by Theorem 3, h(x) =

=

No) (x)

-

By Theorem 2, the equation which determines the

stability of the zero solution of (1.4.1) is u = uh(u) + au3 + buh2(u) = (a+c)u3 + 0(u5).

Thus the zero solution of (1.4.1) is asymptotically stable if a + c <

and unstable if

0

If

a + c > 0.

a + c = 0

we have to obtain a better approximation to Suppose that fi(x)

=

0(x4).

Then

m(x) = cx2 + cdx4 h(x) = cx

2

+ cdx

4

a + c = 0.

(M4)(x) = c(x)

-

6

h.

cx2 + c(x)

=

cdx4 + 0(x6).

where

Thus, if

so by Theorem 3,

(M4)(x) = 0(x6)

then + 0(x

4(x)

Let

then

The equation that governs the sta-

).

bility of the zero solution of (1.4.1) is u = uh(u) + au3 + buh2(u) = (cd+bc2)u5 + 0(u7).

Hence, if

a + c = 0, then the zero solution of (1.4.1) is cd + bc2 < 0

asymptotically stable if cd + bc2 > 0.

If

cd + bc2 =

better approximation to Exercise 1.

h

0

and unstable if

then we have to obtain a

(see Exercise 1). a + c = cd + bc2

Suppose that

=

0

in Example 1.

Show that the equation which governs the stability of the zero solution of (1.4.1) is Exercise 2.

u = -cd2u7 + 0(u9).

Show that the zero solution of (1.2.2) 1% asymp-

totically stable if

a <

0

and unstable if

a

n.

1.4.

Examples

Exercise 3.

Suppose that in equation (1.3.2), n =

2q > p +

1

and

solution of (1.3.2) is asymptotically stable if p

IyIq)

Show that the zero

is non-zero.

a

so that

1

f(x,y) = axp + O(IxIp+l +

Suppose also that

A = 0.

where

7

a <

and

0

is odd, and unstable otherwise.

Consider the system

Example 2.

x = ex

-

x3 + xy

Y = -Y +

Y2 - x2

(1.4.2)

where

c

The object is to study small

is a real parameter.

solutions of (1.4.2) for small

lei.

The linearized problem corresponding to (1.4.2) has eigenvalues

-1

and

c.

This means that the results given

in Section 3 do not apply directly.

However, we can write

(1.4.2) in the equivalent form

x = ex

- x

y= -y+y

3

+ xy

2

- x

When considered as an equation on

2

(1.4.3)

]R3

ex

the

Thus the linearized problem correspond-

(1.4.3) is nonlinear.

ing to (1.4.3) has eigenvalues

-1,0,0.

The theory given in

Section 3 now applies so that by Theorem 1, dimensional centre manifold

y = h(x,c),

To find an approximation to

h

(1.4.3) has a two

Ixl

< 61,

+ x2

4(x,c) - -x2, (M4)(x,e) = O(C(x,c))

homogeneous cubic in

x

and

Iel

< a2'

set

(M4)(x,c) - mx(x,e)[ex-x3+xm(x,e)] + m(x,e) 'then, if

term in

c.

By Theorem 3,

-

where

02(x,e).

C

is a

INTRODUCTION TO CENTRE MANIFOLD THEORY

1.

8

h(x,c) - -x2 + O(C(x,c)) .

Section 2.6).

Note also that

h(O,c) = 0

(see

By Theorem 2 the equation which governs small

solutions of (1.4.3) is u = cu

-

2u3 + O(IuIC(u,e))

(1.4.4)

The zero solution

(u,c)

= (0,0)

of (1.4.4) is stable so

the representation of solutions given by Theorem 2 applies here.

For

-62 < e < 0

u = 0

the solution

of the first

equation in (1.4.4) is asymptotically stable and so by Theorem 2 the zero solution of (1.4.2) is asymptotically stable. For

0 < c < 62, solutions of the first equation in

(1.4.4) consist of two orbits connecting the origin to two small fixed points.

Hence, for

0 < c < 62

the stable mani-

fold of the origin for (1.4.2) forms a separatrix, the unstable manifold consisting of two stable orbits connecting the origin to the fixed points.

Exercise 4. w +

Study the behavior of all small solutions of

+ ew + w3 = 0

Example 3.

for small

C.

Consider the equations

y = -y + (y+c)z

ei = where

c

> 0

is small and

y-

(1.4.5)

(Y+1) z

0 < c < 1.

The above equations

arise from a model of the kinetics of enzyme reactions [33]. If

e = 0, then (1.4.5) degenerates into one algebraic equa-

tion and one differential equation. equation we obtain

Solving the algebraic

Examples

1.4.

9

z=

Y

(1.4.6)

Y

and substituting this into the first equation in (1.4.5) leads to the equation _

y

where

A =

1

(1.4.7) 1

c.

-

Using singular perturbation techniques, it was shown in [331 that for

c

sufficiently small, under certain condi-

tions, solutions of (1.4.5) are close to solutions of the degenerate system (1.4.6), (1.4.7).

We shall show how centre

manifolds can be used to obtain a similar result. t = CT.

Let to

t

by

'

We denote differentiation with respect

and differentiation with respect to

by

T

'.

Equation (1.4.5) can be rewritten in the equivalent form

Y' w'

where 1,

f(y,w)

Ef(Y,w)

_ -w + y2 - yw + ef(Y,w)

= -y + (y+c)(y-w)

and

(1.4.8) has a centre manifold

proximation to

h

w = h(y,c).

0 (Y,e) = y2

z.

By Theorem

To find an ap-

set

(MO)(Y,E) = Emy(Y,E)f(Y,m) + 0(Y,E) If

w = y -

(1.4.8)

- Acy

then

- y2

(Mm)(Y,c)

+ y (Y, e)

-

O(1Y13 + IcI3)

ef(Y, m) so

that by Theorem 3, h(Y,e) - y2

-

aey + O(IY13 +

1el3).

By Theorem 2, the equation which determines the asymptotic behavior of small solutions of (1.4.8) is

-

10

INTRODUCTION TO CENTRE MANIFOLD THEORY

1.

= ef(u,h(u,e))

u'

or in terms of the original time scale u = f(u,h(u,c)) _ -A(u-u2) + O(Ieul

Again, by Theorem 2, if z(0)

a

+

(1.4.9)

lul3).

is sufficiently small and

y(0),

are sufficiently small, then there is a solution

of (1.4.9) such that for some

u(t)

y > 0,

y(t) = u(t) + 0(e Yt/E)

z(t) = y(t) - h(y(t),c) + O(e yt/e).

(1.4.10)

Note that equation (1.4.7) is an approximation to the equation on the centre manifold.

Also, from (1.4.10), z(t) = y(t)

-

y2(t), which shows that (1.4.6) is approximately true.

The above results are not satisfactory since we have to assume that the initial data is small.

In Chapter 2, we show

how we can deal with more general initial data. briefly indicate the procedure involved there.

Here we If

y0 # -1,

then

(y,w,e) _ (y0,y0(l+y0) 1,0)

is a curve of equilibrium points for (1.4.8). pect that there is an invariant manifold (1.4.8) defined for and with

h(y,c)

a

small and

Thus, we ex-

w = h(y,c)

0 < y < m

for

(m = 0(1)),

close to the curve w = y2 (1+y)- 1.

(1.4.11)

For initial data close to the curve given by (1.4.11), the stability properties of (1.4.8) are the same as the stability properties of the reduced equation 6 - f(u.h(u,e)).

1.5.

Bifurcation Theory

1.5.

Bifurcation Theory

11

Consider the system of ordinary differential equations

w = F (w, c)

F(O,c) s 0 where

w EIRn+m c =

say that

0

and

is a p-dimensional parameter.

c

is a bifurcation point for (1.5.1) if the

qualitative nature of the flow changes at in any neighborhood of e2

We

c =

c = 0; that is, if

there exist points

0

and

c1

such that the local phase portraits of (1.5.1) for

E = C1

c = c2 are not topologically equivalent.

and

Suppose that the linearization of (1.5.1) about

w =

0

is

w = C(e)w.

If the eigenvalues of then, for small

all have non-zero real parts

C(0)

10, small solutions of (1.5.1) behave like

solutions of (1.5.2) so that point.

(1.S.2)

c = 0

is not a bifurcation

Thus, from the point of view of local bifurcation

theory the only interesting situation is when

C(0)

has eigen-

values with zero real parts.

Suppose that parts and

m

C(0)

has

n

eigenvalues with zero real

eigenvalues whose real parts are negative.

are assuming that

C(0)

We

does not have any positive eigen-

values since we are interested in the bifurcation of stable phenomena.

Because of our hypothesis about the eigenvalues of (:(0)

we can rewrite (1.5.1) as

12

INTRODUCTION TO CENTRE MANIFOLD THEORY

1.

X = Ax + f(x,y,c) (1.5.3)

By + g(x,y,c)

E= where

0

x EIRn, y EIRm, A

is an

matrix whose eigen-

n x n

is an

values all have zero real parts, B

m x m matrix

whose eigenvalues all have negative real parts, and g

and

f

vanish together with each of their derivatives at

(x,y,E) _ (0,0,0).

By Theorem 1, (1.5.3) has a centre manifold 1xI

< 61,

IEI

< d2.

y = h(x.E),

By Theorem 2 the behavior of small solu-

tions of (1.5.3) is governed by the equation u - Au + f(u,h(u,c),c) (1.5.4) E = 0.

In applications

n

is frequently

very useful reduction.

1

or

2

so this is a

The reduction to a lower dimensional

problem is analogous to the use of the Liapunov-Schmidt procedure in the analysis of static problems.

Our use of centre manifold theory in bifurcation problems follows that of Ruelle and Takens [57] and of Marsden and McCracken [51].

For the relationship between centre

manifold theory and other perturbation techniques such as amplitude expansions see [14]. We emphasize that the above analysis is local.

In

general, given a parameter dependent differential equation it is difficult to classify all the possible phase portraits. For an example of how complicated such an analysis can be, see [661 where a model of the dynamic behavior of a continuous stirred tank reactor is studied.

The model con%i%tt of

is

1.6.

Comments on the Literature

13

parameter dependent second order system of ordinary differential equations.

The authors show that there are 3S pos-

sible phase portraits! 1.6.

Comments on the Literature Theorems 1-3 are the simplest such results in centre

manifold theory and we briefly mention some of the possible generalizations. (1)

The assumption that the eigenvalues of the lin-

earized problem all have non-positive real parts is not necessary. (2)

The equations need not be autonomous.

(3)

In certain circumstances we can replace 'equilibrium

point' by 'invariant set'. (4)

Similar results can be obtained for certain classes

of infinite-dimensional evolution equations, such as partial differential equations.

There is a vast literature on invariant manifold theory [1,8,22,23,27,28,30,32,34,35,42,44,45,48,51].

For applications

of invariant manifold theory to bifurcation theory see [1,14, 17,18,19,24,31,34,36,37,38,47,48,49,51,57,65).

For a simple

discussion of stable and unstable manifolds see (22, Chapter 13) or (27, Chapter 3]. In Chapter 2 we prove Theorems 1-3.

Our proofs of

Theorems 1 and 2 are modeled on Kelly (44,45).

Theorem 3 is

a special case of a result of Henry [34] and our proof follows his.

The method of approximating the centre manifold in

Theorem 3 was essentially used by Hausrath (32) in his work on stability in critical cases for neutral functional differential equations.

Throughout Chapter 2 we use methods that

generalize to infinite dimensional problems in an obvious way.

CHAPTER 2

PROOFS OF THEOREMS

2.1.

Introduction

In this chapter we give proofs of the three main The proofs are essentially

theorems stated in Chapter 1.

applications of the contraction mapping principle.

The pro-

cedure used for defining the mappings is rather involved, so we first give a simple example to help clarify the technique. The proofs that we give can easily be extended to the corresponding infinite dimensional case; indeed essentially all we have to do is to replace the norm space by the norm 2.2.

11.11

1.1

in finite dimensional

in a Banach space.

A Simple Example

We consider a simple example to illustrate the method that we use to prove the existence of centre manifolds. Consider the system x1 = x2,

where (0,0).

g

is smooth and

x2 = 0, k = -Y + g(xl,x2),

g(xl,x2) = 0(x2+x2)

as

(2.2.1)

(xl,x2) +

We prove that (2.2.1) has a local centre manifold.

1 e

2.2.

A Simple Example

*:IR2 IR be

Let

such that

is

p(xl,x2) = 1

Define

the origin.

a

C

for

in a neighborhood of

(xl,x2)

G(xl,x2) =

by

G

function with compact support

(xl,x2)g(xl,x2).

We

prove that the system of equations

xl = x2, x2 = 0, Y = -y + G(xl,x2), has a centre manifold

y = h(xl,x2), (xl,x2)

Since

E IR2.

in a neighborhood of the origin, this

G(xl,x2) = g(xl,x2)

y = h(xl,x2), x2 + x2 <

proves that

(2.2.2)

6

for some

6,

is a

local centre manifold for (2.2.1).

The solution of the first two equations in (2.2.2) is xl(t) = z1 + z2t, x2(t) = z2, where y(t) = h(xl(t),x2(t))

xi(0) = zi.

If

is a solution of the third equation in

(2.2.2) then

= -h(zI+z2t,z2) + G(zl+z2t,z2). (2.2.3)

ft-h(z

To determine a centre manifold for (2.2.2) we must single out a special solution of (2.2.3). ,ill

is small for

x2, solutions of the third equation in (2.2.2)

and

xl

G(xl,x2)

Since

behave like solutions of the linearized equation

y = -y.

l'he general solution of (2.2.2) therefore contains a term like

e-

t.

As

t + -, this component approaches the origin

perpendicular to the I,; tangent to the

mate the

a-t

zl,z2

zl,z2

plane.

Since the centre manifold

plane at the origin we must elim-

component, that is we must eliminate the com-

l-onent that approaches the origin along the stable manifold .is

t -

To do this we solve (2.2.3) together with the

kondition lim t y -.

h(zl+z2t,z2)et

-

0.

(2.2.4)

16

PROOFS OF THEOREMS

2.

Integrating (2.2.3) between

and

and using

0

(2.2.4) we obtain 0

h(zl,z2) = f- esG(zl+z2s,z2)ds. m

1

By construction, y = h(zl,z2)

Using the fact that

(2.2.2). that

G(xl,x2)

that is,

has compact support and

G

has a second order zero at the origin it h(zl,z2)

follows that

2.3.

is an invariant manifold for

has a second order zero at the origin;

is a centre manifold.

h

Existence of Centre Manifolds In this section we prove that the system

x = Ax + f (x, y) (2.3.1)

y = By + g(x,y) has a centre manifold.

A

values of

As before

x EIRn, y EIRm, the eigen-

have zero real parts, the eigenvalues of

have negative real parts and

f

and

are

g

C2

B

functions

which vanish together with their derivatives at the origin. Equation (2.3.1) has a local centre manifold

Theorem 1. y = h(x), Proof:

lxi

< 6, where

h

is

C2.

As in the example given in the previous section, we

prove the existence of a centre manifold for a modified equation.

when define

p:IRn + [0,11

Let lxi

F

<

1

and

and

G

y,(x)

be a = 0

Cm

when

function with lxi

>

2.

For

,y(x)

c >

=

1

0

by

F(x,y) = f(x*(E),y),

G(x,y) - g(xP(E),y)

The reason that the cut-off function

p

is only a function of

Existence of Centre Manifolds

2.3.

x

17

is that the proof of the existence of a centre manifold

generalizes in an obvious way to infinite dimensional problems.

We prove that the system x = Ax + F(x,y) (2.3.2)

By + G(x,y)

has a centre manifold Since

F

and

y = h(x) ,

agree with

G

f

x EIRn, for small enough and

g

c.

in a neighborhood of

the origin, this proves the existence of a local centre manifold for (2.3.1). For

functions

for

p >

and

0

h:IRn +]Rm

let

0

be the set of Lipschitz

X

with Lipschitz constant

h(0) = 0.

and

xEIRn

pl >

pl,

With the supremum norm

Ih(x)I

< p

11-11, X

is a complete space. For

h E X

and

x0 EIRn, let

x(t,x0,h)

be the solu-

tion of

x = Ax + F(x,h(x)), x(O,xO,h) = x0. The bounds on

and

F

exists for all

t. r

(Th)(x0)

If

h

=

h

ensure that the solution of (2.3.3)

enough, T

by

0 1-

e- Bs G(x(s,x0,h),h(x(s,x0,h)))ds.

We prove that for

is a contraction on

k(c)

with

h

(2.3.4)

is a centre mani-

p,pl, and

c

small

X.

Using the definitions of tinuous function

Th

We now define a new function

is a fixed point of (2.3.4) then

fold for (2.3.2).

(2.3.3)

F

and

k(O) = 0

G, there is a consuch that

18

2.

IF(x,y)I +

IG(x,y)I

ck(c),

IF(x,y)

- F(x',y')I c k(c)[Ix-x'I + Iy-y'II,

IG(x,y)

-

S,C

c.

such that for

< CesslyI.

Since the eigenvalues of

all have zero real parts, for

A

there is a constant

0

(2.3.6)

M(r)

such that for

x E IRn

s E IR,

and

IeAsxl

M(r)erlsllxl.

<

Note that in general, M(r) + m

shall suppose that

p <

We

from now on.

c

x0 E]Rn, then using (2.3.6) and the estimates on

If

h, we have from (2.3.4) that

ITh(x0)I

Now let F

r + 0.

G(x(s,x0,h),h(x(s,x0,h))) and similar terms.

involving

and

as

(2.3.7)

p < c, then we can use (2.3.5) to estimate terms

If

on

<

Iy'I

y E IRm, le- Bsyl

r >

IyI,

all have negative real

B

parts, there exist positive constants

each

with

y, y' E]Rm

and all

Since the eigenvalues of

s < 0 and

(2.3.5)

G(x',y')I c k(c)[Ix-x'I + Iy-y'I],

x, x' E]Rn

for all

G

PROOFS OF THEOREMS

and

< CS lck(c).

x0, x1 E]Rn.

(2.3.8)

Using (2.3.7) and the estimates

h, we have from (2.3.3) that for

r > 0

and

t < 0, Ix(t,x0,h)

- x(t,xl,h)I

+ (l+pl)M(r)k(c)

M(r)e-rtlx0-x1I

<

0 I

e

r(s-t)Ix(s,x0,h) -

it

By Gronwall's inequality, for

t

< 0,

x(s,xl,h)Ids.

Reduction Principle

2.4.

19

M(r)Ixl-x0Ie-Yt,

- x(t,x1,h)I <

Ix(t,x0,h)

where

y = r + (l+pl)M(r)k(c).

on

and

G

ITh(x0)

if

c

and

(2.3.9)

Using (2.3.9) and the bounds

h, we obtain from (2.3.4) - Th(xl)I < C(M(r) +pl)k(c)(s-y) r

are small enough so that hl, h2 E X

Similarly, if ITh1(x0)-Th2(x0)I

1Jx0-x1I

(2.3.10)

0 > y.

x0 E]Rn, we obtain

and

+(1+pi)M(r)k(c)r-1(S-y)

< Ck(c)(S

1]

(2.3.11)

IIh1-h2II By a suitable choice of

p,p1,c

(2.3.8), (2.3.10) and (2.3.11) that X.

T

and

r, we see from

is a contraction on

This proves the existence of a Lipschitz centre manifold

for (2.3.2).

To prove that

contraction on a subset of ferentiable functions.

h X

is

C1

we show that

is a

T

consisting of Lipschitz dif-

The details are similar to the proof

given above so we omit the details.

To prove that

h

is

C2

we imitate the proof of Theorem 4.2 on page 333 of (22]. 2.4.

Reduction Principle

The flow on the centre manifold is governed by the n-dimensional system u = Au + f(u,h(u)).

(2.4.1)

In this section we prove a theorem which enables us to relate

the asymptotic behavior of small solutions of (2.3.1) to solutions of (2.4.1).

We first prove a lemma which describes the stability properties of the centre manifold.

20

PROOFS OF THEOREMS

2.

Let

Lemma 1.

I(x(0),y(0))I and

C1

u

Proof:

t

Let

(x(0),y(0))

Then there exist positive

sufficiently small.

such that

Iy(t)

for all

be a solution of (2.3.2) with

(x(t),y(t))

- h(x(t))I < C1e-utly(0)

- h(x(0))I

> 0.

be a solution of (2.3.2) with

(x(t),y(t))

sufficiently small.

Let

z(t)

- y(t) - h(x(t)),

then by an easy computation i = Bz + N(x,z)

(2.4.2)

where

N(x,z) = h'(x)[F(x,h(x))

- F(x,z+h(x))] + G(x,z + h(x)) - G(x,h(x)).

Using the definitions of

and

F

there is a continuous function that

IN(x,z)I

< 6(c)Izi

if

and the bounds on

G

with

6(c)

< c.

IzI

6(0) = 0

h,

such

Using (2.3.6) we ob-

tain, from (2.4.2), t

Iz(t)I < Ce-stlz(0)I

+

C6(c)I0

e-'(t-s)Iz(s)Ids

and the result follows from Gronwall's inequality.

Before giving the main result in this section we make some remarks about the matrix

A.

Since the eigenvalues of

A

all have zero real parts, by a change of basis we can put

A

in the form A = A

+ A2 1

where

A2

is nilpotent and

A t le 1 xI

Since

A2

-

IxI.

(2.4.3)

is nilpotent, we can choose the basis such that

Reduction Principle

2.4.

IA2xI

where

21

< (S/4)IxI,

(2.4.4)

is defined by (2.3.6).

S

We assume for the rest of this section that a basis has been chosen so that (2.4.3) and (2.4.4) hold. Theorem 2.

(a)

Suppose that the zero solution of (2.4.1) is

stable (asymptotically stable) (unstable).

Then the zero

solution of (2.3.1) is stable (asymptotically stable) (unstable).

Suppose that the zero solution of (2.3.1) is

(b)

stable.

sufficiently small.

(x(O),y(O))

tion

be a solution of (2.3.1) with

(x(t),y(t))

Let

Then there exists a solu-

of (2.4.1) such that as

u(t)

m,

t

x(t) - u(t) + 0(e-Yt) (2.4.5)

0(e-Yt)

y(t) = h(u(t)) + where

is a constant depending only on

y > 0

Proof:

B.

If the zero solution of (2.4.1) is unstable then by

invariance, the zero solution of (2.3.1) is unstable.

From

now on we assume that the zero solution of (2.3.1) is stable. We prove that (2.4.5) holds where of (2.3.2) with and

G

f

and

is a solution

sufficiently small.

I(x(0),y(0))I

are equal to

(x(t),y(t))

g

Since

F

in a neighborhood of the

We divide the proof into two

origin this proves Theorem 2. steps. I.

