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. 1. Title. II. Series: Applied mathematical sciences (SpringerVerlag New York Inc.); v. 35. QA1.A647 vol. 35 [QA613] 510s [516'.07]814431 AACR2
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© 1981 by SpringerVerlag New York Inc. Printed in the United States of America 9H7654121
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.
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.
bifurcation theory is concerned with topological changes in the nature of the solutions of differential equations as parameters are varied.
orbits from an equilibrium point as a parameter crosses a critical value.
centre manifold theory reduces the dimension of the system under investigation.
theory plays the same role for dynamic problems as the LiapunovSchmitt procedure plays for the analysis of static solutions.
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.
essentially technical.
simplest such theory, for example our equations are autonomous.
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 their use by a few simple examples.
the theorems which were stated in Chapter 1, and Chapter 3 contains further examples.
line Hopf bifurcation theory for 2dimensional systems.
Section 3 of Chapter 3 we apply this theory to a singular perturbation problem which arises in biology.
Chapter 6 we apply the same theory to a system of partial differential equations.
tion problem in the plane with two parameters.
results in this chapter are new and, in particular, they confirm a conjecture of Takens [64].
dependently of the rest of the notes.
the theory of Chapter 4 to a 4dimensional system.
ter 6, we extend the centre manifold theory given in Chapter 2 to a simple class of infinite dimensional problems.
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.
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 Z976GOZ94.
TABLE OF CONTENTS Page CHAPTER 1. 1.1. 1. 2. 1. 3. 1.4. 1. 5. 1. 6.
CHAPTER 2. 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7.
2.8.
CHAPTER 3. 3.l. 3.2. 3.3. 3.4.
CHAPTER 4. 4.l. 4.2. 4.3.
4.4. 4.5. 4.6. 4.7. 4.8. 4.9.
CHAPTER 5. 5.l. 5.2. 5.3. 5.4.
Introduction Motivation. Centre Manifolds Examples . . . . . Bifurcation Theory Comments on the Literature
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.
EXAMPLES . . . . . . . .
Rate of Decay Estimates in Critical Cases. Hopf Bifurcation . . . . . ..... . Hopf Bifurcation in a Singular Perturbation Problem. . . . . . . . . . . . . . . . Bifurcation of Maps . . . . . . . . . . .
Introduction . Preliminaries. Scal ing. . . The Case £1 > 0 The Case £ < 0 More Scalink . . ..... Completion of the Phase Portraits. Remarks and Exercises . . . . . Quadratic Nonlinearities . . .
Introduction . Reduction to a Second Order Equation Calculation of Linear Terms. . Calculation of the Nonlinear Terms .
Page CHAPTER 6. 6.1. 6.2. 6.3. 6.4.
INFINITE DIMENSIONAL PROBLEMS. Introduction . . Semigroup Theory Centre Manifolds Examples
INDEX . . .
1.1.
manifold theory for finite dimensional systems and give some simple examples to illustrate their application. 1.2.
look at a simple example. x
is a constant.
we can easily show that the zero solution of (1.2.1) is asymptotically stable if and only if
O.
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
O.
1.
an abstraction of the idea of uncoupled equations. A curve, say
x= if the solution lies on the curve h(x(t)).
manifold.
zero solution of (1.2.1), the only important equation is .
x = ax , that is, we only need study a first order equation on a particular invariant manifold. The theory that we develop tells us that equat ion (1. 2.1) has an invariant manifold
x ... O.
of the zero solution of (1.2.2) can be proved by studying a first order equation. au
O.
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
O.
is a solution Clf (1. 2.2)
x(t) = u(t)(l+o(l)), t ......
1.3.
1.3.
We first recall the definition of an invariant manifold for the equation (1.3.1)
x E mn.
O.
is an invariant manifold. Consider the system x
have negative real parts.
O.
= o.
since if we restrict initial data to tend to zero.
centre manifold. In general, if (1.3.2) and if
h(O) • 0, h' (0) • O.
place of local centre manifold if the meaning is clear.
1.
x = Ax.
That is, the equation on the centre manifold determines the asymptotic behavior of solutions of the full equation modulo exponentially decaying terms.
are nonzero.
are proved in Chapter 2. Theorem 1. y=h(x),
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) ,yeO)) 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 x(t)
u(t)
t +
Then there exists a solution +
00,
O(e Yt ) f1.:L S)
yet) • h(u(t)) + O(e yt)
1.4.
Examples
where
y > 0
5
is a constant.
If we substitute
yet)
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'(O)
=
0
(1.3.6) h(O)
0,
is the system to be solved for the centre manifold.
This is impossible, in general, since it is equivalent to solving (1.3.2).
The next result, however, shows that, in prin
ciple, the centre manifold can be approximated to any degree of accuracy. For functions
~: mn
+
mm
Cl
which are
in a neigh
borhood of the origin define (M~)(x)
~
I
(x) [Ax + f (x, ~ (x))] 
Note that by (1.3.6), (Mh)(x) Theorem 3. origin in
Let mn
Suppose that as as 1.4.
x + 0,
~ into
l
be a
C
mm
with
x + 0,
B~
 g (x, ~ (x)) .
O. mapping of a neighborhood of the HO) = 0
and q
(M~)(x) = O(lxl )
Ih(x)  Hx) I
(x)
~' (0)
where
O.
q > 1.
Then
= O(lxlq).
Examples We now consider a few simple examples to illustrate the
use of the above results. Example 1.
Consider the system 2 3 x = xy + ax + by x 2 2 y " y + cx + dx y.
(1.4.1)
1.
6
INTRODUCTION TO CENTRE MANIFOLD THEORY
By Theorem 1, equation (1.4.1) has a centre manifold To approximate
if
cx
4
(MCP) (x) =
cP (x)
2
h (x) •
we set
2
If
h
y
4
+ O(x ).
By Theorem 2, the equation which determines the
stability of the zero solution of (1.4.1) is u
Thus the zero solution of (1.4.1) is asymptotically stable if a + c
<
0
and unstable if
a + c
>
O.
If
a + c
we have to obtain a better approximation to
\)I (x)
=
0
then
h.
O. Let
bility of the zero solution of (1.4.1) is u
Hence, if
2 5 uh(u) + au 3 + buh 2 (u) = (cd+bc)u + O(u 7 ) .
a +
0, then the zero solution of (1.4.1) is 2 asymptotically stable if cd + bc < 0 and unstable if cd + bc 2 > O. If cd + bc 2 = 0 then we have to obtain a C =
better approximation to Exercise 1.
h
(see Exercise 1). a + c = cd + bc 2 = 0
Suppose that
in Example 1.
Show that the equation which governs the stability of the zero solution of (1.4.1) is U = _cd 2u 7 + O(u 9 ). Exercise 2.
Show that the zero solution of (1.2.2) h
totically stable if
a
<
0
and unstable if
A
•
0,
IIsyml'
1.4.
Examples
7
Exercise 3.
Suppose that in equation (1.3.Z), n = 1 so that Suppose also that f(x,y) = ax P + o(lxl p + l + Iylq)
A = O. where
Zq
~
P
+
1
and
a
is nonzero.
Show that the zero
solution of (1.3.Z) is asymptotically stable if
P
a
<
a
and
is odd, and unstable otherwise.
Example Z.
where
e:
Consider the system x3 Z
x
e:x

Y
y
+ Y
+

is a real parameter.
xy Z x
(1.4.Z)
The object is to study small
solutions of (1.4.Z) for small
1e:1.
The linearized problem corresponding to (1.4.Z) has eigenvalues
1
and
e:.
This means that the results given
in Section 3 do not apply directly.
However, we can write
(1.4.Z) in the equivalent form x3 Z
x
e:x

Y
y
+ Y
e:
O.
+
xy Z x
When considered as an equation on (1.4.3) is nonlinear.
(1.4.3)
lR
3
the
e:x
term in
Thus the linearized problem correspond
ing to (1.4.3) has eigenvalues
1,0,0.
The theory given in
Section 3 now applies so that by Theorem 1, (1.4.3) has a two dimensional centre manifold
y
To find an approximation to
h
h(x,e:), Ixl <
13i'
le:I < 13 Z '
set Z
e:) .
C
is a
Then, if
cp(x,e:) • x Z
homogeneous cubic in
(M
and
e:.
=
O(C(x,e:))
By Theorem 3,
where
1.
8
h(x,e:)
=
INTRODUCTION TO CENTRE MANIFOLD THEORY
_x Z + O(C(x,e:)).
Section Z.6).
Note also that
h(O,e:)
=
0
(see
By Theorem Z the equation which governs small
solutions of (1.4.3) is
u
e:u  Zu
£
o.
The zero solution
3
+ O(luIC(u,e:)) (1.4.4)
(u,e:)
=
(0,0)
of (1.4.4) is stable so
the representation of solutions given by Theorem Z applies here.
For
oZ
<
e:
0
<
the solution
u
=
0
of the first
equation in (1.4.4) is asymptotically stable and so by Theorem Z the zero solution of (1.4.Z) is asymptotically stable. For
0
<
e:
<
0Z' solutions of the first equation in
(1.4.4) consist of two orbits connecting the origin to two small fixed points.
Hence, for
0
<
e:
<
0z
the stable mani
fold of the origin for (1.4.Z) forms a separatrix, the unstable manifold consisting of two stable orbits connecting the origin to the fixed points. Exercise 4.
Study the behavior of all small solutions of 3 + e:w + w = 0 for small e:.
w+ w
Example 3.
where
e:
>
Consider the equations
0
y
y + (y+c)z
e:z
y  (y+l) z
is small and
(1.4.5)
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
1.4.
Examples
9
z =
Ly+l
(1.4.6)
and substituting this into the first equation in (1.4.5) leads to the equation
2l
y
where
A
=
(1.4.7)
l+y
1  c.
Using singular perturbation techniques, it was shown in [33) that for
E
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. Let to
t
t
=
ET.
We denote differentiation with respect
and differentiation with respect to
by
T
by
Equation (1.4.5) can be rewritten in the equivalent form
where 1,
=
f(y,w)
d(y,w)
w'
w + y2  yw + Ef(y,w)
E
0
I
y + (y+c)(yw)
(1.4.8) has a centre manifold
proximation to (M
y'
w
w
=
=y
 z.
h(y,E).
By Theorem
To find an ap
set 2
E
=
=
h
and
(1.4.8)
y
2
 AEY
then
(M
=
O(lyl
3
3
+ lEI)
so
that by Theorem 3,
My Theorem 2, the equation which determines the asymptotic
hchavior of small solutions of (1.4.8) is
10
1.
INTRODUCTION TO CENTRE MANIFOLD THEORY
= e:f(u,h(u,e:))
u'
or in terms of the original time scale u
= f(u,h(u,e:)) =
2
,,(uU ) +
Again, by Theorem 2, if z(O)
e:
O(Ie:ul
+
3 luI ).
(1.4.9)
is sufficiently small and
y(O),
are sufficiently small, then there is a solution
of (1.4.9) such that for some
y
>
0,
yet)
u(t)
z (t)
yet)  h(y(t),e:)
+
u(t)
O(e yt/e:) +
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) '" yet) 
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.
Here we
briefly indicate the procedure involved there.
If
Yo
r
1,
then
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,e:)
e:
small and
0
~
y
w <
=
m
Thus, we exh(y,e:) (m
for
= 0(1)),
close to the curve (1.4.11)
w
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 u •
f u hUE
1.5.
Bifurcation Theory
1.5.
Bifurcation Theory
11
Consider the system of ordinary differential equations
w = F(w,e:)
(1.5.1)
F(O,e:) :: 0 where
w
n m and
say that
e:
is a pdimensional parameter.
e:
€ lR +
= 0 is a bifurcation point for (1.5.1) if the
qualitative nature of the flow changes at in any neighborhood of e:
2
We
e:
=
e:
=
0; that is, if
there exist points
0
and
such that the local phase portraits of (1.5.1) for
e: = e: l
e: = e: 2 are not topologically equivalent. Suppose that the linearization of (1.5.1) about and
w
0
is
w= If the eigenvalues of then, for small
C(e:)w.
C(O)
all have nonzero real parts
le:l, small solutions of (1.5.1) behave like
solutions of (1.5.2) so that point.
(1.5.2)
e:
= 0 is not a bifurcation
Thus, from the point of view of local bifurcation
theory the only interesting situation is when
C(O)
has eigen
values with zero real parts. Suppose that parts and
m
C(O)
has
n
eigenvalues with zero real
eigenvalues whose real parts are negative.
are assuming that
C(O)
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 CeO)
we can rewrite (1.5.1) as
12
1.
where
x
€
ffin, y
€
INTRODUCTION TO CENTRE MANIFOLD THEORY
X
Ax
+
f(x,y,E)
y
By
+
g(X,y,E)
E:
0
ffim, A
is an
n
x
values all have zero real parts, B
n
(1.5.3)
matrix whose eigen
is an
m
x
m matrix
whose eigenvalues all have negative real parts, and g
f
and
vanish together with each of their derivatives at
(X,y,E)
=
(0,0,0).
By Theorem I, (1.5.3) has a centre manifold Ixl <
°1 ,
lEI < 02'
y
=
h(x,E),
By Theorem 2 the behavior of small solu
tions of (1.5.3) is governed by the equation
Ii
Au
E
O.
+
f(u,h(u,E) ,E) (1.5.4)
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 [66] where a model of the dynamic behavior of a continuous stirred tank reactor is studied.
The model
con~l~t'
of "
1.6.
Comments on the Literature
13
parameter dependent second order system of ordinary differential equations.
The authors show that there are 35 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 different
ia1 equations.
Throughout Chapter 2 we use methods that 111m..;,
in
lin
nh";n,,c.
CHAPTER 2 PROOFS OF THEOREMS
2.1.
Introduction In this chapter we give proofs of the three main
theorems stated in Chapter 1.
The proofs are essentially
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.
I· I
in finite dimensional
11'11 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 (2.2.1) where (0,0).
g
g(x l ,x 2) We prove that (2.2.1) has a local centre manifold. is smooth and
Z.Z.
A Simple Example
Let
1jJ: lR
Z
15
... lR
such that
1jJ(x l ,x Z) the origin. Define
..
be a
C
for
1
G by
function with compact support in a neighborhood of
(xl,x Z)
1jJ(xl'x Z)g(xl'x Z) .
G(xl'xZ)
We
prove that the system of equations
y
0,
(Z.Z.Z)
has a centre manifold G(Xl,X Z)
=
g(xl,x Z)
Since in a neighborhood of the origin, this
Z
Z
y = h(xl,x Z)' xl + X z < 0 local centre manifold for (Z.Z.l).
proves that
for some
0, is a
The solution of the first two equations in (Z.Z.Z) is Xl(t) = zl + zzt, xZ(t) = zz, where
yet)
=
h(xl(t) ,xz(t))
xi(O) = zi'
If
is a solution of the third equation in
(Z.Z.Z) then
To determine a centre manifold for (Z.Z.Z) we must single out :I
special solution of (Z. Z. 3). and
.ill
G(xl,x Z)
Since
is small for
xz' solutions of the third equation in (Z.Z.Z)
behave like solutions of the linearized equation
y
=
yo
rhe general solution of (Z.Z.Z) therefore contains a term I ike e t As t ..... , this component approaches the origin perpendicular to the I~
tangent to the
Inate the
e
t
zl'zZ
zl'zZ
plane.
Since the centre manifold
plane at the origin we must elim
component, that is we must eliminate the com
"onent that approaches the origin along the stable manifold .IS
t ...
DO.
To do this we solve (Z.Z.3) together with the
\onJition lim t+oo
h(ZI+zzt,zz)e
t
•
o.
(Z.Z.4)
16
2.
Integrating (2.2.3) between
PROOFS OF THEOREMS
and
0
and using
(2.2.4) we obtain
By construction, y = h(zl,z2)
is an invariant manifold for
(2.2.2).
G has compact support and
Using the fact that
that
G(x l ,x 2) has a second order zero at the origin it follows that h(zl,z2) has a second order zero at the origin;
that is, h 2.3.
is a centre manifold.
Existence of Centre Manifolds In this section we prove that the system :ic
Ax
+
f(x,y)
y
By
+
g(x,y)
(2.3.1)
has a centre manifold. values of
As before .x
€
ffin, y
€
ffim, the eigen
A
have zero real parts, the eigenvalues of B 2 have negative real parts and f and g are C functions
which vanish together with their derivatives at the origin. Theorem 1. y
=
Equation (2.3.1) has a local centre manifold 2 hex), Ixl < 0, where h is C .
Proof:
As in the example given in the previous section, we
prove the existence of a centre manifold for a modified equation. when define
Let
~: ffin ~ [0,1)
Ixl < 1 F
and F(x,y)
and
~(x)
be a
o
COO
when
~(x)
function with Ixl ~ 2.
For
£
=1
> 0
G by x f(xH£"),y),
G(x,y)·
The reason that the cutoff function
~
x
g(x~(£),y).
is only
II
function of
2.3.
x
Existence of Centre Manifolds
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
has a centre manifold Since
F
and
x
Ax
+
F(x,y)
y
By
+
G(x,y)
y
(2.3.2)
= h(x), x
G agree with
f
€
and
ffi n , for small enough g
g.
in a neighborhood of
the origin, this proves the existence of a local centre manifold for (2.3.1). For
p > 0
and
PI > 0 let X be the set of Lipschitz functions h: ffin ... ffim with Lipschitz constant PI' I hex) I < p for x € ffin and h (0) = O. With the supremum norm 11'11, X is a complete space. For
h
X and
€
Xo
€
n
ffi , let
x(t,xO,h)
be the solu
tion of x The bounds on
F
exists for all
=
Ax
and t.
+
h
F(x,h(x)), x(O,xO,h)
= xo.
(2.3.3)
ensure that the solution of (2.3.3)
We now define a new function
Th
by (2.3.4)
If
h
is a fixed point of (2.3.4) then
fold for (2.3.2). enough, T
We prove that for
is a contraction on
keg)
with
is a centre mani
P'PI' and
g
small
X.
Using the definitions of tinuous function
h
F
k(O)
and
G, there is a con
= 0 such that
2.
18
PROOFS OF THEOREMS
IF(x,y)1 + IG(x,y)1 < e:k(e:), (2.3.5)
IF(x,y)  F(x',y')1 < k(e:)[lxx'l + lyy'll, IG(x,y)  G(x',y')1 < k(e:)[lxx'l + lyy'll, for a11
x, x'
€
JRn
and a11
€
y, y'
Since the eigenvalues of
JRm
Bs
and
r s
0
>
€
S,C
such that for
yl < CeSs Iyl.
Since the eigenvalues of each
IYI, Iy'l < e:.
B all have negative real
parts, there exist positive constants s < 0 and y € JRm, Ie
with
(2.3.6)
a11 have zero real parts, for
A
there is a constant
M(r)
such that for
x
€
JRn
JR, (2.3.7)
Note that in general, M(r) If
~
as
00
r
~
O.
P < e:, then we can use (2.3.5) to estimate terms
involving
G(x(s,xO,h) ,h(x(s,xO,h))) and similar terms.
We
shall suppose that p < e: from now on. If Xo € JRn , then using (2.3.6) and the estimates on G and
h, we have from (2.3.4) that (2.3.8) Now let
on
F
and
x o ' xl
€
n JR .
Using (2.3.7) and the estimates
h, we have from (2.3.3) that for
r
>
0
and
t < 0,
Ix(t,xO,h)  x(t,xl,h) I ~ M(r)ertlxoxil + (l+Pl)M(r)k(e:)
Ito
er(st) Ix(s,xo,h)  x(s,xl,h) Ids.
By Gronwall's inequality, for
t
~
0,
2.4.
Reduction Principle
19
(2.3.9) where
y =
r + (l+Pl)M(r)k(E).
Using (2.3.9) and the bounds
on
G and
h, we obtain from (2.3.4)
if
E and
r
are small enough so that hI' h2 E X
Similarly, if
S
Xo
and
By a suitable choice of
E
and
(2.3.8), (2.3.10) and (2.3.11) that
T
>
y. n
lR , we obtain
r, we see from
is a contraction on
x.
This proves the existence of a Lipschitz centre manifold l for (2.3.2). To prove that h is C we show that T is a
contraction on a subset of ferentiable functions.
X
consisting of Lipschitz dif
The details are similar to the proof
given above so we omit the details.
To prove that
h
is
2 C
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
2.
Lemma 1.
Let
(x(t) ,yet))
!(x(O) ,yeO)) I C l
and
t >
Proof:
Let
(x(O) ,yeO))
be a solution of (2.3.2) with
sufficiently small.
Then there exist positive
such that
~
for all
PROOFS OF THEOREMS
o. (x(t) ,yet))
be a solution of (2.3.2) with
sufficiently small.
Let
z(t)
=
yet)  h(x(t)),
then by an easy computation
z = Bz
+ N(x,z)
(2.4.2)
where N (x, z)
h' (x) [F(x,h(x))  F(x,z+h(x))l + G(x,z + hex))  G(x,h(x)).
Using the definitions of
F
and
there is a continuous function that
IN(x,z)1 < o(£)lzl
if
G and the bounds on 0(£)
with
Izl < £.
0(0)
h,
= 0 such
Using (2.3.6) we ob
tain, from (2.4.2),
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
= Al + A2
where
A2
is nilpotent and (2.4.3)
Since
A2
is nilpotent, we can choose the basis such that
2.4.
Reduction Principle
21
(2.4.4) where
is defined by (2.3.6).
~
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.
Let
(x(O),y(O)) tion
y
Proof:
be a solution of (2.3.1) with
sufficiently small.
u(t)
where
(x(t),y(t))
Then there exists a solu
of (2.4.1) such that as
> 0
x (t)
u(t)
y (t)
h(u(t))
t ... DO,
0 (e yt)
+
(2.4.5)
O(e yt )
+
is a constant depending only on
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
I (x(O),y(O)) I
G are equal to
f
and
origin this proves Theorem 2.
(x(t),y(t))
is a solution
sufficiently small. g
Since
F
in a neighborhood of the We divide the proof into two
$teps. I.
Let
ficiently small.
uo
€
lR
Let
n
and u(t)
Zo
€
lR
m
with
I (uO'zO) I
suf
be the solution of (2.4.1) with
u(O) • u O' We prove that there exists a solution (x(t),y(t)) of (2.3.2) with yeO)  h(x(O)) • Zo and x(t)  u(t),
22
2.
yet)  h(u(t)) II.
exponentially small as
=
(xO,zO)
where
ffin+m
xo
Let
I.
(x(t) ,yet))
a solution of (2.4.1).
into
x(O).
ficiently small, we prove that
u(t)
~
t
00.
By Step I we can define a mapping
neighborhood of the origin in S(uO,zO)
PROOFS OF THEOREMS
(xO'zO)
S
from a
ffin+m
For
by
I (xO,zO) I
suf
is in the range of S.
be a solution of (2.3.2) and Note that i f
is suffici
u(O)
ently small,
u = Au
+ F(u,h(u))
since solutions of (2.4.1) are stable.
=
~(t)
h(x(t)),
x(t)  u(t). Bz +
M where
N
(2.4.6) Let
z(t) .. yet)

