Applied Mathematical Sciences I Volume 35
Jack Carr
Applications of Centre Manifold Theory
SpringerVerlag New York Heidelberg
Berlin
Jack Carr Department of Mathematics HeriotWatt 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 197879"Pref. 1. Manifolds (Mathematics) 2. Bifurcation theory. I. Title. II. Series: Applied mathematical sciences (SpringerVerlag New York Inc.); v. 35. QA1.A647 vol. 35 [QA613] 510s [516'.07] 814431 AACR2
All rights reserved.
No part of this book may be translated or reproduced in any form without written permission from SpringerVerlag. 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 SpringerVerlag 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 197879.
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 LiapunovSchmitt 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 2dimensional 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 4dimensional 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 MalletParet 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 2976G0294.
Jack Carr December 1980
TABLE OF CONTENTS Page CHAPTER 1. 1.1.
INTRODUCTION TO CENTRE MANIFOLD THEORY
Introduction
.
.
.
. .
. .
. .
. .
.
.
.
.
.
.
.
. .
. .
. .
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5 11
.
.
.
.
.
.
.
.
.
13
.
.
.
.
.
.
.
.
14
Bifurcation Theory Comments on the Literature
CHAPTER 2. 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7.
.
. .
PROOFS OF THEOREMS
.
.
.
.
.
.
.
CHAPTER 3.
.
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. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2.8.
.
.
.
.
EXAMPLES
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3.1. 3.2. 3.3.
Rate of Decay Estimates in Critical Cases. Hopf Bifurcation Hopf Bifurcation in a Singular Perturbation .
.
.
.
3.4.
Bifurcation of Maps .
.
.
.
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.
.
1
.
1.5. 1.6.
.
.
.
Motivation
Centre Manifolds
.
. .
1.2. 1.3. 1.4.
Examples
.
.
1 1 3
14 14 16 19 25 28
.
30 33
37
.
.
37
.
39
.
.
.
.
.
.
.
.
.
.
.
.
44
.
.
.
.
.
.
.
.
.
50
BIFURCATIONS WITH TWO PARAMETERS IN TWO SPACE DIMENSIONS . . . . . . . . . . . .
.
.
54
Introduction
.
.
.
Preliminaries .
.
.
.
.
>0
.
<
.
Scaling . The Case The Case
.
.
e c
.
0
. .
. . .
.
.
.
.
.
.
.
.
54
.
.
.
.
.
.
.
.
.
.
.
.
57
. . .
. .
. . .
.
. .
. .
.
.
.
. .
.
.
. .
. .
. .
.
.
.
.
.
.
.
.
.
64 64 77
.
.
.
.
.
78
.
.
.
.
.
.
.
.
.
More Scalin Completion of the Phase Portraits. Remarks and Exercises. Quadratic Nonlinearities
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
APPLICATION TO A PANEL FLUTTER PROBLEM Introduction Reduction to a Second Order Equation Calculation of Linear Terms. Calculation of the Nonlinear Terms .
.
.
.
.
.
.
80 81 83 88 88
89 93 95
Page INFINITE DIMENSIONAL PROBLEMS.
.
.
.
.
.
.
.
97
6. 1 .
Introduction
.
.
.
.
.
97
6.2. 6.3.
Semigroup Theory Centre Manifolds
.
.
97
CHAPTER 6.
6.4.
Examples
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
117
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
120
REFERENCES
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
136
INDEX .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
141
.
.
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 nonzero.
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 ndimensional 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 nonzero.
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)[exx3+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)(yw)
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(uu2) + 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 pdimensional 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 nonzero 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 LiapunovSchmidt 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 13 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 nonpositive 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 infinitedimensional 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 13.
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)
fth(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
at
zl,z2
zl,z2
plane.
Since the centre manifold
plane at the origin we must elim
component, that is we must eliminate the com
lonent 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 cutoff 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
1111, 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)[Ixx'I + Iyy'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)[Ixx'I + Iyy'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)ertlx0x1I
<
0 I
e
r(st)Ix(s,x0,h) 
it
By Gronwall's inequality, for
t
< 0,
x(s,xl,h)Ids.
Reduction Principle
2.4.
19
M(r)Ixlx0IeYt,
 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)(sy) r
are small enough so that hl, h2 E X
Similarly, if ITh1(x0)Th2(x0)I
1Jx0x1I
(2.3.10)
0 > y.
x0 E]Rn, we obtain
and
+(1+pi)M(r)k(c)r1(Sy)
< Ck(c)(S
1]
(2.3.11)
IIh1h2II 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 ndimensional 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 < C1eutly(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 < Cestlz(0)I
+
C6(c)I0
e'(ts)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(eYt) (2.4.5)
0(eYt)
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(ts) 
(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 Izlz21]
IN($1,z1)
 N($2,z2)I
k(c)[Iz11101021 +
IR($1)z1)
 R($2,z2)I
k(c)[1z1z2l + 141$21]
(2.4.10)
From (2.4.7), t
+ Ck(e)j
CIzOIeOt
<
Iz(t)I
eO(ts)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, eat + <
IT,(t)I
k(e)1W(eas
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
<
eS(ts)Iw(s)Ids
+ Ck(c)IO
Clk(c)IIO102IIeSt
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 < 211010211 + k(c)fm(I01(s)
Izl(s)
+
where
a <