Let

ficiently small. u(0) - u0.

u0 E]Rn Let

and u(t)

z0 E]Rm

with

I(u0,z0)I

be the solution of (2.4.1) with

We prove that there exists a solution

of (2.3.2) with

y(0)

-

suf-

h(x(0))

- z0

and

x(t)

-

(x(t),y(t)) u(t),

22

PROOFS OF THEOREMS

2.

y(t)

exponentially small as

- h(u(t))

By Step I we can define a mapping

II.

neighborhood of the origin in _ (x0)z0)

S(u0,z0)

where

IRn+m

I.

Let

(x(t),y(t))

from a

S

by

IRn+m

into

x0 = x(0).

ficiently small, we prove that

For

I(x0,z0)l

suf-

is in the range of S.

(x0)20)

be a solution of (2.3.2) and

a solution of (2.4.1).

u(t)

t + m

Note that if

is suffici-

u(O)

ently small,

u - Au + F(u,h(u))

(2.4.6)

since solutions of (2.4.1) are stable. h(x(t)), $(t) - x(t)

- u(t).

Let

z(t) - y(t)

Then by an easy computation

t - Bz + N($+u,z)

(2.4.7)

AO + R($,z) where

-

(2.4.8)

is defined in the proof of Lemma 1 and

N

R($,z) - F(u+$,z+h(u+$)) - F(u,h(u)).

We now formulate (2.4.7), (2.4.8) as a fixed point problem.

For

functions

a > 0, K > 0, let with

[0,m) +IRn

$:

X

be the set of continuous

I,(t)eatl

< K

for all

If we define

II$II - sup{14(t)eatl: t > 0}, then

plete space.

Let

(u0,z0)

z(t)

Define

u(0) - u0.

be the solution of (2.4.7) with

T$

is a com-

be sufficiently small and let

be the solution of (2.4.6) with let

X

t > 0.

Given

u(t)

$ E X

z(0) - z0.

by

A1(t-s) -

(T$)(t) -

e

[A20(s) + R(0(s),z(s))1ds.

(2.4.9)

t

We solve (2.4.9) by means of the contraction mapping principle.

If

$

is a fixed point of

T, then retracing our

Reduction Principle

2.4.

steps we find that

23

We can take

is a solution of (2.3.2). S

+ 4(t), y(t) - z(t) + h(x(t))

x(t) = u(t)

to be as close to

a

as we please at the cost of increasing

and shrinking

K

the neighborhood on which the result is valid. however, we take

and

K - 1

2a -

S

where

0

For simplicity is defined by

(2.3.6).

F,G,h

Using the bounds on

and the fact that

N($,0) - 0, there is a continuous function k(0) - 0 Izil

]Rn

such that if

and

01,02 E

with

k(c)

zl,z2 EIRm

with

< c, then Izl-z21]

IN($1,z1)

- N($2,z2)I

k(c)[Iz11101-021 +

IR($1)z1)

- R($2,z2)I

k(c)[1z1-z2l + 141-$21]

(2.4.10)

From (2.4.7), t

+ Ck(e)j

CIzOIe-Ot

<

Iz(t)I

e-O(t-s)Iz(s)Ids

0

where we have used (2.3.6) and (2.4.10).

By Gronwall's in-

equality sl t

CIz0Ie

Iz(t)I

where

S1 =

0

-

Ck(e).

(2.4.11)

From (2.4.9), if

c

is sufficiently

small, e-at + <

IT,(t)I

k(e)1W(e-as

CIzOIe- sls)ds

+

< e-

it

at

where we have used (2.4.3), (2.4.4), (2.4.10) and (2.4.11). Hence

T

maps

Now let

X

into

X.

01,02 E X

and let

ing solutions of (2.4.7) with w(t) - zl(t)

-

z2(t).

zl,z2

zi(0) - z0.

be the correspond-

We first estimate

From (2.4.7) and (2.4.10),

24

2.

t I w ( t )I < C k ( c )

t s) [Izl(s)II01(s)-02(s)I

e

I

PROOFS OF THEOREMS

+ Iw(s)lids.

0

Using (2.4.11), t

Iw(t) I

where

C1

Iw(t) I

<

e-S(t-s)Iw(s)Ids

+ Ck(c)IO

Clk(c)IIO1-02IIe-St

is a constant, so that by Gronwall's inequality

=

Using (2.4.4) and (2.4.12), for

sufficiently small,

c

ITO1(t)-T02(t)I < 21101-0211 + k(c)fm(I01(s)

Izl(s)

+

where

a <

-

02(s)I

-

z2(s)I)ds < a101-0211

1.

The above analysis proves that for each ficiently small, T

has a unique fixed point.

neighborhood of the origin in ]Rn+m

Define

x0 = u0 + 0(0). z0, S

Since

and

u0

T: X x U i X

We prove that

is continuous.

(x0,20)

is a con-

where

depends continuously on

0

is a

U

z0.

S(u0,z0)

by

S

If

suf-

This proves that the fixed point

tinuous uniform contraction.

depends continuously on

(u0,z0)

then it is easy to re-

peat the above analysis to show that

II.

(2.4.12)

Iz1(t)-z2(t)I < C1k(c)II01-0211 e Slt.

S

u0

and

is one-to-one, so

that by the Invariance of Domain Theorem (see [11] or [601) S

is an open mapping.

the range of

S

S(0,0) = 0, this proves that

is a full neighborhood of the origin in

Proving that to proving that if and

Since

S

initial values for

is one-to-one is clearly equivalent

u0 + 00(0) = ul + 01(0)

00(0) = 01(0).

If x

]Rn+m

u0 + 00(0) and

y

then

- ul + 41(0)

u0 = u1 then the

are the same, so that by

Approximation of the Centre Manifold

2.5.

2S

uniqueness of solution of (2.3.2), u0(t) + 00(t) = ul(t) + for all

01(t)

(2.4.6) with

t

0, where

>

is the solution of

ui(t)

Hence, for

ui(0) = ui.

t >

u0(t) - ul(t) = O1 (t)

0,

00(t).

-

Since the real parts of the eigenvalues of limlul(t) - u0(t)lect = m for any c tam at u0(0). Also, 10i(t)l < efor all

from (2.4.13) that

S

>

are all zero,

unless

0

t

A

(2.4.13)

>

u1(0) _

It now follows

0.

is one-to-one and this completes the

proof of the theorem.

Approximation of the Centre Manifold

2.5.

For functions

0:IRn +IRm

which are

C1

in a neigh-

- BO (x)

- g(x,0(x))

borhood of the origin define = O'(x)[Ax + f(x,0(x))]

(MO) (x)

Suppose that

Theorem 3.

(Mo)(x) = 0(1x1q)

as

x + 0

lh(x)

Proof:

6: Rn

Let

= 0, 0'(0) = 0

0(0)

-

where

q > 1.

Rm be a continuously differentiable

N(x) = 6' (x) [Ax + F(x,6(x))]

F

and

N(x) = O(1xjq)

-

BO(x)

0(x)

= O(x)

for

as

(2.5.1)

Note that

x + 0.

a contraction mapping -

- G(x,6(x)),

are defined in Theorem 1.

G

In Theorem 1, we proved that

Sz - T(z+e)

x + 0,

Set

small.

where

Then as

= 0(Ixlq)

O(x)l

function with compact support such that lxj

and that

T: X + X.

0; the domain of

h

was the fixed point of

Define a mapping S

S

by

being a closed subset

PROOFS OF THEOREMS

2.

26

Since

Y c X.

is a contraction mapping on

T

contraction mapping on Y -

{ z E X :

l z ( x )

If we can find a

S

is a

let

for all

< K j x j q

I

such that

K

K > 0

For

Y.

X, S

maps

x E I R n}.

Y

into

Y

then we

will have proved the theorem.

We first find an alternative formulation of the map z E Y

For

x(t,x0)

let

S.

be the solution of

ii = Ax + F(x,z(x) + 6(x)),

x(O,xO) - x(0).

(2.5.2)

From (2.3.4) 0

(T(z+6))(x0) -

e-BsG(x(s,x0),z(x(s,x0))

-

+ O(x(s,x0)))ds.

f

r

-6(x0)

0

-I-

d[e-BsO(x(s,x0))]ds u-s

1 1

_

0 (e-Bs[BO(x(s,x0)) O(x(s,x0))]ds.

-

Writing

B6 (x)

x

for

x(s,x0)

- ds (x)

as

etc., from (2.5.1) and (2.5.2)

= B6 (x) - 6' (x) [Ax + F (x, z (x) + 6 (x)) ] _ -N(x)

- G(x,6(x)) + 6' (x)[F(x,6)

- F(x,z(x) + 6(x))]. Using

Sz - T(z+6) (Sz)(x0) =

where

x(s,x0)

-

0 r-

6

and the above calculations e-BsQ(x(s,x0),

z(x(s,x0)))ds

(2.5.3)

is the solution of (2.5.2) and

Q(x,z) - G(x,6+z) - G(x,6)

- N(x) + 6'(x)[F(x,6) (2.5.4)

F(x,6+z)].

Approximation of the Centre Manifold

2.5.

We now show that By choosing for all

N(x) = O(Ixlq)

Since

x E]Rn.

< Cllxlq,

IQ(x,o)I

IQ(x,z)I_

= IN(x)I

We can estimate

IQ(x,z)

constants of

and

function

F

G.

with

k(c)

IzI

+

+

< E

x + 0,

as

(2. S. S)

x E]Rn

- Q(x,o)I

IQ(x,z)

(2.5.6) IQ(x,z)

- Q(x,0)I

- Q(x,0)I.

in terms of the Lipschitz

Using (2.3.5), there is a continuous

k(0) - 0, such that IQ(x,z)

for

I0(x)I

- Q(x,0)I

< k(c)IzI

(2.5.7)

Using (2.S.S), (2.5.6), (2.5.7), for

< c.

K > 0.

Now

is a constant.

C1

for some

Y

into

suitably, we may assume that

6

IN(x)I

where

Y

maps

S

27

z E Y

x E fn, we have that

and

IQ(x,z)I

< C11xIq + k(c)Iz(x)I (2. S.8)

< (C1 + Kk(c)) IxIq.

Using the same calculations as in the proof of Theorem 1, if is the solution of (2.5.2), then for each

x(t,x0)

there is a constant

M(r)

Ix(t,x0)I

where

r > 0,

such that

< M(r)Ix0Ie Yt,

t

<

(2.5.9)

0

y - r + 2M(r)k(c).

Using (2.3.6), (2.5.8) and (2.5.9), if

z

E Y,

C(C1+Kk(c))(M(r))q(S-gy)-llxOIq

I($z)(x0)I

provided

c

-

<

and

r

are small enough so that

0

-

C2Ix0Iq

qy >

0.

28

2.

By choosing

large enough and

K

c

small enough, we have

and this completes the proof of the theorem.

that

C2 < K

2.6.

Properties of Centre Manifolds In general (2.3.1) does not have a unique centre

(1)

For example, the system

manifold.

PROOFS OF THEOREMS

x - -x3, y = -y, has the

two parameter family of centre manifolds

y - h(x,cl,c2)

where c

h(x,cl,c2) =

exp(

1

and

h

), x > 0

1

x

- 2

),

x < 0.

are two centre manifolds for

h1

- h1(x) - O(jxjq)

x + 0

as

q > 1. If

(2)

(441.

-2

x - 0

(2.3.1), then by Theorem 3, h(x) for all

x

0

c2 exp(-

However, if

1

If

f

f

and

and

g

Ck, (k > 2), then

are

h

is

Ck

are analytic, then in general (2.3.1)

g

does not have an analytic centre manifold, for example, it is easy to show that the system x - -x3,

(2.6.1)

y = -y + x2

does not have an analytic centre manifold (see exercise (1)). (3)

Centre manifolds need not be unique but there are

some points which must always be on any centre manifold. example, suppose that

of (2.3.1) and let (2.3.1). if

r

(x0,y0)

y - h(x)

is a small equilibrium point be any centre manifold for

Then by Lemma 1 we must have

Similarly,

y0 - h(x0).

is a small periodic orbit of (2.3.1), then

on all centre manifolds.

For

r

must lie

2.6.

Properties of Centre Manifolds

Suppose that

(4)

29

(x(t),y(t))

is a solution of (2.3.1)

which remains in a neighborhood of the origin for all

t > 0.

An examination of the proof of Theorem 2, shows that there is a solution

u(t)

of (2.4.1) such that the representation

(2.4.5) holds.

In many problems the initial data is not arbit-

(5)

rary, for example, some of the components might always be Suppose

nonnegative.

S c7Rn+m

with

defines a local dynamical system on

0 E S

and that (2.3.1)

It is easy to check,

S.

that with the obvious modifications, Theorem 2 is valid when (2.3.1) is studied on Exercise 1.

S.

Consider X

-x 3

y

Suppose that (2.6.1) has a centre manifold h

is analytic at

x = 0.

an+2 = nan

x.

E

n-2

anxn

a2n+1 '

Show that

for all

0

n - 2,4,..., with

for

y - h(x), where

Then

h(x) for small

(2.6.1)

-y + x2.

a2 = 1.

n

and that

Deduce that

(2.6.1) does not have an analytic centre manifold.

Exercise 2 (Modification of an example due to S. J. van Strien [63]). each

If

f

and

r, (2.3.1) has a

Cr

g

are

Cm

functions, then for

centre manifold.

However, the

size of the neighborhood on which the centre manifold is defined depends on

r.

The following example shows that in

general (2.3.1) does not have a f

and

g

are analytic.

Cm

centre manifold, even if

30

PROOFS OF THEOREMS

2.

Consider -ex - x3,

x

Suppose that (2.6.2) has a for

lxi

< 6,

h(x,(2n)-1)

is

Choose

< 6.

jej

centre manifold n > 6-l.

y = h(x,c)

Then since

x, there exist constants

in

Cw

C'

(2.6.2)

0.

Y = -Y + x2,

such that

al,a2.... ,a2n

h(x,(2n)

1)

-

2n

alxi + O(x2n+1) i=1

for

lxi

small enough.

Show that

ai - 0

for odd

and

i

n > 1,

that if

(2i)(2n)-1)a2i

(1 -

= (2i-2)a2i-2' i - 2,...,n (2.6.3)

a2

0.

Obtain a contradiction from (2.6.3) and deduce that (2.6.2) does not have a Exercise 3.

odd, that is

C

centre manifold.

Suppose that the nonlinearities in (2.3.1) are

that (2.3.1) has a centre manifold [The example

-h(-x).

Prove

f(x,y) = -f(-x,-y), g(x,y) - -g(-x,-y). y = h(x)

with

h(x) -

-x3, y = -y, shows that if

any centre manifold for (2.3.1) then

h(x)

-h(-x)

h

is

in

general.]

2.7.

Global Invariant Manifolds for Singular Perturbation Problems To motivate the results in this section we reconsider

Example 3 in Chapter 1.

In that example we applied centre

manifold theory to a system of the form

31

Global Invariant Manifolds

2.7.

Y'

- Ef(Y,w)

w' = -w + y2

f(0,0) = 0.

where

yw + Ef(Y,w)

-

(2.7.1)

Because of the local nature of our results

on centre manifolds, we only obtained a result concerning small initial data.

v - -w(l+y) + y2,

Let

S(T), where

s

s'(T) = 1 + y(T); then we obtain a system of the form Y' = Egl(Y,v)

gi(0,0) -

where

v'

= -v + Eg2(Y,v)

E,

-0

0,

i = 1,2.

(2.7.2)

Note that if

y j -1, then

is always an equilibrium point for (2.7.2) so we ex-

(y,0,0)

pect that (2.7.2) has an invariant manifold fined for

-1 < y < m, say, and

Theorem 4.

Consider the system

c

v - h(y,c)

de-

sufficiently small.

x' - Ax + Ef (x,y, E) Y' - By + Eg(x,Y,E) E' = 0 where

x E]Rn, y E]Rm

also that Let

f,g

m > 0.

are

and

Then there is a

invariant manifold

and

> 0

y - h(x,E),

jxj

p:]Rn + [0,1)

be a

if

< m

and

if

by

< m,

Ic

<

6, with

g.

Let

C

such that (2.7.3) has an

is a constant which depends on

C

Proof: lxi

Suppose

f(0,0,0) - 0, g(0,0,0) - 0. 6

Ih(x,E)l < Cici, where

m,A,B,f

are as in Theorem 1.

A,B

with

C2

(2.7.3)

y(x) -

0

C_ lxi

function with > m + 1.

Define

p(x) - 1 F

and

PROOFS OF THEOREMS

2.

32

F(x,y,c) - ef(xip(x),y,e),

G(x,y,c) = eg(xi(x),y,e).

We can then prove that the system x' = Ax + F(x,y,c) (2.7.4)

y' = By + G(x,y,c)

y = h(x,e), x E]Rn, for

has an invariant manifold

IcI

suf-

The proof is essentially the same as that

ficiently small.

given in the proof of Theorem 1 so we omit the details. If

Remark.

x = (x1,x2,...9xn)

the existence of

for

h(x,c)

then we can similarly prove mi < xi < mi.

The flow on the invariant manifold is given by the equation u'

- Au + cf(u,h(u,c)).

(2.7.5)

With the obvious modifications it is easy to show that the stability of solutions of (2.7.3) is determined by equation (2.7.5) and that the representation of solutions given in (2.4.5) holds.

Finally, we state an approximation result. Theorem 5.

$:]Rn+l i]Rm

Let

J(M$)(x,e)1 < Cep integer, C

for

jxl

satisfy

< m where

$(0,0) =

and

is a positive

p

is a constant and

(M$)(x,e) - Dx$(x,e)[Ax + ef(x,$(x,e))]

- B$(x,e) -

Then, for

0

1xI

< m,

jh(x,e)

for some constant

C1.

-

$(x,e)l < C1ep

eg(x,$(x,e)).

Centre Manifold Theorems for Maps

2.8.

33

Theorem 5 is proved in exactly the same way as Theorem 3 so we omit the proof.

For further information on the application of centre manifold theory to singular perturbation problems see Fenichel [24] and Henry [34].

Centre Manifold Theorems for Maps

2.8.

In this section we briefly indicate some results on centre manifolds for maps.

We first indicate how the study

of maps arises naturally in studying periodic solutions of differential equations.

Consider the following equation in

]Rn

A = f(x,A) where

is a real parameter.

A

A = A0, (2.8.1) has a periodic solution

for

-

with period

define

P(A)

let

cross section of neighborhood of

be some point on

y

through

y

y

in

I'(a)(z) - x(s), where x(0)

=

z

and

s

> 0

Then

U.

P(A).

y, let

and let

y

for

y

small is to consider the Poincare map

A01

Suppose that

y

One way to study solutions of (2.8.1) near

1'.

A

is smooth and

f

(2.8.1)

U

To

be a local

be an open

U1

P(A): U1 i U

is defined by

is the solution of (2.8.1) with

x(t)

is the first time

hits

x(t)

U.

(See [51] for the details). If

has a fixed point then (2.8.1) has a periodic

P(A)

,,rhit with period close to

point of order n,(P(A)kz /

T. z

If

for

P(A) 1

has a periodic

< k < n

-

1

and

I'r'(a)n - z) then (2.8.1) has a periodic solution with period ,lose to

nT.

If

P(a)

preserves orientation and there is a

,losed curve which is invariant under

P(a)

then there exists

34

PROOFS OF THEOREMS

2.

an invariant torus for (2.8.1). If none of the eigenvalues of the linearized map lie on the unit circle then it can be shown that

P'(A0)

has essentially the same behavior as

P(a) Ix

Hence in this case, for

small.

-

X01

solutions of (2.8.1) near

IA

for

a0I

-

small,

have the same behavior as when

y

If some of the eigenvalues of

A - a0.

P(A0)

P'(A0)

lie on the

unit circle then there is the possibility of bifurcations taking place.

In this case centre manifold theory reduces

the dimension of the problem.

As in ordinary differential

equations we only discuss the stable case, that is, none of the eigenvalues of the linearized problem lie outside the unit circle.

n+m

n+m +

Let

T: IR

have the following form:

IR

T(x.Y) - (Ax + f(x,y), By + g(x,y)) where

that each eigenvalue of value of and

and

x E]Rn, y E]Rm, A

B

A

(2.8.2)

are square matrices such

B

has modulus

has modulus less than

and each eigen-

1

1, f

and

g

are

C2

and their first order derivatives are zero at the

f,g

origin.

Theorem 6. T.

tion jxl

There exists a centre manifold

More precisely, for some h: ]Rn + ]Rm

< c

and

with

c > 0

there exists a

h(0) = 0, h'(0) - 0

(xl,yl) - T(x,h(x))

In order to determine

h

implies

C2

for

func-

such that

yl - h(xl).

we have to solve the equation

(x1,h(xl)) - T(x,h(x)) By (2.8.2) this is equivalent to

h:]Rn + ]Rm

2.8.

Centre Manifold Theorems for Maps

35

h(Ax + f(x,h(x))) = Bh(x) + g(x,h(x)). For functions

$: IR

n

i Dm

(M$)(x) = $(Ax + f(x,h(x)))

by

- B$(x) - g(x,h(x))

Mh - 0.

so that

Theorem 7.

01(0) = 0 Then

M$

define

Let

$: IR

and

n

i

Dm

be a

(M$)(x) = O(Ixlq)

h(x) = $(x) + O(Ixlq)

as

C1

map with

as

x i 0

$(0) = 0,

for some

q > 1.

x i 0.

We now study the difference equation xr+l = Axr + f(xr''r) (2.8.3)

yr+l = Byr +

As in the ordinary differential equation case, the asymptotic behavior of small solutions of (2.8.3) is determined by the flow on the centre manifold which is given by (2.8.4)

ur+l = Aur + f(ur,h(ur)). Theorem 8.

Suppose that the zero solution of (2.8.4) is

(a)

stable (asymptotically stable) (unstable).

Then the zero solu-

tion of (2.8.3) is stable (asymptotically stable) (unstable). Suppose that the zero solution of (2.8.3) is

(b)

stable.

Let

(xr,yr)

be a solution of (2.8.3) with

sufficiently small.

Then there is a solution

such that

< KBr

r

where

Ixr K

-

and

url B

and

ur

of (2.8.4)

Iyr - h(ur)I < KBr

are positive constants with

(x1,yl)

B <

for all 1.

The proof of Theorem 6 and the stability claim of Theorem 8 can be found in [30,40,51].

The rest of the asser-

tions are proved in the same way as the ordinary differential

2.

36

PROOFS OF THEOREMS

equations case. Exercise 4.

Show that the zero solution of xr+l = -xr + taxi + xryr _

yr+1

1

2

2 yr - axr + xryr

is asymptotically stable if

a > 0.

CHAPTER 3 EXAMPLES

3.1.