Then by an easy computation N(~+u,z)
(2.4.7)
+ R(~,z)
(2.4.8)
is defined in the proof of Lemma 1 and 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.
a > 0, K > 0, let
For
functions
~: [0,00) ~ffin
If we define
II ~ II
plete space.
Let
with
z(t)
Define
T~
(uO'zO)
for all
X
t > O.
is a com
be sufficiently small and let u(O) = u ' O
be the solution of (2.4.7) with
Given z(O)
=
~
E X
z00
e 1
(t)
[A Hs) + R(~(s) ,z(s))lds. 2
(2.4.9)
We solve (2.4.9) by means of the contraction mapping principle.
If
u(t)
by
A (ts) (T~)
I~(t)eatl ~ K
sup{ I ~ (t) eat I: t ~ O}, then
be the solution of (2.4.6) with let
X be the set of continuous
~
is a fixed point of
T,
th~n
retracing our
2.4.
Reduction Principle
steps we find that
23
= u(t)
x(t)
is a solution of (2.3.2). a
+ ~(t),
We can take
yet) a
= z(t)
to be as close to
as we please at the cost of increasing
K and shrinking
the neighborhood on which the result is valid.
K=1
however, we take
and
= a where
2a
h(x(t»
+
For simplicity
a
is defined by
(2.3.6) • Using the bounds on N(~,O)
k (0)
=
F,G,h
and the fact that
= 0, there is a continuous function k(e:) 0
such that if
~1'~2 E lR
n
with
and
I z.1 I < e:, then IN(~l'zl)
 N(~2,z2)1 < k(e:)[lzlll~1~21
IR(~l'zl)
 R(~2,z2)1 ~ k(e:)[lz l  z 2 1
+
Iz l  z 2 11 (2.4.10)
+
1~1~211.
From (2.4.7), Iz(t)1
~
Clzoleat
+
Ck(e:)( ea(ts)lz(s)lds
o
where we have used (2.3.6) and (2.4.10).
By Gronwall's in
equality (2.4.11) where
al
a  Ck(e:).
From (2.4.9), if
e:
IS
sufficiently
small,
IT~(t) I ~
t at
f""
k(e:) t (e
+
as
+
C1zole
a1 s )ds
~ e 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
~1'~2
X.
E X and let
ing solutions of (2.4.7) with wet)
= zl(t)  z2(t).
zi (0)
zl,z2
= zOo
be the correspondWe first estimate
From (2.4.7) and (2.4.10),
2.
24
PROOF5 OF THEOREM5
Us ing (2.4.11),
where
Cl
is a constant, so that by Gronwall's inequality
Iw(t)1
=
Iz l (t)z2(t)1 ~ Cl k(e:)II
Using (2.4.4) and (2.4.12), for
where
ex
<
e:
 f3 t
(2.4.12)
1
sufficiently small,
1.
The above analysis proves that for each ficiently small, T
has a unique fixed point.
neighborhood of the origin in ffin+m
tinuous uniform contraction. depends continuously on
x0
Define
= U o +
zo' 5
U
by
5 5ince
is continuous.
5(u O'ZO)
U is a
T: X x U
~
X is a con
This proves that the fixed point and
o
If
suf
then it is easy to re
peat the above analysis to show that
II.
(uO,zO)
zO°
= (xO,zO)
where
depends continuously on
We prove that
5
o and is onetoone, so U
that by the Invariance of Domain Theorem (see [11) or [60)) 5
is an open mapping.
the range of
5
5ince
5(0,0) = 0, this proves that
is a full neighborhood of the origin in ffin+m.
Proving that
5
is onetoone is clearly equivalent
+
U
o
2.5.
Approximation of the Centre Manifold
25
~O(t)
uniqueness of solution of (2.3.2), uO(t) + ~l(t) I
for all
(2.4.6) with
t > 0, where ui(O)
=
ui .
ui(t)
=
ul(t) +
is the solution of
Hence, for
uO(t)  u l (t)
=
~l
~
t
0, ~O(t).
(t) 
(2.4.13)
Since the real parts of the eigenvalues of A are all zero, e:t for any e: > 0 unless ul(O) = limlul(t)  uo(t)le t+oo at Also, l~i(t)1 < efor all t > O. It now follows U (0) . o from (2.4.13) that
S
is onetoone and this completes the
proof of the theorem. 2.5.
Approximation of the Centre Manifold For functions
~: mn + mm
Cl
which are
in a neigh
borhood of the origin define (M~)
Theorem 3.
(x)
=
~'(x)
[Ax + f(x,Hx))l  BHx)  g(x,Hx)).
Suppose that
(M~) (x)
=
Proof:
Let
0 (I x I q)
as
6: mn +mm
~(O)
x + 0
0,
=
where
~'(O)
small.
N(x) where N(x)
=
and that Then as
x + 0,
be a continuously differentiable 6(x)
=
~(x)
for
Set
6' (x) [Ax + F(x,6(x))1  B6(x)  G(x,6(x)),
F
=
0
q > 1.
function with compact support such that
Ix I
=
and G are defined in Theorem 1. O(lxl q ) as x + o. In Theorem 1, we proved that
a contraction mapping
T: X + X.
Sz • T(z+O)  0; the domain of
h
Note that
was the fixed point of
Define a mapping S
(2.5.1)
S
by
being a closed subset
2.
26
Y c X.
Since
T
is a contraction mapping on
contraction mapping on Y
= {z
E X:
PROOFS OF THEOREMS
Y.
For
K
Iz(x) I < Klxl q
If we can find a
K such that
>
0
for all S
maps
X, S
is a
let x Y
E JR
n
}.
into
Y then we
will have proved the theorem. We first find an alternative formulation of the map For
z E Y x
let
x(t,x ) O
S.
be the solution of
Ax + F(x,Z(x) + 6(x)),
x(o,x ) = x(O). o
(2.5.2)
From (2.3.4)
Now
o