02(s)I

z2(s)I)ds < a1010211
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)II010211 e Slt.
S
u0
and
is onetoone, 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 onetoone 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 onetoone 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) 
eBsG(x(s,x0),z(x(s,x0))

+ O(x(s,x0)))ds.
f
r
6(x0)
0
I
d[eBsO(x(s,x0))]ds us
1 1
_
0 (eBs[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 eBsQ(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(Sgy)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
n2
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 > 6l.
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 
= (2i2)a2i2' 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(xy)
y  y

f(xy). 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 t1
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
t1/2
7
where
1
is the solution of w  w3,
then
ju(t)s3ds.
+ Ct3/2 + 0(t5/2)
(3.1.7)
is a constant, we have that u(t) 
t1/2 + o(t1/2).
(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
w1(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 t1/2 + constant + 0(1).
Substituting (3.1.11) into (3.1.10) and using (3.1.7), t1/2
u(t)
t3/2[(a+3)ln t + C]
4/7
where
C
+
o(t3/2) (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 oneparameter 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 coordinates 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 nonzero, 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.
JaKlll/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 nonzero
speed and that
A(0) = with
m0
nonzero.
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(1x)
)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
nonnegative.
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(1F(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(1x 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(xx0)  x0y 3x2 + a0 
Then assuming
.
ip
z
is nonzero,
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
i1
lw
 y2)z 
ylyz
f3(w,Y,z,E) e ylb0y  ylyz N(e,Y) 
p263
+ 3i1x062  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)iy1

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
(1x0)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)i1
By the implicit
0.
sufficiently
c, x0  x0(0), b0  b0(0)
6(e,x0,b0) = 6(0)
trace(J(c)) = 2 i1x0 
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)11
8
6=6(e)
(f x0)
is sufficiently small.
3
R(e)
For

6
0,
<
Hence, the eigenvalues of
cross the imaginary axis with nonzero speed at Let
and
[6y1x0
+ (2Y2y1)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)x1Y2]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(m1(e)z1,
1(e)y1,e)Y1L1(c)y1zI
3.3. Hopf Bifurcation in a Singular Perturbation Problem
49
where w2(c)
=
Y1b0(c)[(6/2),y1
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 nonzero then
K(0)
To calculate
will be nonzero 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)