Rate of Decay Estimates in Critical Cases In this section we study the decay to zero of solutions

of the equation

i + i + f(r) = 0 where

f

is a smooth function with f(r) = r3 + ar5 + 0(r7)

where

a

is a constant.

as

r + 0,

(3.1.2)

By using a suitable Liapunov func-

tion it is easy to show that the zero solution of (3.1.1) is asymptotically stable.

However, because

f'(0)

= 0, the rate

of decay cannot be determined by linearization.

In [10] the rate of decay of solutions was given using techniques which were special to second order equations.

We

show how centre manifolds can be used to obtain similar results.

We first put (3.1.1) into canonical form.

Let

x = r + i, y = i, then x

-f(x-y)

y - -y

-

f(x-y). 37

(3.1.3)

3.

38

By Theorem 1 of Chapter 2, y = h(x).

EXAMPLES

(3.1.3) has a centre manifold

By Theorem 2 of Chapter 2, the equation which

determines the asymptotic behavior of small solutions of (3.1.3) is

u - -f(u - h(u)). Using (3.1.2) and

(3.1.4)

h(u) - 0(u2), u = -u3 + 0(u4).

(3.1.5)

Without loss of generality we can suppose that the solution u(t)

of (3.1.5) is positive for all

t > 0.

Using L'Hopital's

rule,

- lim t-1

-1 - lim

tHence, if

w(t)

u

J

u(t) - w(t+o(t)).

w(t) - 1 C

w(0) - 1,

(3.1.6)

Since

t-1/2

7

where

1

is the solution of w - -w3,

then

ju(t)s-3ds.

+ Ct-3/2 + 0(t-5/2)

(3.1.7)

is a constant, we have that u(t) -

t-1/2 + o(t-1/2).

(3.1.8)

To obtain further terms in the asymptotic expansion of u(t), we need an approximation to

h(u).

(M$)(x) - -$'(x)f(x-$(x)) + $(x) If

$(x) = -x3

then

To do this, set + f(x-$(x)).

(M$)(x) - 0(x5)

Theorem 3 of Chapter 2, h(x) - -x3 + 0(x5). into (3.1.4) we obtain

so that by

Substituting this

Hopf Bifurcation

3.2.

39

u = -u3 Choose

so that

T

grating over

[T,t]

(a+3)u5 + 0(u7)

-

u(T) a 1.

(3.1.9)

Dividing (3.1.9) by

u3, inte-

and using (3.1.8), we obtain rt

w-1(u(t))

- t + constant + (3+a) ft u

(3.1.10)

T

where

w

is the solution of (3.1.6).

Using (3.1.8) and

(3.1.9), ft

ftu2(s)ds

-

T

T

u s ds + ft 0(u4(s))ds u(s)

T

(3.1.11)

- -ln t-1/2 + constant + 0(1).

Substituting (3.1.11) into (3.1.10) and using (3.1.7), t-1/2

u(t)

t-3/2[(a+3)ln t + C]

4/7

where

C

+

o(t-3/2) (3.1.12)

is a constant.

If

(x(t),y(t))

is a solution of (3.1.3), it follows

from Theorem 2 of Chapter 2 that either

(x(t),y(t))

tends to

zero exponentially fast or

x(t) = tu(t), y(t) = Tu3(t) where 3.2.

u(t)

is given by (3.1.12).

Hopf Bifurcation There is an extensive literature on Hopf Bifurcation

[1,17,19,20,31,34,39,47,48,51,57,59] so we give only an outline of the theory.

Our treatment is based on [19].

Consider the one-parameter family of ordinary differential equations on

]R 2

40

3.

z = f(x,a), such that

f(0,a) =

0

xE

IR2

sufficiently small

for all

that the linearized equation about y(a) ± iw(a)

where

EXAMPLES

x =

a.

Assume

has eigenvalues

0

y(O) = 0, w(0) = w0 + 0.

We also assume

that the eigenvalues cross the imaginary axis with nonzero speed so that

Since

y'(0) + 0.

y'(0) + 0, by the implicit

function theorem we can assume without loss of generality that

y(a)

= a.

By means of a change of basis the differen-

tial equation takes the form (3.2.1)

i = A(a)x + F(x,a), where r

-w(a)

a

A(a) = 1 w (a)

a

F(x,a) = O(lx12).

Under the above conditions, there are periodic solutions of (3.2.1) bifurcating from the zero solution. precisely, for

a

More

small there exists a unique one parameter

family of small amplitude periodic solutions of (3.2.1) in exactly one of the cases (i)

a = 0, (iii) a > 0.

a < 0, (ii)

However, further conditions on the nonlinear terms are required to determine the specific type of bifurcation. Exercise

1.

Use polar co-ordinates for xl = axl - wx2 + Kxl(x2

x2)

+

x2 = wxl + ax2 + Kx2(x2 + x2)

to show that case (i) applies if plies if

K <

0.

K >

0

and case (iii) ap-

Hopf Bifurcation

3.2.

41

To find periodic solutions of (3.2.1) we make the substitution

x1 = Cr cos 0, x2 = er sin 0, a + ca, where

is a function of

c

(3.2.2)

After substituting (3.2.2)

a.

into (3.2.1) we obtain a system of the form

r = e[ar + r2C3(O,ac)I + e2r3C4(O,ac) + O(e3) (3.2.3) 6

= m0 + O(e) .

We now look for periodic solutions of (3.2.3) with and of

near a constant

r 0

C3 and

If

r0.

c - 0

are independent

C4

and the higher order terms are zero then the first

equation in (3.2.3) takes the form c[ar + $r2]

+

e2Kr3.

Periodic solutions are then the circles

(3.2.4)

r = r0, where

is a zero of the right hand side of (3.2.4).

r0

We reduce the

first equation in (3.2.3) to the form (3.2.4) modulo higher It turns

order terms by means of a certain transformation. out that the constant

B

is zero.

Under the hypothesis

K

is non-zero, it is straightforward to prove the existence of periodic solutions by means of the implicit function theorem. The specific type of bifurcation depends on the sign of so it is necessary to obtain a formula for Let

K

K.

F(x,a) = [F1(xl,x2,a), F2(xl,x2,a)]T

and let

F(xiIx2)a) = BZ(xl,x2,a) + B3(xl,x2,a) + 0(x4+x4) (3.2.5) where

Bi

(xl,x2).

is a homogeneous polynomial of degree

i

in

Substituting (3.2.2) into (3.2.1) and using (3.2.5)

we obtain (3.2.3) where for

i

- 3,4,

42

EXAMPLES

3.

Ci(6,a) _ (cos 0)B1i _1(cos 6, sin 6,a) (3.2.6)

+ (sin 0)Bi_1(cos 6, sin 6,a). There exists a coordinate change

Lemma 1.

r - r + cul(r,6,a,c) + c2u2(r,6,a,c) which transforms (3.2.3) into the system r = car +

(3.2.7)

- m0 + 0(c)

6

where the constant

c2 r3K + O(c3)

K

is given by

2n

K -

[C4(6.0)

(1/2n)f

-

(3.2.8)

w01C3(6.0)D3(6.0)]dO

0

where

C3

and

are given by (3.2.6) and

C4

D3(6,0) - (cos 6)BZ(cos 6, sin 6,0)

(sin 6)B2(cos 6, sin 6,0).

The coordinate change is constructed via averaging.

We

refer to [19] for a proof of the lemma. If

then we must make further coordinate changes.

K = 0

We assume that

K + 0

from now on.

Recall that we are looking for periodic solutions of (3.2.7) with

and

c + 0

gests that we set

r

near a constant

a = -sgn(K)c

and

r0 =

This sug-

r0.

The next

IKJ -1/2.

result gives the existence of periodic solutions of r - c2[-sgn(K)r + r3K] + 0(c3)

(3.2.9)

e'W0+0(c) with

r

Lemma 2. c

-

r0

small.

Equation (3.2.9) has a unique periodic solution for

small and

r

in a compact region either for

c

>

0

(when

Hopf Bifurcation

3.2.

K<

0) or

c

0

<

43

K > 0) .

(when

Also

r = r0 + 0(c), 6(t,c) - mot + 0(c) and the period of the solution

t(c) -

(21r/wo)

is given by

i(c)

+ 0(c).

The periodic solution is stable if

K <

0

and unstable if

K > 0. Lemma 2 is proved by a simple application of the impli-

We again refer to [19] for a proof.

cit function theorem.

Lemma 2 also proves the existence of a one parameter We cannot immedi-

family of periodic solutions of (3.2.1).

ately assert that this family is unique however, since we may have lost some periodic solutions by the choice of scaling, i.e. by scaling

a + ac

and by choosing

c - -sgn(K)a.

In order to justify the scaling suppose that x1 = R cos 6, x2 = R sin 6

bifurcating from

is a periodic solution of (3.2.1)

x1 - x2 - 0.

Then

satisfies

R

R = aR + 0(R2). When

attains its maximum,

R

justifies the scaling

a + ac.

= 0

so that

This

a - 0(R).

A similar argument applied to

periodic solutions of (3.2.7) justifies the choice of

c.

defined by (3.2.8) is

Theorem.

Suppose that the constant

nonzero.

Then (3.2.1) has a unique periodic solution bifur-

cating from the origin, either for a <

0

(when

K > 0).

If

K

a > 0

(when

K <

x - R cos 6, y = R sin 6,

periodic solution has the form

0) or

then the

44

3.

JaK-lll/2

R(t,a) _

EXAMPLES

+ O(JaI)

9(t,a) - w0t + 0(Jajl/2)

with period

t(a) _ (2n/w0) + O(Iaj 1/2).

tion is stable if

K <

and unstable if

0

The periodic soluK >

Finally, we note that since the value of only on the nonlinear terms evaluated at

0.

K

depends

a - 0, when apply-

ing the above theorem to (3.2.1) we only need assume that the eigenvalues of

A(a)

cross the imaginary axis with non-zero

speed and that

A(0) = with

m0

non-zero.

3.3.

Hopf Bifurcation in a Singular Perturbation Problem In this section we study a singular perturbation prob-

lem which arises from a mathematical model of the immune reThe equations are

sponse to antigen [521.

[x3 + (a

ez

a m

6(1-x)

)x + b -]

-

-

a

- ylab

(3.3.1)

ylab + y2b where a

and

e,6,yl,y2

are positive parameters.

In the above model

represent certain concentrations so they must be

b

non-negative.

Also, x

measures the stimulation of the system

and it is scaled so that

jxj

< 1.

The stimulation is assumed

to take place on a much faster time scale than the response so that

c

is very small.

3.3. Hopf Bifurcation in a Singular Perturbation Problem

45

The above problem was studied in [52J and we briefly outline the method used by Merrill to prove the existence of periodic solutions of (3.3.1).

Putting

c =

in the first

0

equation in (3.3.1) we obtain x3 + (a

Solving (3.3.2) for tain

-

+ b

4)x

m 0.

-

as a function of

x

a

(3.3.2)

and

b

we ob-

and substituting this into the second equa-

x = F(a,b)

tion in (3.3.1) we obtain 6(1-F(a,b))

- ylab

a

-

(3.3.3)

2

-Ylab +

Using

Y2

b.

as the parameter, it was shown that relative to a

6

certain equilibrium point place in (3.3.3).

(a0,b0)

a Hopf bifurcation takes

By appealing to a result in singular per-

turbation theory, it was concluded that for

c

sufficiently

small, (3.3.1) also has a periodic solution.

We use the theory given in Chapter 2 to obtain a similar result. Let

b0 + 0

then

be a fixed point of (3.3.1).

(x0,a0,b0)

and

a0 = Y2/Y1

x0,b0

satisfy

Y

3

x0 + (Y1

-

If

2)x0 + b0 -

1

= 0,

Y

6(1-x 0)

2

- Y2

- Y2b0 = 0.

1

Recall also that for the biological problem, we must have b0 > 0 b0

and

jx0I

< 1.

We assume for the moment that

satisfy (3.3.4) and these restrictions.

these solutions are considered later. for the rest of this section.

We let

x0

The reality of a0 = Y2/Yl

and

46

3.

Let where

ip

=

EXAMPLES

y = a - a0, z - b - b0, w - -i(x-x0) - x0y 3x2 + a0 -

Then assuming

.

ip

z

is non-zero,

Ew = g(w,Y,z,E) (3.3.5)

Y - f2(w,Y,z,E) i - f3(w,Y,z,E) where

g(w,Y,z,E) - fl(w,Y,z,E)

- Ex0f2(w,y,z,E)

- Ef3(w,Y,z,E)

fl(w,Y,z,E) - -ipw + N(w+x0Y+z,Y) -1x0

f2(w,Y,z,E) - (2

-

1

- ylb0)y + (. +

2

i-1

lw

- y2)z -

ylyz

f3(w,Y,z,E) e -ylb0y - ylyz N(e,Y) -

-p-263

+ 3i-1x062 - ye.

In order to apply centre manifold theory we change the time scale by setting respect to

s

by

'

t = Es.

We denote differentiation with

and differentiation with respect to

t

Equation (3.3.5) can now be written in the form

by

w'

- g(w,Y,z,E)

y' = Ef2(w,y,z,E) (3.3.6)

Suppose that

* > 0.

z'

= Ef3(w,y,z,E)

E'

= 0.

Then the linearized system corresponding

to (3.3.6) has one negative eigenvalue and three zero eigenvalues.

fold

By Theorem 1 of Chapter 2, (3.3.6) has a centre mani-

w = h(y,z,c).

By Theorem 2 of Chapter 2, the local be-

havior of solutions of (3.3.6) is determined by the equation

3.3. Hopf Bifurcation in a Singular Perturbation Problem

47

y' = Ef2(h(y,z,E),y,z,E) (3.3.7)

z' = Ef3(h(y,z,E),y,z,E)

or in terms of the original time scale y - f2(h(y,z,E),y,z,E) (3.3.8)

i = f3(h(y,z,E),y,z,E).

We now apply the theory given in the previous section to show that (3.3.8) has a periodic solution bifurcating from the origin for certain values of the parameters.

The linearization of the vector field in (3.3.8) about

y - z = 0 irs given by J(E) -

>y

2

1x0

-

1

*-1

- Ylb0

-

2

Y2

0

-Y1b0

If (3.3.8) is to have a Hopf bifurcation then we must have

and 2 *-I - Y2 > 0.

trace(J(E)) - 0

sis, we must also have that

x0,b0

with

ip

1x01

< 1, b0 > 0

and

From the previous analy-

are solutions of (3.3.4)

We do not attempt to ob-

> 0.

tain the general conditions under which the above conditions are satisfied, we only work out a special case. Lemma.

Let

6(E), x0(E), b0(E) >

Then for each

Y1 < 2Y2.

0, 6(E)iy-1

-

such that

E > 0, there exists

< 2x0(E) < 1, b0(E) > 0,

0

2Y2 > 0, trace (J(E)) = 0

and (3.3.4) is

satisfied. Proof:

Fix

Y1

and

Y2

with

Y1 < 2Y2.

If

x0,b0,6

isfy the second equation in (3.3.4) then trace(J(0)) -

Y

6 T[x0>V

1

-

1 2Y

(1-x0)J.

sat-

48

It is easy to show that there is a unique satisfies of

EXAMPLES

3.

trace(J(O)) =

We now obtain

x0(0).

i >

Clearly

0.

and

b0(0)

x0(0) E (0,-) 0

6(0)

that

for this choice as the unique

solution of (3.3.4) and an easy computation shows that b0(0)

6(0) >

0,

>

and

0

function theorem, for small, there exists

6(0)i-1

By the implicit

0.

sufficiently

c, x0 - x0(0), b0 - b0(0)

6(e,x0,b0) = 6(0)

trace(J(c)) = 2 i-1x0 -

After substituting

2Y2 >

-

1

+ O(e)

such that

- y1b0 + O(e) = 0.

6 = 6(e,x0,b0)

into (3.3.4), another

application of the implicit function theorem gives the reThis completes the proof of the lemma.

sult.

From now on we fix

Y1

and

Y2

with

yl < 2Y2.

the same calculations as in the lemma, for each with

and

a

6

x0(e,6), b0(e,6)

-

6(c)

of (3.3.4).

Writing

x0 = x0(e,6(e))

Y16(e)1-1

8

6=6(e)

(f x0)

is sufficiently small.

3

R(e)

For

-

6

0,

<

Hence, the eigenvalues of

cross the imaginary axis with non-zero speed at Let

and

[6y1x0

+ (2Y2-y1)x0 - 12y1x0] + O(e) c

6

6, we have that

a6(trace(J(6)))

if

and

c

sufficiently small there is a solution

as a function of

trace(J)

Using

y1 = R(e)z, z1 - m(c)y

6

J(6)

- 6(c).

where

(y1b0(e))-1/2 + O(e), m(c) = [(6/2)x-1-Y2]1/2 + O(e).

= 6(c), (3.3.8) in these new coordinates becomes Y1m- 1

y1 ` -w0z1

-

(e)y1z1 (3.3.9)

z1 = w0y1 + m(e)(6/2)*-1h(m-1(e)z1,

1(e)y1,e)-Y1L-1(c)y1zI

3.3. Hopf Bifurcation in a Singular Perturbation Problem

49

where w2(c)

=

Y1b0(c)[(6/2),y-1

Y2]

+ O(c).

To apply the results of the last section, we need to calculate the how

K(0)

K(c)

associated with (3.3.9).

can be calculated; if

is non-zero then

K(0)

To calculate

will be non-zero also.

K(0)

K(c)

we need to know

the quadratic and cubic terms in (3.3.9) when we have to find

We shall show

=

c

0.

Thus,

modulo fourth order terms.

h(y,z,0)

Let

(3.3.10)

(MO)(Y,z) ` -g(,(Y,z),Y,z,O)

Then by Theorem 2 of Chapter 2, if we can find (MO)(Y,z) = O(y4+z4)

then

0

such that

h(Y,z,0) = 0(Y,z) + O(y4+z4).

Suppose that (3.3.11)

0 _ 02 + 03 where

0j

is a homogeneous polynomial of degree

j.

Substi-

tuting (3.3.11) into (3.3.10) we obtain

(Mo)(Y,z) = *02(Y,z)

-

3iy-1x0(x0y+z)2 + Y(XOY+z) + O(IYI3 + IzJ3).

Hence, if 02(Y,z) = 3iy-2x0(Xoy+z)2 then

(Mo)(y,z)

(3.3.10) with

=

O(IyJ3 + jzJ3).

g2(y,z)

1y(x0y+z) -

Substituting (3.3.11) into

given by (3.3.12), we obtain

(Mo)(Y,z) _ ip03(Y,z) + i-2(XOY+z)3

+ Y02(Y,z) + O(Y4+z4). Hence,

(3.3.12)

6iy-1x0(XOY+z)02(Y,z)

50

3.

h(y,z)

02(y,z)

i-3(x0y+z)3 + 6iy-2x0(xOY+z)02(Y,z)

-

*-1y02(Y,z)

where

+ O(Y4+z4)

is defined by (3.3.12).

02(y,z)

yl

and

If

y2.

K(0)

can now be cal-

The sign of

culated as in the previous section. depend on

EXAMPLES

will

K(0)

is non-zero then we can

K(0)

apply Theorem 1 of Section 2 to prove the existence of periodic solutions of (3.3.1).

The stability of the periodic

solutions is determined by the sign of zero then we have to calculate

K(0).

K(0)

If

is

K(e).

Bifurcation of Maps

3.4.

In this section we give a brief indication of some reFor more details and

sults on the bifurcation of maps.

references the reader is referred to the book by Iooss [40]. Let let

be a neighborhood of the origin in

V

V -,]Rm

F

with

u E]Rq

]Rm

and

be a smooth map depending on a parameter

F(0) =

for all

0

If bifurcation is to

V.

take place then the linearized problem must be critical.

has simple complex eigenvalues

We

assume that

F6

with

- 1, A(0) 0 ±1, while all other eigenvalues of

FO'

IA(0)I

A(0), a(0)

By centre manifold theory

are inside the unit circle.

all bifurcation phenomena take place on a two-dimensional manifold so we can assume

m =

2

without loss of generality.

We first consider the case when sional parameter and we assume that

u

a(u)

is a one-dimen-

crosses the unit

circle, that is 0

A(u)

when

u = 0.

U 1-1

If

An 0

1

for

n -

3

or

4

then in general

Fu

has an

Bifurcation of Maps

3.4.

51

invariant closed curve bifurcating from an = 1, n > 5, then in general

over, if

any period points of order However, if eral

n

More-

does not have

Fu

bifurcating from

[40].

0

is a two-dimensional parameter, then in gen-

u

has periodic points of order

Fu

[57,58].

0

for

n

in a small

u

The method of proof used in [4] and [64] re-

region [4,64].

lies on approximating

by the time one map of a certain

Fu

differential equation; we outline a more direct proof. Suppose that - A(u)w + O(IwI2),

Fu (w)

where n >

has eigenvalues

A(0)

w E1R2, u E

with

a(u), a(u)

positive integer

p; the case

an(0) - 1,

n - 2p +

For definiteness we assume that

5.

IR2

n = 2p + 4

for some

is treated in the

Then by changing coordinates [40], Fu

same way.

3

has the

normal form 2k

Fn(z) - An(u)z + U where

z

k-1 a k(u)l11

+ b(u)z2p+2 + R(z,z,u)

z = x + iy, R(z,z,u) - O(jzj 2p+3)

and

ak(u), b(u)

are complex numbers.

We assume that after a change of para-

meter, An(u) -

where

that

al(0)

point of

Fu

1

-

and

u

u - ul + iu2.

are nonzero.

b(0)

If

We also assume z = reie

is a fixed

then

h(r,O,u) = 0 where ul + iu2 By redefining generality that

J1k+ b()r2eie + O(r22).

-

u + u(a1(u))-1, we can assume without loss of aI(u) -

1.

Given

0 E

[0,27T), ul >

0

and

52

u2

EXAMPLES

3.

small enough, the equation Re(h(r,e,u)) - 0

has a unique (small) solution

with

r - r(ul'u2,O)

(3.4.1)

r(ul,u2,0) - ul/2(1 + O(u))

Substituting this into the equation Im(h(r,e,u)) =

(3.4.2)

0

we obtain + O(r2P+2)

Im(b(u)e-in9)r2p+1

Im(ak(u))r2k +

u2 = k=2 Let

J2k

Im(a(u))r2k.

v2(ul,u2,0

o f parameter

We now make the change

= ul + i(u2 - u2).

c = el + ic

By (3.4.1),

2

the map

is one-to-one for

u -+ c

u

Equation

small enough.

(3.4.2) now takes the form e2 - Im(b(u)e-in9)e12p+l)/2(1 + O(el/2 + Ie2I)).

Since

b(0) 0 0, by the implicit function theorem the above

equation has Ie2l

n

distinct solutions

< Cei2p+1)/2

fixed points for If

where u

C

0r(e), r - 1,...,n, for

is a constant.