f
_00
d
lS[e
 Bs
6(x(s,x O)))ds
d 6(x(s,x )))ds. e Bs [B6(x(s,x O))  ds O Writing
x
for
x(s,x O)
etc., from (2.5.1) and (2.5.2)
B6(x)  6' (x) [Ax + F(x,z(x) + 6(x)))
B6(x)
N(x)  G(x,6(x)) + 6' (x) [F(x,6)  F(x,Z(x) + 6(x))). Using
= T(z+6)
Sz
 6
and the above calculations (2.5.3)
where
x(s,x O)
Q(x,z)
=
is the solution of (2.5.2) and
G(x,6+z)  G(x,6)  N(x) + 6'(x)[F(x,6) F(x,6+z)).
(2.5.4)
2.5.
We now show that By
27
Approximation of the Centre Manifold
choosing
S
maps
Y
into
Y
for some
K
>
O.
suitably, we may assume that le(x) 1 < g x ElRn. Since N(x) = O(lxl q ) as x'" 0,
for all
e
(2.5.5) where
C 1
is a constant.
Now
IQ(x,z) 1 < IQ(x,O) 1 IN(x) 1
+
IQ(x,z)  Q(x,O) 1
We can estimate
IQ(x,z)  Q(x,O) 1
constants of
and
function
F
keg)
with
G. k(O)
(2.5.6)
IQ(x,z)  Q(x,O) I·
+
in terms of the Lipschitz
Using (2.3.5), there is a continuous =
0, such that (2.5.7)
IQ(x, z)  Q(x, 0) 1 oS. keg) 1z 1 for and
Izl oS. g. Using (2.5.5), (2.5.6), (2.5.7), for x E lR n , we have that IQ(x,z) 1 oS. C11xlQ
+
z E Y
keg) Iz(x) 1
(2.5.8)
oS. (C 1 + Kk ( g)) 1 x 1 Q •
Using the same calculations as in the proof of Theorem 1, if x(t,x O)
is the solution of (2.5.2), then for each
there is a constant
M(r)
r
>
0,
such that (2.5.9)
where
y = r
+
2M(r)k(g).
Using (2.3.6), (2.5.8) and (2.5.9), i f
provided
£
and
r
are small enough so that
z E Y,
a 
qy >
o.
28
2.
By choosing ~
K large enough and
small enough, we have
£
that
C2
2.6.
Properties of Centre Manifolds
K and this completes the proof of the theorem.
(1) manifold.
PROOFS OF THEOREMS
In general (2.3.1) does not have a unique centre For example, the system
x
=
x 3 ,
two parameter family of centre manifolds
y
y= =
y, has the
h(x,c ,c 2) l
where c
1 2 l exp ( 2" x ),
0
1 2 c 2 exp(  2" x ), However, if
hI
and
g
are
0 <
o.
q > 1. (2)
[441.
and
> 0
are two centre manifolds for q (2.3.1), then by Theorem 3, hex)  hl(x) = O(lxl ) as x ~ 0 for all
h
x x x
If
If
f
f and
g
Ck , (k ~ 2), then
h
is
Ck
are analytic, then in general (2.3.1)
does not have an analytic centre manifold, for example, it is easy to show that the system 3
x,
x
Y
=
y + x
2
(2.6.1)
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
(xO'YO)
y
hex)
For
is a small equilibrium point be any centre manifold for
Then by Lemma 1 we must have
Similarly, O is a small periodic orbit of (2.3.1), then r must lie
on all centre manifolds.
Yo = h(x ).
2.6.
Properties of Centre Manifolds
(4)
Suppose that
29
(x(t),y(t))
is a solution of (2.3.1)
which remains in a neighborhood of the origin for all
t
>
O.
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. (5)
In many problems the initial data is not arbit
rary, for example, some of the components might always be nonnegative. Suppose S cffi n +m with 0 € S and that (2.3.1) defines a local dynamical system on
S.
It is easy to check,
that with the obvious modifications, Theorem 2 is valid when (2.3.1) is studied on Exercise 1.
S.
Consider x
3 x,
Y=
2
y + x .
(2.6.1)
Suppose that (2.6.1) has a centre manifold h
is analytic at
x = O.
a n+2
= na n
x.
2 n=2
anx
n = 2,4, ... , with
for
n
a 2n + l = 0
Show that
h(x), where
Then
hex) for small
y
for all 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 Cr
r, (2.3.1) has a
g
are
COO
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
~eneral (2.3.1) does not have a
rand
g
are analytic.
COO
centre manifold, even if
2.
30
PROOFS OF THEOREMS
Consider 3 x =  e:X  x ,
y +
y
Suppose that (2.6.2) has a for
o.
Ixl < 0, le:I <
h(x,(2n)
1
)
is
00
C
e:
= O.
(2.6.2)
centre manifold y = h(x,e:) n> 0 1 . Then since
COO
Choose
in
2
x ,
x, there exist constants
such that hex, (2n)
1
2n )
=
L
a.x i 1
i=l for
Ixl
that i f
small enough.
Show that
o
a.
1
for odd
i
and
n > 1, (2i) (2n)
(1
1
)a 2i
(2i2)a
_ , i 2i 2
2, ... ,n
(2.6.3)
Obtain a contradiction from (2.6.3) and deduce that (2.6.2) does not have a Exercise 3. odd, that is
COO
centre manifold.
Suppose that the nonlinearities in (2.3.1) are f(x,y)
=
f(x,y), g(x,y)
that (2.3.1) has a centre manifold [The example
hex) .
•
x
y
=
g(x,y).
= hex)
with
Prove
hex)
3
= x , y = y, shows that if h is
any centre manifold for (2.3.1) then
hex)
~
hex)
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
2.7.
31
Global Invariant Manifolds y'
e:f(y,w)
w'
w + y2  yw + Ef(y,w)
E'
where
0

= O.
f(O,O)
(2.7.1)
Because of the local nature of our results
on centre manifolds, we only obtained a result concerning small initial data.
= SeT), where
v = w(l+y) + y2, s
Let
= 1 + yeT); then we obtain a system of the form
S'(T)
(2.7.2)
where
gi(O,O)
(y,O,O)
=
E'
0
0, i
= 1,2.
Note that i f
is always an equilibrium point for (2.7.2) so we ex
pect that (2.7.2) has an invariant manifold fined for
1
Theorem 4.
where
<
Y
<
f,g O.
>
y'
By + Eg(X,y,E)
E'
0
<
m,A,B,f
and
are
y
=
cIEI, where
Proof:
Let
if
<
by
de
(2.7.3)
and A,B are as in Theorem 1. C2 with f(O,O,O) 0, g(O,O,O)
Then there is a
Ih(x,E) I
Ixl
= h(y,E)
sufficiently small.
Ax + Ef(x,y, E)
x E mn, y E mm
m
E
x'
invariant manifold
G
m, say, and
v
Consider the system
also that Let
y F 1, then
6
h(x,E), C
>
0
Suppose =
O.
such that (2.7.3) has an
Ixl < m, lEI < 0, with
is a constant which depends on
g. 1jI: mn
m and
+
[0,1)
tlt(x)
.0
be a if
ex>
function with
C Ixl
>
m + 1.
Define
1jI(x) F
=1 and
32
2.
F(x,y,E)
=
e:f(x1jJ(X),y,E),
G(X,y,E)
PROOFS OF THEOREMS
Eg(X1jJ(X),y,E).
We can then prove that the system Xl
Ax
+
F(x,y,E)
yl
By
+
G(X,y,E)
has an invariant manifold ficiently small.
(2.7.4)
= h(x,E), x E mn , for
y
lEI
suf
The proof is essentially the same as that
given in the proof of Theorem I so we omit the details. Remark.
If
x
=
(x ,x 2 '" .,x n ) then we can similarly prove l m. < x.1 < m1.. the existence of h(x,E) for 1 The flow on the invariant manifold is given by the
equation U
I
= Au
+
Ef(u,h(u, E)) .
(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.
~: mn + l +mm
Let
I(M~)(x,E)1 < CE P
integer, C (M~)
(X,E)
for
Ixl ~m
~(O,O) = 0 and
satisfy where
p
is a positive
is a constant and = DX~(X,E)
[Ax
+ Ef(x,~(x,E))l

B~(x,E)
 Eg(x,Hx, E)) . Then, for
Ixl ~ m, Ih(x,E)  Hx,E)1 < CIE P
for some constant
CI .
2.8.
Centre Manifold Theorems for Maps
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).
2.8.
Centre Manifold Theorems for Maps 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 mn
x= where for T.
f
is smooth and
A is a real parameter.
Suppose that
A ' (2.8.1) has a periodic solution y with period o One way to study solutions of (2.8.1) near y for
define
small is to consider the Poincare map PP)
let
y
cross section of neighborhood of (z)
=
be some point on through
y
y
in
U.
x(s), where
=z
and
(See
[51)
for the details).
If
s
peA)
>
0
y
peA).
y, let
and let
U l
To
U be a local be an open
Then
P(A): Ul + U is defined by is the solution of (2.8.1) with
x(t)
x(o)
(l
(2.8.1)
A
IA  Aol
1'(,\)
f(x,A)
is the first time
x(t)
hits
U.
has a fixed point then (2.8.1) has a periodic
rhi t wi th per iod close to
point of order n, (pp)k z
r
T. z
If for
P P)
has a periodic
1 < k < n  1
and
l,np)Z = z) then (2.8.1) has a periodic solution with period , lose to
nT.
If
PP)
preserves orientation and there is a
, lused curve which i5 invariant under
PP)
then there exists
2.
34
PROOFS OF THEOREMS
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' (AO)
has essentially the same behavior as
pP.)
IA 
AO I
small.
Hence in this case, for
solutions of (2.8.1) near
A = AO'
for small,
have the same behavior as when
y
If some of the eigenvalues of
PI(AO)
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. T: ~n+m
Let
+
~n+m
(Ax + f(x,y), By + g(x,y))
T(x,y)
x E ~n, y E~m, A and
where
that each eigenvalue of value of and
have the following form:
B
f,g
(2.8.2)
B are square matrices such
A has modulus
has modulus less than
1
1, f
and each eigenand g are C2
and their first order derivatives are zero at the
origin. Theorem 6. T. tion
There exists a centre manifold
More precisely, for some h:
Ixl < e:
~n + ~m
and
with
(xl'Yl)
h (0) =
e: > 0
~n + ~m
there exists a
C2
for func
= 0, h' (0) = 0 such that
T(x,h(x))
In order to determine
h:
h
By (2.8.2) this is equivalent to
implies
Yl
=
h(x l ).
we have to solve the equation
2.8.
Centre Manifold Theorems for Maps
h(Ax + f(x,h(x))) For functions
cp: IR
Mh =
Theorem 7.
o
CP' (0) Then
hex)
= Bh(x) + g(x,h(x)).
+ IRm define
MCP
by
cp(Ax + f(x,cp(x)))  BCP(x)  g(x,cp(x))
(MCP) (x) so that
n
35
o.
Let and
=
cp:IRn+IRm bea (MCP)(x) = O(lxl q )
cp(x) + O(lxl q )
as
Cl
map with
as
x + 0
CP(O)=O,
for some
q> 1.
x + O.
We now study the difference equation xr+l
Ax
Yr+l
BY r + g(xr,y r ) .
r + f(xr,y r )
(2.8.3)
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) Theorem 8.
(a)
Suppose that the zero solution of (2.8.4) is
stable (asymptotica11y stable) (unstable).
Then the zero solu
tion of (2.8.3) is stable (asymptotica11y stable) (unstable). (b) stable.
Let
Suppose that the zero solution of (2.8.3) is (xr'Yr)
be a solution of (2.8.3) with
of (2.8.4) r such that IX r  uri 5 KSr and IY r  h(u r ) I 5 KSr for a11 r where K and S are positive constants with S < 1.
sufficiently small.
Then there is a solution
(xl'Yl)
u
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
equations case. Exercise 4.
Show that the zero solution of x
r
is asymptotically stable if
a
>
O.
PROOFS OF THEOREMS
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 r + where
f
+ fer)
=
(3.1.1)
0
is a smooth function with fer)
where
r
a
r
3
+
is a constant.
ar
5
+
7 OCr )
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 resuIts. We first put (3.1.1) into canonical form. x
= r
+
r,
y
Let
= r, then x 
f(xy) (3.1.3)
y •  y  f (x  y) . "!>7
3.
38
EXAMPLES
By Theorem 1 of Chapter 2, (3.1.3) has a centre manifold y
= h(x).
By Theorem 2 of Chapter 2, the equation which
determines the asymptotic behavior of small solutions of (3.1.3)
is feu  h(u)).
Using (3.1.2) and
(3.1.4)
h(u) u
(3.1.S)
Without loss of generality we can suppose that the solution u(t)
of (3.1.S) is positive for a11
t > O.
Using L'Hopital's
rule, . u 1 1 = 1 1m ~ = lim t t~oo
Hence, if
wet)
u
t~oo
u(t)
=
1
is the solution of _w 3 ,
then
fU (t) s  3 ds.
w(t+o(t)).
w(O)
1,
(3.1.6)
Since (3.1.7)
where
C
is a constant, we have that (3.1. 8)
To obtain further terms in the asymptotic expansion of u(t), we need an approximation to (Mcp)(x) If
cp(x)
=
h(u).
To do this, set
cp'(x)f(xcp(x)) + cp(x) + f(xcp(x)).
= x 3 then
Theorem 3 of Chapter 2, hex) = _x 3 + O(x S). into (3.1.4) we obtain
so that by Substituting this
3.2.
Hopf Bifurcation
39
(3.1.9)
u
Choose
T
so that
grating over
[T, t)
1 w (u(t)) where
w
u(T)
=
= 1.
Dividing (3.1.9) by
u 3 , inte
and using (3.1. 8), we obtain
t + constant + (3+a)
is the solution of (3.1.6).
f:
u 2 (s)ds
(3.1.10)
Using (3.1.8) and
(3.1.9), t
fTU
2
ft
(s)ds
T
=
U(()~s u s
+ ft O(u 4 (s))ds T
(3.1.11)
In t 1 / 2 + constant + 0(1).
Substituting (3.1.11) into (3.1.10) and using (3.1.7), 3/2 1 t 1 / 2 __ t _ _ [(a+3)ln t + C) + 0(t 3 / 2 ) (3.1.12)
u(t)
12 where
412
C
is a constant.
If
(x (t) ,y(t))
is a solution of (3.1.3), it follows
from Theotem 2 of Chapter 2 that either
(x(t) ,yet))
tends to
zero exponentially fast or x (t) where 3.2.
u(t)
= ±u(t),
yet)
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 ~2,
3.
40
EXAMPLES
f(x,a), such that
f(O,a)
for all
0
=
sufficiently small
that the linearized equation about Yea)
±
iw(a)
where
yeO)
= 0,
x
=
0
Assume
has eigenvalues
Wo + O.
w(O)
a.
We also assume
that the eigenvalues cross the imaginary axis with nonzero speed so that
+ O.
y'(O)
Since
y'(O)
+ 0,
by the implicit
function theorem we can assume without loss of generality that
yea)
=
a.
By means of a change of basis the differen
tial equation takes the form
:it
A(a)x
+
(3.2.1)
F(x,a),
where A(a)
F(x,a)
=
2
O(lxl ).
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, (ii) a
=
0, (iii) a
>
However, further conditions on the nonlinear terms are required to determine the specific type of bifurcation. Exercise
1.
Use polar coordinates for 2
aX l  wX 2
+
Kxl(x l
wX
+
Kx (x 2 2 l
l
+
aX
2
to show that case (i) applies if plies i f
K < O.
K
>
0
and case (iii) ap
O.
3.2.
Hopf Bifurcation
41
To find periodic solutions of (3.2.1) we make the substitution Xl where
E
Er sin 8, a
Er cos 8, x 2
=
is a function of
Ea,
+
(3.2.2)
After substituting (3.2.2)
a.
into (3.2.1) we obtain a system of the form r
dar
+
2
r C3 (8,aE)1
+
2 3
E r C4 (8,aE)
+
3
O(E )
We now look for periodic solutions of (3.2.3) with and of
r 8
E
(3.2.3)
+
0
C3 and C4 are independent and the higher order terms are zero then the first near a constant
rOo
If
equation in (3.2.3) takes the form r
E [ar
+
2
Br 1
+
2
3
E Kr .
(3.2.4)
r = r O' where rO is a zero of the right hand side of (3.2.4). We reduce the
Periodic solutions are then the circles
first equation in (3.2.3) to the form (3.2.4) modulo higher order terms by means of a certain transformation. out that the constant
B is zero.
It turns
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
K. and let
Let
where
B~1
(xl'x 2 ).
K
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
• 3,4,
42
EXAMPLES
3.
1
(cos e)Bi_l(cOs e, sin e,a) +
Lemma 1.
(3.2.6)
2
(sin e)Bi_l(cos e, sin e,a).
There exists a coordinate change 
r
=
r
+
£ul(r,e,a,£)
+
£ 2u 2 (r,e,a,£)
which transforms (3.2.3) into the system
r
Ear
Wo where the constant
+
£ 23 r K
+
0(£ 3 ) (3.2.7)
+
0(£)
K is given by (3.2.8)
where
C3
D3 (e,0)
C4
and =
are given by (3.2.6) and
(cos e)B~(cOS e, sin e,O)  (sin e)B~(COS e, sin e,O).
The coordinate change is constructed via averaging.
We
refer to [19) for a proof of the lemma. If
K
= 0 then we must make further coordinate changes.
We assume that
K
+0
from now on.
Recall that we are looking for periodic solutions of (3.2.7) with
£
+
gests that we set
0 a
and
r
near a constant
rOo
This sug
= sgn(K)£ and rO = \K\1/2.
The next
result gives the existence of periodic solutions of
(3.2.9)
with
r  rO
Lemma 2. £
small.
Equation (3.2.9) has a unique periodic solution for
small and
r
in a compact region either for
£
>
0
(whrn
3.2.
Hopf Bifurcation
K < 0) or
E < 0
43
(when
K > 0).
and the period of the solution
Also
T(E)
The periodic solution is stable if
is given by
K< 0
and unstable if
o.
K >
Lemma 2 is proved by a simple application of the implicit function theorem.
We again refer to [19) for a proof.
Lemma 2 also proves the existence of a one parameter family of periodic solutions of (3.2.1).
We cannot immedi
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
+
aE
and by choosing
E
=
sgn(K)a.
In order to justify the scaling suppose that Xl
= R cos e, x 2
bifurcating from
e
R sin xl
= x2
O.
aR
R When
is a periodic solution of (3.2.1)
+
a
+
aE.
R satisfies
O(R 2 ) .
R o
R attains its maximum,
justifies the scaling
Then
so that
a
= O(R).
This
A similar argument applied to
periodic solutions of (3.2.7) justifies the choice of
E.
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
x
K defined by (3.2.8) is
a > 0
(when
K < 0) or
= R cos e, y = R sin e, then the
periodic solution has the form
44
3.
with period
R(t,a)
laK l l l / 2 + O(lal)
6(t,a)
l 2 Olot + 0(lal / ) 2 + 0(lal l / ).
= (2~/OlO)
,(a)
tion is stable if
K< 0
EXAMPLES
The periodic solu
and unstable if
K > O.
Finally, we note that since the value of only on the nonlinear terms evaluated at
a
=
K depends
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(O)
[0
OlO] 0
Olo
with 3.3.
nonzero. 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 response to antigen [52].
1) b  ~] [ 3 + ( aZx+ x ~
EX
where a
and
E,Ii'Yl'Y2 b
The equations are
1
a
2" 1i(1x)  a  ylab
b
ylab + Y2b
are positive parameters.
(3.3.1)
In the above model
represent certain concentrations so they must be
nonnegative.
Also, x
measures the stimulation of the system
and it is scaled so that
Ixl
~
1.
The stimulation is assumed
to take place on a much faster time scale than the response so that
E
is very small.
3.3. Hopf Bifurcation in a Singular Perturbation Problem
45
The above problem was studied in [52] and we briefly outline the method used by Merrill to prove the existence of periodic solutions of (3.3.1).
=
e:
Putting
in the first
0
equation in (3.3.1) we obtain x Solving (3.3.2) for tain
x
=
F(a,b)
3
x
1 Z)x + b