3iy1x0(x0y+z)2 + Y(XOY+z) + O(IYI3 + IzJ3).
Hence, if 02(Y,z) = 3iy2x0(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) + i2(XOY+z)3
+ Y02(Y,z) + O(Y4+z4). Hence,
(3.3.12)
6iy1x0(XOY+z)02(Y,z)
50
3.
h(y,z)
02(y,z)
i3(x0y+z)3 + 6iy2x0(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 nonzero 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 twodimensional 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 onedimen
crosses the unit
circle, that is 0
A(u)
when
u = 0.
U 11
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 twodimensional 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
k1 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)ein9)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 onetoone for
u + c
u
Equation
small enough.
(3.4.2) now takes the form e2  Im(b(u)ein9)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
(n2)/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 twodimensional 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 twodimensional 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. 333348],
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 coordinates 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 25.
Sections
38 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 nonzero 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 = lEla1 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
lal1/261T.
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 = slat1/2 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 35 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) _ [(z21) +
g(c1)
for
0 (c2/2)(lz4)]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) = 61[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. 337338] 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 + (y1l) 2(yl+l)2  (c1)2(c+l)2. (4.4.32) Thus, for
c
close to
1
the closed curves are approximately
y2 + 2(yll) 2  2(c1)
(4.4.33)
2.
So, instead of equation (4.4.29) we consider the equation G(1,6,c) = (c1)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(y3lyl)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  (8b4)JO'Ji  5(Ji)2 + 4b(Jo,) 2.
By (4.5.3), bJ0 > J.
Hence, for
b > 2,
S > (J,)2b1(8b45b+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 = 1Ela111/2h1, 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  h4.
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(h1,6)
u
and that as
h
yh2Q(h1)6
=
=
0, h2Q(h1)

+
(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
ra1A(e)
x = B(c)x, where
a3f1(0,0) ax
x  x
a3f2(0,0)
3
3X3
will be onetoone 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 =
IaI1/2(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
a1 d
=
l
y1,y2
e1
x1
1
2
=
6
Y1,
x2 =
a 1/263 l
l
Y2,
tlal1/261
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(2c3), (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 67.
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 46 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
lpp01
small, by centre manifold theory, the local behavior
Irr01
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
lpp01
and
Irr0l
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
[(b1b2)2
4(a1+a2)]1/2
+
X3,4
and a calculation shows that they have negative real parts and nonzero 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 onedimensional 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(zz),
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 = Slx.
Then
(5.1.1) can be written in the form
where
F(y,I',p)
and where
jy
= By + S1(AA0)SY + F(Y,r,P)
=
Slf(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
Irr0I
(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
Ipp0I
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(Irr0I +IpPOI) 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 nonzero, 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 nonzero 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 + S1(AAO)S, it is easily seen that
J
4
x 4
(5.3.2) matrix
is a nonzero
multiple of the Jacobian of the mapping given by (5.3.1). Hence
is nonzero, 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). S1
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(2a1b1b2) t22 = mat
t14 = m2c(bl+b2) t24 = mc m =
(al+a2blb2)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(2a2bZ4a1)
>
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((IP)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(ts))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.
(entan),
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)  eWtT(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.
h1[T(t+h)w  T(t)w] = h1[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)h1[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 h1[T(t)w  T(t=h)]w  T(th)h1[T(h)w

w]
shows that (6.2.6) holds for the twosided 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
t1w(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 eAtT(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 HilleYosida Theorem.
A necessary and sufficient
Theorem 1 (HilleYosida 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
h1(T(h)I)R(A)w = h'[(e
Ah 
w E Z.
Use the identity
1)j eAtT(t)wdt 0
h 
eah
eAtT(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 HilleYosida 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 HilleYosida 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 HilleYosida 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(ts)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(ts)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)
n0
where
U0(t)  T(t)
and rt
T(s)AUn(ts)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 HilleYosida 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 skewselfadFor
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 (LumerPhillips 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 (XIC) 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 nonempty set
is closed.
C
(X > 0: range of
XI