Thus

in a region of width ON

n Fu

has

(n-2)/2

A3(0) = 1, then for one parameter families of maps,

in general, there are bifurcating fixed points of order and no invariant closed curves [40,41). is more complicated.

The case

A4(0)

3

= 1

In this case the map can be put into

the normal form Fu(z) - a(u)z + b(u)z2z + c(u)z3 + O(IzIS)

Bifurcation of Maps

3.4.

where

a(u) - i + O(u).

fixed points of order cate from

Fu

If

then in general either

u E IR

or an invariant closed curve bifur-

4

0; the actual details depend on

[40,41,681.

c(u)

53

If

a(u), b(u)

and

is a two-dimensional parameter then

u

is studied by investigating bifurcations of the differen-

tial equation

A(u)z + B(u)z2z + C(u)z3 where

A(u), B(u)

one map

C(u)

and

are chosen such that the time

of (3.4.3) satisfies

T(u)

Fu (z)

- iT(u) (z)

The bifurcation diagrams given in seem only to be conjectures. studied in [541

parameters

(3.4.3)

and

O(Iz15) [41 for equation (3.4.3)

Bifurcations of (3.4.3) are

A(u) - u E

in the case

Re(u)

-

with bifurcation

Re(B(u)).

We note that the above bifurcation results do not apply if the map has

special properties.

For example, F U

could represent the phase flow of a periodic Hamiltonian system so that

Fu

is a symplectic map.

is due to Moser [531:

The following result

if

F

is a smooth symplectic map with

F(0) - 0, then in general

F

has infinitely many periodic

points in every neighborhood of the origin if the linearized map

F'(0)

has an eigenvalue on the unit circle.

For an ap-

plication of this result to the existence of closed geodesics on a manifold see [461.

CHAPTER 4

BIFURCATIONS WITH TWO PARAMETERS IN TWO DIMENSIONS

4.1.

Introduction

In this chapter we consider an autonomous ordinary differential equation in the plane depending on a two-dimensional parameter

We suppose that the origin

c.

point for all

x = 0

is a fixed

More precisely, we consider

c.

f(x,E),

z

x E ]R2,

El,E2) E ]R2,

a

(4.1.1)

f(O,E) E 0.

The linearized equation about

x = 0

is

z = A(c)x,

and we suppose that

A(0)

has two zero eigenvalues.

ject is to study small solutions of (4.1.1) for

The ob-

(E1,E2)

in

a full neighborhood of the origin.

More specifically, we wish

to divide a neighborhood of

into distinct components,

such that if

e,-

c = 0

are in the same component, then the phase

portraits of (4.1.1)E and (4.1.1)- are topologically equivalent.

We also want to describe the behavior of solutions for

each component.

The boundaries of the components correspond S4

4.1.

Introduction

55

to bifurcation points.

Since the eigenvalues of that either (i)

A(O)

are both zero we have

A(O)

is the zero matrix, or (ii)

A(O)

has

a Jordan block,

There is a distinction between (i) and (ii) even for the study Under generic assumptions, in case (ii),

of fixed points.

equation (4.1.1) has exactly 2 fixed points in a neighborhood of the origin.

For case (i) the situation is much more com-

plicated [21,29].

Another distinction arises when we consider the eigenvalues of

A(c).

values of

A(c)

bifurcation.

We would expect the nature of the eigento determine (in part) the possible type of

If

then the eigenvalues of

A(c)

are always real so we do not

expect to obtain periodic orbits surrounding the origin.

On

the other hand if (4.1.2)

then the range of the eigenvalues of

of the origin in , that is, if ber then

A(e)

has an eigenvalue

A(e)

is a small complex num-

z z

for some

We shall assume from now on that (4.1.2).

is a neighborhood

A(e)

C.

is given by

56

BIFURCATIONS WITH TWO PARAMETERS

4.

Takens [64,65] and Bodganov [see 3] have studied norTakens shows, for example,

mal forms for local singularities.

that any perturbation of the equation 2

2

xl = x2 + x1,

is topologically equivalent to 2

x2 = E1 +

1 = x2 + xl,

for some

E1,E2.

E 2

2

x

-

xl

1

There are certain difficulties in applying

these results since we must transform our equation into normal form, modulo higher order terms. In [47, p. 333-348],

under the assumptions that 2 a

Kopell and Howard study (4.1.1) A(E)

is given by (4.1.2) and that

f

-2 2(0,0) + 0 ax1

where

f2

is the second component of

f.

Their approach con-

sists of a systematic use of scaling and applications of the implicit function theorem.

In this chapter, we use the same techniques as Kopell and Howard to study (4.1.1) when the nonlinearities are cubic. Our results confirm the conjecture made by Takens [64] on the bifurcation set of (4.1.1).

Similar results are given in [4] together with a brief outline of their derivation.

The results on quadratic nonlinearities are given in Section 9 in the form of exercises. be found in Kopell and Howard [47].

Most of these results can

4.2.

Preliminaries

4.2.

Preliminaries

57

Consider equation (4.1.1) where (4.1.2). is

A(e)

is given by

We also suppose that the linearization of

f(x,e)

A(e)x, and that f(O,e) = 0,

f(x,e) = -f(-x,e).

(4.2.1)

The object is to study the behavior of all small solutions of (4.1.1) for

in a full neighborhood of the origin.

a

Equa-

tion (4.1.1) is still too general, however, so we shall make some additional hypotheses on the nonlinear terms.

Set

f = (fl,f2)T 3

3

- f2(0,0)'

axlx2

axl

(Hl)

a+

0

(H2)

$+

0

f2(0,0

s =

3

f1(0,0) = 0. (H3)

ax3 1

(Hl) implies that for small or

3

fixed points.

c, (4.1.1) has either

1

Under (Hl), it is easy to show that by

a change of co-ordinates in (4.1.1), we can assume (H3) (see Remark 1).

We assume (H3) in order to simplify the computa-

tions.

Under (Hl) we can prove the existence of families of periodic orbits and homoclinic orbits.

Under (Hl)-(H3) we can

say how many periodic orbits of (4.1.1) exist for fixed The sign of

$

E.

will determine the direction of bifurcation

and the stability of the periodic orbits, among other things. From now on we assume (Hl)-(H3).

58

4.

Level Curves of

BIFURCATIONS WITH TWO PARAMETERS

H(yl,y2) a (y2/2) -

(y2/2) + (y4/4).

Figure 1

E1

Bifurcation Set for the Case

a < 0,

8 <

0.

Figure 2

The main results are given in Figures 2-5.

Sections

3-8 of this chapter show how we obtain these pictures. pictures for variables

$ >

0

are obtained by using the change of

x2 + -x2, E2 + -E2

and

t + -t.

The

Preliminaries

4.2.

59

E2

3

2

E1 1

Bifurcation Set for the Case

a > 0,

8

<

0.

Figure 3 The cases ferent.

a >

and

0

a <

0

are geometrically dif-

The techniques involved in each case are the same

and we only do the more difficult case a >

0

a < 0.

The case

is left for the reader as an exercise. From now on we assume

a <

0

in addition to (Hl)-(H3).

Note that this implies that locally, (4.1.1) has point for

E1 <

0

and

3

fixed points for

1

E1 > 0.

fixed

60

4.

BIFURCATIONS WITH TWO PARAMETERS

REGION 1

REGION 2

REGION 3

REGION 4

Phase Portraits for the Case Figure 4

a < 0,

a <

0.

4.2.

Preliminaries

61

ON L1

REGION 5

ON L2

Figure 4 (cont.)

62

4.

BIFURCATIONS WITH TWO PARAMETERS

REGION 6

Figure 4 (cont.)

4.2.

Preliminaries

63

REGION 1

REGION 2

REGION 3

REGION 4

Phase Portraits for the Case

Figure S

a > 0,

$

<

0.

BIFURCATIONS WITH TWO PARAMETERS

4.

64

Scaling

4.3.

We scale the variables in equation (4.1.1) so that the first components of the non-zero fixed points are given by To do this we introduce parameters

±1 + 0(E).

yl,y2

variables

and a new time

u,6, scaled

by the relations

t

62l0111/2

6 = lEla-1 1/2, E2 = lall/26u, xl = 6yl, x2 = t =

to a region of the form 2

(constant)E 2}. say

The

{(El,E2):

belongs

1Ell

< E0, El I are assumed to lie in a bounded set,

yi

A further discussion of the scaling is given

< M.

lyil

lal-1/26-1T.

in a neighborhood of the origin, (E1,E2)

(u,6)

For

Y29

in Section 6.

After scaling (4.1.1) becomes Yl = Y2 + 62g1(u,6,Y) (4.3.1)

y2 = sgn(E1)Y1 +

uy2

- y1 + 6yyly2 + 62g2(u,6,Y)

where the dot means differentiation with respect to y = slat-1/2 on the

gi

and

gi(u,6,y) = 0(1).

depends on

M,u,6.

t,

The size of the bounds

We write

t

for

T

from now

on.

The cases

4.4.

The Case

With

E1 >

0

and

E1 <

are treated separately.

0

El > 0

E1 > 0, equation (4.3.1) becomes

Yl = Y2 + 62g1(u,6,Y) (4.4.1) 2

Y2 = y1 + zY2 - y1 + 6yyly2 Let

H(yl,y2) -

(y2/2)

+

6

(y2/2)

+

92(u,6,Y) (y4/4).

Then along

The Case

4.4.

65

0

El >

solutions of (4.4.1), (4.4.2)

H(Y1,Y2) = VY2 + 6yy2Y2 + 0(62). Note that for

u = 6 = 0, H

is a first integral of (4.4.1).

The level curves of of eight if Figure 1).

H(yl,y2) = b

consist of a figure

b = 0, and a single closed curve if For

b

through the point

> 0, the curve

b

H(yl,y2) = b

yl = 0, y2 = (2b)1'2.

For

>

0

(see

passes b

0

>

and

6

sufficiently small, we prove the existence of a function u = u1(b,6) = -yP(b)6 + 0(62) with

u = ul(b,6)

such that for

>

0,

(4.4.1)

has a periodic solution passing through the

yl = 0, y2 = (2b)1'2, and with

point

b

ul(0,6), (4.4.1)

has a figure of eight solutions.

u,6, the number of periodic solutions of

For fixed

(4.4.1), surrounding all three fixed points, depends upon the number of solutions of u = ul(b,6) = -yP(b)6 + 0(62). P(b) - m

We prove that

P'(b) <

such that

b1 >

0

for

b > b1.

b - m

as 0

for

and that there exists b < b1

These properties of

(4.4.3)

P(b)

and

P'(b) > 0

determine the number

of solutions of (4.4.3).

Suppose, for simplicity of exposition, that -yP(b)6

and that

b3 > b1

such that

y < 0.

If

0

ul(b,6)

< b2 < bl, then there exists

ul(b2,6) = ul(b3,6).

Hence, if

ul(b216), then (4.4.1) has two periodic solutions, one passing through through

yl = 0, y2 = (2b2)1'2, the other passing

yl - 0, y2 =

(2b3)1"2.

If

u > ul(0,6), then (4.4.1)

has one periodic solution surrounding all three fixed points.

66

BIFURCATIONS WITH TWO PARAMETERS

4.

u = u(b1,6), then the periodic solutions coincide.

Finally, if

In Figure 4, the periodic solutions

surrounding all

three fixed points in regions 3-5 correspond to the periodic solutions of (4.4.1) which are parametrized by

Similarly the "inner" periodic solutions in region 5

b > b1.

u = ul(b,6), 0 < b < bl.

are parametrized by

space corresponds to the curve

(el,e2)

in

u = ul(b,6),

Similarly the curve

L2

The curve

L1

u = ul(0,6).

corresponds to the curve

u = ul(bl,6)

(see Figure 2).

In general

is not identically equal to

ul(b,6)

-yP(b)6, but the results are qualitatively the same.

example, we prove the existence of a function 0(6), such that if

satisfy

u,6

bl(6) = bl +

u = ul(bl(6),6), then equa-

tion (4.4.3) has exactly one solution.

The curve

which is mapped into the curve

u = ul(bl(6),6)

For

L2

in

space corresponds to the points where the two

(el,e2)

periodic solutions coincide. If

b < 0, then the set of points for which

H(yl,y2) = b points

consists of two closed curves surrounding the and

(-1,0)

(1,0).

the point

0 < c < 1, we prove the

such

u = u2(c,6) = -yQ(c)6 + 0(62)

existence of a function that for

For

u = u2(c,6), (4.4.1) has a periodic solution surrounding (1,0)

and passing through

Using

yl = c, y2 = 0.

= -f(-x), this proves the existence of a periodic solu-

f(x)

tion surrounding the point

(-1,0)

and passing through

yl = -c, y2 = 0.

We also prove that 6

>

0

and suppose

u

Q'(c) > 0

for

0

< c <

1.

Let

satisfies

u2(0,6) < -sgn(y)u <

u2(1,6)

(4.4.4)

el >

The Case

4.4.

Then the equation

67

0

has exactly one solution.

u = P2 (c,6)

Hence, for fixed

satisfying (4.4.4), equation (4.4.1)

u,6

has exactly one periodic solution surrounding region in

space corresponding to (4.4.4)

(u,6)

into region 4 in For

Lemma 1.

(1,0).

The

is mapped

space (see Figure 2).

(el,e2)

sufficiently small, there exists a function

6

u = u(6) = -(4/5)6y + 0(62)

such that when

u(6), (4.4.1)

has a homoclinic orbit. Proof:

Let

S(u,6), U(u,6)

folds of the fixed point exist, since for [271.

Let

These manifolds

in (4.4.1).

be the value of

H(u,6,+)

when

H(yl,y2)

(0,0)

sufficiently small, (0,0)

u,6

y2 = 0, yl >

hits

be the stable and unstable mani-

H(yl,y2)

Similarly, H(u,6,-)

0.

S(u,6)

hits

is a saddle

when

U(u,6)

is the value of

y2 = 0, yl > 0.

H(u,6,±)

are

well defined since stable and unstable manifolds depend continuously on parameters. Let

I(u,6,+)

the portion of y2 = 0

to

denote the integral of

U(u,6)

with

y2 = 0, yl > 0.

yl > 0, y2 >

0

H(yl,y2)

from

yl =

Then

H(u,6,+) = I(u,6,+). Similarly, I(u,6,-) the portion of to

y2 = 0, yl

>

(4.4.5)

denotes the integral of

S(u,6)

with

0, so that

over

yl > 0, y2 <

0

H(yl,y2)

from

over

yl = y2 = 0

H(u,6,-) = I(u,6,-).

Equation (4.4.1) has a homoclinic orbit (with

yl > 0)

if and only if H(u,6,+)

- H(u,6,-) = 0.

(4.4.6)

We solve (4.4.6) by the implicit function theorem.

BIFURCATIONS WITH TWO PARAMETERS

4.

68

Using (4.4.2) and (4.4.5), H(u,6,+)

f (iy2

+

y6y2y2)dt + O(u2 + 62

(4.4.7)

where the above integral is taken over the portion of yl - y2 = 0

from

Similarly,

y2 = 0, yl > 0.

to

(1YZ + y6y2y2)dt + O(u2 + 62),

where the integral is taken over the portion of yl = Y2 =

to

0

U(0,0)

(4.4.8) S(0,0)

from

Using (4.4.7), (4.4.8) and

Y2 = 0, yl > 0.

U(0,0) = -S(0,0), we obtain H(u,6,+) = -H(u,6,-) + O(u2 + 62)

(4.4.9)

Using (4.4.7) and (4.4.9), y2dt > 0,

- H1(0,0,-)) - 2f

. (H1(0,0,+)

so that by the implicit function theorem, we can solve (4.4.6) for

u

as a function of

u - u(6).

6, say

to get an approximate formula for

u(6).

We now show how

We can write equa-

tion (4.4.6) in the form of+y2dt

+

y6f+ y2Y2dt + O(u2 + 62) = 0.

Hence, using (4.4.1) and

U=

yly2dyl

y6f +

4 (yl/2)]1/2, we obtain

y2 = +[yl2

+ 0(6 2 )

=

- 4

5 + 0(6 2 )

y2dy1

This completes the proof of Lemma 1.

Lemma 1 proves the existence of a homoclinic orbit of (4.1.1) when

(E1,E2)

E2 -

lies on the curve -(4/5)lal

1

8E1

L1

+ O(Ei/2).

given by

The Case

4.4.

el >

69

0

f(x,e) = -f(-x,e), when

Using

lies on

(el,e2)

L1, equa-

tion (4.1.1) has a figure of eight solutions.

We now prove the existence of periodic solutions of In the introduc-

(4.4.1) surrounding all three fixed points.

tion to this section, we stated that (4.4.1) has a periodic solution passing through

yl = 0, y2 = (2b) 1/2

for any

In Lemma 2, we only prove this for "moderate" values of The reason for this is that in (4.3.1) the only for

b.

are bounded

In Section 6, we show that by

in a bounded set.

y

gi

b > 0.

a simple modification of the scaling, we can extend these results to all

b

Then for

b > 0.

Fix

Lemma 2.

> 0.

< b < b

0

ently small, there exists a function such that if

0(62)

u = ul(b,6)

b

ul(b,6)

u

suffici-

-yP(b)6 +

yl = 0, y2 =

(2b)1/2.

the periodic solution tends to the figure of

0

-

6

in (4.4.1), then (4.4.1) has

a periodic solution passing through As

and

eight solutions obtained in Lemma 1. Proof:

H(u,6,b,+)

Let

be the value of

orbit of (4.4.1) which starts at sects

y2 = 0.

y2 = -(2b)

1/2

Similarly, H(u,6,b,-)

is the value of

H(u,6,b,+) I(u,6,b,+)

if and only if

- H(u,6,b,-) = 0.

denote the integral of

the portion of the orbit of (4.4.1) with yl

-

0, y,

-

yl = 0,

Then (4.4.1) has a periodic solution passing

yl = 0, y2 - (2b) 1/2

Let

inter-

is integrated backwards in time until it inter-

y2 = 0.

through

when the

yl - 0, y2 = (2b) 1/2

when the orbit of (4.4.1) which starts at

H(yl,y2)

sects

H(yl,y2)

(2b)1/2

and finishing at

(4.4.10)

H(yl,y2)

over

y2 > 0, starting at y2 - 0, yl >

0.

4.

70

is defined by integrating backwards

Similarly, I(u,6,b,-) in time.

BIFURCATIONS WITH TWO PARAMETERS

Thus,

H(u,6,b,±) = b + I(11,6,b,±).

(4.4.11)

Using (4.4.2) and (4.4.11), H(u,6,b,+) - b +

J(uy2

+

y6y2y2)dt

+

0(u2+62)

(4.4.12)

where the above integral is taken over the portion of the orbit of (4.4.1) with to

u = 6 = 0

from

yl = 0, y2 = (2b)

1/2

where

yl = c, y2 = 0

4b = c4

-

(4.4.13)

2c2.

Similarly, I(u,6,b,-) = -I(u,6,b,+) + O(u2 + 62), so

that equation (4.4.10) may be written in the form J(uy2

+

y6y2y2)dt + O(u2 + 62) = 0.

(4.4.14)

Hence by the implicit function theorem, we can solve (4.4.14) to obtain

u = -yP(b)6 + 0(62)

where 2

P(b) =

d

(4.4.15)

yly2 yl y2dy1

In order to prove that the periodic solution tends to the figure of eight solutions as

b + 0, we prove that

H(u,6,b,±) - H(u,6,±)

as

b - 0.

(4.4.16)

This does not follow from continuous dependence of solutions on initial conditions, since as

b + 0

periodic solution tends to infinity.

the period of the

The same problem occurs

in Kopell and Howard [47, p. 339] and we outline their method. For

yl

and

y2

small, solutions of (4.4.1) behave like

The Case

4.4.

Cl >

71

0

solutions of the linearized equations.

The proof of (4.4.16)

follows from the fact that the periodic solution stays close to the solution of the linearized equation for the part of the solution with

small and continuous dependence on

(yl,y2)

initial data for the rest of the solution.

P(b) + -

Lemma 3.

The integrals in (4.4.15) are taken over the curve

Proof:

c

(y1/2) + 2b]1/2, from

In

y2

b + m.

as

is defined by (4.4.13).

yl = c

to

0

where

Thus, JO(b)P(b) = Jl(b)

Jc w2i(w2

Ji(b) =

yl =

(w4/2)

-

where

2b)1/2dw. (4.4.17)

+

0

Substituting

w = cz

in (4.4.17) we obtain r1

Ji(b) = c2

(cz)2ig(z)dz J

where

g(z) _ [(z2-1) +

g(c-1)

for

0 (c2/2)(l-z4)]1'2.

0 < z < 1, we have that

positive constant

such that

D2

Lemma 4.

There exists PI(b) >

g(z) <

JO(b) < Dlc3

for some

Similarly, there exists a positive

D1.

constant

Since

Jl(b) b1 >

0

for

b > b1.

b < b1

and

Proof:

It is easy to show that

0

> D2c5.

The result now follows.

such that

P'(b) <

P'(b) +

as

by Lemma 3 it is sufficient to show that if

0

for

b + 0.

P'(bl) =

0

Hence, then

P"(bl) > 0. Let

r(w) _ [w2

-

(4.4.17) with respect to

(w4/2)

c

Integrating by harts in

J

.J0

2b]1/2.

Differentiating

we obtain

b

Ji

+

0

2i

r w

dw.

we obtain

(4.4.18)

BIFURCATIONS WITH TWO PARAMETERS

4.

72

c JO

= 1

4

2

wr w

(4.4.19)

dw.

0

Also

J

1c r2 w

=

w4 2 r w

[w2

(c

w =

0 r(w)

0

O

+ 2b]

(4.4.20)

dw.

Similarly, rc

1c

3J1 =

w =

iw,6ww4

3

0

0

Using (4.4.18) terms of

rwj

[w2

(w4/2)+2b]dw. (4.4.21)

-

(4.4.21) we can express

-

and J.

JI

2 rww

and

J0

J1

in

A straightforward calculation yields

3J0 = 4bJ0 + J1 (4.4.22)

15J1 = 4bJ6 + (4+12b)Ji.

Suppose that

P'(b1) = 0.

- P(b1)J0"(b1).

J111(b1)

JO(b1)P"(b1) _

Then

Using (4.4.22) we obtain

4b1(4b1+l)[J"(b1)

- P(b1)J3(b1)] (4.4. 23)

J6(b1) [P2(b1) + 8b1P(b1) Hence

P"(b1)

has the same sign as P2(b1) + 8b1P(b1)

Since

- 4b1] .

P'(b1) = 0

we have that

-

(4.4.24)

4b1.

Ji(b1) = P(b1)J;(b1).