+ (a

1 Z
as a function of
O. a
(3.3.2) and
b
we ob
and substituting this into the second equa
tion in (3.3.1) we obtain 1
2"
a
o(lF(a,b))  a  ylab (3.3.3)
b
Using
0
as the parameter, it was shown that relative to a
certain equilibrium point place in (3.3.3).
(aO,b ) O
a Hopf bifurcation takes
By appealing to a result in singular per
e:
turbation theory, it was concluded that for
sufficiently
small, (3.3.1) also has a periodic solution. We use the theory given in Chapter 2 to obtain a similar result. Let bO
+0
then
(xO,aO,b O) aO
= x
be a fixed point of (3.3.1).
Y2!Yl
3
o
+
and
Y2 Yl
( 
1 2)x
xO,b O 0
+
b
O
1 Y2  0 (lx )   Y b 2 0 Yl 2 0
If
satisfy 1  2
=
0, (3.3.4)
=
O.
Recall also that for the biological problem, we must have bO > 0 b
O
and
IXol
~ 1.
We assume for the moment that
satisfy (3.3.4) and these restrictions.
these solutions are considered later. for
th~
rest of this section.
We let
Xo
The reality of aO
=
Y2!Yl
and
3.
46
Let where
1jJ
=
EXAMPLES
y = a  a O' z = b  b O' w = 1jJ (xx O)  xoY  z 2 1 3x O + a O  2· Then assuming 1jJ is nonzero, e:w
g(w,y,z,e:) f 2 (w,y,z,e:)
y
(3.3.5)
where g(w,y,z,e:)
fl(w,y,z,e:)  e:x Of 2 (w,y,z,e:)  e:f 3 (w,y,z,e:)
=
1jJw + N(w+xOY+z,y) Ii
f 2 (w,y,z,e:)
(21jJ
1
Ii
Xo  1  ylbO)y + (2 1jJ
1
 Y2)z
+
f 3 (w,y,z,e:) N(e,y)
=
1jJ
= ylbOY  YlYz 2 3
e
+ 31jJ
1
xOe
2
 yeo
In order to apply centre manifold theory we change the time scale by setting respect to
s
by
t
= e:s.
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'
e:f 2 (w,y,z,e:)
z,
e:f (w,y,z,e:) 3 O.
e:' Suppose that
1jJ
>
O.
(3.3.6)
Then the linearized system corresponding
to (3.3.6) has one negative eigenvalue and three zero eigenvalues. fold
w
By Theorem 1 of Chapter 2, (3.3.6) has a centre manih(y,z,e:).
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
y'
e:fZ(h(y,z,e:) ,y,z,e:)
z'
e:f 3 (h(y,z,e:) ,y,z,e:)
47
(3.3.7)
or in terms of the original time scale y
fZ(h(y,z,e:) ,y,z,e:)
Z
f 3 (h(y,z,e:) ,y,z,e:).
(3.3.8)
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
is given by cS
2"
1jJ
1

yz ]
J (e:)
+ 0
(e:) .
o If (3.3.8) is to have a Hopf bifurcation then we must have cS 1 trace(J(e:)) = 0 and 2" 1jJ  yz > O. From the previous ana1ysis, we must also have that with
Xo ,b O are solutions of (3.3.4)
1, b O > 0 and 1jJ > O. We do not attempt to obtain the general conditions under which the above conditions
IXol
<
are satisfied, we only work out a special case. Lemma.
Let
Y1 < ZyZ'
6(e:), xO(e:), bO(e:) tjJ
>
0, cS(e:)1jJ
1
Then for each
such that
e:
>
0, there exists
0 < ZxO(e:) < 1, bO(e:) > 0,
 ZyZ > 0, trace (J(e:)) = 0
and (3.3.4) is
~atisfied.
Proof:
Fix
and
YZ with isfy the second equation in (3.3.4) then cS 1 trace(J(O)) • ![x 01jJ
If
sat
48
3.
EXAMPLES
It is easy to show that there is a unique satisfies of
trace(J(O)) = O.
xO(O).
We now obtain
~
Clearly bO(O)
and
> 0
0(0)
that for this choice as the unique
solution of (3.3.4) and an easy computation shows that b (0) O
> 0,
and
0 (0) > 0
function theorem, for small, there exists
o(OH
1
 ZYZ > O.
By the implicit
e:, xo  xO(O), b O  bO(O) sufficiently o(e:,xO,b ) = 0(0) + O(e:) such that O
After substituting
0 = o(e:,xO,b ) into (3.3.4), another O application of the implicit function theorem gives the resuIt.
This completes the proof of the lemma. From now on we fix
and
with
YZ
Using
the same calculations as in the lemma, for each with
e:
and
0  o(e:)
xO(e:,o), bO(e:,o) trace (J)
and
0
sufficiently small there is a solution
of (3.3.4).
as a function of
Writing
xo
xO(e:,o(e:))
and
0, we have that 1
I
a
e:
ylo(e:H 3 a1"(trace(J(o))) o=o(e:) = 4(1x ) [6y l x O O
if
e:
is sufficiently small.
Hence, the eigenvalues of
cross the imaginary axis with nonzero speed at Let
For
0
Yl =
9,
(e:) z, zl = m(e:)y
0 = o(e:).
where
o(e:), (3.3.8) in these new coordinates becomes wOz l  ylm wOYl
1
(e:)ylzl
+ m(e:)(o/Z)~
1
hem
~l
(e:)zl'
J(o)
3.3. Hopf Bifurcation in a Singular Perturbation Problem
49
where
To apply the results of the last section, we need to calculate the how
K(O)
K(E)
associated with (3.3.9).
can be calculated; if
will be nonzero also.
K(O)
To calculate
is nonzero then K(O)
h(y,z,O)
K(E)
we need to know
the quadratic and cubic terms in (3.3.9) when we have to find
We shall show
E
=
O.
Thus,
modulo fourth order terms.
Let (M<j»(y,z)
=
g(<j>(y,z),y,z,O)
Then by Theorem 2 of Chapter 2, if we can find (M<j>)(y,z)
= O(y4+z4)
then
(3.3.10) <j>
such that
= <j>(y,z) + O(y4+z4).
h(y,z,O)
Suppose that (3.3.11) where
<j>. J
is a homogeneous polynomial of degree
j.
Substi
tuting (3.3.11) into (3.3.10) we obtain (M<j» (y,z)
= 1jJ<j>2(Y'z)  31jJ 1 xO(xoY+z) 2 + y(xoY+z) + 0 ( 1y 1 3 +
3
1z 1 ) .
Hence, if (3.3.12) then
(M<j>)(y,z)
(3.3.10) with (M<j»(y,z)
= o(
3 lyl 3 + IzI ).
<j>2(Y'z)
given by (3.3.12), we obtain
1jJ<j>3(Y'z) + 1jJ
2
(xoY+z)
4
4
+ y<j>2(Y'z) + O(y +z ). lienee,
Substituting (3.3.11) into
3
3.
50
CP2(y,z)  ljI
h (Y. z)
3
(xoy+z)
3
+ 6lj1
2
EXAMPLES
x O(x Oy+z)CP2(y,z)
 ljIlYCP2(Y'Z) + O(y4+Z4) where
CP2(y,z)
is defined by (3.3.12).
culated as in the previous section. depend on
and
can now be cal
The sign of
K(O)
will
is nonzero then we can
K (0)
If
K(O)
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
If
is
K(O)
K(E).
zero then we have to calculate 3.4.
K(O).
Bifurcation of Maps In this section we give a brief indication of some re
sults on the bifurcation of maps.
For more details and
references the reader is referred to the book by Iooss [40). Let let
F~:
q ~ Em
V be a neighborhood of the origin in mm and V + mm be a smooth map depending on a parameter
with
F (0) ~
0
~.
for all
If bifurcation is to
take place then the linearized problem must be critical. assume that with Fa
Fb
A(O), X(O)
has simple complex eigenvalues I, A(O)
IA(O)I
We
±l, while all other eigenvalues of
~
are inside the unit circle.
By centre manifold theory
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
~
A(~)
is a onedimencrosses the unit
circle, that is
<&1 A(~) 1 ~ for
n '" 3
or
0 4
when
~ =
O.
then in general
F
II
hi.
In
3.4.
51
Bifurcation of Maps
invariant closed curve bifurcating from An = I, n ~ 5, then in general
over, if
any period points of order However, if eral
0
]..I
n
[57,58). F ]..I
More
does not have
bifurcating from
0
[40).
is a twodimensional parameter, then in gen
F
has periodic points of order n for ]..I in a small ]..I region [4,64). The method of proof used in [4) and [64) ren lies on approximating F by the time one map of a certain ]..I differential equation; we outline a more direct proof. Suppose that 2 2 wElR,]..IElR,
F (w)
]..I
where
A(O)
n > 5.
has eigenvalues
A(]..I), I(]..I)
For definiteness we assume that
positive integer same way.
p; the case
with
An(O) = I,
n = 2p + 3
for some
n = 2p + 4
is treated in the Then by changing coordinates [40), Fn]..I has the
normal form n p 2k 2 2 A (]..I)z + z L ak(]..I)lzl + b(]..I)zP+ + R(z,z,]..I) k=l where
z = x + iy, R(z,z,]..I) = O(lzI2p+3)
and
ak(]..I), b(]..I)
are complex numbers.
We assume that after a change of para
meter, An(]..I) =
where
that
al(O)
point of
1
]..I
and
]..I =
+ i]..l2'
]..11
are nonzero.
b(O)
If
We also assume i8 is a fixed z = re
then h(r,8,]..I)
0
where h(r,8,]..I) =
]..11
I ak(]..I)r 2k + b(]..I)r2p+lein8
+ i]..l2 
+ o(r 2p + 2 ) .
k=l
]..I ... ]..I(a l (]..I)) 1 , we can assume without loss of generality that a 1 (]..I) 1. Given 8 E [0,211) , ]..11 > 0 and By redefining
.
3.
52
EXAMPLES
small enough, the equation
~2
Re(h(r,e,~))
has a unique (small) solution
= 0
r = r(~I'~2,e)
with (3.4.1)
Substituting this into the equation Im(h(r,e,~))
=0
(3.4.2)
we obtain
I Im(ak(~))r2k + Im(b(~)eine)r2p+1 + o(r2p +2). ~2(~I'~2,e) = I Im(ak(~))r2k. We now make the change k=2
~2 Let
=
k=2
of parameter the map
E
= EI
+ iE2
=
~l + i(~2  ~2)'
E is onetoone for
~ +
~
By (3.4.1),
small enough.
Equation
(3.4.2) now takes the form
Since
b(O)
equation has
~
0, by the implicit function theorem the above n
IE I < CE(2P+I)/2 2
1
fixed points for If A3 (0)
er(E), r = 1, ... ,n, for is a constant. Thus Fn has
distinct solutions '
where
~
C
in a region of width
~
O(~~n2)/2).
= 1, then for one parameter families of maps,
in general, there are bifurcating fixed points of order 3 1 and no invariant closed curves [40,41]. The case A4 (0) is more complicated. the normal form
In this case the map can be put into
3.4.
Bifurcation of Maps
=
a(~)
where
O(~).
i +
fixed points of order cate from c(~)
F~
53
~
If 4
Em
then in general either
or an invariant closed curve bifur
0; the actual details depend on
[40,41,68].
If
~
a(~),
b(~)
and
is a twodimensional parameter then
is studied by investigating bifurcations of the differen
tial equation (3.4.3) A(~),
where one map
B(~)
T(~)
and
C(~)
are chosen such that the time
of (3.4.3) satisfies
The bifurcation diagrams given in [4] for equation (3.4.3) seem only to be
conjec~ures.
studied in [54]
in the case
parameters
Re(~)
and
Bifurcations of (3.4.3) are A(~)
= ~
E
¢
with bifurcation
Re(B(~)).
We note that the above bifurcation results do not apply if the map has
special properties.
For example, F
~
could represent the phase flow of a periodic Hamiltonian system so that
F
~
is a symplectic map.
is due to Moser [53]: F(O)
The following result
if
F
is a smooth symplectic map with
= 0, then in general
F
has infinitely many periodic
points in every neighborhood of the origin if the linearized map
F'(O)
has an eigenvalue on the unit circle.
For an ap
plication of this result to the existence of closed geodesics on a manifold see [46].
CHAPTER 4 BIFURCATIONS WITH TWO PARAMETERS IN TWO DIMENSIONS
4.1.
Introduction In this chapter we consider an autonomous ordinary dif
ferential equation in the plane depending on a twodimensional parameter
E.
We suppose that the origin
point for all
E.
X
x
=
0
is a fixed
More precisely, we consider f(x,E),
x E1R
z,
E
=
(El,E Z) E1R
z,
(4.1.1)
f(O,E)  O. The linearized equation about
x
x and we suppose that
A(O)
=
0
is
A(E)x,
has two zero eigenvalues.
ject is to study small solutions of (4.1.1) for
The ob
(El'E Z)
in
a full neighborhood of the origin.
More specifically, we wish
to divide a neighborhood of
into distinct components,
such that if
E,E
E
=
0
are in the same component, then the phase
portraits of (4.1.1)E and (4.1.1)E are topologically equivalent.
We also want to describe the behavior of solutions for
each component.
The boundaries of the components correspono
4.1.
Introduction
55
to bifurcation points. Since the eigenvalues of that either (i)
A(O)
A(O)
are both zero we have
is the zero matrix, or Cii)
A(O)
has
a Jordan block,
A(O)
~
= [:
l
There is a distinction between (i) and (ii) even for the study of fixed points.
Under generic assumptions, in case (ii),
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(E).
values of
A(E)
bifurcation.
We would expect the nature of the eigento determine (in part) the possible type of
If
I El l
A(E)
0 E2
0
then the eigenvalues of
A(E)
J
are always real so we do not
expect to obtain periodic orbits surrounding the origin.
On
the other hand if A(E) =
[0
1
El
E2
then the range of the eigenvalues of of the origin in ber then
A(E)
~,that
is, if
has an eigenvalue
z
j A(E)
is a neighborhood
is a small complex numz
for some
We shall assume from now on that (4.1. 2) •
(4.1.2)
A(E)
E. is given by
56
4.
BIFURCATIONS WITH TWO PARAMETERS
Takens [64,65) and Bodganov [see 3) have studied normal forms for local singularities.
Takens shows, for example,
that any perturbation of the equation
is topologically equivalent to
for some
E ,E 2 . There are certain difficulties in applying l these results since we must transform our equation into nor
mal form, modulo higher order terms. In [47, p. 333348), under the assumptions that
where
f2
Kopell and Howard study (4.1.1) A(E)
is given by (4.1.2) and that
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.Z.
Preliminaries
4.Z.
Preliminaries
57
Consider equation (4.1.1) where (4.1.Z). 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
0,
f(X,E)
=0
f(X,E).
(4.Z.1)
The object is to study the behavior of all small solutions of (4.1.1) for
E
in a full neighborhood of the origin.
Equa
tion (4.1.1) is still too general, however, so we shall make some additional hypotheses on the nonlinear terms. f
=
(f1'fZ) , 3
ex
= a 3 fZ(O,O),
3
8
aX l
(HI)
ex
f
0
(HZ)
8
f
0
(H3)
3
a ::z
fZ (0,0) •
axlx Z
O.
(HI) implies that for small or
Set
T
fixed points.
E, (4.1.1) has either
1
Under (HI), 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 (HI) 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
8
E.
will determine the direction of bifurcation
and the stability of the periodic orbits, among other things. From now on we assume (Hl)(H3).
4.
58
BIFURCATIONS WITH TWO PARAMETERS
Level Curves of Figure 1
2
1
Bifurcation Set for the Case
a < 0,
e
<
o.
Figure 2 The main results are given in Figures 25.
Sections
38 of this chapter show how we obtain these pictures. pictures for variables
e>
0
are obtained by using the change of and
t
+
to
The
4.2.
Preliminaries
59
3
2 ~~ £1
Bifurcation Set for the Case
a > 0, 8 < O.
Figure 3 The cases ferent.
a > 0
and
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 < O.
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
£1 < 0
and
3
fixed points for
1
£1 > O.
fixed
60
4.
BIFURCATIONS WITH TWO PARAMETERS
REGION 1
REGION 2
~.
REGION 3
.....~,
REGION 4
Phase Portraits for the Case Figure 4
a
<
0,
e
<
o.
4.2.
Preliminaries
61
~
REGION 5
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
a > 0, 8
<
o.
4.
64
4.3.
BIFURCATIONS WITH TWO PARAMETERS
Scaling We scale the variables in equation (4.1.1) so that the
first components of the nonzero fixed points are given by ±l
O(E).
+
variables
To do this we introduce parameters Yl'YZ
and a new time
t
Ia I l/Z a~, xl t
=
~,a,
scaled
by the relations ay l ,
=
X
 ZI I l / ZY ' z  a a Z
lal l /2a l t.
in a neighborhood of the origin, (E l , EZ) belongs to a region of the form {(E l , EZ) : IEll < EO' El < Z (constant)E Z}· The y.1 are assumed to lie in a bounded set, say IYi l < M. A further discussion of the scaling is given
For
(~,
a)
in Section 6. After scaling (4.1.1) becomes
(4.3.1) YZ where the dot means differentiation with respect to Y
= alai liZ
on the
g.
1
gi(~,a,y)
and
depends on
=
M,~,a.
t,
The size of the bounds
0(1).
We write
for
t
t
from now
on. The cases 4.4.
The Case With >'1 YZ Let
El > 0
and
El < 0
are treated separately.
El~
El > 0, equation (4.3.1) becomes YZ Yl
+
a
Z
gl(~,a,y)
+ ~YZ
H(Y I ,Y Z)
3 Yl
+
Z . (YZ/2)
Z ayylyz
+
a
 (y/Z)
+
Z
(4.4.1)
Z
gz(~,a,y).
4
(y/4) .
Then alon).!
4.4.
The Case
El > 0
65
solutions of (4.4.1), (4.4.Z) ~
Note that for
= 0 = 0, H
The level curves of of eight i f Figure 1) .
is a first integral of (4.4.1). H(yl,yZ) = b
consist of a figure
b = 0, and a single closed curve i f
b > 0
(see
b > 0, the curve
H(yl'YZ) = b passes through the point Yl = 0, YZ = (Zb)l/Z For b > 0 and sufficiently small, we prove the existence of a function ~
For
~l(b,o)
=
with
~
point
=