is open and closed and use
H)
is
C
the HilleYosida 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 < Mewt, 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(ts)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(ts)
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(ts)
s
f
h1
we first Let
sufficiently small.
Then
the set
{A(U(tsh)  U(ts))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(ts). Then for IIwII = 1,
Let
Ilg(s+h)w  g(s)wII < IIA(U(tsh)  U(ts)) (wwk) II  U(ts))wkll
+ IIA(U(tsh)
Hence if
is chosen such that
6
IIA(U(tsh)  U(ts))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(ts).
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) + (IP)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(IP)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
A1
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 =
AB
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
a2IB
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 < eat,
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, A1 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 skewselfadjoint 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 < Met, 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) = 2n1 m(cos mt  cos amt)sin mx (6.2.19)
v(x,t) = ml 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(ts)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
tT
.
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 Cinvariant 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
aebt,
B  (IP)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) ' (IP)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(eYt) (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(tT)g(x(T),Y(T))dT,
y(t) = U(t)h(x0) + 1
so it is sufficient to prove that t
lim+ i
U(tT)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 selfadjoint 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)
Qwlw2 Since
C
 f(wl)
is the sum of a skewselfadjoint 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 ±
(54n2)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 = (IP)C.
Then we can
write (6.4.2) in the form sql = PN(sgl+y) (6.4.4)
y  By + (IP)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 onedimensional 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(t1/2).
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)

(IP)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)
(IP)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 nonlinear 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 + [14E]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(w2a2(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 + (IP)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 twodimensional 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 ± (1n2tia)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  q32q4.
Q  [g1,g2,q3,q4}, then we can rewrite
6.4.
Examples
129
(6.4.18) as i = Q1CQz
Q1N(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 + (IP)Q 1N(Q z) where
B  (IP)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
Q1 = (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
PQ1N(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 + (Bl)u + A2v + 2Auv + u2v vt = 6Dvxx  Bu  A2v

2Auv  u2v 
BA1u2
+
BA1u2
(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 + BA1
Z = (H0(011))2.
We study (6.4.23) on
D
W =
C]
2
A2
(B1)
+
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
(B1ir2D) (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 nonzero 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
Q1C1(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)Q1C1(e)QM1(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 = QM1(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(QM1(0)r
+ h(s,O))  [P1,P2ITR(rl,r2) + 0(jr1I
+
I r2I3)
L34
INFINITE DIMENSIONAL PROBLEMS
6.
where p1  3n(l+p2)1(1P),
j(l+p2)1(l+p)(A2+B)
p2
R(r1,r2) a a1r1 + a2r1r2 + a3r2 BA1
al 
a2  2(A2+B)1(B3/2A2+ABA1).
2B1/2,