Using

(4.4.22) we obtain 5P2(b1) + 8b1P(b1) Using

-

4P(b1)

-

(4.4.25)

4b1 = 0.

b1 > 0, it is easy to show that (4.4.25) implies that

P(b1) < 1. sign as required.

Using (4.4.24) and (4.4.25), P"(b1)

P(b1)

-

P2(b1).

This proves that

has the same

P"(b1) >

0

as

e1 >

The Case

4.4.

For

Lemma S.

73

0

b1(6) =

sufficiently small there exist

6

b1 + 0(6), b2(6) = b2 + 0(6), where

P(0) = P(b2), with the

following properties: Let

6

(i)

if

0.

>

u = u1(b1(6),6), then the equation (4.4.26)

u = 'l(b,6)

has exactly one solution. y <

If

(ii)

and

0

u1(b1(6),6) < u < ul(b2(6)P6)p

then equation (4.4.26) has exactly two solutions with b4

b3(6), b4(6)

b3(6) = b3 + 0(6), b4(6) = b4 + 0(6), where

and

b3

are solutions of ly 1.

P(b) = -ud

A similar result holds if y < 0

If

(iii)

y >

and

has exactly one solution

(4.4.27)

0.

> ul(b2(6),6), then (4.4.26)

u

b5(6) = b5 + 0(6), where

b5

is

the unique solution of (4.4.27). y < 0

If

(iv)

P(0)

By Lemma 4, there exists and

P'(b2) >

ul(0,6)], for

z(6)

+

of

b1(6)

and

z2).

= 0(6)

b2(6) = b2 + z(6)

Set

0.

+ 0

6

P'(b2)z + 0(161 exists

u < u1(b1(6),6), then (4.4.27)

A similar result holds for

has no solutions. Proof:

and

0.

such that

0

g(z,6) = 6-1[111(b2+z,6)

g(z,O) = 0.

P(b2) _ -

g(z,6) _

Then

By the implicit function theorem there

such that then

b2 >

y >

g(z(6),6) =

0.

u1(b2(6),6) = u1(0,6).

is proved in a similar way.

follows from the properties of

Hence, if The existence

The rest of the lemma

P(b).

We now prove the existence of periodic solutions of (4.4.1) currnunding a jingle fixed point.

BIFURCATIONS WITH TWO PARAMETERS

4.

74

Lemma 6.

0 < c < 1

For

and

sufficiently small, there

6

such that if

u = u2(c,6) _ -yQ(c)6 + 0(62)

exists

u - u2(c,6), then (4.4.1) has a periodic solution passing through

yl - c, y2 - 0.

As

c -

0

the periodic orbits tend

to the homoclinic orbit obtained in Lemma 1. Proof:

Let

H(u,6,c,+)

be the value of

orbit of (4.4.1) starting at y2 - 0, y2 > H(yl,y2)

yl - c, y2 -

Similarly, H(u,6,c,-)

0.

H(yl,y2)

when the

intersects

0

is the value of

when the orbit of (4.4.1) starting at

yl = c,

is integrated backwards in time until it intersects

y2 = 0 y2 - 0.

Equation (4.4.1) will have a periodic orbit passing

through

yl - c, y2 - 0

if and only if (4.4.28)

H(u,6,c,+) - H(u,6,c,-) - 0.

Using the same method as in Lemma 2, we can rewrite equation (4.4.28) as

G(u,6,c) - j(uy2 + y6y2y2)dt + O(u2 + 62)

(4.4.29)

- 0

where the above integral is taken over the curve y2 = r(yl) = [y1 from

yl = c, y2 - 0

to

-

(y4l /2) +

y2 = 0

(c4/2)-c2]1/2,

Thus, for fixed

By (4.4.29)

again.

au G(0,0,c) - f y2dt >

(4.4.29) uniformly in

c

however, since as

hand side of (4.4.31) tends to

0.

(4.4.31)

0.

we can solve (4.4.29).

c

(4.4.30)

We cannot solve c -

1

the right

We use a method similar to

that used by Kopell and Howard in [47, p. 337-338] to obtain u2(c,6)

for

0

<

c

<

1.

The Case

4.4.

c1 >

75

0

Equation (4.4.1) has a fixed point at

sufficiently small we make a

(1,0) + O(IiiI,I61).

For

change of variables

yj = hj(yl)y2,u,6)

u,6

point is transformed into

(yl,y2) -

so that the fixed

(yl,y2) = (1,0).

An easy calcula-

tion shows that if we make this transformation, then the only change in (4.4.1) is in the functions

and

gl

We sup-

g2.

pose that the above change of variables has been made and we write

y,

y,.

for

The curves

H(yl,y2) = H(c,0)

can be written in the

form

4H(Y1,Y2)+1 - 2y2 + (y1-l) 2(yl+l)2 - (c-1)2(c+l)2. (4.4.32) Thus, for

c

close to

1

the closed curves are approximately

y2 + 2(yl-l) 2 - 2(c-1)

(4.4.33)

2.

So, instead of equation (4.4.29) we consider the equation G(1,6,c) = (c-1)-2G(c,u,6) - 0. If we prove that

G(u,6,c)

(3/3u)_(0,0,c)

that

is bounded for

(4.4.34)

< c < 1

0

is bounded away from zero for

then we can solve (4.4.34) uniformly in

and

0 < c < 1,

c.

Now

H(Y1,Y2) = VY2 + 6(yy12 y + 6Y2g2 + 6(y3l-yl)g1]. Since the fixed point is at

and below by quadratic forms in 4H(yl,y2) + 1 in

y2

O(juj +

and 161)

is bounded above

(1,0), H(yl,y2) y2

and

yl

-

1.

By (4.4.32),

is bounded above and below by quadratic forms yl

-

1.

Hence there exist functions

such that

gi(u,6)

BIFURCATIONS WITH TWO PARAMETERS

4.

76

g2(u,6)

gl(u,6) < at ln(4H(yl,y2) + 1)

Integrating the above inequality over the curve given by (4.4.30) we obtain exp(gl(11,6)T) < [4H(u,6,c,+)+1]/[4H(c,0)+1] (4.4.35)

< exp(g2(u,6)T), where

is a bound for the time taken to trace the orbit.

T

Using (4.4.29), (4.4.32) and (4.3.35) we see that The fact that

is bounded.

(3/3v)-(c,0,0)

from zero follows easily from (4.4.32). 6

sufficiently small and

to obtain

6(11,6,c)

is bounded away

This proves that for

< c < 1, we can solve (4.4.32)

0

u - u2(c,6) - -yQ(c)6 + 0(62)

where

J0(c)Q(c) = J1(c), d Ji(c) - 1

w2ir(w)dw c

and

r(w)

is defined by (4.4.30) and

r(d) - 0, d > c.

The fact that the periodic solution tends to the homoclinic orbit is proved in the same way as the corresponding result in Lemma 2.

f(x) = -f(-x), Lemma 6 proves the existence of

Using

periodic solutions of (4.4.1) surrounding Lemma 7. Proof:

4b - c4

for

Q'(c) > 0

We write -

prove that

2c2.

Q,J0

Since

Q1(b) <

0

0

and

< c < 1. J1

(db/dc)

for

as functions of < 0

for

Thus, if

0

-1 < 4b < 0.

procedure as in Lemma 4, we find that equation (4.4.22).

(-1,0).

Q'(bl)

b

< c < 1, we must

Following the same

J0,J0,, J1,Ji -

0

where

then

satisfy

The Case

4.5.

77

0

c1 <

5Q2(bl) + 8b1Q(bl) Since

4Q(bl)

-

4b1 = 0.

(4.4.36)

-1 < 4b1 < 0, the roots of (4.4.36) are less than

Q'(bl) - 0

Hence, if if

-

Q'(bl) -

8b1Q(bl)

0

then

Q(bl) < 1.

plies that

Q"(bl) <

shows that

Q1(b) <

Since

0.

4b1

-

< 1, this im-

Q(bl)

Q(-1/4) - 1, Q(O) - 4/5, this

-1 < 4b <

for

0

Also, from (4.4.23),

has the same sign as

Q"(bl)

Using (4.4.36) and

Q2(bl).

-

then

1.

This completes the

0.

proof of the lemma.

4.5.

The Case With

El

El <

0

< 0, equation (4.3.1) becomes

yl = Y2 + 6g1(u,6,Y) (4.5.1)

Y2 - -Yl + uY2 - yl + 6Yyly2 + 62g2(v,6,Y) Let

H1(yl'y2) = (y2/2) + (y1/2) + (y4l/4).

Then along solu-

yZ + 6yy2y2 + 0(62).

tions of (4.5.1), H1 -

Using the same

methods as in the previous lemmas, we prove that (4.5.1) has a periodic solution passing through and only if

y1 - c > 0, y2 = 0, if

u = u3(c,6) - -yR(c)6 + 0(62), where

JO(c)R(c) = J1(c), J1(c) -

c r

w2ir(w)dw,

0

r(w) = [2b

-

(w4/2)

-

w211/2, 4b

In order to prove that for fixed

-

u,6, equation (4.5.1)

has at most one periodic solution we prove that monotonic. Lemma 8.

R'(c)

>

0

for

c > 0.

c4 + 2c2.

R

is strictly

78

BIFURCATIONS WITH TWO PARAMETERS

4.

We write

Proof:

R,JO,J1

ent to prove that

as functions of

R'(b) >

b > 0.

for

0

It is suffici-

b.

Using the same methods as before, we show that 3J0 - 4bJ6 - J1 (4.5.2)

15J1 - (12b+4)Ji - 4bJ6 3wwdw

- 4bJ6 - 4Ji.

(4.5.3)

to

Now

has the same sign as

R'

where

S

S - 15[JiJO - J0,J1].

By (4.5.2), S - (8b-4)JO'Ji - 5(Ji)2 + 4b(Jo,) 2.

By (4.5.3), bJ0 > J.

Hence, for

b > 2,

S > (J,)2b-1(8b-4-5b+4)

Similarly, if

0 < b < 2, then

-

3(Ji)2 > 0.

S > 3b2(Jo)2 > 0.

This com-

pletes the proof of Lemma 8. 4.6.

More Scaling

In Section 4 we proved the existence of periodic solutions of (4.4.1) which pass through the point Lemma 3 indicates that we may take

to be as large as we

b

However our analysis relied on the fact that the

please. are

(0,(2b)1/2).

and this is true only for

0(1)

y

in a bounded set.

Also, our analysis in Sections 4 and S restricts region of the form

{(E1,E2):

1c11

gi

< EO, E1 <

E1,E2

(constant)

to a c

}

To remedy this we modify the scaling by setting 6 = 1Ela-111/2h-1, where 0 < 4h < same.

1.

h

is a new parameter with, say,

The other changes of variables remain the

Note that if

u,6

lie in a full neighborhood of the

.

4.6.

More Scaling

origin then

79

(c1,e2)

(xl,x2)

and

lie in a full neighbor-

hood of the origin.

After scaling, (4.1.1) becomes Yl - Y2 + 62g1(11,6,h,y) (4.6.1)

y2 - h2sgn(el)Y1 + 11Y2

0 < 4h <

For

for

0(1)

(y4/4).

c1 > 0.

gi

are

Let

H2(Y1OY2) _ (Y2/2)

-

(h2y2/2)

+

Then along solutions of (4.6.1), H2 - 11y2 +

y6y2y2 + 0(112 + 62). 1

sufficiently small, the

11,6

in a bounded set.

y

Let

and

1

yl + 6yyiy2 + 62g2(i,6,h,Y)

-

Following the same procedure as in Sec-

2

tion 4, we find that (4.6.1) has a periodic solution passing through

yl = 0, y2 = 1, if and only if =

11

where

-yh2P(b)6

=

h2111(b,6)

0(62),

+

An easy computation shows that as

2b - h-4.

h

0,

J0K = 2 J1,

h2P(b) = K + 0(h2), where r1

Ji = fl w 0

h, (4.6.1) has a periodic solution passing

Thus, for small through

yl = 0, y2 - 1

if and only if

In particular, when

0(h2161 + 62).

11

- -yK6 +

el - 0,

E2 > 0, B < 0,

(4.1.1) has a periodic solution passing through IaI(-BK)-1c2

x2 = For

<

c1

xl - 0,

+ 0(c3/2)

0, a similar analysis shows that (4.6.1) has

a periodic solution passing through

yl = 1, y2 - 0, if and

only if h2u2(h-1,6)

u

and that as

h

-yh2Q(h-1)6

=

=

0, h2Q(h-1)

-

+

(K//f)

+ 0(h2).

0(62),

BIFURCATIONS WITH TWO PARAMETERS

4.

80

Completion of the Phase Portraits

4.7.

Our analysis in the previous sections proves the existence of periodic and homoclinic orbits of (4.1.1) for certain regions in

(el,e2)

We now show how to complete the

space.

pictures.

Obtaining the complete phase portrait in the different regions involves many calculations.

However, the method is

the same in each case so we only give one representative y < 0, then the "outer" periodic

We prove that if

example.

in region 5 is stable.

orbit

We use the scaling given in

Section 3.

Since we are in region 5, ul(0,d) Fix

with

and

u

d

and let

b2 > bl(d).

solution

b2

u = ul(b,d)

be the solution of

Then we have to prove that the periodic

of (4.4.1) passing through

r

> u > ul(bl(d),d).

yl = 0, y2 = (2b2)1/2

is stable. Let

b3

be the solution of

with

u = ul(b,d)

b3 < bl(d).

Then there is a periodic solution passing through

yl = 0, y2 =

(2b3)1/2.

In fact, we prove that any solution

of (4.4.1) starting "outside" this periodic solution (and inside some bounded set) tends to Let

b > b3.

Let

r

c(b,+)

the orbit of (4.4.1) starting at hits

y2 =

0.

Similarly, c(b,-)

the orbit of (4.1.1) starting at integrated backwards until it hits

as

t - m.

be the value of yl = 0, y2 =

yl

(2b) 1/2

is the value of

yl

yl = 0, y2 = (2b) 1/2 y2 = 0.

-f(-x,c), the solution passing through

Using

when first

when is

f(x,c)

_

(0,(2b)1/2), spirals

inwards or outwards according to the sign of

c(b,+) + c(b,-).

Using the calculation in Lemma 2, c(b,+) + c(b,-)

has the

4.8.

Remarks and Exercises

81

H(u,d,b,+) - H(u,d,b,-)

same sign as

which in turn has the

same sign as

S = ul(b2,d)

Using the properties of positive if

b < b2

ul(b,b)

- ul(b,6).

given in Lemma 5,

and negative if

b > b2.

S

is

The result now

follows.

4.8.

Remarks and Exercises

Consider equation (4.1.1) under the hypothesis

Remark 1.

that the linearization is given by is defined by (4.1.2).

Make the change of variables

B(c) =

I

ra-1A(e)

x = B(c)x, where

a3f1(0,0) ax

x - x

a3f2(0,0)

3

3X3

will be one-to-one for

Using the fact that

A(c)

and

r =

The map

where

Suppose also that (4.2.1) and (Hl)

hold.

-

x = A(c)x

A(e)

and

B(c)

c

sufficiently small.

commute, it is easy to

show that the transformed equation satisfies all the above hypotheses and that in addition it enjoys (H3). lar, if the transformed equation is x = F(x,e) then a3F1(0,0)

ax

0

a3F2(0,0)

a3f2(0,0)

ax -3

ax

1

3 1

In particu-

BIFURCATIONS WITH TWO PARAMETERS

4.

82

ax

-

ax

a3

+

We have assumed that

Remark 2.

3a3f1(0,0)

a3f2(0,0)

a3F2(0,0)

f(x,e) = -f(-x,e).

If we

assume that this is true only for the low order terms then we would obtain similar results. get a homoclinic orbit with curve

L1

and

L1

we would

Similarly on another

x1 > 0.

we would get a homoclinic orbit with

Li

general

For example, on

x1 <

0.

would be different although they would

Li

have the same linear approximation

u = -(4/5)6y.

Suppose that we only assume (Hl) and (H3).

Remark 3.

In

Then

we can still obtain partial results about the local behavior of solutions.

For example, in Lemma 2 we did not use the

hypothesis

nonzero.

B

Hence, for each

a periodic solution through for some

e1

and

e2

with

b > 0, (4.1.1) has

x1 = 0, x2 =

IaI-1/2(2b)l/2c1 However, we cannot say

e1 > 0.

anything about the stability of the periodic orbits and we cannot say how many periodic orbits (4.1.1) has for fixed and

e1

62.

Exercises (1)

that for

Suppose that (61,62)

ing orbit.

(Hint:

a

and

B

are negative.

in regions 5 and 6,

Prove

(4.1.1) has a connect-

Use the calculations in Lemma 1 and

Lemma 6.) (2)

(61,62)

the point

Suppose that

is in region 3. (0,0)

a

Let

in (4.1.1).

and U

g

are negative and that

be the unstable manifold of

Prove that

gion of attraction of the periodic orbit.

U

is in the re-

4.9.

Quadratic Nonlinearities

Suppose that

(3)

83

is positive and

a

is negative.

8

Show that the bifurcation set and the corresponding phase portraits are as given in Figures 3 and S. 4.9.

Quadratic Nonlinearities

In this section we discuss the local behavior of soluMost

tions of (4.1.1) when the nonlinearities are quadratic.

of the material in this section can be found in [47]. Suppose that the linearization of (4.1.1) is where

x = A(e)x

is given by (4.1.2) and that

A(e)

2

fl(0,0)

f(O,e) = 0,

0,

1 2

2

a -

ax2

0,

f2(O,O)

8

ax1ax2 f2(O,O)

, 0.

1

8 > 0, a < 0; the results for the other

We also assume that

cases are obtained by making use of the change of variables,

t - -t, e2

-e2, x1 -. ±x1, x2 -. +x2,

-1.

u,6, scaled variables

Introduce parameters and a new time

by the relations

t

1/2

a-1 d

=

l

y1,y2

e1

x1

1

2

=

6

Y1,

x2 =

a 1/263 l

l

Y2,

tlal-1/26-1

e2 = 'all/26u,

We assume that the case

e1 >

0

t =

in what follows, see Exercise 8 for

e1 < 0.

After scaling, (4.1.1) becomes y1 = y2 + 0(62) Y2 ' Y1 + "Y2

(4.9.1) 2

-

y1 + 6YYly2 + 0(6) 2

BIFURCATIONS WITH TWO PARAMETERS

4.

84

where the dot means differentiation with respect to yi

lie in a bounded set and

y - a a,-1/2.

Note that by

making a change of variable we can assume that fixed point of (4.9.1) for

p,6

t, the

is a

(1,0)

sufficiently small.

The object of the following exercises is to show that the bifurcation set is given by Figure 6 and that the associated phase portraits are given by Figure 7. Exercises. (4)

Let

H(y1,Y2)

=

(Y2/2)

-

(y2/2)

Show

(y /3).

+

that along solutions of (4.9.1),

(5)

Prove that there is a function

0 < c < 1, such that if

u = u1(c,6)

is a periodic solution through homoclinic orbit if (6)

(c,0)

u = ul(c,6),

in (4.9.1) then there if

0

< c <

and a

1

c = 0.

Prove that

u1(c,6) - -yP(c)6 + 0(62)

where

J0(c)P(c) = J1(c), (c1

Ji(c) = Jc R(w) _

[w2

w1R(w)dw,

-

(2/3)w3 + 2b] 1/2,

6b - c2(2c-3), (7)

for

0

Prove that

< c < 1.

R(c1) - 0,

P(0) = (6/7), P(l) =

Deduce that for fixed

most one periodic solution.

c1 > c.

1

and

(To prove that

P'(c) > 0

>

0

u,6, (4.9.1) has at P'(c) > 0

use the same techniques as in the proof of Lemma 4. ternative method of proving

P'(c)

we can

An al-

is given in [47].)

4.9.

85

Quadratic Nonlinearities

If

(8)

el < 0, then after scaling (4.1.1) becomes

Yl ° Y2 + 0(62) 2

2

y2 = -yl + PY2 - yl + y6Y1Y2 + 0(6 ) Put

yl s z

-

Then

1, u = p - y6.

a y2 + 0(62)

y2 = z + PY2

-

z2 + 6yzy2 + 0(62)

which has the same form as equation (4.9.1) and hence transforms the results for

eI >

0

into results for the case

E1 < 0. (9)

Show that the bifurcation set and the correspond-

ing phase portraits are as given in Figures 6-7.

E1

Bifurcation Set for the Case Figure 6

a <

0,

8 > 0.

86

BIFURCATIONS WITH TWO PARAMETERS

4.

REGION 1

REGION 2

Phase Portraits for the Case

a < 0, (For the phase 8 > 0. portraits in regions 4-6 use the transformations in Exercise 8.)

Figure 7

4.9.

Quadratic Nonlinearities

87

ON

L

REGION 3

Figure 7 (cont.)

CHAPTER 5

APPLICATION TO A PANEL FLUTTER PROBLEM

5.1.

Introduction

In this chapter we apply the results of Chapter 4 to a particular two parameter problem.

The equations are

z = Ax + f(x)

(5.1.1)

where

x -

[xl,x2,x3,x4]T,

r

f(x) = [f 1(x),f2(x),f3(x),f4(x)]T,

0

1

0

0

al

b1

c

0

0

0

0

1

-c

0

a2

b2

1

A --

f1(x) = f3(x) = 0, f2(x) = xlg(x), f4(x) = 4x3g(x),

2g(x) = -n4(kx2 + axlx2 + 4kx2 + 4ax3x4), c = 8p, 3 bj

-

a

j

-[an4j4

-n2j2[n2j2 + r],

+ T 6]; as

a - 0.005,

6

= 0.1,

Reduction to a Second Order Equation

5.2.

k > 0, a >

0

are fixed and

p,r

89

The above

are parameters.

system results from a two mode approximation to a certain partial differential equation which describes the motion of a thin panel.

Holmes and Marsden [36,38] have studied the above equaBy numerical

tion and first we briefly describe their work. calculations, they find that for -2.237r2, the matrix

A

p = p0 = 108, r = r0 =

has two zero eigenvalues and two Then for

eigenvalues with negative real parts.

and

lp-p01

small, by centre manifold theory, the local behavior

Ir-r01

of solutions of (5.1.1) is determined by a second order equaThey then use some results

tion depending on two parameters.

of Takens [64] on generic models to conjecture that the local behavior of solutions of (5.1.1) for

lp-p01

and

Ir-r0l

small can be modelled by the equation

u + au + bu + u2u + u3 = for

and

a

b

0

small.

has been proved by Holmes

Recently, this conjecture [37], in the case

a = 0, by reducing the equation on the

centre manifold to Takens' normal form.

We use centre mani-

fold theory and the results of Chapter 4 to obtain a similar result.

5.2.

Reduction to a Second Order Equation The eigenvalues of

where the

di

A

are the roots of the equation

a4 + d1A3 + d2a2 + d3A + d4 =

0

are functions of

If

zero eigenvalue% then

d3 - d4 -

r

0.

and

p.