= yP(b)o
~l(b,o)
+
Z
0(0)
such that for
0
b > 0, (4.4.1)
has a periodic solution passing through the
Yl = 0, YZ = (Zb)l/Z, and with
~ = ~1(0,0), (4.4.1)
has a figure of eight solutions. For fixed
the number of periodic solutions of
~,o,
(4.4.1), surrounding all three fixed points, depends upon the number of solutions of (4.4.3) We prove that b
l for
> 0
PCb)
such that
~
as
00
b
P'(b) < 0
~
for
b > b l . These properties of of solutions of (4.4.3).
00
and that there exists b < b PCb)
l
and
P' (b) > 0
determine the number
Suppose, for simplicity of exposition, that y <
yP(b)o
and that
b 3 > bl
such that
~
=~l(bZ'o),
o.
~l(b,o)
=
0 < b Z < b l , then there exists ~l(bZ'o)  ~1(b3'0). Hence, if If
then (4.4.1) has two periodic solutions, one
0, YZ = (Zbz)l/Z, the other passing (Zb ) l/Z. I f ~ >~1(0,0), then (4.4.1) through YI = 0, YZ 3 has one periodic solution surrounding all three fixed points.
passing through
Yl
4.
66
~ =
Finally, if
~(bl'o),
BIFURCATIONS WITH TWO PARAMETERS
then the periodic solutions coincide.
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
b > b l . Similarly the "inner" periodic solutions in region 5 are parametrized by ~ = ~l(b,o), 0 < b < b l . The curve Ll in (El'E Z) space corresponds to the curve ~ = ~1(0,0). Similarly the curve
LZ
~ =
corresponds to the curve
~l(bl'o)
(see Figure Z). ~l
In general
(b,o)
is not identically equal to
yP(b)o, but the results are qualitatively the same. example, we prove the existence of a function ~,o
0(0), such that if
satisfy
~
=
~ = ~l(bl(o)
,0)
bl(o)
~l(bl(o)
tion (4.4.3) has exactly one solution.
For bl
=
+
,0), then equa
The curve
which is mapped into the curve
LZ
in
(El,E ) space corresponds to the points where the two Z periodic solutions coincide. If H(Yl'YZ) points
b
<
b
=
0, then the set of points for which consists of two closed curves surrounding the
(1,0)
and
(1,0). ~
existence of a function that for
~ = ~Z(c,o),
the point f(x)
=
(1,0)
=
c
<
1, we prove the
<
= yQ(c)o
~Z(c,o)
+
z
such
0(0 )
and passing through
Yl
c, yz
=
=
O.
Using
f(x), this proves the existence of a periodic solu
c, Yz
=
(1,0)
and passing through
O.
We also prove that o > 0
=
0
(4.4.1) has a periodic solution surrounding
tion surrounding the point Yl
For
and suppose
~
~z(O,o)
Q' (c) > 0
for
0
<
c
<
1.
Let
satisfies <
sgn(y)~ <
~2(1
,0).
(4.4.4)
4.4.
The Case
a
El >
Then the equation Hence, for fixed
67
~ =
has exactly one solution.
~Z(c,o)
satisfying (4.4.4), equation (4.4.1)
~,o
has exactly one periodic solution surrounding region in
(~,o)
(1,0).
The
space corresponding to (4.4.4) is mapped
into region 4 in
(El,E Z)
Lemma 1.
sufficiently small, there exists a function
For
0
space (see Figure Z).
~ = ~(o) = (4/5)oy + O(oZ)
~ = ~(o), (4.4.1)
such that when
has a homoclinic orbit. Proof:
Let
S(~,o),
U(~,O)
folds of the fixed point exist, since for [Z7]. hits
Let yz
H(yl'YZ)
(0,0)
in (4.4.1).
These manifolds
sufficiently small, (0,0)
~,o
H(~,o,+)
be the value of
0, Yl > O.
Similarly,
S(~,O)
when
be the stable and unstable mani
hits
yz
H(yl'YZ)
U(~,O)
when
is the value of
H(~,o,)
=
is a saddle
0, Yl > O.
H(~,o,±)
are
well defined since stable and unstable manifolds depend continuously on parameters. Let
I(~,o,+) U(~,O)
the portion of YZ
=
a to YZ
denote the integral of
=
with
0, Yl > O.
Yl > 0, YZ > Then
H(~,o,+)
Similarly, I(~,o,) the portion of to
=
with
over
a from Yl
(4.4.5)
I(~,o,+).
denotes the integral of
S(~,O)
H(Yl'YZ)
H(Yl'Y Z)
Yl > 0, YZ < a from H(~,o,) = I(~,o,).
Yz = 0, Yl > 0, so that Equation (4.4.1) has a homoclinic orbit (with
over
Yl = Yz = Yl > 0)
if and only if H(~,o,+)

H(~,o,)
=
O.
(4.4.6)
We solve (4.4.6) by the implicit function theorem.
a
4.
68
BIFURCATIONS WITH TWO PARAMETERS
Using (4.4.2) and (4.4.5), 2
f
22
= + (~Y2 + YOYIY2)dt +
H(~,o,+)
O(~
2
2 + 0 ),
(4.4.7)
where the above integral is taken over the portion of from
Yl
=
Y2 = 0
H(~,o,)
=
to
L(~Y~
Similarly,
0, Yl > O.
Y2
2 + YOY 2Y2)dt + I
O(~
2 + 0 2) ,
where the integral is taken over the portion of Yl
= Y2
=
0
to
Y2 = 0, Yl > O.
U(O,O)
(4.4.8) 5(0,0)
from
Using (4.4.7), (4.4.8) and
U(O,O) = 5(0,0), we obtain (4.4.9) Using (4.4.7) and (4.4.9),
so that by the implicit function theorem, we can solve (4.4.6) for
~
as a function of
0, say
to get an approximate formula for
=
~
~(o).
~(o).
We now show how
We can write equa
tion (4.4.6) in the form
Hence, using (4.4.1) and
Y2
YOf+
J+
Y2dy 1
This completes the proof of Lemma 1. Lemma 1 proves the existence of a homoclinic orbit of (4.1.1) when
(£1'£2)
lies on the curve
Ll
given by
4.4.
The Case
Using
El > 0
69
= f(x,E), when
f(x,E)
(El,E Z)
lies on
Ll , equa
tion (4.1.1) has a figure of eight solutions. We now prove the existence of periodic solutions of (4.4.1) surrounding all three fixed points.
In the introduc
tion to this section, we stated that (4.4.1) has a periodic 0, Y = (Zb)l/Z for any Z In Lemma Z, we only prove this for "moderate" values of
solution passing through
=
Yl
The reason for this is that in (4.3.1) the only for
Y
in a bounded set.
gi
b >
o.
b.
are bounded
In Section 6, we show that by
a simple modification of the scaling, we can extend these resuIts to all Lemma Z.
b > O.
Fix
b >
o.
Then for
0 < b < band
ently small, there exists a function O(oZ)
such that if
]..I
=
]..11 (b, 0)
b
~
=
]..11 (b,o)
suffici=
yP(b)o +
in (4.4.1), then (4.4.1) has
a periodic solution passing through As
]..I
0
0, Y = (Zb)l/Z. Z the periodic solution tends to the figure of
0
Yl
=
eight solutions obtained in Lemma 1. Proof:
Let
H(]..I,o,b,+)
be the value of
orbit of (4.4.1) which starts at sects
yz = O.
y
1 = Similarly, H(]..I,o,b,)
H(yl,y ) when the Z 0, y = (Zb)l/Z interZ is the value of
H(yl'YZ) when the orbit of (4.4.1) which starts at Yl = 0, YZ = _(Zb)l/Z is integrated backwards in time until it intersects YZ = o. Then (4.4.1) has a periodic solution passing through
Yl
=
0, YZ
=
(Zb)l/Z
H(]..I,o,b,+) Let
I(]..I,o,b,+)