Using (6.4.27), al = (r2D0) 1(1n4D2) 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 nonzero.
To reduce (6.4.29) to the form given in Section 9, Chapter 4,we make the substitution (6.4.30)
p  (Ip1p21A(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].)
(See, for example, Exercises 12 in Section 3.4 of
REFERENCES
1]
:2]
R. Abraham and J. Marsden, Foundations of Mechanics, 2nd ed., Benjamin, New York (1978).
R. A. Adams, Sobolev Spaces, Academic Press, New York (1975).
I. Arnold, Bifurcations in versal families, Russian Math. Surveys 27, 54123 (1972).
[3]
V.
[4]
V.
[5]
J. F. G. Auchmuty and G. Nicolis, Bifurcation analysis of nonlinear reactiondiffusion equations, Bull. Math. Biology 37, 323365 (1975).
[6]
A. V. Balakrishnan, A lied Functional Analysis, Applications of Mathematics, Vol. pringer er ag, Berlin and New York (1976).
[7]
J. M. Ball, Stability theory for an extensible beam, J. Diff. Egns. 14, 399418 (1973).
[8]
J. M. Ball, Saddle point analysis for an ordinary differential equation in a Banach space and an application to dynamic buckling of a beam, Nonlinear Elasticity, J. Dickey (ed.), Academic Press, New York
[9]
J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc. 63, 370373 (1977).
I. Arnold, Loss of stability of selfoscillations close to resonance and versal deformations of equivariant vector fields, Functional Analysis and its Appl. 11, No. 2, 110 (1977).
[10] J. M. Ball and J. Carr, Decay to zero in critical cases of second order ordinary differential equations of Duffing type, Arch. Rat. Mech. Anal. 63, 4757 (1976).
136
References
137
[11]
M. Berger, Nonlinearity and Functional Analysis, Academic Press, New or
[12]
J. Boa and D. S. Cohen, Bifurcation of localized disturbances in a model biochemical reaction, SIAM J. Appl. Math. 30, 123135 (1976).
[13]
J. Carr and NajiAlAmood, Rate of decay in critical cases, J. Math. Anal. and Appl. 75, 242250; II, Ibid., 293300
[14]
J. Carr and R. G. Muncaster, The application of centre manifolds to amplitude expansions, submitted to SIAM J. Appl. Math.
[15]
J. Carr and M. Z. M. Malhardeen, Beck's Problem, SIAM J. Appl. Math. 37, 261262 (1979).
[16]
J. Carr and M. Z. M. Malhardeen, Stability of nonconservative linear systems, S rin er Lecture Notes in Math., Vol. 799, Springer er ag, Berlin and New York
MO).
[17]
N. Chafee, The bifurcation of one or more closed orbits from an equilibrium point of an autonomous differential system, J. Diff. Egns. 4, 661679 (1968).
[18]
N. Chafee, A bifurcation problem for a functional differential equation of finitely retarded type, J. Math. Anal. Appl. 35, 312348 (1971).
[19]
S. N. Chow and J. MalletParet, Integral averaging and bifurcation, J. Diff. Egns. 26, 112159 (1977).
[20]
S. N. Chow and J. MalletParet, Fuller index and Hopf bifurcation, J. Diff. Egns. 29, 6685 (1978).
[21]
S. N. Chow, J. MalletParet, and J. K. Hale, Applications of generic bifurcation, I, Arch. Rat. Mech. Anal. 59, 159188 (1975); II, Ibid. 62, (1976).
[22]
E. A. Coddington and N. Levinson, Theory of Ordinar Differential Equations, McGrawHill, New York (1955).
[23]
N. Fenichel, Persistence and smoothness of invariant manifolds for flows, IndianaUniv. Math. J. 21, 193226
(1971).
[24]
N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Diff. Egns. 31, 5398 (1979).
[25]
M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory, Commun. Pure Appl. Math. 32, 2198 (1979).
[26]
M. Golubitsky and D. Schaeffer, Bifurcation analysis near a double eigenvalue of a model chemical reaction, MRC Rep. 1859, Univ. of Wisconsin (1978).