(5.2.1)

A

has two

A calculation shows that

90

APPLICATION TO A PANEL FLUTTER PROBLEM

S.

d3 - d4 - 0, then

if

a1a2 + c2

=

0

(5.2.2)

a1b2 + b1a2 - 0 or in terms of

and

r

p,

4n4(n2+r)(4n2+r) (16an4+6p1/2)(n2+r)

+

64

+

p2 =

(5.2.3)

0

4(an2+6p1/2)(4n2+r) - 0.

We prove that (5.2.3), (5.2.4) has a solution From (5.2.3) we can express

r - r0,

in terms of

p

(5.2.4)

p0'

p

Sub-

r.

stituting this relation into (5.2.4) we obtain an equation H(r) = 0.

Calculations show that

rl - -(2.225)r2 some

r2 = -(2.23),r2, so that

and

r0 E (r2,r1).

H(r1) < 0, H(r2) > 0 H(r0) =

where 0

for

Further calculations show that (5.2.3),

(5.2.4) has a solution

r0,p0

with

107.7 < p0 < 107.8.

In the subsequent analysis, we have to determine the sign of various functions of know

r0

and

r0

and

p0.

Since we do not

exactly we have to determine the sign of

p0

these functions for

r0

and

p0

in the above numerical

ranges.

When

r = r0, p = p0, the remaining eigenvalues of

are given by (b1+b2) t

[(b1-b2)2

4(a1+a2)]1/2

+

X3,4

and a calculation shows that they have negative real parts and non-zero imaginary parts.

We now find a basis for the appropriate eigenspaces when

r = r0, p - p0.

Solving

Av1 -

0

we find that

A

5.2.

Reduction to a Second Order Equation

91

vl - [1,0,-al/c,0]T.

A

The null space of

A2v2 - 0

is in fact one-dimensional so the can-

must contain a Jordan block.

A

onical form of

(5.2.5)

Solving

we obtain Av2 = v1.

v2 = [0,1,-b1/c,-a1/c]T,

The vectors

and

vi

A

eigenspace of

(5.2.6)

form a basis for the generalized

v2

corresponding to the zero eigenvalues.

Similarly, we find a (real) basis for the

by the eigenvectors corresponding to Az - A3z, we find that

space and

A3

is spanned by

V

2v3 - z + z,

V

A4.

and

v3

Let

p = p0.

A

3

S =

v4

where

(5.2.7)

- b1b2 + a2.

denote the matrix

A0

Let

2

Solving

2v4 = i(z-z),

z - [1,A3,w,A3w]T we - b

spanned

[vl,v2,v3,v4]

A

when

where the

by (5.2.5), (5.2.6) and (5.2.7) and set

I'

= r0

vi

and

are defined

y = S-lx.

Then

(5.1.1) can be written in the form

where

F(y,I',p)

and where

jy

= By + S-1(A-A0)SY + F(Y,r,P)

=

S-lf(Sy), 0

1

0

0

0

0

0

0

0

0

pl

p2

0

0

-p2

pl

aS - Pi + ip2, pl < 0, p2 t 0.

1

JI

(5.2.8)

92

APPLICATION TO A PANEL FLUTTER PROBLEM

S.

Then for

and

Ir-r0I

(5.2.8) has a centre manifold

y3 = hl(yl'y2'r'p)'

The flow on the centre manifold is gov-

h2(yl,y2,r,p).

y4 =

sufficiently small

Ip-p0I

erned by an equation of the form yl

]

=

0

[

1

0

Y2

][

Y1

E(r,P)

[

y2

0

Y1 y2

(5.2.9)

+ N(Y1,Y2,r,P) where

E(r,p)

is a

2

x

2

matrix with

E(r0,p0) - 0

contains no linear terms in

N(yl,y2,r,p)

yl

or

y2.

show that there is a nonsingular change of variables (c1,c2)

for

close to

(r,p)

change of variables

and a

(r0,p0)

r,p

and

We (r,p)

dependent

such that the lin-

(yl,y2) i (yl'y2)

earized equation corresponding to (5.2.9) is yl

0

1

Y2

E1

c2

The transformation

el = 0, c2 = 0.

and

(5.2.10)

y2

is of the form

(yl,y2) i (yl'y2)

Identity + O(Ir-r0I +Ip-POI) into

yl

r - rot

p

= p0

is mapped

After these transformations, (5.2.9)

takes the form

yl Y2

a

0

E1

1

c2

'l

+ N(y1,Y2,r0,P0)

Y2 (S. 2.11)

+ N(Y1,Y2,r,P)

where we have dropped the bars on the will contain no linear terms in N(yl'y2'r0,p0) = 0.

yl

yi.

or

y2

The function

N

and

Since the nonlinearities in (5.1.1) are

cubic, the same will be true of

N

and

A.

93

Calculation of Linear Terms

5.3.

Let

N(Y1,Y2,ro,PO) _ [N1(Y1,Y2,ro,PO), N2(Y1,Y2,r0,P0)]T (5.2.12) and let 3

al = -s N2(o,o,ro,PO) ayl 3

ay3 N1(o,o,ro,PO)

(5.2.13)

1

3

S =

N2(o,o,r0,PO)

a

aylay2

8 = 3r + S. Using the results in Chapter 4 (see, in particular, Remark 1 in Section 8), if

a1

and

0

are non-zero, we can deter-

mine the local behavior of solutions of (5.2.11).

By Theorem

2 of Chapter 2, this determines the local behavior of solutions of (5.2.8).

Calculation of the Linear Terms

5.3.

From (5.2.1), trace(A) = d3(r,p), det(A) = d4(r,p) where

d3(r,P) = 4n2(n2+r) (ant+r)

+

69 p2

d4(r,P) - n2(16an4+6p1/2)(n2+r)

+

4n2(air2+6P1/2)(4n2+r)

Calculations show that the mapping (r,p)

has non-zero Jacobian at Define the matrix

(d3(r,P),d4(r,P)) (r,p)

C(r,p)

= (ro,po). by

(5.3.1)

94

APPLICATION TO A PANEL FLUTTER PROBLEM

S.

C(r,p) = and let

+ E(r,p)

be the value of the Jacobian of the mapping

J

(r,p) -. (trace(C(r,p)), det(C(r,p))), evaluated at

By considering the

(r0,p0).

B + S-1(A-AO)S, it is easily seen that

J

4

x 4

(5.3.2) matrix

is a non-zero

multiple of the Jacobian of the mapping given by (5.3.1). Hence

is non-zero, so by the implicit function theorem we

J

E1 = -det(C(r,p)), E2 ' trace(C(r,p))

can use

cation parameters.

Approximate formulae for

as our bifurE1

and

E2

can easily be found if so desired. C(r,p) - [cij],

Let

M =

[:11

yl

-M

Y2

:12] Y1

Y2

J

Then the linearized equation corresponding to (5.2.9) is

Note that e1 = E2 =

M 0.

yl

0

1

y2

E1

E2

yl Y2

is equal to the identity matrix when

5.4.

Calculation of the Nonlinear Terms

5.4.

Calculation of the Nonlinear Terms

95

We now calculate the nonlinear term in (5.2.9) when r - rot p - p0.

Since the nonlinearities in (5.1.1) are

cubic, the centre manifold has a "cubic zero" at the origin. Using

x - Sy, on the centre manifold xl - Yl + 03,

(5.4.1)

bl

-a

yl - E Y2 + 03,

x3 where

y2 + 03

x2

x4

al

c Y2 + 03

03 - 0(IY1I3 + IY2I3). S-1

Let

-

[t..]

and let

Fj(Y1,Y2) - fj(Y1,Y2,hl(Y1,Y2,r0,PO)). Then using the notation introduced in (5.2.12),

N1(Y1,Y2,r0,PO) ' t12F2(Y1,Y2) + t14F4(Y1,Y2) (5.4.2)

N2(Y1,Y2,r0,PO) - t22F2(Y1,Y2) + t24F4(Y1,Y2) Using (5.4.1), F2(Y1,Y2) ' -

n4

al 8ka1b1 2 2 3 2 3 [kyl + 4k(c) yl + ayly2 + yly2 +

c 4aal

2

Y1Y2]

+ terms in -4c 1

F4(Yl,y2) =

4 F2(Yl,y2) + 2bcn rkyiy2

+ terms in where and

2

yly2 + 05

05 - 0(Iyl1S +

yly2 IY21

and ).

+ 4k( c1)2 YlY21

y3 + 05

Note also that since

p - pot (5.2.2) holds.

From (5.2.13) and (5.4.2),

r - r0

96

APPLICATION TO A PANEL FLUTTER PROBLEM

5.

a,

-T- k(l + 4 (c) 2(t22 - 74

al

a

4

r

_

- 4 ac a 4

k(1 + 4(c)2)(t12

(a +

-

2

4aa1

8kalbl

_n4

c t14)

al

)(t22 -

+

t24)

c

4

2ir4

2 t24) + -3b1kt24(c 2+4a1).

Routine calculations show that t12 = m2b2(2a1-b1b2) t22 = mat

t14 = m2c(bl+b2) t24 = -mc m =

(al+a2-blb2)-1

< 0.

Using (5.2.2) and numerical calculations, we find that a

so that of

k

al

t22

-

4

t12

-

4

and

in

r

c t24 = m(4a1+a2) >

ac

t14 a

are negative.

0

b1m2(2a2-bZ-4a1)

>

0

Similarly, the coefficient

is

7 a1b1m(12a1+a2) < 0. c

Hence

a1

and

a

are negative and the local behavior of solu-

tions of (5.1.1) can be determined using the results of Chapter 4.

CHAPTER 6 INFINITE DIMENSIONAL PROBLEMS

6.1.

Introduction

In this chapter we extend centre manifold theory to a class of infinite dimensional problems.

For simplicity we

only consider equations of the form

w = Cw + N(w), where

Z

is a Banach space, C

continuous semigroup on

Z

w(O) E Z is the generator of a strongly

and

N:

Z - Z

is smooth.

In the

next section we give a brief account of semigroup theory.

For additional material on semigroup theory see [6,43,50,55, 561.

For generalizations to other evolution systems see

[34,51).

6.2.

Semigroup Theory

In earlier chapters we studied centre manifold theory for the finite dimensional system w - Cw + N(w).

(6.2.1)

The most important tools for carrying out this program were:

97

98

INFINITE DIMENSIONAL PROBLEMS

6.

The solution of the linear problem

(1)

of

exp(PCt), exp((I-P)Ct), where

= Cw

and estimates

is the projection

P

onto the space associated with eigenvalues of

with

C

zero real parts.

The variation of constants formula

(2)

rt

w(t) = exp(Ct)w(O) +

exp(C(t-s))N(w(s))ds J

0

for the solution of (6.2.1) and Gronwall's inequality.

To carry out this program for partial differential equations we have to study ordinary differential equations in We first study linear problems.

infinite dimensional spaces. Let

be a Banach space and

Z

from some domain

of

D(C)

into

Z

a linear operator

C

We wish to solve the

Z.

problem w = Cw,

t > 0,

(6.2.2)

w(0) = w0 E Z.

Suppose that for each

w0 E Z

unique solution

w(t).

term solution).

The solution

w0

and we write

mapping from

into

solution of (6.2.2)

w(t+s) = T(t+s)w0 Hence

(We define later the meaning of the

-

Z

that

with

T(t)

T(t)w0 + w0

is a function of

Then

T(0) = I.

as

and

If

w(t)

is a

s > 0, w(t)

w(0) - w(s).

If solutions depend continu-

T(t)wn

T(t)w

must be bounded. t - 0+.

t

is a linear

T(t)

is a solution of (6.2.2) with

T(t+s)w0 = T(t)T(s)w0.

Z, so that

w(t)

(6.2.3), then for any

ously on initial data then in

the above equation has a

w(t) = T(t)w0.

Z

(6.2.3)

whenever

wn - w

Finally, we require

6.2.

Semigroup Theory

99

A one parameter family

Definition.

bounded linear operators from

T(t), 0 < t <

into

Z

of

is a (strongly

Z

continuous) semigroup if T(O) = I,

(i)

T(t+s) = T(t)T(s), t,s > 0

(ii)

IIT(t)w - wII- 0

(iii)

Example 1.

Let

tors from

t - 0

as

for every

w E Z.

denote the set of bounded linear opera-

L(Z)

into

Z

(semigroup property),

and let

Z

C E L(Z).

Define

T(t)

by

0

and it

n

T(t) = eCt

L

Cn.

n=0

The right hand side converges in norm for each

t >

is easy to verify conditions (i), (ii), (iii).

Thus

eCt

defines a semigroup. Example 2.

Let

be a Banach space of uniformly continuous

Z

bounded functions on

with the supremum norm.

[0,m)

(T(t)f)(8) = f(O+t),

f E Z,

8

> 0,

t >

Define

0.

Conditions (i) and (ii) are obviously satisfied and since IIT(t)f - f11 = sup{If(0+t)

(iii) is satisfied. Definition. T(t)

Hence

f(8)I:

-

T(t)

8

> 0} - 0

as

t - 0+,

forms a semigroup.

The infinitesimal generator

C

of the semigroup

is defined by Cz = lim+

T(t)z -

t+0

whenever the limit exists. of all elements also say that

z E Z C

z

t

The domain of

C, D(C), is the set

for which the above limit exists.

generates

T(t).

We

100

INFINITE DIMENSIONAL PROBLEMS

6.

In Example 1 the infinitesimal generator is

C E L(Z)

while in Example 2 the infinitesimal generator is

D(C) - {f: f' E Z}.

(Cf) (e) = f' (e) , Exercise 1.

If

C E L(Z)

Let

Z - R2

prove that

IleCt

-

111- 0

as

t - 0. Exercise 2.

(e-ntan),

T(t)(an) =

Prove that generator

II T (t) - III

Exercise 3.

by

Z

with infinitesimal

given by C(an) = (-nan),

Does

Z - Z

(an) E Z, t > 0.

is a semigroup on

T(t)

T(t):

and define

t - 0+?

as

0

-

D(A) - {(an): (nan) E Z).

Prove that if

is a semigroup then

T(t)

IIT(t)II < Mewt,

w > 0, M > 1.

for some constants

(6.2.4)

t > 0,

(Hint:

Use the uniform

boundedness theorem and property (iii) to show that is bounded on some interval

[O,e].

IIT(t)11

Then use the semigroup

property.) Exercise 4. Il eCt Il < M Exercise S.

Find a

2

for all t > Let

which satisfies

C

x

2

0

matrix

such that if

then M > 1.

be the generator of a semigroup

IIT(t)II < Mewt, t > 0.

is the generator of the semigroup 11 S(t) 11 < M

C

for all t > 0.

Prove that

S(t) - e-WtT(t)

T(t)

C - mI

and that

Semigroup Theory

6.2.

Exercise 6.

If

w E Z, T(t)w

is a semigroup, prove that for each

T(t)

is a continuous function from T(t)

Let

101

into

[0,m)

be a semigroup with generator

Z.

Now

C.

h-1[T(t+h)w - T(t)w] = h-1[T(h) - I]T(t)w (6.2.5) -

If

w E D(C)

T(t)Cw

then the right side of (6.2.5) converges to h - 0+.

as

h - 0+

verges as

T(t)h-1[T(h)w - w].

Thus the middle term in (6.2.5) conand so

T(t)

maps

D(C)

into

D(C).

Also

the right derivative satisfies dT t w = CT(t)w - T(t)Cw,

t > 0, w E D(C).

t

(6.2.6)

Similarly, the identity h-1[T(t)w - T(t=h)]w - T(t-h)h-1[T(h)w

-

w]

shows that (6.2.6) holds for the two-sided derivative. Exercise 7.

For

w E Z, prove that

w(t) E D(C)

where

t

T(s)w ds,

w(t) = J0

and that

t-1w(t) - w

dense in

Z.

Exercise 8.

If

as

t + 0+.

Deduce that

D(C)

is

wn E D(C), then from (6.2.6) t

T(s)Cwnds.

T(t)wn - wn = J0

Use this identity to prove that

C

is closed.

Equation (6.2.6) shows that if the generator of a semigroup

w0 E D(C)

T(t), then

solution of the Cauchy problem (6.2.2)

-

and

w(t) = T(t)w0 (6.2.3).

C

is

is a

In applica-

102

INFIMITE DIMENSIONAL PROBLEMS

6.

tions to partial differential equations, it is important to know if a given operator is the generator of a semigroup. see the problems involved, suppose that T(t), that is

traction semigroup

To

generates the con-

C

IIT(t)II <

1

for all

t > 0.

The Laplace transform

J e-AtT(t)w dt,

R(a)w =

(6.2.7)

0

exists for

Re(A) > 0

and

is in some sense

T(t)

exp(Ct), we expect

C, that is, (XI

resolvent of

for

1

IIR(A)II< A

verse Laplace transform, that is, given -

C)-1

semigroup?

to be the

R(A)

The main problem is in finding the in-

and is easy to prove.

(XI

exists for

A >

0,

say, is

C C

such that

the generator of a

The basic result is the Hille-Yosida Theorem.

A necessary and sufficient

Theorem 1 (Hille-Yosida Theorem).

condition for a closed linear operator

C

with dense domain

to generate a semigroup of contractions is that each is in the resolvent set of

I

Since

This is indeed the case

C)-1.

-

A > 0.

IR(A)

=

I I

I

I

C

( A I - C ) - 1 11

A > 0

and that

for


(6.2.8)

A > 0.

The reader is led through the proof of the above theorem in the following exercises. Exercise 9. A > 0

define

Let

T(t)

R(A)

be a contraction semigroup and for

as in (6.2.7) for

h-1(T(h)-I)R(A)w = h-'[(e

Ah -

w E Z.

Use the identity

1)j e-AtT(t)wdt 0

h -

eah

e-AtT(t)w dt]

1

0

to prove that

R(A)

maps

Z

into

D(C)

and that

6.2.

Semigroup Theory

103

- C)R(A) = I.

(XI

Prove also that for that

Suppose that

is a closed linear operator on

C

C(X) E L(Z)

is in the resolvent

(0,m)

with (6.2.8) satisfied.

C

and deduce

C.

with dense domain such that

set of

- C)w = w

R(X) (XI

is the resolvent of

R(X)

Exercise 10. Z

w E D(C) ,

define

A > 0

For

by C)-1

C(X) = XC(XI

-

= X2(XI

C)-1

-

-

XI.

Prove that

C(a)w - Cw

(i)

(ii)

as

Il exp (tC(X)) II < 1

w E D(C),

for

A

and

Ilexp(tC(X))w - exp(tC(u))wII < tlIC(X)w

for t > 0,

X,u >

and

0

- C(u)wII

w E Z.

Use the above results to define

by

T(t)

T(t)w = lim exp(C(A)t)w aim

and verify that with generator Exercise 11.

is a semigroup of contractions on

T(t) C.

Let

Z

=

R2

complex numbers such that Define

Z

and let

{an}

be a sequence of

a = sup{Re(an): n = 1,2,...} <

by

C

C(an) = (an an ),

D(C) = {(an): (an an) E Z}.

generates

Use the Hille-Yosida Theorem to prove that

C

a contraction semigroup and deduce that

generates a semi-

group

T(t)

with

IIT(t)II < eat

for

C

t > 0.

-

aI

104

INFINITE DIMENSIONAL PROBLEMS

6.

The above version of the Hille-Yosida Theorem char-

Remark.

acterizes the generators of semigroups which satisfy eWt

(see Exercise 5).

IIT(t)II <

A similar argument gives the

characterization of the generators of all semigroups. Remark.

Let

be a closed operator with dense domain such

C

that all real nonzero II(AI

-

C)_111.'

of a group

Example 3. space

H

T(t)z - z

and Let

with

Then

A.

C

and

is the generator

C

t EIR, that is, T(0) = I, T(t+s) = T(t)T(s)

T(t) ,

all t,s E IR

for such

1

IAI

are in the resolvent set of

A

t + 0 for each z E Z.

as

be a selfadjoint operator on a Hilbert

C

C <

0 .

Then

11 (XI

so by the Hille-Yosida Theorem, C

-

C)

1 1

1

<

A

1

for

A > 0,

generates a contraction

semigroup.

Example 4.

Let

A = iC

on a Hilbert space I

I (AI -A)_111 < Let

semigroup

H.

where

Then for all nonzero real

so that

IAI

f E C([0,T];Z) T(t)

on

Z.

is a selfadjoint operator

C

A

A,

generates a unitary group.

and let

C

be the generator of a

Then formally the solution of

w = Cw + f(t),

0 < t < T,

w(0) = w0,

(6.2.9)

is given by the variation of constants formula t

w(t) = T(t)w0 +

T(t-s)f(s)ds.

I

(6.2.10)

0

If

f

is continuously differentiable and

is easy to check that

w(t)

w0 E D(C)

defined by (6.2.10) satisfies

(i)

w(t)

is continuous for

(ii)

w(t)

is continuously differentiable for

0 < t < T,

then it

0 < t < T,

Semigroup Theory

6.2.

105

w(t) E D(C)

(iii)

for

0

<

t < T

and (6.2.9) is sat-

isfied.

A function

which satisfies (i), (ii), (iii) is

w(t)

said to be a strong solution of (6.2.9).

It is easy to check

that such a solution is unique in this class. or

f

is only continuous, then in general

(6.2.10) is not in

w0 t D(C)

If

w(t)

defined by

so (6.2.9) does not make sense.

D(C)

We

now define what we mean by a solution in this case. A function

Definition.

<w(t),v>

is a weak solution

and if for every

w(0) = w0

of (6.2.9) if function

w E C([0,T];Z)

v E D(C*), the

is absolutely continuous on

is the adjoint of

C*

ing between

and

C

<

,

and

a.e.

dt<w(t),v> = <w(t),C*v> + where

[0,T]

>

denotes the pair-

and its dual space.

Z

The proof of the following theorem can be found in 191. The unique weak solution of (6.2.9) is given by

Theorem 2. (6.2.10).

A semigroup

if in addition the map

tic semigroup > 0.

-

T(t)z

In particular, if

properties.

t

t

T(t), then

Thus if

is real analytic on

(0,W)

Analytic semigroups have many additional

z E Z.

for each

is said to be an analytic semigroup

T(t)

f

T(t)

C

is the generator of an analy maps

Z

into

D(C)

for each

is continuously differentiable, then

w(t)

defined by (6.2.10) is a strong solution of (6.2.9) for each w0 E Z

if

T(t)

is an analytic semigroup.

An example of a

generator of an analytic semigroup is a nonpositive selfadjoint operator and in general, solutions of parabolic equations

106

INFINITE DIMENSIONAL PROBLEMS

6.

are usually associated with analytic semigroups.