i f and only i f
H(]..I,o,b,)
and finishing at
(4.4.10)
O.
denote the integral of
the portion of the orbit of (4.4.1) with Yl • 0, Y L  (2b)1/Z
=
H(yl,y Z)
over
YZ > 0, starting at YZ • 0, Yl
> O.
4.
70
Similarly, in time.
BIFURCATIONS WITH TWO PARAMETERS
is defined by integrating backwards
I(~,o,b,)
Thus, b +
H(~,o,b,±)
I(~,o,b,±).
(4.4.11)
Using (4.4.Z) and (4.4.11),
where the above integral is taken over the portion of the orbit of (4.4.1) with to
Yl = c, YZ
=
0
~
0 = 0
=
from
Yl
0, Yz
=
=
where 4b
Similarly,
(Zb) l/Z
(4.4.13)
I(~,o,b,)
=
+
I(~,o,b,+)
O(~
Z + 0 Z), so
that equation (4.4.10) may be written in the form (4.4.14) Hence by the implicit function theorem, we can solve (4.4.14) to obtain
~ = yP(b)o + O(oZ)
where
f yiYZdYl
P (b)
(4.4.15)
J YZdYl In order to prove that the periodic solution tends to the figure of eight solutions as H(~,o,b,±)
+
b
0, we prove that
+
H(~,o,±)
as
b
+
O.
(4.4.16)
This does not follow from continuous dependence of solutions on initial conditions, since as
b
+
periodic solution tends to infinity.
0
the period of the
The same problem occurs
in Kopell and Howard [47, p. 339] and we outline their method. For
Yl
and
YZ
small, solutions of (4.4.1) behave like
4.4.
The Case
El > 0
71
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
(yl,yZ)
small and continuous dependence on
initial data for the rest of the solution. Lemma 3.
PCb)
+
as
00
b
+
00
Proof:
The integrals in (4.4.lS) are taken over the curve Z Zb]l/Z, from Yl = 0 to Yl = c where YZ = [Yl  (yilZ) + c is defined by (4.4.13) . Thus, JO(b)P(b) = J 1 (b) where
w = cz
Substituting
in (4.4.17) we obtain J. (b)
=
c Z fl (cz) Zig(z)dz 0
1
[(zZ_l)
where g(z) g(c l ) for
o
<
z
~
(c Z/Z) (1_z4) ]l/Z.
+
1, we have that
Since
JO(b)
~
Dlc
~
g(z) 3
for some
positive constant
Dl . Similarly, there exists a positive such that Jl(b) ~ DZC S The result now follows.
constant
DZ
Lemma 4.
There exists
b
<
b
and
l
Proof:
bl > 0
pI (b) > 0
for
such that
pI (b) < 0
for
b > b . l
It is easy to show that
pI (b)
+
00
by Lemma 3 it is sufficient to show that if
as
b
+
O.
pI (b l ) = 0
Hence, then
pit (b ) > O.
l
Let
r(w)
=
[w Z  (w 4 /Z)
(4.4.17) with respect to
b
in~:
hy pans in
Zi ~ dw.
fo r
1
.10
Zb]l/Z.
Differentiating
we obtain c
J!
Intcgrat
+
l
w)
we obtain
(4.4.18)
4.
72
J Also J
r
r2 (w) 0 rCw) dw
O
=
BIFURCATIONS WITH TWO PARAMETERS
O
f:
f:
[w 2
4 2 (w w 2 dw. rCw)
(4.4.19)
r tW4w) / 22
(4.4.20)
+ 2b] dw.
Similarly,
Using (4.4.18)  (4.4.21) we can express JI
terms of
J O and
Jl
in
A straightforward calculation yields
o
(4.4.22) l5J l Suppose that
4bJ O + (4+l2b)Ji'
P'(bl)
J],(b l )  P(bl)J'O(b l ).
=
O.
Then
JO(bl)PIt(b l )
Using (4.4.22) we obtain
4b l (4b l +l) [J],(b l )  P(bl)J'O(b l )] =
Hence
PIt(b l )
(4.4.23)
JO(b l ) [p2(b l ) + 8b l P(b l )  4b ]· l
has the same sign as (4.4.24)
Since
pI (b l ) = 0 we have that (4.4.22) we obtain
Ji (b l )
=
P(bl)JO(b l ).
Using
(4.4.25) Using P(b l )
b l > 0, it is easy to show that (4.4.25) implies that Using (4.4.24) and (4.4.25), PIt(b ) has the same l P (b l )  P2 (b ). This proves that pit (b l ) > 0 as l
< l.
sign as required.
4.4.
The Case
Lemma S.
For
El > 0
0
73
sufficiently small there exist
b l + 0(0), bZ(o)
b Z + 0(0), where
=
prO)
=
bl(o)
P(b Z)' with the
following properties: Let
0 > O.
(i)
if
\l
=
\ll(bl(o),o), then the equation (4.4.Z6)
has exactly one solution. (ii)
If
y <
0
and
\ll(bl(o),o)
<
\l
<
\ll(bZ(o),o),
then equation (4.4.Z6) has exactly two solutions
b 3 (0), b 4 (0) with b 3 (0) = b 3 + 0(0), b 4 (0) = b 4 + 0(0), where b 3 and b 4 are solutions of
PCb)
(4.4.Z7)
A similar result holds i f
(iii)
If
y
and
0
<
has exactly one solution
O.
y >
\l > \ll(bZ(o),o), then (4.4.Z6) bS+O(o), where
bS(o)
bS
is
the unique solution of (4.4.Z7). (iv)
If
y < 0
P (0)
<
\ll(bl(o),o), then (4.4.Z7) y > O.
By Lemma 4, there exists and
pt (b Z) > O.
\l1(0,0)], for
0
f
z(o) = 0(0)
Set and
0
P'(bZ)z + 0(101 + zZ). exists
\l
A similar result holds for
has no solutions. Proof:
and
b Z > 0 such that P (b Z) g(z,o) = 0 1 [\ll(bZ+z,o) 
g(z,O)
=
O.
Then
g(z,o)
By the implicit function theorem there
such that
g(z(o),o) = O.
Hence, if
bZ(o) = b Z + z(o) then \ll(bZ(o),o) = \l1(0,0). The existence of b l (0) is proved in a similar way. The rest of the lemma follows from the properties of
P(b).
We now prove the existence of periodic solutions of (4.4.1) slIrrlllllHl i n~: a single fi xeJ po into
4.
74
Lemma 6.
For
~
exists
0
c
<
~Z(c,o)
=
~ = ~Z(c,o),
1
< =
and
0
sufficiently small, there
yQ(c)o + O(oZ)
such that if
then (4.4.1) has a periodic solution passing
Yl = c, YZ = O.
through
BIFURCATIONS WITH TWO PARAMETERS
As
~
c
0
the periodic orbits tend
to the homoclinic orbit obtained in Lemma 1. Proof:
H(~,o,c,+)
Let
be the value of
=
0, )Z > O.
H(Yl'YZ) YZ
0
YZ
O.
o
Yl = c, YZ
orbit of (4.4.1) starting at YZ
H(Yl'YZ)
Similarly, H(~,o,c,)
when the
intersects
is the value of
when the orbit of (4.4.1) starting at
Yl
= c,
is integrated backwards in time until it intersects Equation (4.4.1) will have a periodic orbit passing
through
Yl
=
c, YZ
=
0
if and only if
H(~,o,c,+)

H(~,o,c,)
(4.4.Z8)
O.
Using the same method as in Lemma Z, we can rewrite equation (4.4.Z8) as G(~,o,c)
=
Z
(~YZ
J
ZZ + YOYlYZ)dt +
O(~
Z
Z + 0 )
=
(4.4.Z9)
0
where the above integral is taken over the curve
from
Yl
c, YZ
=0
to
YZ
=0
a G(O,O,c) ajJ Thus, for fixed
c
=
again.
J Yzdt> Z
By (4.4.Z9)
we can solve (4.4.Z9).
(4.4.Z9) uniformly in
c
however, since as
hand side of (4.4.31) tends to
O.
(4.4.31)
O.
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 ~Z(c,o)
for
0 < c < 1.
4.4.
The Case
El > 0
75
Equation (4.4.1) has a fixed point at (1,0) + O(I\lI,lal).
For
\l,a
(Yl'YZ)
=
sufficiently small we make a
= hj(Yl'YZ,\l,a) so that the fixed point is transformed into (Yl,YZ) = (1,0). An easy calculachange of variables
Yj
tion shows that if we make this transformation, then the only change in (4.4.1) is in the functions
gl
and
We sup
pose that the above change of variables has been made and we write H(c,O)
can be written in the
form
z
z
Zy Z + (Yll) (Yl+l) Thus, for
c
close to
1
z
=
z
z
(cl) (c+l) . (4.4.3Z)
the closed curves are approximately
Z Z Z YZ + Z(Yll) = Z(cl) .
(4.4.33)
So, instead of equation (4.4.Z9) we consider the equation
=
G(\l,a,c) If we prove that that
(Cl)ZG(c,\l,a)
G(\l,a,c)
(a/a\l)G(O,O,c)
= O.
is bounded for
0
(4.4.34) <
c
<
is bounded away from zero for
then we can solve (4.4.34) uniformly in
1
and 0
<
c
<
1,
c.
Now
Since the fixed point is at
(1,0), H(Yl'Y Z) is bounded above and below by quadratic forms in YZ and Yl  1. By (4.4.3Z), 4H(Yl'YZ) + 1 in
YZ
o( I \ll
and +
161)
is bounded above and below by quadratic forms Yl  1.
Hence there exist functions
such that
qi(\l,a) =
4.
76
Integrating the above
BIFURCATIONS WITH TWO PARAMETERS
inequ~lity
over the curve given by
(4.4.30) we obtain exp(ql(~,o)T)
< [4H(~,0,c,+)+11/[4H(c,0)+11
(4.4.35)
where
T
is a bound for the time taken to trace the orbit.
Using (4.4.29), (4.4.32) and (4.3.35) we see that is bounded.
(a/a~)G(c,O,O)
The fact that
from zero follows easily from (4.4.32). o
sufficiently small and
to obtain JO(c)Q(c)
and
r(w)
~
~2(c,0)
=
=
G(~,o,c)
is bounded away
This proves that for
0 < c < 1, we can solve (4.4.32) yQ(c)o + 0(0 2) where
= J l (c),
is defined by (4.4.30) and
red) = 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. Using
f(x)
= fex), Lemma 6 proves the existence of
periodic solutions of (4.4.1) surrounding Lemma 7.
Q' (c)
>
0
fo r
(1,0).
0 < c < 1.
Proof:
We write Q,J and J l as functions of b where O 4 2 4b = c  2c . Since (db/dc) < 0 for 0 < C < 1, we must
prove that
Q' (b) < 0
for
1 < 4b < O.
procedure as in Lemma 4, we find that equation (4.4.22).
Thus, i f
Following the same
JO,J O' Jl,Ji
Q'(b l ) • 0
then
satisfy
4.5.
The Case
77
El < 0
(4.4.36) 1 < 4b l < 0, the roots of (4.4.36) are less than 1. Hence, i f Q'(b l ) = 0 then Q(b l ) < 1. Also, from (4.4.Z3), Since
Q' (b l ) = 0 then Q"(b l ) has the same sign as 4b l 8b l Q(b l )  QZ (b l )· Using (4.4.36) and Q(b l ) < 1, this im
if
plies that
Q" (b l ) < O.
shows that
Q' (b) < 0
Since for
Q(1/4)
1, Q(0)
1 < 4b < O.
=
4/5, this
This completes the
proof of the lemma. 4.5.
El~
The Case With
El < 0, equation (4.3.1) becomes YZ
Yl
°Zgl(\l,o,y)
+

3 Yl
Let
+
(yilZ)
tions of (4.5.1),
Z YZ
Yl
Yz
\lYZ
+
HI
=
+
+
Z oyylyz +
Z Z oYYlYZ
+
Z g Z(\l , ,y) .
°
(4.5.1)
°
(yi/ 4). Then along soluZ + 0(0 ). Using the same
methods as in the previous lemmas, we prove that (4.5.1) has a periodic solution passing through and only i f J O(c)R(c)
\l
= \l3(c,0)
=
yR(c)o
+
Yl = c > 0, YZ O(OZ) , where
0, i f
= J 1 (c) , Ji(c) = r(w)
c z' 0 w lr(w)dw,
I
[Zb  (w 4 /Z)  WZ]l/Z, 4b
In order to prove that for fixed
=
monotonic. R' (c) > 0
for
c > O.
c4
+
Zc Z.
\l,0, equation (4.5.1)
has at most one periodic solution we prove that
Lemma 8.
=
R
is strictly
BIFURCATIONS WITH TWO PARAMETERS
4.
78
Proof:
We write
R,JO,J l
ent to prove that
R'(b)
as functions of 0
>
for
b
b.
It is suffici
O.
>
Using the same methods as before, we show that
lSJ l
(12b+4)J
(4.S.2)
1  4bJb
c 3w4dw
fo rtWJ Now
R'
(4.S.3)
has the same sign as
S
where
S
By (4.S.2), S
=
(8b4)J'J'
o
By (4. S. 3) , bJ b > J
1·
1
=
S(J,)2 + 4b(J,)2 1
Hence, for
0
b
S > (J )2 b l(8b4Sb+4)
1
Similarly, i f
0 < b < 2, then
~
=
2, 3(J,)2 > O. 1
S > 3b 2 (Jb)2 > O.
This com
pletes the proof of Lemma 8. 4.6.
More Scaling In Section 4 we proved the existence of periodic solu
tions of (4.4.1) which pass through the point Lemma 3 indicates that we may take please. are
b
(O,(2b)1/2).
to be as large as we
However our analysis relied on the fact that the
0(1)
and this is true only for
y
gi
in a bounded set.
Also, our analysis in Sections 4 and S restricts
El ,E 2 to a region of the form {(E l ,E 2 ): [E l [ ~ EO' El ~ (constant) E~}. To remedy this we modify the scaling by setting I) [E a l [1/2 h l, where h is a new parameter with, say, l o < 4h ~ 1. The other changes of variables remain the same.
Note that if
~.I)
lie in a full
n~ighborhood
of the
4.6.
More Scaling
origin then
79
(El,E Z)
and
(xl'x Z)
lie in a full neighbor
hood of the origin. After scaling, (4.1.1) becomes
YZ For
0 < 4h < 1
0(1)
for
Y
Let
and
sufficiently small, the
~,o
gi
are
in a bounded set.
El
>
O.
Let
HZ(Yl'YZ)
=
Z
(YZ/Z)

~Y~
Then along solutions of (4.6.1), HZ
O(~Z
oZ).
+
Z Z (h yl/Z)
+
+
Following the same procedure as in Sec
tion 4, we find that (4.6.1) has a periodic solution passing through
Yl
= 0, YZ = I, if and only if h
~
where
Zb
hZP(b)
=K+
=
Z
~l
(b,o)
= yh ZP(b)o
+
Z 0(0 ),
h 4 .
An easy computation shows that as Z O(h ), where JOK = 12 J l , J. 1
Thus, for small
h ~ 0,
= fl WZi(1_w4)1/Zdw. 0
h, (4.6.1) has a periodic solution passing
yKo + Yl = 0, YZ = 1 i f and only if ~ O(hZ[o[ + o Z) . In particular, when El = 0, E > 0, 8 < 0, Z (4.1.1) has a periodic solution passing through xl = 0, through
X
z=
[a[(8K)lE For
z
+
O(E~/Z).
El < 0, a similar analysis shows that (4.6.1) has
a periodic solution passing through
Yl
= I,
YZ
=
0, if and
only if
~ = hZ~Z(hl.o) = _yhZQ(hl)o and t hat
II
s
h
• (), 1/ Q(h  1) 
( KI fl)
+
+ 0 ( h Z) .
O(oZ),
4.
80
4.7.
BIFURCATIONS WITH TWO PARAMETERS
Completion of the Phase Portraits Our analysis in the previous sections proves the exist
ence of periodic and homoclinic orbits of (4.1.1) for certain (El,E Z)
regions in
space.
We now show how to complete the
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 example.
We prove that if
y < 0, then the "outer" periodic
We use the scaling given in
in region 5 is stable.
orbit
Section 3. Since we are in region 5, \ll(O,c) > \l > \ll (b (c) ,c). l Fix
and
\l
with
c
and let
b Z > bl(c).
solution
r
bZ
\l = \ll(b,c)
be the solution of
Then we have to prove that the periodic Yl = 0, YZ = (Zb Z) l/Z
of (4.4.1) passing through
is stable. Let b3
<
b3
\l = \ll (b,c)
be the solution of
with
bl(c).
Then there is a periodic solution passing through
0, YZ
(Zb 3 )1/Z.
In fact, we prove that any solution
of (4.4.1) starting "outside" this periodic solution (and inside some bounded set) tends to
r
Let
as
t
~
00
Yl when (Zb)l/Z first Yl = 0, YZ =
b > b 3 · Let c (b, +) the orbi t of (4.4.1) starting at
be the value of
hits
is the value of
YZ = O.
Similarly, c (b ,  )
the orbit of (4.1.1) starting at integrated backwards until it hits
Yl
0, YZ
YZ
f(X,E), the solution passing through
=
O.
Yl when (Zb)l/Z is
Using
f(X,E)
(O,(Zb)l/Z), spirals
inwards or outwards according to the sign of Using the calculation in Lemma Z, c(b,+) +
c(b,+) + c(b,).
~(b,)
has the
4.8.
Remarks and Exercises
H(~,o,b,+)
same sign as