REFERENCES
58
27]
J. K. Hale, Ordinary Differential Equations, WileyInterscience, New York 1
28]
J. K. Hale, Critical cases for neutral functional differential equations, J. Diff. Egns. 10, 5982 (1970).
29]
J. K. Hale, Generic bifurcation with applications, Nonlinear Analysis and Mechanics: HeriotWatt Symposium Ced. R. J. Knops) Vol. 1, Pitman, London (1977).
30]
P. Hartman, Ordinary Differential Equations, Second Edition, P. artman
31]
P. Hassard and Y. H. Wan, Bifurcation formulae derived from center manifold theory, J. Math. Anal. Appl. 63(1), 297312 (1978).
32]
A. R. Hausrath, Stability in the critical case of pure imaginary roots for neutral functional differential equations, J. Diff. Egns. 13, 329357 (1973).
33]
F. G. Heinenken, H. M. Tsuchiya and R. Aris, On the mathematical status of the pseudosteady state hypothesis of biochemical kinetics, Math. Biosci. 1, 95113 (1967).
:34]
D. Henry, Geometric Theor of Parabolic Equations, to be published by pringer er ag.
:35]
M. Hirsch, C. Pugh and M. Schub, Invariant Manifolds, Springer Lecture Notes in Math., o 5B3, SpringerVerlag, Berlin and New York (1977). .
[36]
P. J. Holmes, Bifurcations to divergence and flutter in flow induced oscillations  a finite dimensional analysis, J. Sound and Vib. 53, 471503 (1977).
[37]
P. J. Holmes, Coupled flutter and divergence in autonomous nonlinear systems, to appear in A.S.M.E. Symposium on Nonlinear Dynamics.
[38]
P. J. Holmes and J. E. Marsden, Bifurcations to divergence and flutter in flow induced oscillations  an infinite dimensional analysis, Automatica 14(4), 367384 (1978).
[39]
E. Hopf, Abzweigung einer periodischen Losung von einer stationaren Losung eines Differential systems, Ber. h. Phys., K1. Sachs Akad. Wiss. Leipzig 94, T2 Mat_
[40]
G. Iooss, Bifurcation of Maps and A lications, NorthHolland Mathematics Studies, Vol. NorthHolland Publishing Company, Amsterdam, New York and Oxford
(1979).
References
139
[41)
G. Iooss and D. D. Joseph, Bifurcation and stability of nTperiodic solutions branching from Tperiodic solutions at points of resonance, Arch. Rat. Mech. Anal. 66, 2, 135172 (1977).
[42)
M. C. Irwin, On the stable manifold theorem, Bull. London Math. Soc. 2, 196198 (1970).
[43)
T. Kato, Perturbation Theory for Linear 0 erators, Springer er ag, Berlin and New or
[44)
A. Kelley, The stable, centerstable, center, centerunstable, and unstable manifolds, J. Diff. Egns. 3, 546570 (1967).
[45)
A. Kelley, Stability of the centerstable manifold, J. Math. Anal. Appl. 18, 336344 (1967).
[46)
W. Klingenberg, Lectures on Closed Geodesics, SpringerVerlag, Berlin and New York
[47)
N. Kopell and L. N. Howard, Bifurcations and trajectories joining critical points, Adv. Math. 18, 306358 (1976).
[48)
0. E. Langford, Bifurcation of periodic solutions into invariant tori: the work of Ruelle and Takens, S rin er Lectures Notes in Mathematics, Vol. 322, pringer er ag, Berlin and New York (1972). pp. 15919Z,
[49)
J. E. Marsden, Qualitative methods in bifurcation theory, Bull. Amer. Math. Soc. 84, 11251147 (1978).
[50)
J. E. Marsden and T. J. R. Hughes, Topics in the mathematical foundations of elasticity, Nonlinear Anal sis and Mechanics: HeriotWatt Symposium (ed. Knops) Vol. itman, London (1978).
[51)
J. E. Marsden and M. F. McCracken, The Hopf Bifurcation and Its A lications, Applied Math. sciences, Vol. 19, pringer er ag, Berlin and New York (1976).
[52)
S. J. Merrill, A model of the stimulation of Bcells by replicating antigen, I., Math. Biosci. 41, 125141 (1978) ; II. Ibid., 41, 143155 (1978).