Hyperbolic

problems, however, do not in general give rise to analytic To see this, suppose that

semigroups.

of a group

and that

T(t)

maps

T(t)

C

is the generator

Z

into

D(C)

By the closed graph theorem, CT(l) E L(Z)

t > 0.

for

and hence

is a bounded linear operator.

C = CT(1)T(-l)

In applications, it is sometimes difficult to prove directly that erator and

C + A

A

is a generator, although

C

is in some sense small relative to

is a genC.

The

simplest such result is the following. Theorem 3. T(t)

and let U(t).

(group)

IIU(t) II

Let

<

be the generator of a semigroup (group)

C

Then

A E L(Z). If

C + A for

IIT(t)II < McWt

Me(-+MIIAII)t

t >

then

0

t > 0.

for

Outline of the Proof of Theorem 3: tor of

generates a semigroup

If

A + C

is the genera-

U(t), then by the variation of constants formula, U(t)

must be the solution of the functional equation t

T(s)AU(t-s)ds.

U(t) - T(t) + J

(6.2.11)

0

Equation (6.2.11) is solved by the following scheme: U(t) -

F

(6.2.12)

Un(t)

n-0

where

U0(t) - T(t)

and rt

T(s)AUn(t-s)ds.

Un+1(t) 0

A straightforward argument shows that (6.2.12) is a semigroup with generator Exercise 12.

U(t)

defined by

A + C.

Prove Theorem 3 in the case

M - 1, w -

0

in

6.2.

Semigroup Theory

the following way:

107

B = A + C

let

operator with dense domain - C)_111

A > IIAII, IIA(AI

exists for (A -

A > IIAII

for

IIAII)-1

<1

so that

so that

(I - A(AI

Hence show that

A > IIAII

Show that for

- D(C).

D(B)

is a closed

B

C)-1)-1

-

II (AI - B)_111

<

and use the Hille-Yosida Theorem.

Example 5 (Abstract Wave Equation).

A

Let

selfadjoint operator on a Hilbert space

be a positive

and let

H

B E L(H).

Consider the equation v + B4 + Av - 0.

Define a new Hilbert space duct

<

Z = D(A1/2) x H

with inner pro-

defined by

>

,

(6.2.13)

<w,w> - (A1/2w1,A1"2w1) + (w2,w2) where

(

,

)

w = [wl,w21.

is the inner product in

and

H

w =

[wl,w21,

Equation (6.2.13) can now be written as a first

order system on

Z,

w = Cw

(6.2.14)

where

C -

D(C) = D(A) x D(A1/2).

If

B -

0

then

C

joint so by Example 4 it generates a group.

is skew-selfadFor

B # 0, C

is the sum of a generator of a group and a bounded operator so by Theorem 3 it generates a group.

As a concrete example of the above, consider the wave equation

is

108

INFINITE DIMENSIONAL PROBLEMS

6.

where

v(x,t) = 0,

x E SI,

vtt + vt - Ov = 0,

is a bounded open subset of

D

with boundary

IRn

This equation has the form (6.2.13) if we set

D(A) _ {w E H: -Ow E H, w = this case

H

T(t)

L2(SI),

I.

In

1

the Sobolev space Let

BSI.

(see [2] for a definition of

x L2(SI)

= H01 (0)

Z

H

801, A = -0, B

on

0

x E BSI,

H0 (S2)).

be a contraction semigroup on Hilbert space

with generator

Then for

C.

s > 0,

IjT(s+t)zII _ IIT(t)T(s)zjl < IjT(t)zjj so that

IIT(t)zII

11T(t) zII 2

is nonincreasing.

is differentiable in t

For

E D(C), v(t)

z

and

v1(t) _ + . Since

v'(0) < 0, we have that + < 0,

A linear operator

C

z

E D(C).

(6.2.15)

with dense domain is said to be dissi-

pative if (6.2.15) holds. Exercise 13.

If

C

is dissipative and is the generator of a

semigroup, prove that the semigroup is a contraction. Theorem 4 (Lumer-Phillips Theorem). operator on the Hilbert space a > 0

the range of

BI

C

-

be a dissipative

C

Suppose that for some

H. is

Let

H.

Then

is the generator

C

of a contraction semigroup. Exercise 14.

(6.2.15) that

Prove the above theorem.

II (XI-C) z II

>

Ali z it

for

deduce from

(Hint:

A>

Use this and the fact that the range of

and

0 BI

-

C

z E D(C). is

II

to

6.2.

Semigroup Theory

deduce that

109

Show that the non-empty set

is closed.

C

(X > 0: range of

XI

-

is open and closed and use

H)

is

C

the Hille-Yosida Theorem.)

There is also a Banach space version of the above

Remark.

result which uses the duality map in place of the inner product.

Let

Exercise 15.

be the generator of a contraction semi-

C

group on a Hilbert space and let D(B)

and

D(C)

IIBwll < alUCwll + bllwll for some

be dissipative with

B

with

a

Use the inequality

0 < 2a < 1.

to show that

(2a + bX 1)Ilwll

IIB(AI - C) lwll <

invertible for large enough

w E D(C)

,

AI

Deduce that

A.

-

B

B + C

-

C

is

generates

a contraction semigroup. Exercise 16.

bert space

Let

be a closed linear operator on a Hil-

C

and suppose that

H

Prove (i) the range of (ii) -

0

C

-

I

-

C

is

H

De-

C

have negative M > 1

The corresponding question in infinite dimen-

w > 0. if

Re(a(C)) < 0 and

= 0.

H.

Ilexp(Ct)II < Met, t > 0, for some

real parts then

w > 0

z

so that by Theorem 4,

If all the eigenvalues of a matrix

sions is:

H,

implies

generates a contraction semigroup on

and

are dissipative.

C*

is closed in h E D(C)

for all

duce that the range of C

I

and

C

C

is the generator of a semigroup

imply that

M > 1?

(a(C)

general, the answer is no.

T(t), does

IIT(t)II < Me-wt, t > 0, for some

denotes the spectrum of

Q.

In

Indeed, it can be shown that if

110

INFINITE DIMENSIONAL PROBLEMS

6.

a < b

then there is a semigroup

such that

sup Re(a(C)) - a

on a Hilbert space

T(t)

IIT(t)II =

and

ebt

[69].

Thus to

obtain results on the asymptotic behavior of the semigroup in terms of the spectrum of the generator, we need additional hypotheses. W t If

e

is the spectral radius of

0

- lim

eWOt

IIT(t)kIIl/k,

then a standard argument shows that for M(w)

such that

IIT(t)II < M(w)eWt.

the asymptotic behavior of of

T(t).

m > m0

there exists

Hence we could determine from the spectral radius

T(t)

However the spectrum of

the spectrum of

T(t), that is

T(t)

is not faithful to

C, that is, in general the mapping relation a(T(t)) - exp(ta(C))

is false.

(6.2.16)

While (6.2.16) is true for the point and residual

spectrum (55]

(with the possibility that the point

0

must

be added to the right hand side of (6.2.16)) in general we only have

continuous spectrum of T(t) Example 6. {(xn) E Z: T(t)

Let

Z -

R2

(nxn) E Z}.

given by

and

Then

T(t)(xn) _

{in: n - 1,2,...,}

exp(t(continuous spectrum of C)). C(xn) - (inxn), D(C) = C

generates the semigroup

(eintxn).

The spectrum of

while the spectrum of

T(l)

C

is

is the unit

circle so (6.2.16) is false.

For special semigroups, e.g., analytic semigroups, (6.2.16) is true (the point zero must be added to the right

hand side of (6.2.16) when the generator is unbounded), but

6.2.

Semigroup Theory

111

this is not applicable to hyperbolic problems.

For the prob-

lem studied in this chapter, the spectrum of the generator consists of eigenvalues and the associated eigenfunctions Thus the asymptotic be-

form a complete orthonormal set.

havior of the semigroup could be determined by direct compuIt seems worthwhile however to give a more abstract

tation.

approach which will apply to more general problems. Let

be the generator of a semigroup

C

Hence the spectrum of

IIT(t)II< McWt. IaI

< exp(ort).

group

U(t).

values of eXt

A E L(Z).

Let

Suppose that there are only

A + C

Q - {A: Re(A) > w}.

in

is an eigenvalue of

consists only of points

be compact.

A E L(Z) U(t)

such that

Proof:

with

{A:

Let

C

IIT(t)II < McWt

A + C

Then

Then if

A E Q,

A

is comemt}

IXI

>

A E Q.

e?t,

T(t)

isolated eigen-

in the set

U(t)

Theorem 5 (Vidav [67], Shizuta [62]). tor of a semigroup

generates a semi-

We prove that if

U(t).

pact then the spectrum,of

lies in the disc

T(t)

A + C

Then

with

T(t)

be the generaand let

generates a semigroup

is compact.

U(t) - T(t)

From Theorem 3, A + C

generates a semigroup

U(t)

which is a solution of the functional equation rt

U(t) - T(t) +

T(s)AU(t-s)ds. I

We prove that the map in norm on in norm.

[O,t].

Since

s

-

(6.2.17)

0

f(s)

= T(s)AU(t-s)

is continuous

Thus the integral in (6.2.17) converges

f(s)

is a compact linear operator and the

set of compact operators in

L(Z)

operator topology, U(t) - T(t)

is closed in the uniform

is compact.

112

INFINITE DIMENSIONAL PROBLEMS

6.

To prove the claim on the continuity of prove that

with

h1 > 0

and

t > s > 0

is continuous in norm.

-r AU(t-s)

s

f

h1

we first Let

sufficiently small.

Then

the set

{A(U(t-s-h) - U(t-s))w: IIwII = 1,

has compact closure.

For

e

>

0

< hl}

it can be covered with a

finite number of balls with radius

wl,w2,...,wn.

IhI

and centres at

a

g(s) = AU(t-s). Then for IIwII = 1,

Let

Ilg(s+h)w - g(s)wII < IIA(U(t-s-h) - U(t-s)) (w-wk) II - U(t-s))wkll

+ IIA(U(t-s-h)

Hence if

is chosen such that

6

IIA(U(t-s-h) - U(t-s))wkll < for

IhI

<

(constant)

k = 1,2,...,n

and

6

The continuity of

E.

E

then f

IIg(s+h)

-

g(s)II <

follows from similar

reasoning applied to f(s+h)

-

f(s) - T(s+h)[g(s+h)-g(s)] + (T(s+h)-T(s))AU(t-s).

This completes the proof of the theorem. Since of

U(t)

- T(t)

U(t)

and

T(t)

is compact, the essential spectrum

coincide [43].

(It is important to note

that we are using Kato's definition of essential spectrum in this claim.)

Hence we have proved:

Corollary to Theorem S. the circle

IxI

=

emt

The spectrum of

U(t)

lying outside

consists of isolated eigenvalues with

finite algebraic multiplicity.

Semigroup Theory

6.2.

113

The above result is also useful for decomposing the space

Suppose that the assumptions of Theorem S hold.

Z.

Since there are only isolated points in the spectrum of outside the circle

IXI

= eWt, there is only a finite number

of points in the spectrum of Re(X) > m +

for each

e

such that

M(6)

>

e

in the halfplane A1,...''n

Let

and let

0

Then we can decompose

PU(t) + (I-P)U(t)

sum

A + C

e > 0.

eigenvalues for some fixed ponding projection.

U(t)

denote these be the corres-

P

U(t)

into the

e

there exists

ii(I-P)U(t)ii < M(6)eN+6)t

Finally, PZ

and for any

6

>

is

finite dimensional and so it is trivial to make further decompositions of

PU(t).

We now show how the above theory can be applied to hyperbolic problems.

Let

product

A

on

H

pact.

(

,

).

Let

be a positive selfadjoint operator is defined on all of

A-1

such that

be a Hilbert space with inner

H

H

and is com-

Consider the equation

v + 2a' + (A+B)v = 0 where

B E L(H)

D(A1/2) x H S.

and

Let

a > 0.

with inner product

be the Hilbert space

Z <

,

>

as defined in Example

We can recast the above equation in the form

w = Cw I

r0

C =

A-B

As in Example 5, C

-2aI

].

is a bounded perturbation of a skew-

selfadjoint operator and so

C

generates a group

S(t).

Let

114

INFINITE DIMENSIONAL PROBLEMS

6.

I

01

aI

IJ

1

U(t) Then

U(t)

S(t)

I

I

0

aI

I

is a group with generator

I

aI L -A

r

0

0

a2I-B

-aI

0

Cl + C2 D(A1/2)

and the compactness of the injection that

C2:

Since

C1

Z + Z

implies

Thus the above theory applies.

is compact.

generates a group

the asymptotic behavior of

-r H

with

U1(t) U(t)

HU1(t)II < e-at,

S(t)) depends

(and hence

on a finite number of eigenvalues.

The above theory applies, for example, to the case H = L2(0), where

is a bounded domain in

fl

D(A) _ {v E H: Av E H elliptic theory, A-1 Example 7.

v - 0

and

on

9c}

IRn, A . -D,

since by standard

is compact.

Consider the coupled set of wave equations Tr

g(x,s)u(s,t)ds

utt + taut - uxx + of

-

By = 0 (6.2.18)

0

vtt + 2avt - vxx .

for 0 < x <

Tr

0

with u = v = 0 at

g(x,s) =

E

n

2

x = 0,Tr, where

sin nx sin ns.

n=1

As in Example 5 we can write (6.2.18) as a first order system

w = Cw

is a bounded perturbation of a skew-selfadjoint operator it generates a on

Z = (H0(0,7r)

x

L2(O,Tr))2.

Since

C

6.2.

group

Semigroup Theory

115

T(t).

An easy computation shows that the eigenvalues of are

-

an = -a ± (a2

an = n2 + (Bn)/(2n2).

a n)

1/2,

For

n = 1,2,..., where a >

and

0

B

C

an = n2

or

small enough the

real parts of all the eigenvalues are negative so we expect that

This formal

w > 0.

IIT(t)II < Me--t, t > 0, for some

argument can be rigorized by applying the above theory. We now suppose that

In this case, for

a = 0.

sufficiently small all the eigenvalues of

are purely ima-

C

By analogy with the finite di-

ginary and they are simple.

mensional situation we expect that some constant

B

> 0, for

IIT(t)II < M, t

We show that this is false.

M.

Consider the following solution of (6.2.18): u(x,t) = 2n-1 m(cos mt - cos amt)sin mx (6.2.19)

v(x,t) = m-l sin mx cos mt

where

m

is an integer and

am

=

m2

+

(Bn)/(2m2).

Note that

the initial data corresponding to (6.2.19) is bounded independently of

in.

Let

tm

2m3n.

Then for large

m,

2

cos mtm - cos amtm = 1 so that

-

cos(n26) + 0(m

Iux(x,tm)I > (constant)m2.

Thus

4)

> constant,

IIT(tm)II

(constant)m2.

The above instability mechanism is associated with the fact that for

a = 0, B # 0, the eigenfunctions of

form a Riesz Basis.

do not

For further examples of the relationship

between the asymptotic behavior of solutions of the spectrum of

C

C see [15,16].

We now consider the nonlinear problem

w = Cw

and

116

INFINITE DIMENSIONAL PROBLEMS

6.

w = Cw + N(w), where

w(0) = w0 E Z,

is the generator of a semigroup

C

N: Z - Z.

(6.2.20)

T(t)

on

Z

and

satisfies the variation of constants

Formally, w

formula rt

w(t) = T(t)w0 +

T(t-s)N(w(s))ds. J

A function

Definition.

w E C([O,T];Z)

of (6.2.20) on

[0,T]

and if for each

v E D(C*)

solutely continuous on

is a weak solution

w(0) = w0;

if

E L1([O,T];Z)

the function

<w(t),v>

is the adjoint of

C*

ing between

Z

is ab-

and satisfies

[0,T]

Ut <w(t),v> = <w(t),C*v> + where

(6.2.21)

to

C

and

<

,

>

a.e.

denotes the pair-

and its dual space.

As in the linear case, weak solutions of (6.2.20) are given by (6.2.21). Theorem 6

[9].

of (6.2.20) on and

w

More precisely:

A function

w:

[0,T]

-.

Z

is a weak solution E L1([O,T];Z)

if and only if

[O,T]

is given by (6.2.21).

As in the finite dimensional case, it is easy to solve (6.2.20) using Picard iteration techniques. Theorem 7

Let

[61].

N:

Z

-.

be locally Lipschitz.

Z

Then

there exists a unique maximally defined weak solution w E C([O,T);Z)

of (6.2.20).

I

Iw(t)

I I

--

Furthermore, if

as

t-T

.

T < W

then

(6.2.22)

As in the finite dimensional case, (6.2.22) is used as a continuation technique.

Thus if for some

w0 E Z, the

6.3.

Centre Manifolds

solution

w(t)

of (6.2.20) remains in a bounded set then

w(t)

exists for all

6.3.

Centre Manifolds Let

Z

117

t > 0.

We consider

be a Banach space with norm

ordinary differential equations of the form w = Cw + N(w),

where

C

semigroup

w(0) E Z,

(6.3.1)

is the generator of a strongly continuous linear and

S(t)

N:

second derivative with

Z

-

has a uniformly continuous

Z

N(0) = 0, N'(0) =

Frechet derivative of

is the

[N'

0

N].

We recall from the previous section that there is a unique weak solution of (6.3.1) defined on some maximal interval

[0,T)

T < W

and that if

then (6.2.22) holds.

As in the finite dimensional case we make some spectral assumptions about (i)

Z = X ® Y

C.

We assume from now on that:

where

is finite dimensional and

X

Y

is closed. (ii)

X

is C-invariant and that if

tion of

to

C

eigenvalues of (iii)

If Y

U(t)

X, then the real parts of the A

are all zero.

is the restriction of

ae-bt,

B - (I-P)C

S(t)

to

Y, then

a,b,

IIU(t)1i< P

is the restric-

U(t)-invariant and for some positive con-

is

stants

Let

A

be the projection on

and for

t > 0. X

x E X, y E Y, let

along

(6.3.2)

Y.

Let

118

INFINITE DIMENSIONAL PROBLEMS

6.

f(x,y) = PN(x+y),

(6.3.3)

g(x,Y) ' (I-P)N(x+Y).

Equation (6.3.1) can be written z

Ax + f(x,y)

Y

By + g(x,y).

(6.3.4)

An invariant manifold for (6.3.4) which is tangent to X

space at the origin is called a centre manifold.

Theorem 8. y = h(x),

There exists a centre manifold for (6.3.4), lxi

< 6, where

h

is

C2.

The proof of Theorem 8 is exactly the same as the proof given in Chapter 2 for the corresponding finite dimensional problem.

The equation on the centre manifold is given by u - Au + f(u,h(u)). In general if

y(O)

is not in the domain of

and consequently Theorem 9.

B

then

y(t)

However, on the centre manifold

will not be differentiable.

y(t) - h(x(t)), and since

(6.3.5)

X

is finite dimensional

x(t),

y(t), are differentiable.

(a) Suppose that the zero solution of (6.3.5) is

stable (asymptotically stable) (unstable).

Then the zero

solution of (6.3.4) is stable (asymptotically stable) (unstable). (b)

stable.

Suppose that the zero solution of (6.3.5) is

Let

(x(t),y(t))

II(x(0),y(0))II

tion

u(t)

be a solution of (6.3.4) with

sufficiently small.

of (6.3.5) such that as

Then there exists a solut -

6.3.

Centre Manifolds

119

x(t) - u(t) + O(e-Yt) (6.3.6)

Y(t) - h(u(t)) + O(e_Yt) where

Y > 0.

The proof of the above theorem is exactly the same as the proof given for the corresponding finite dimensional result.

Using the invariance of

and proceeding formally

h

we have that

h'(x)[Ax + f(x,h(x))] - Bh(x) + g(x,h(x)).

(6.3.7)

To prove that equation (6.3.7) holds we must show that is in the domain of

B.

x0 E X

Let

the domain of

B

h(x)

To prove that

be small.

h(x0)

is in

it is sufficient to prove that

U(t)h(x0) - h(x0) lim ti0+

exists.

Let

(6.3.4) with

t

x(t), y(t) = h(x(t)) x(0) = x0.

differentiable.

be the solution of

As we remarked earlier, y(t)

is

From (6.3.4) r0 t

U(t-T)g(x(T),Y(T))dT,

y(t) = U(t)h(x0) + 1

so it is sufficient to prove that t

lim+ i

U(t-T)g(x(T),y(T))dT 0

exists.

This easily follows from the fact that

strongly continuous semigroup and h(x0)

is in the domain of

B.

g

is smooth.

U(t)

Hence

is a

120

INFINITE DIMENSIONAL PROBLEMS

6.

Let

Theorem 10.

origin in

into

X

be a

0

map from a neighborhood of the

C1

such that

Y

Suppose that as

O(x) E D(B).

0(0) - 0, 0'(0) = 0

and

x _ 0, (Mo)(x) = O(jxjq),

q > 1, where

_ 0' (x) [Ax + f (x, 0) 1 - BO (x)

(MO) (x)

x - 0, IIh(x) - O(x)

Then as

- 9 (x, 0 (x) )

= O(Ixlq).

11

The proof of Theorem 10 is the same as that given for the finite dimensional case except that the extension of

6: X -r Y

domain of

0

must be defined so that

8(x)

B.

Examples

6.4.

Example 8.

Consider the semilinear wave equation

vtt + vt - vxx - v + f(v) ` 0, (x,t) E v= where as

is in the

is a

f

v -r

0.

C3

0

at

(0,Tr) x (0,°°)

(6.4.1)

x - O,Tr

function satisfying

f(v) - v3 + 0(v4)

We first formulate (6.4.1) as an equation on a

Hilbert space.

Let Q

Q - (d/dx) 2

+

I, D(Q) - H2(0,Tr) f1 H1(O,Tr).

is a self-adjoint operator.

Let

Z - H1(0,Tr)

x

Then

L2(0,Tr),

then (6.4.1) can be rewritten as

w = Cw + N(w)

(6.4.2)

where w2

Cw

N(w)

Qwl-w2 Since

C

- f(wl)

is the sum of a skew-selfadjoint operator and a

Examples

6.4.

121

bounded operator, C Clearly

N

is a

C3

generates a strongly continuous group. map from

The eigenvalues of

into

are

an =

Z.

[-1 ±

(5-4n2)1/2]/2.

and all the other eigenvalues have real part less

ai = 0

than

C

Z

0.

The eigenspace corresponding to the zero eigen-

value is spanned by

where

ql

1

ql(x)

Js in x.

I

0

To apply the theory of Section 3, we must put (6.4.2) We first note that

into canonical form.

Cq2 - -q2

where

1

q2(x)

]sin x

and that all the other eigenspaces are spanned by elements of the form

an sin nx, n > 2, an EIR2.

other eigenvectors are orthogonal to X - span(gl), V - span(gl,g2), Y Z - X ® Y.