81
H(~,o,b,)
which in turn has the
same sign as
~l(b,o)
Using the properties of positive if
b < b
2
given in Lemma 5, S
and negative if
b > b . 2
is
The result now
follows. 4.8.
Remarks and Exercises
Remark 1.
Consider equation (4.1.1) under the hypothesis
that the linearization is given by is defined by (4.1.2). hold.
r
The map
x
~
x
A(E)
x
=
B(E)X, where
and
a =
=
will be onetoone for
Using the fact that
where
Suppose also that (4.2.1) and (HI)
Make the change of variables
B(E) = I  ra lA(E)
x = A(E)X
A(E)
and
B(E)
E
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
o
In particu
BIFURCATIONS WITH TWO PARAMETERS
4.
8Z
3 3a fl(O,O)
3
+
Remark Z.
We have assumed that
aX l
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
L'1
For example, on xl > O.
and
L'1
we would
Similarly on another
we would get a homoclinic orbit with
general
xl < O.
In
would be different although they would ~ =
have the same linear approximation Remark 3.
Ll
(4/S)oy.
Suppose that we only assume (HI) and (H3).
Then
we can still obtain partial results about the local behavior of solutions.
For example, in Lemma Z we did not use the
hypothesis
nonzero.
8
a periodic solution through for some
and
b > 0, (4.1.1) has l Z 0, X = I a I / (Zb) l/ZE z I However, we cannot say
Hence, for each xl
with
anything about the stability of the periodic orbits and we cannot say how many periodic orbits (4.1.1) has for fixed
El
and Exercises (1) that for ing orbit.
Suppose that (El,E Z) (Hint:
a
and
8
are negative.
Prove
in regions 5 and 6, (4.1.1) has a connectUse the calculations in Lemma 1 and
Lemma 6.) (Z)
Suppose that
a
(E , E ) is in region 3. Let l Z the point (0,0) in (4.1.1).
and
8
are negative and that
U be the unstable manifold of Prove that
gion of attraction of the periodic orhit.
U
is in the re
4.9.
Quadratic Nonlinearities
(3)
Suppose that
a
83
is positive and
8
is negative.
Show that the bifurcation set and the corresponding phase portraits are as given in Figures 3 and 4.9.
s.
Quadratic Nonlinearities In this section we discuss the local behavior of solu
tions of (4.1.1) when the nonlinearities are quadratic.
Most
of the material in this section can be found in (47). Suppose that the linearization of (4.1.1) is where
A(E)
x
A(E)
is given by (4.1.Z) and that f(O,E)
We also assume that
0,
0,
8 > 0, a < 0; the results for the other
cases are obtained by making use of the change of variables, t
~
t, EZ ~ E Z ' xl ~ ±xl' Introduce parameters
and a new time
t 1
We assume that the case
X
z~ ~,o,
+x Z. scaled variables
Yl'YZ
by the relations Ela Ill/Z ,
El > 0
in what follows, see Exercise 8 for
El < O.
After scaling, (4.1.1) becomes
(4.9.1)
X
BIFURCATIONS WITH TWO PARAMETERS
4.
84
where the dot means differentiation with respect to t, the Yi lie in a bounded set and Y = Blal l/Z . Note that by making a change of variable we can assume that fixed point of (4.9.1) for
(1,0)
is a
sufficiently small.
~,o
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
Show
that along solutions of (4.9.1), . H (5)
o<
c
~
Z
= ~yz
Z oYYlYZ
+
=
~
~l(c,o)
is a periodic solution through homoclinic orbit if
JO(c)P(c)
Z
0(0 ).
Prove that there is a function
1, such that if
(6)
+
c
Prove that
=
~
=
~l(c,o),
in (4.9:1) then there
(c,O)
if
0 < c < 1
and a
O. ~l
YP(c)o
(c, 0)
O(oZ)
+
where
= J l (c), wi R(w)dw, R(w)
(7) for
Prove that
0 < c < 1.
[w Z _ (Z/3)w 3
P(O)
=
Zb]1/2,
+
(6/7), P (1)
Deduce that for fixed
most one periodic solution.
= 1 and P' (c)
~,o,
(To prove that
P' (c)
>
0
0
(4.9.1) has at P' (c)
use the same techniques as in the proof of Lemma 4. ternative method of proving
>
>
0
we can
An al
is given in [47] .)
4.9.
Quadratic Non1inearities
Put
E:1 < 0, then after scaling (4.1.1) becomes
If
(8)
z

85
1,
Y1
Y2
Y2
Y1
II
II
z
Y2
Y2
z
2 0(0 )
+
 yo. + +

llY2
+
2 Y1
+
YOY1 Y2
+
2 0(0 ).
Then
2 0(0 )

llY2
z2
+
oyzY2
+
0(0 2 )
which has the same form as equation (4.9.1) and hence transforms the results for
E:l
>
0
into results for the case
E:1 < O. (9)
Show that the bifurcation set and the correspond
ing phase portraits are as given in Figures 67.
4
3
5 6
L
Bifurcation Set for the Case Figure 6
a < 0, B
>
O.
86
4.
BIFURCATIONS WITH TWO PARAMETERS
REGION 1
REGION 2
Phase Portraits for the Case a < 0, B > O. (For the phase 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.
ic = Ax
+
The equations are
f(x)
(5.1.1)
where T [flex) ,fZ(x) ,f 3 (x) ,f 4 (x)] ,
f(x)
o A
c
fl (x)
f 3 (x) = 0,
fZ(x)
Xlg(x),
f4 (x)
4x 3g(x) ,
Zg(x)
'J[ 4 (kx Z l
c = b.
J
8p
3'
+
1
0
0
bl
c
0
0
0
1
0
aZ
bZ
axlx Z
4kx;
_'J[ZjZ['J[ZjZ
a.
J
_ [Cl'J[4 j 4
+
+
IP
0];
+
+
4ax 3x 4 ) ,
fl,
Cl • 0.005,
/)
0.1,
5.2.
k
>
Reduction to a Second Order Equation
0, a
>
0
are fixed and
p,r
89
are parameters.
The above
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 equation and first we briefly describe their work. calculations, they find that for 2
2.23n , the matrix
p
=
Po
108, r
=
~
rO
A has two zero eigenvalues and two
eigenvalues with negative real parts. !rrO!
~
By numerical
Then for
!pPO!
and
small, by centre manifold theory, the local behavior
of solutions of (5.1.1) is determined by a second order equation depending on two parameters.
They then use some results
of Takens [64] on generic models to conjecture that the local behavior of solutions of (5.1.1) for
!pPO!
and
!rrO!
small can be modelled by the equation
for
a
and
b
small.
Recently, this conjecture [37], in the case
a
has been proved by Holmes
= 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
A
are the roots of the equation (5.2.1)
where the
ui
are functions of
zero eiIlCHlvlI!U(,s then
u3
rand
• d 4 • O.
p.
If
A
has two
A calculation shows that
5.
90
d
d3
if
APPLICATION TO A PANEL FLUTTER PROBLEM
0, then
4
2 a 1a 2 + c
a b
1 2
or in terms of
and
r
0 (5.2.2)
+ b a 1 2
0
p,
4 2 2 4'1f ('If +r) (4'1f +r) + 64 p2
9
=
4 1/2 2 2 1/2 2 (16o'lf +t5P ) ('If +r) + 4 (O'lf +t5p ) (4'1f +r)
0
(5.2.3)
=
We prove that (5.2.3), (5.2.4) has a solution From (5.2.3) we can express
P
(5.2.4)
O. r = r
in terms of
' O
P = PO'
r.
Sub
stituting this relation into (5.2.4) we obtain an equation H(r) r
1
=
some
=
Calculations show that
O.
(2.225)'If
2
and
r
2
=
H (r 1) < 0, H (r 2) > 0 where 2 (2.23)'If , so that H (r 0) = 0 for
Further calculations show that (5.2.3) ,
rO E (r 2 ,r 1 ) •
(5.2.4) has a solution
rO'PO
with
107.7 < Po < 107.8.
In the subsequent analysis, we have to determine the sign of various functions of know
and
rO
rO
and
PO'
Since we do not
exactly we have to determine the sign of
Po
these functions for
rO
and
Po
in the above numerical
ranges.
r
When
Po, the remaining eigenvalues of
are given by
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
=
r
O
'
P
=
PO'
Solving
Av 1 • 0
we find that
A
5.2.
Reduction to a Second Order Equation
v
The null space of onical form of 2
A v2
0
(5.2.5)
=
1
A
91
is in fact onedimensional so the can
A must contain a Jordan block.
Solving
we obtain (5.2.6)
The vectors
vI
eigenspace of
and A
v2
form a basis for the generalized
corresponding to the zero eigenvalues.
Similarly, we find a (real) basis for the by the eigenvectors corresponding to Az
A3 z , we find that
=
z
Let P
=
PO'
AO
Let
z
+
z,
=
v3
V
spanned
A4 • Solving and v 4 where
= i(zz),
2v 4
(5.2.7)
=
denote the matrix S
and
A3
is spanned by
V
2v 3
space
[vl,v2,v3,v41
A when where the
by (5.2.5), (5.2.6) and (5.2.7) and set
y
r
= rO
vi
and
are defined
= Slx.
Then
(5.1.1) can be written in the form
Y where
F(y,r,p)
By
=
S
1
B
and where
A3 • PI
+
S
+
1
(AAO)Sy
+
F(y,r,p)
f(Sy) , 0
1
0
0
0
0
0
0
0
0
PI
P2
0
0
P2
PI
=
iP2' PI
<
0, P2
t
O.
(5.2.8)
92
5.
Then for
APPLICATION TO A PANEL FLUTTER PROBLEM
Irrol
and
Ippol
(5.2.8) has a centre manifold Y4
Y3
sufficiently small h l (Yl'Y2,r,p),
=
= h 2 (Yl'Y2,r,p). The flow on the centre manifold is gov
erned by an equation of the form
(5.2.9)
where
E(r,p)
N(Yl'Y2,r,p)
is a
2 x 2
matrix with
E(rO'PO)
contains no linear terms in
0
=
We
or
show that there is a nonsingular change of variables (£1'£2)
for
(r,p)
close to
(rO'PO)
(Yl'Y2) ~ (Yl'Y2)
change of variables
and
and a
r,p
(r,p)
~
dependent
such that the lin
earized equation corresponding to (5.2.9) is
(5.2.10)
The transformation Identity into
+
o(lrrol
(Yl'Y2) ~ (Yl'Y2) +
0, £2 = O.
Ippol)
and
r
is of the form =
r O' p
= Po
is mapped
After these transformations, (5.2.9)
takes the form
(5.2.11)
where we have dropped the bars on the will contain no linear terms in N(Yl'Y2,r O'PO)
=
O.
Yl
Yi' or
Y2
The function
N
and
Since the nonlinearities in (5.1.1) are
cubic, the same will be true of
Nand
N.
5.3.
93
Calculation of Linear Terms
Let
and let
(5.2.13)
B
B.
= 3r +
Using the results in Chapter 4 (see, in particular, Remark 1 and B are nonzero, we can deterl mine the local behavior of solutions of (5.2.11). By Theorem in Section 8), if
a
2 of Chapter 2, this determines the local behavior of solutions of (5.2.8). 5.3.
Calculation of the Linear Terms From (5.2.1), trace (A)
=
d (r,p), det(A) 3
where 2 2 2 64 2 4'1f ('If +r) (41f +r) + 9" p
Calculations show that the mapping (5.3.1) has nonzero Jacobian at Define the matrix
(f,p) C(f,p)
=
(fO'PO)' by
5.
94
APPLICATION TO A PANEL FLUTTER PROBLEM
c(r,p)
and let
J
evaluated at +
+
E(r,p)
be the value of the Jacobian of the mapping (r,p)
B
[ 0 10J
(5.3.2)
(trace(C(r,p)), det(C(r,p))),
~
(rO'PO)'
By considering the
Sl(AAO)S, it is easily seen that
J
4 x 4
matrix
is a nonzero
multiple of the Jacobian of the mapping given by (5.3.1). Hence
J
can use
is nonzero, so by the implicit function theorem we £1
=
det(C(r,p)), £2
cation parameters.
= trace(C(r,p))
Approximate formulae for
as our bifur£1
and
£2
can easily be found if so desired. Let
c(r,p)
=
[C
ij ],
" [ :11 :12 ]
[;:] "[ ~:l Then the linearized equation corresponding to (5.2.9) is
Note that
M 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
=
r O' P
PO'
=
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 + °3' x 2 = Y2 + °3 a b l l cYl  c: Y2 + °3' x 4
Xl
where
Yl
(5.4.1)
3 03 = 0(IY l 1 + 1Y213). Let Sl = [t .. ] and let 1J
Then using the notation introduced in (5.2.12),
t 12 F 2 (Y l 'Y2)
+
t 14 F 4 (Yl'Y2) (5.4.2)
t 22 F 2 (Yl'Y2)
where and
+
terms in
+
terms in
2 YlY2
S °5 • O(IYlI + 1Y215) . P • PO' (5.2.2) holds.
and
3 Y2
+
+
t 24 F 4 (Yl'Y2)'
°5
Note also that since
From (5.2.13) and (5.4.2),
5.
96
APPLICATION TO A PANEL FLUTTER PROBLEM
Routine calculations show that t12
m2b (2a b b ) l l 2 2
t22
ma 2
t14
2 m c (b l +b 2 )
t24
mc
Using (5.2.2) and numerical calculations, we find that
and
so that of
k
in
Hence
B
and
r
are negative.
Similarly, the coefficient
is
B 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)€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, 56].
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 Cw + N(w).
(6.2.1)
The most important tools for carrying out this program were: 97
6.
98
INFINITE DIMENSIONAL PROBLEMS
w= Cw
The solution of the linear problem
(1)
of
exp(PCt), exp((IP)Ct), where
P
and estimates
is the projection
onto the space associated with eigenvalues of
C with
zero real parts. The variation of constants formula
(2)
J:
wet) = exp(Ct)w(O) +
exp(C(ts))N(w(s))ds
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 infinite dimensional spaces. Let
We first study linear problems.
Z be a Banach space and
from some domain
D(C)
of
Z into
Cw,
t > 0,
C
a linear operator
Z.
We wish to solve the
problem
w=
w(O) = Suppose that for each
Wo
€
Wo Z
unique solution
wet).
term solution).
The solution
Wo
and we write
mapping from
into
(6.2.3)
Z.
€
the above equation has a
(We define later the meaning of the wet)
T(t)w O'
wet)
Z
(6.2.2)
Z
with
is a function of
Then T(O)
T(t)
= I.
solution of (6.2.2)  (6.2.3), then for any w(t+s) Hence
T(t+s)w T(t+s)w O
o
ously on initial data then in that
Z, so that T(t)wO ~
T(t)
Wo
as
If
wet) s
>
is a
0, wet) w(O) = w(s).
If solutions depend continu
T(t)w n
~
T(t)w
must be bounded. t ~ 0+.
and
is a linear
is a solution of (6.2.2) with
= T(t)T(s)w O'
t
whenever
wn
~
Finally, we require
w
6.2.
Semigroup Theory
99
A one parameter family
Definition.
bounded linear operators from
Z
T(t), 0
into
Z
2
t <
~,
of
is a (strongly
continuous) semigroup if (i)
T(O) = I,
(ii)
T(t+s)
(iii)
T(t)T(s), t,s > 0
IIT(t)w  wll'" 0
Example 1.
Let
tors from
Z
L(Z)
into
as
(semigroup property),
t ... 0+
for every
w € Z.
denote the set of bounded linear opera
Z
and let
T(t)
=
e
C € L(Z).
y
Ct
n=O
Define
T(t)
by
~ n.
The right hand side converges in norm for each
t > 0
is easy to verify conditions (i),
Thus
(ii), (iii).
and it e Ct
defines a semigroup. Example 2.
Let
Z
be a Banac!l space of uniformly continuous [O,~)
bounded functions on (T(t)f) (8)
=
with the supremum norm.
f(8+t),
f
€
Z, 8
~
Define
0, t > O.
Conditions (i) and (ii) are obviously satisfied and since IIT(t)f  fll
=
sup{lf(8+t)  f(8) I: 8 ~ O} ... 0
(iii) is satisfied. Definition. T (t)
Hence
T(t)
as
t ... 0+,
forms a semigroup.
The infinitesimal generator
C
of the semigroup
is defined by T(t)z  z , I Im+ t t ... O
Cz whenever the limit exists. of all elements also say that
z € Z C
The domain of
C, D(C), is the set
for which the above limit exists.
generates
T(t).
We
6.
100
INFINITE DIMENSIONAL PROBLEMS
In Example 1 the infinitesimal generator is
C E L(Z)
while in Example 2 the infinitesimal generator is
= f' (8),
(Cf) (8) Exercise 1.
If
C
E
L(Z)
D(C)
=
{f: f'
prove that
II e
Ct
E