[53)
J. Moser, Proof of a generalized form of a fixed point theorem, S rin er Lecture Notes in Math., Vol. 597, Springer er ag, Berlin and New York 977).
[54)
A. I. Neishtadt, Bifurcation of the phase pattern of an equation system arising in the problem of stability 4 resonance. loss of selfoscillations close to 1 P.M.M. 42, 896907 (1978). :
[55]
A. Pazy, Semigroups of linear operators and applications to partial differential equations, University of Maryland Lecture Notes, Vol. 10, (1974).
REFERENCES
40
56]
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, a a ointness, Academic Press, New or (1975).
57]
D. Ruelle and F. Takens, On the nature of turbulence, Comm. Math. Phys. 20, 167192 (1971).
:58]
R. J. Sacker, On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations, New York Univ. IMMNYU, 333 (1964).
[59]
D. H. Sattinger, Topics in Stability and Bifurcation The pry, Springer Lecture Notes in Math., Vol. ,
pierVerlag, Berlin and New York (1973). [60]
J. T. Schwartz, Nonlinear Functional Analysis, Gordon and Breach Science Publishers, New York (1969).
[61]
I. Segal, Nonlinear semigroups, Ann. Math. 78, 339364
[62]
Y. Shitzuta, On the classical solutions of the Boltzmann equation, to appear in Commun. Pure Appl. Math.
[63]
S. J. van Strien, Center manifolds are not 166, 143145 (1979).
[64]
F. Takens, Forced Oscillations and Bifurcations, ij suniversiteit, Communication Math. Institute, Utrecht (1974).
[65]
F. Takens, Singularities of vector fields, Publ. Inst. Hautes Etudes Sci. 43, 47100 (1974).
[66]
D. A. Vaganov, N. G. Samoilenko and V. G. Abramov, Periodic regimes of continuous stirred tank reactors, Chem. Engrg. Sci. 33, 11331140 (1978).
[67]
I. Vidav, Spectra of perturbed semigroups with applications to transport theory, J. Math. Anal. Appl., 30, 264279 (1970).
[68]
Y. H. Wan, Bifurcation into invariant tori at points of resonance, Arch. Rat. Mech. Anal. 68, 343357 (1978).
[69]
J. Zabczyk, A note on Co semigroups, Bulletin de L'Academie Polonaise des Sciences, Serie e3 ssciences aastr. et p ys., Z3, 895898 (197S). mat
(1963).
C", Math. Z.
INDEX
Analytic semigroup, 105, 106,
Contraction mapping, 14, 17, 19, 22, 2426, 102104, 108109
110
Asymptotic behavior, 4, 812 19, 29, 32, 3739, 46, 89, 114115, 118, 122123, 126
Coupled equations, 1 Dissipative operator, 108109
128
Asymptotic stability, 1, 4, 68, 21, 3536, 118, 122, 124, 126
Eigenvalues, 3, 7, 1113, 16, 18, 20, 25, 34, 40, 44, 4647, 50, 5355, 8991, 98, 109, 111112, 114115, 117, 120, 124125, 127128, 131132
2,
Averaging, 42
Equilibrium point, 8, 13, 28, 55, 57, 59, 6567, 69, 7375, 82, 84, 126, 131
Autonomous equations, 13 Bifurcation (see also Hopf bifurcation), 1112, 4041, 43, 47, 50, 5253, 5559, 82, 8485, 94, 127, 131, 134135
Evolution equation, 13, 97 First integral, 65 Generator, 97, 99134
Cauchy problem, 101 Geodesics, 53
Center manifold Gronwall's inequality, 18, 20, 2324, 98
approximation of, 2, 57, 9, 13, 25, 32, 35, 38, 49, 120, 122123 definition,
Group, 106107, 113114, 124
Hamiltonian system, 53
3
existence of, 14, 67, 9, 1217, 2934, 38, 46, 89, 93, 97, 117118, 122, 124125, 127129, 132 flow on, 26, 8, 10, 19, 2122, 29, 32, 38, 4647, 89, 92, 119, 126, 128129,
HilleYosida theorem, 102104, 107, 109
Homoclinic orbit, 57, 6768, 7475, 80, 82, 84
12, 35, 118132
Hopf bifurcation, 3950
properties of, 1920, 2830,
Implicit function theorem, 4041, 43, 48, 52, 56, 6768, 70, 73, 94, 132
95, 118
Infinitesimal generator (see Chemical reactions, 8, 12,
generator)
130131
Closed graph theorem, 106
Instability, 2, 4, 67, 21, 35, 4344, 115, 118
Compact operator, 111112,
Invariance of domain theorem, 24
114
Invariant manifold, 23, 13, 16, 3034
Continuation, 116
141
.42
Invariant set, 13, 3334, 5153, 117
INDEX
Separatrix, 8
Skewselfadjoint operator, 107, Jacobian, 3, 9394, 132
113, 120
Laplace transform, 102
Spectral radius, 110
Liapunov function, 37
Spectrum, 109113, 115, 117
LiapunovSchmidt procedure,
Stability, 2, 4, 6, 8, 19, 2122, 35, 4344, 50, 57, 118
12
Linearization, 7, 1113, 15, 34, 37, 40, 44, 4647, 50, 5354, 57, 71, 8182, 88, 9294, 127
Stable manifold, 3, 8, 13, 15,
LumerPhillips theorem, 108
Symplectic mapping, 53
Neutral functional differential equations, 13
Uncoupled equations, 1
67
Strong solution, 104105
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, 4045, 47, 5051, 55, 57, 6567, 6971, 7380, 82, 84, 127 Perturbation, 9, 12, 30, 33, 4445, 113114 Phase portraits, 1113, 54, 6063, 80, 82, 8487, 126 Poincare map, 33 Rate of decay, 2, 4, 10, 37, 124
Reduction of dimension, 12, 19, 89
Resolvent, 102103 Saddle point, 67 Selfadjoint operator, 104105, 107, 113, 120
Semigroup, 97134