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

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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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14

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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3.4.

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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1

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1.5. 1.6.

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Motivation

Centre Manifolds

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1.2. 1.3. 1.4.

Examples

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1 1 3

14 14 16 19 25 28

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50

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

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54

Introduction

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Preliminaries .

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>0

.


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


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

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




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


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


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
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

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

(when

K
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,


0,

8




and

0

a


0

a < 0.

The case

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

a
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)


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


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)

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)


0

for

b > b1.

b < b1

and

Proof:

It is easy to show that

0

> D2c5.

The result now follows.

such that

P'(b)
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.

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
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
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

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,...}
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
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
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


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


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