The projection

P:

In particular, all ql

and

q2.

Let

span(q 2) ® V1, then X

Z

is given by

wl P

=

1

w2

+ w2)gl

(6.4.3)

L

where n

w. =

2

w. (6) sin e do. 1

Let

w = sql + y, s EIR, y E Y

0

and

B = (I-P)C.

Then we can

write (6.4.2) in the form sql = PN(sgl+y) (6.4.4)

y - By + (I-P)N(sgl+y)

122

INFINITE DIMENSIONAL PROBLEMS

6.

By Theorem 8, (6.4.4) has a centre manifold h(O) = 0, h'(0) - 0, h:

(-6,6) - Y.

y - h(s),

By Theorem 9, the equa-

tion which determines the asymptotic behavior of solutions of (6.4.4) is the one-dimensional equation sql - PN(sgl + h(s)).

(6.4.5)

Since the nonlinearities in (6.3.4) are cubic, h(s) - 0(s3), so that rn

s = n

f(s+0(s3))sin 46 d9 1

0

or

s = -33 s3 + 0(s4).

(6.4.6)

Hence, by Theorem 9, the zero solution of (6.4.4) is asymptotically stable.

Using the same calculations as in Section 1 of

Chapter 3, if

s(O) > 0

s(t)

Hence, if

v(x,t)

then as

=

t) 1/2 + o(t-1/2).

is a solution of (6.4.1) with

small, then either

vt(x,O)

(

t - -,

v(x,t)

(6.4.7)

v(x,O),

tends to zero exponen-

tially fast or v(x,t) - ±s(t)sin x + 0(s3) where

s(t)

(6.4.8)

is given by (6.4.7).

Further terms in the above asymptotic expansion can be calculated if we have more information about f(v) = v3 + av5 + 0(v7)

approximation to

(MO)(s)

h(s)

as

v - 0.

f.

Suppose that

In order to calculate an

set

` O'(s)PN(sgl+0(s)) - BO(s)

-

(I-P)N(sgl+0(s)) (6.4.9)

6.4.

where

Examples

123

To apply Theorem 10 we choose

0:IR + Y.

(Mo)(s) = 0(s5).

then

O(s) = 0(s 3)

If

MO(s) = -BO(s)

so that

O(s)

(I-P)N(sgl) + 0(s5)

-

(6.4.10)

- s3g2 -

-BO(s)

0

]q + o(s5)

1 1

where

= sin 3x.

q(x)

If s

(s)

ag2s3 +

-

1

]qs3

(6.4.11)

s2

then substituting (6.4.11) into (6.4.10) we obtain

- S3 0 2

(MO) (s)

=

ag2s3

qs3

+

-

s3g2

-

801+02

Hence, if

q +

0(s5)

1

a - 3/4, 01 - 1/32, 02

0, then

MO(s) - 0(s5),

so by Theorem 10 3

h(s) =

1

g2s3 + 3

]qs3 + 0(s5).

(6.4.12)

0

Substituting (6.4.12) into (6.4.5) we obtain -3s3

s

=

_

4

213 + 5a s5 + 0 s7 2T y a (

1

)

The asymptotic behavior of solutions can now be found using the calculations given in Section 1 of Chapter 3. Example 9.

In this example we apply our theory to the equa-

tion 1

vtt + vt + vxxxx - [B + (2/r4)J (vs(s,t))2dslvxx = 0, 0

(6.4.13)

with

v - vxx -

v(x,0), vt(x,0),

at

0 0

x - 0,1

< x <

1.

and given initial conditions

Equation (6.4.13) is a model for

124

INFINITE DIMENSIONAL PROBLEMS

6.

the transverse motion of an elastic rod with hinged ends, v being the transverse deflection and

B

a constant.

The

above equation has been studied by Ball [7,8] and in particua =

lar he showed that when

the zero solution of

-Tr2

However, in this case the

(6.4.13) is asymptotically stable.

linearized equations have a zero eigenvalue and so the rate of decay of solutions depends on the non-linear terms.

In

[13] the rate of decay of solutions of (6.4.13) was found

Here we discuss the behavior

using centre manifold theory.

of small solutions of (6.4.13) when

is small.

B + Tr2

As in the previous example we formulate (6.4.13) as an We write

ordinary differential equation.

for time derivatives.

vatives and

Let

Z = H2(0,1) f1 H1(0,1) x L2(0,1), wl

for space deri-

'

Qv =

-V,,,, + By",

w2

w=

C

w=

N (w)

_

[f:1]

Qww2

w2

2w

1

Tr

0

)

Then we can write (6.4.13) as w = Cw + N(w). It is easy to check that group on If

and that

Z

A

C

generates a strongly continuous

is a

N

u(x)

map from

C"

is an eigenvalue of

trivial solution

(6.4.14)

Z

Z.

then we must have a non-

C

of

u,,,, - Bu" + (),+A2)u

into

=

0

u(0) = u"(0) = u(1) = u"(1) = 0.

Examples

6.4.

125

An easy computation then shows that C

is an eigenvalue of

if and only if

2A =

Let

-1

[1

- 4 (n4Tr4+Bn2Tr2) ] 1/2.

2A1(E) = -1 + [1-4E]1/2, A2(E) = -1

the rest of the

1

ql(x)

sin Trx, 1

A1(E), and all

-

and

ql

1

sin Trx

(E)

2

are spanned by elements orthogonal to

An(E) ql

where

q2

q2(x) _

while the eigenspaces corresponding to

and

(E) for

n > 2

q2.

X = span(gl), V = span(gl,g2), Y = span(g2) 0 V

Let

Z = X 0 Y

then

an(E),

The eigenspaces corresponding to

are spanned by

A2(E)

and

are

C

have real parts less than zero for

an(E)

sufficiently small.

al(E)

±

Then the eigenvalues of

E = r20 + Tr4.

where

E

A

and the projection

P:

Z - X

is given by

wl

= (XI(E)

PI

-

)2(E)) 1(w2-a2(E)WI)gl

W2 I

where 1

wj = 2 j0 wj (9)sin re d9. Let

w = sql + y, where

s EIR

and

y E Y.

Then we

can write (6.4.14) in the form Sgl = A1(E)sg1 + PN(sgl+y) (6.4.15)

= By + (I-P)N(sgl+y) = 0.

(6.4.15) has a centre manifold

By Theorem 8, IsI

<

6,

IEI

<

CO.

Using

is the second component of

y = h(s,E),

h(s,E) = O(s2+JESI), if N(w)

then

N2(w)

126

INFINITE DIMENSIONAL PROBLEMS

6.

1

N2(sgl+h(s,e))(x) _

2 [JO s2Tr4cos2Trede]Tr2s sin Trx O(s4 + ics31)

+

-s3sin

=

Hence

Trx +

O(s4 + Ies3I).

PN(sgl+h(s,e)) _ (-s3+O(s4+1Cs31))gl, so that by

Theorem 9, the asymptotic behavior of small solutions of (6.4.15) is determined by the equation s - al(e)s - s3

O(les3I

+

+

Is41).

(6.4.16)

We can now determine the asymptotic behavior of small solutions of (6.4.14).

For

asymptotically stable.

0

<

For

e

<

6, solutions of (6.4.14) are

-6 < c<0, the unstable manifold

of the origin consists of two stable orbits connecting two fixed points to the origin. Exercise 17.

For

(See Figure 1.)

c = 0, show that equation (6.4.16) can be

written as s = -s3 - 3s5

+

O(s7)

Phase Portrait for Small Negative Figure 1

c

Examples

6.4.

127

Consider the equation

Example 10.

utt + tut - uxx + a2v + f(u,v) =

vtt + 2vt - vxx - u + g(u,v) (x,t) E (0,Tr)

for

(O,Tr)

x

f(u,v), g(u,v)

where

0

(6.4.17) 0

u - v - 0 at x - 0,7r,

with

have a second order zero at

u = v - 0.

a = 2, we show that the linearized problem has two purely

For

imaginary eigenvalues while all the rest have negative real We then use centre manifold theory to reduce the prob-

parts.

lem of bifurcation of periodic solutions to a two-dimensional problem.

w - (u,v,u,S)T, then we can write (6.4.17) as

Let

w = Cw + N(w) on

(6.4.18)

Let

Z - (H0(O,Tr))2 x (L2(0,n))2.

uxx + av 2 -v

If C

is an eigenvalue of

u

where

a2 + 2A + u - 0

arise in this way.

values of

- u

then

A

and all the eigenvalues of

-1 ± (1-n2tia)1"2, C

n - 1,2,.... are

all the rest have negative real parts.

(Re A1(2)) > al(a)

zero speed.

C

An easy calculation shows that the eigen-

a - 2, the eigenvalues of

so that

is an eigenvalue of

are given by

C

A

For

A

xx

It

and

a1(a)

al - i, al

.

-i, while

Also

0

cross the imaginary axis with non

is now trivial to apply centre manifold theory

128

INFINITE DIMENSIONAL PROBLEMS

6.

to conclude that for

a -

small, the behavior of small

2

solutions is determined by an equation of the form

r

a

1l

L

1

a

where

s

J(s,a) - O(s2).

is a real parameter and

E ]R2, a

(6.4.19)

Is + J(s,a)

s

To apply the theory in Section 2, Chapter 3, we need to calculate the quadratic and cubic terms in

To do this we

J(s,O).

need to put (6.4.18) into canonical form and to calculate the centre manifold when

a - 2.

On the subspace

From now on we let

{r sin nx: r E]R4}

can be represented by the matrix

C

n

L

1

0

0

0

0

1

-4

-2

0

-n2

1

Cn

for

n = 1,2,...

.

C

-2 J

0

are given by the eigenvalues

To put (6.4.18) into canonical

form we first find a basis which puts form.

C

where

Cn

0

Note that the eigenvalues of of

the operator

0

-n2

a - 2.

C1

into canonical

Calculations show that if

-2

ql

0

0

q2

0 0

1

-2

-1

0

q3

-1

-2 2

-2 0

q4

4

L

then Let

Clgl - -q2, Clg2 w - Qz

where

1J

ql, Clg3 = -2q3 + q4, Clg4 - -q3-2q4.

Q - [g1,g2,q3,q4}, then we can rewrite

6.4.

Examples

129

(6.4.18) as i = Q-1CQz

Q-1N(Qz)

+

(6.4.20)

s1,s2 E IR}, V = rl,r2 E]R}, Y = V 0 [X ® V]1 where ,y(x) = sin x. Let

X=

and the projection

Z = X 0 Y

P

on

along

X

Then is given by

Y

wl w2

Pw = 0 0

2

w. =

fn

wj(e)sin a de. 1

z -

Let

J

0

+ y, si EIR, y E Y, then we can

[s1,s200,0]T

write (6.4.20) in the form 0

1

-1

0

Is

+

(6.4.21)

y = By + (I-P)Q 1N(Q z) where

B - (I-P)Q 1CQ,

By Theorem 8, From

w2 - z2

-

w = Qz Let

F(z)

J

Suppose that O(1zI4)

where

J1(s), J2(s)

we have that

w1 = -2z1

[s1,s21T.

h(s).

y -

2z4

and

Q-1 = (tij),

t13

t14

L t23

t24

f(-2z1 - 2z4, z2

-

z3)

2z4, z2

-

z3)

g(-2z1

F(z) a F3(z) + O(IzI 4) F3

=

s

(6.4.21) has a centre manifold

z3.

G(z)

and

11,1,0,0]T,y

and

G3

and

-

G(z) = G3(z) +

are homogeneous cubics.

denote the first two components of

Then if

130

INFINITE DIMENSIONAL PROBLEMS

6.

on the centre manifold, r

n

J1(s) - n J

F3(slsin 6, s2sin 6,0,0)sin 6 d6 + O(Isl 4) 0

with a similar expression for

J2(s).

Hence, on the centre

manifold 0

1

s1

1

0

s2

+

[Jl(s) (6.4.22) J2(s)

and we can apply the theory given in Section 2 of Chapter 3. If the constant

associated with (6.4.22) is zero (see

K

Section 2 of Chapter 3 for the definition of

K) then the

above procedure gives no information and we have to calculate higher order terms. If

F(z) = F2(z) + F3(z) + O(1z14) where

G3(z) + O(Izl 4)

F2

and

G2

and

G(z) - G2(z) +

are homogeneous quad-

ratics, then the calculation of the nonlinear terms is much more complicated.

On the centre manifold, z1 - s1iy + O(s2),

z2 = s2 + O(s2), z3 - O(s2), and order

s2

PQ-1N(Qz). to

h(s).

z4 = O(s2).

The terms of

make a contribution to the cubic terms in Hence, we need to find a quadratic approximation This is straightforward but rather complicated so

we omit the details. Example 11.

Consider the equations

ut - Duxx + (B-l)u + A2v + 2Auv + u2v vt = 6Dvxx - Bu - A2v

-

2Auv - u2v -

BA-1u2

+

BA-1u2

(6.4.23)

u = v - 0 at x - 0,1, where

A,B,6,D

are positive.

The above equations come from

a simplified model of a chemical reaction with

u + A

and

Examples

6.4.

131

as the chemical concentrations [5,12].

v + BA-1

Z = (H0(011))2.

We study (6.4.23) on

D

W =

C]

2

A2

(B-1)

+

dx

C-

,

d

Set

9D

B

d

2

- A2

dx

N(w) -

(2Auv + u2v + BA lug)

1

C then we can write (6.4.23) as w - Cw + N(w).

(6.4.24)

We analyze the situation in which for some value of the parameters

A,B,9,D, C

the restriction of block.

C

has two zero eigenvalues such that to the zero eigenspace has a Jordan

The bifurcation of static solutions where

zero eigenvalues and the restriction of

C

has two

to the zero eigen-

C

space is zero, has been studied in [25,26].

On the subspace

the operator

{r sin nirx, r E R2}

C

can be represented by the matrix

rn2n2D C

n

+ B

The eigenvalues of

C

1

A2

=

-B

for

-

n = 1,2,...

.

C

-9Dn2n2

-

A2

are given by the eigenvalues of

Cn

We suppose that two of the eigenvalues of

are zero while all the rest have negative real parts.

For simplicity we assume that the eigenvalues of If

C1

C1

is to have two zero eigenvalues then

are zero.

132

INFINITE DIMENSIONAL PROBLEMS

6.

trace(C1) - B -

1

det(C1) - A2B -

- rr2D - A2 - Oir2D

0

(B-1-ir2D) (A2+Oir2D)

= 0.

(6.4.25)

We make the following hypotheses: AO,BO,9O,D0

There exists

(Hl)

such that (6.4.25) is satis-

fied and the real parts of the rest of the eigenvalues of

For

(H2)

are negative.

C

in a neighborhood of

A,B,0,D

parametrize

trace(C1)

AO,BO,9O,DO, we can by

det(C1)

and

trace(C1) = E2, det(C1) _ -E1.

(6.4.26)

The first hypothesis is satisfied, for example, if

D0 >

0

and 00 = 1,

If we vary ping at

B

and

A0

0

,r2 DO,

and keep

B0 - (1 + rr2D0)2.

A - A0, D = DO, then the map-

(B,9) + (det(C1), trace(C1)) B = BOB

0

= 00

if

AO,BO,00

(6.4.27)

has a non-zero Jacobian are given by (6.4.27), so

by the implicit function theorem, (H2) is satisfied.

In

order to simplify calculations we assume (6.4.27) from now on.

Let then

X = {siy: s

Z = X ® Y.

i

2

,

Y = X1

where

iy(x)

= sin rrx,

By Theorem 8, the system if

Cw + N (w)

e=

0

has a centre manifold

h: (neighborhood of

where we have written

c _ (E1,E2).

X x]R2)

Y,

On the centre manifold,

the equation reduces to s = C1(E)s + l(siy + h(s,E))

(6.4.28)

6.4.

Examples

where

133

(sl,s21T

s =

and r0 1

Ni(z) =

Ni(z(6))sin n9 d9.

2 1

We treat the linear and nonlinear parts of (6.4.28) separately. If

then

where

p =

C2(0)g2 = (A2+B)ql.

Let

[-p,l]T

ql = [1,P]T, q2 =

C1(0)g1 =

0

and

Q = (g1,g2]'

then

A2+B

0

Q 1C1(0)Q = 0

Let

Q-1C1(e)Q = T = (tip) 1

0

and let 1

0

M(e) = t11

Then for

+ 0(e)

2

0

t12

A +B

is nonsingular and

sufficiently small, M(c)

c

0

=

1

0 M(e)Q-1C1(e)QM-1(E)

_

trace(T)

[-detT)

by (6.4.26). r 0 r

Let 1

I

L

el

0

1

det(C1)

trace(C1)

0

1

el

e2

s = QM-1(e)r, then (6.4.28) becomes

_ r + M(e)Q 1N(QM 1(e)r,

+ h(s,c)). (6.4.29)

e2

To check the hypotheses of Section 9 of Chapter 4 we only need calculate the nonlinear terms when the fact that

c = 0.

Using

h(s,0) = 0(s2), routine calculations show that

M(0)Q 1N(QM-1(0)r

+ h(s,O)) - [P1,P2ITR(rl,r2) + 0(jr1I

+

I r2I3)

L34

INFINITE DIMENSIONAL PROBLEMS

6.

where p1 - 3n(l+p2)-1(1-P),

j(l+p2)-1(l+p)(A2+B)

p2

R(r1,r2) a a1r1 + a2r1r2 + a3r2 BA-1

al -

a2 - 2(A2+B)-1(B3/2A-2+A-BA-1).

2B1/2,

-

Using (6.4.27), al = (r2D0) 1(1-n4D2) We assume that

n2D0 + 1.

zero if

Note also that since

so that

n2D0 }

1 + p =-(n2D0)

1,

is non-

a1

from now on.

1

we have that

p2

is non-zero.

To reduce (6.4.29) to the form given in Section 9, Chapter 4,we make the substitution (6.4.30)

p - (I-p1p21A(e))r.

Substituting (6.4.30) into (6.4.29) and using the above calculations we obtain 0

1

p + F(p,c)

Lel a2F2(0,0) a =

atep

(6.4.31)

e2

a2F2(0,0) a

B -

ap- p - = 2alp1 + a2p2

1

-=0

a2F1(0,0) apl

We have already checked that of

DO,B

is nonzero [B

For most values

a + 0.

is only zero when

tion of a certain algebraic equation].

If

D0

is a solu-

a +

0

then we

can apply the theory given in Section 9, Chapter 4 to obtain the bifurcation set for (6.4.23).

If

B -

0

then the theory

given in Section 9, Chapter 4 still gives us part of the

6.4.

Examples

135

bifurcation set; the full bifurcation set would depend on higher order terms. Remark.

If we vary

9

in (6.4.23) the theory given in Sec-

tion 3 does not apply since the map even defined on the whole space.

(9,v)

-

evxx

is not

However, it is easy to

modify the results of Section 3 to accommodate the above situation. [34].)

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INDEX

Analytic semigroup, 105, 106,

Contraction mapping, 14, 17, 19, 22, 24-26, 102-104, 108-109

110

Asymptotic behavior, 4, 8-12 19, 29, 32, 37-39, 46, 89, 114-115, 118, 122-123, 126

Coupled equations, 1 Dissipative operator, 108-109

128

Asymptotic stability, 1, 4, 6-8, 21, 35-36, 118, 122, 124, 126

Eigenvalues, 3, 7, 11-13, 16, 18, 20, 25, 34, 40, 44, 46-47, 50, 53-55, 89-91, 98, 109, 111-112, 114-115, 117, 120, 124-125, 127-128, 131-132

2,

Averaging, 42

Equilibrium point, 8, 13, 28, 55, 57, 59, 65-67, 69, 73-75, 82, 84, 126, 131

Autonomous equations, 13 Bifurcation (see also Hopf bifurcation), 11-12, 40-41, 43, 47, 50, 52-53, 55-59, 82, 84-85, 94, 127, 131, 134-135

Evolution equation, 13, 97 First integral, 65 Generator, 97, 99-134

Cauchy problem, 101 Geodesics, 53

Center manifold Gronwall's inequality, 18, 20, 23-24, 98

approximation of, 2, 5-7, 9, 13, 25, 32, 35, 38, 49, 120, 122-123 definition,

Group, 106-107, 113-114, 124

Hamiltonian system, 53

3

existence of, 1-4, 6-7, 9, 12-17, 29-34, 38, 46, 89, 93, 97, 117-118, 122, 124-125, 127-129, 132 flow on, 2-6, 8, 10, 19, 21-22, 29, 32, 38, 46-47, 89, 92, 119, 126, 128-129,

Hille-Yosida theorem, 102-104, 107, 109

Homoclinic orbit, 57, 67-68, 7475, 80, 82, 84

12, 35, 118132

Hopf bifurcation, 39-50

properties of, 19-20, 28-30,

Implicit function theorem, 4041, 43, 48, 52, 56, 67-68, 70, 73, 94, 132

95, 118

Infinitesimal generator (see Chemical reactions, 8, 12,

generator)

130-131

Closed graph theorem, 106

Instability, 2, 4, 6-7, 21, 35, 43-44, 115, 118

Compact operator, 111-112,

Invariance of domain theorem, 24

114

Invariant manifold, 2-3, 13, 16, 30-34

Continuation, 116

141

.42

Invariant set, 13, 33-34, 51-53, 117

INDEX

Separatrix, 8

Skew-selfadjoint operator, 107, Jacobian, 3, 93-94, 132

113, 120

Laplace transform, 102

Spectral radius, 110

Liapunov function, 37

Spectrum, 109-113, 115, 117

Liapunov-Schmidt procedure,

Stability, 2, 4, 6, 8, 19, 2122, 35, 43-44, 50, 57, 118

12

Linearization, 7, 11-13, 15, 34, 37, 40, 44, 46-47, 50, 53-54, 57, 71, 81-82, 88, 92-94, 127

Stable manifold, 3, 8, 13, 15,

Lumer-Phillips theorem, 108

Symplectic mapping, 53

Neutral functional differential equations, 13

Uncoupled equations, 1

67

Strong solution, 104-105

Uniform boundedness theorem, Nilpotent, 20

Normal form, S2, 56, 89

100

Unstable manifold, 8, 13, 67, 82, 126

Parabolic equations, 105

Weak solution, 105, 116 Partial differential equations, 13, 98, 102 Periodic solutions, 28, 33, 40-45, 47, 50-51, 55, 57, 65-67, 69-71, 73-80, 82, 84, 127 Perturbation, 9, 12, 30, 33, 44-45, 113-114 Phase portraits, 11-13, 54, 60-63, 80, 82, 84-87, 126 Poincare map, 33 Rate of decay, 2, 4, 10, 37, 124

Reduction of dimension, 12, 19, 89

Resolvent, 102-103 Saddle point, 67 Selfadjoint operator, 104105, 107, 113, 120

Semigroup, 97-134