Z}. as
I II ... 0
t ... O. Exercise 2.
Prove that generator
Does
R,2
Z
Let
T(t)
is a semigroup on
T (t) : Z ... Z by
Z with infinitesimal
given by
IIT(t)  I II'" 0
Exercise 3.
and define
as
t ... O+?
Prove that if IIT(t) 112. Me
for some constants
w
~
T(t) wt
,
is a semigroup then t > 0,
0, M ~ 1.
(6.2.4)
(Hint:
Use the uniform
boundedness theorem and property (iii) to show that is bounded on some interval
[0,£1.
IIT(t) II
Then use the semigroup
property. ) Exercise 4. Find a 2 x 2 matrix C such that if Ct II e II 2. M for a11 t > 0 then M > 1. Exercise 5.
Let
which satisfies
C be the generator of a semigroup T(t) wt IIT(t) 112. Me , t > O. Prove that C  wI
is the generator of the semigroup IIS(t)II2.M
fora11
t>O.
Set)
K
ewtT(t)
and that
6.2.
Semigroup Theory
Exercise 6.
If
w E Z, T(t)w Let
101
T(t)
is a semigroup, prove that for each
is a continuous function from T(t)
[0,)
be a semigroup with generator
into
C.
Now
h 1 [T(h)  I]T(t)w
h 1 [T(t+h)w  T(t)wl
(6.2.5)
T(t)h 1 [T(h)w  wl. If
w E D(C)
T(t)Cw
then the right side of (6.2.5) converges to h ~ 0+.
as
Z.
Thus the middle term in (6.2.5) conand so
verges as
T (t)
maps
D (C)
into
D (C) •
Also
the right derivative satisfies
dTJ~)W
=
CT(t)w
Similarly, the h
1
= T(t)Cw,
t
>
0, w E D(C).
(6.2.6)
ide~tity
[T(t)w 
T(th)h
T(t~h)lw
1
[T(h)w  wl
shows that (6.2.6) holds for the twosided derivative. Exercise 7.
For
w E Z, prove that wet)
and that dense in
~
w
wet) E D(C)
where
J: T(s)w ds,
as
Deduce that
D(C)
is
and
C
Z.
Exercise 8.
If
wn E D(C), then from (6.2.6) T(t)w
n
 w = Jt T(s)Cw ds. nOn
Use this identity to prove that
C
is closed.
Equation (6.2.6) shows that if the generator of a semigroup
Wo E D(C)
T(t), then
wet)
is
= T(t)w O is a
solution of the Cauchy problem (6.2.2)  (6.2.3).
In applica
102
6.
INFIMITE DIMENSIONAL PROBLEMS
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 traction semigroup
T(t), that is
The Laplace transform R(A)w exists for T(t)
Re(A) > 0
and
is in some sense
resolvent of
IS,
and is easy to prove.
generates the con
~
IIT(t) II
1
for a11
eAtT(t)w dt,
IIR(A) II < AI
(AI  C)
1
.
t > O.
(6.2.7)
for
exp (Ct), we expect .
C, that
I:
C
A > O.
RCA)
Since
to be the
This is indeed the case
The main problem is in finding the in
verse Laplace transform, that is, given
C
A > 0, say, is
exists for
semigroup?
The basic result is the HilleYosida Theorem.
Theorem 1 (HilleYosida Theorem).
C
such that
(AI  C) 1
the generator of a
A necessary and sufficient
condition for a closed linear opera tor
C
with dense domain
to generate a semigroup of contractions is that each is in the resolvent set of
= II
IIRCA)II
To
A> 0
and that
C
(AIC)lll ~ AI
A > O.
for
(6.2.8)
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)
hl(T(h)I)R(A)w
be a contraction semigroup and for
as in (6.2.7) for Ah l h [ (e  1)  e
to prove that
R(A)
maps
Z
Ah
into
I:
w E Z.
Use the identity
eAtT(t)wdt
J: eAtT(t)w dt] D(C)
and that
6.2.
Semigroup Theory
103
(AI  C)RCA) = I.
w E D(C). R(A) (AI  C)w = wand deduce
Prove also that for that
R(A)
is the resolvent of
Exercise 10.
Suppose that
C
C. is a closed linear operator on
Z with dense domain such that set of
(O.~)
C with (6.2.8) satisfied.
C (A) E L(Z)
is in the resolvent
For
A
0
>
define
by AC (AI  C)l
C CA)
Prove that (i)
(ii)
CCA)w
+
Cw
as
A
Ilexp(tC(A)) II ~ 1
+
~
for
w E D(C).
and
Ilexp(tCCA))w  exp(tC(ll))wll ~ tlICCA)w  C(ll)wll t
for
A.ll
> O.
> 0
and
w E Z.
Use the above results to define T(t)w
T (t)
by
lim exp (C CA) t)w A+~
and verify that with generator Exercise 11.
T(t)
is a semigroup of contractions on
C. Let
Z =.Q,2
complex numbers such that Define
Z
and let
{an}
be a sequence of
a = sup{Re(a ): n n
1.2 .... } <
~.
C by
Use the HilleYosida Theorem to prove that
C  aI
a contraction semigroup and deduce that at group T (t) with IIT(t) II ~ e for t
generates a semi
C >
O.
generates
104
6.
Remark.
INFINITE DIMENSIONAL PROBLEMS
The above version of the HilleYosida Theorem char
acterizes the generators of semigroups which satisfy wt IIT(t) II.::. e (see Exercise 5). A similar argument gives the characterization of the generators of all semigroups. Remark.
Let
C be a closed operator with dense domain such
that all real nonzero A are in the resolvent set of C and II (AI  C)III < IAI l for such A. Then C is the generator of a group all
t,s
and
€ JR
Example 3. space
T(t), t
Let
H with
€ JR,
T(t)z
that is, T(O) = I, T(t+s) ~
z
as
t
~
0
T(t)T(s) ,
for each
z
€
z.
C be a selfadjoint operator on a Hilbert C < O.
Then
II(AI  c)III < AI
for
A> 0,
so by the HilleYosida Theorem, C generates a contraction semigroup. Example 4.
Let
A
on a Hilbert space
iC
=
H.
_A)III < IAI l
liP.!
Let semigroup
f
€
where
so that
on
Z.
A,
A generates a unitary group. and let
C be the generator of a
Then formally the solution of
Cw + f ( t),
w
is a selfadjoint operator
Then for all nonzero real
C([O,T] ;Z)
T(t)
C
0 < t < T,
w( 0 )
=
w0 '
(6.2.9)
is given by the variation of constants formula wet) If
f
=
T(t)w O +
f~
is continuously differentiable and
is easy to check that
wet)
(6.2.10)
T(ts)f(s)ds.
Wo
€
D(C)
defined by (6.2.10) satisfies
(i)
wet)
is continuous for
(ii)
wet)
is continuously differentiable for
o<
t < T,
then it
0.::. t < T,
6.2.
Semigroup Theory
(iii)
lOS
wet) € D(C)
for
0 < t < T
and (6.2.9) is sat
isfied. A fUnction
wet)
which satisfies (i),
said to be a strong solution of (6.2.9).
(ii),
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
D(C)
(iii) is
Wo t
If
wet)
D(C)
defined by
so (6.2.9) does not make sense.
We
now define what we mean by a solution in this case. Definition.
A function
of (6.2.9) if function
<wet) ,v> d
dt<w(t),v> where
=
w(O)
C*
w € C([O,T] ;Z)
Wo
and if for every
Z
v € D(C*), the
is absolutely continuous on =
<w(t),C*v> +
is the adjoint of
ing between
is a weak solution
C
and
[O,T]
and
a.e.
<, >
denotes the pair
and its dual space.
The proof of the following theorem can be found in 19]. Theorem 2.
The unique weak solution of (6.2.9) is given by
(6.2.10). A semigroup
T(t)
if in addition the map for each
z € Z.
properties.
o.
t
~
T(t)z
T(t), then
Thus if
is real analytic on
(0,00)
Analytic semigroups have many additional
In particular, if
tic semigroup t >
is said to be an analytic semigroup
f
T(t)
C
is the generator of an analy
maps
Z
into
D(C)
for each
is continuously differentiable, then
wet)
defined by (6.2.10) is a strong solution of (6.2.9) for each
Wo
€ 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
6.
106
INFINITE DIMENSIONAL PROBLEMS
are usually associated with analytic semigroups.
Hyperbolic
problems, however, do not in general give rise to analytic semigroups. of a group t
>
C
O.
To see this, suppose that T(t)
and that
T(t)
maps
C
is the generator
Z
into
D(C)
By the closed graph theorem, CT(l) E L(Z)
CT(l)T(l)
for
and hence
is a bounded linear operator.
In applications, it is sometimes difficult to prove directly that
C + A is a generator, although
erator and
is in some sense small relative to
A
C is a genC.
The
simplest such result is the following. Theorem 3. T(t)
and let
(group) IIU(t) II
Let
U(t). <
C be the generator of a semigroup (group) A E L(Z). If
Me(w+MllAll)t
Then
IIT(t) II for
<
C + A generates a semi group wt Me for t > 0 then
t > O.
Outline of the Proof of Theorem 3: tor of
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 U(t) = T(t) +
I
to T(s)AU(ts)ds.
(6.2.11)
Equation (6.2.11) is solved by the following scheme: co
U(t)
L Un(t)
(6.2.12)
n=O where
UO(t) = T(t)
and
Un+l(t) =
I:
T(s)AUn(ts)ds.
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
Mal, w • 0
in
6.2.
Semigroup Theory
the following way:
107
let
B
=A
C
+
operator with dense domain
D(B)
;\ > IIAII, IIA(AI  C) 111 < 1
so that
exists for (;\  IIAID 1
;\ > IIAII. for
so that D(C).
K
is a closed
Show that for
(I  A(;\I  C)l)l II (AI  B) 111 ~
Hence show that
;\ > IIAII
B
and use the Hi1leYosida Theorem.
Example 5 (Abstract Wave Equation).
Let
selfadjoint operator on a Hilbert space
A H
be a positive B E L(H).
and let
Consider the equation
v
+
Bv
Define a new Hilbert space duct
< , >
+
Av
o.
=
Z = D(AI/2) x H with inner pro
defined by l 2 l 2 ) <w,w>  (A / wI' A / wI
where
(,)
W= [wl ,w2 ].
(6.2.13)
+
is the inner product in
( w ,w ) 2 2
Hand
w
= [w l ,w 2 ],
Equation (6.2.13) can now be written as a first
order system on
Z, Cw
(6.2.14)
where
D(C)
= D(A)
l 2 x D(A / ).
If
B
=
0
then
C
joint so by Example 4 it generates a group.
is skewselfadFor
B
r
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
6.
108
Vtt + v t  /::'v = 0,
x
INFINITE DIMENSIONAL PROBLEMS
n,
€
v(x,t) = 0,
x
an,
€
is a bounded open subset of :rn. n
with boundary an. 2 This equation has the form (6.2.13) if we set H = L (n)' where
n
D(A) = {w this case
H, w = 0 on an}, A =  /::, , B = I. In H~(n) x L2 (n) (see [2] for a definition of
H: /::'w
€
Z =
the Sobolev space Let
T(t)
H with generator
€
1
HO(n)). be a contraction semigroup on Hilbert space C.
Then for
11T(s+t)zll so that
II T(t) z II
II ret) zl12
IIT(t)T(s)zll ~ IIT(t)zll
is nonincreasing.
is differentiable in
V'(O)
t
For
z
D(C), vet)
€
and
+ .
v I (t)
Since
=
s > 0,
~
0, we have that
+ A linear operator
~
0,
z
€
D(C).
(6.2.15)
C 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
B> 0
the range of
BI  C
H. is
Let
C
be a dissipative
Suppose that for some H.
Then
C
is the generator
of a contraction semigroup. Exercise 14. (6.2.15) that
Prove the above theorem. II(AIC)zll ~ ~Izll
for
(Hint: A> 0
Use this and the fact that the range of
and
B1  C
deduce from z is
D(C).
€
If
10
6.2.
Semigroup Theory
deduce that
C
109
is closed.
{A > 0: range of
AI  C
Show that the nonempty set is
H}
is open and closed and use
the HilleYosida Theorem.) Remark.
There is also a Banach space version of the above
result which uses the duality map in place of the inner product. Exercise 15.
Let
C be the generator of a contraction semi
group on a Hilbert space and let D(B)
:::>
D(C)
be dissipative with
and IIBwl1 < allCwl1
for some
B
a
with
IIB(AIC)lwll < (2a
+
bllwll,
0 < 2a < 1. +
Use the inequality
bAl)llwll
invertible for large enough
w € D(C)
to show that A.
Deduce that
AI  B  C B
C
+
is
generates
a contraction semigroup. Exercise 16. bert space
Let H
C be a closed linear operator on a Hil
and suppose that
Prove (i) the range of (ii)
=
0
for all
duce that the range of C
I  C
C
and
is closed in h € D(C)
I  C
C*
is
H
are dissipative. H,
implies
Z =
O.
De
so that by Theorem 4,
generates a contraction semigroup on
H.
If all the eigenvalues of a matrix C have negative wt real parts then II exp (Ct) II .::. Me , t ~ 0, for some M > 1 and
w > O.
sions is:
The corresponding question in infinite dimenif
Re(cr(C)) < 0 w > 0
and
C
is the generator of a semigroup T(t), does imply that IIT(t)II'::'Me wt , t ~ 0, for some
M > I?
(cr(C)
general, the answer is no.
denotes the spectrum of
C).
In
Indeed, it can be shown that if
6.
110
a < b
INFINITE DIMENSIONAL PROBLEMS
then there is a semigroup
such that
sup Re(a(C))
=
a
and
T(t)
on a Hilbert space bt IIT(t) II = e [691. Thus to
obtain results on the asymptotic behavior of the semigroup in terms of the spectrum of the generator. we need additional hypotheses. is the spectral radius of
If
T(t). that is
lim IIT(t)k I11/k.
k+ao
then a standard argument shows that for w > Wo there exists wt M(w) such that IIT(t) II ~ M(w)e . Hence we could determine the asymptotic behavior of of
T(t).
However the spectrum of
the spectrum of
from the spectral radius T(t)
is not faithful to
C. that is. in general the mapping relation a(T(t))
is false.
T(t)
=
exp(ta(C))
(6.2.16)
While (6.2.16) is true for the point and residual
spectrum [551 (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.
Let
Z = 9,2
and
=exp(t(continuous
spectrum of C)).
C(xn ) = (inx n ), D(C) = C generates the semigroup
{(xn ) E Z: (nx n ) E Z}. Then T(t) given by T(t) (x ) = (eintx ). The spectrum of C is n n {in: n = 1.2 •...• } while the spectrum of T(l) 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 form a complete orthonormal set.
Thus the asymptotic be
havior of the semigroup could be determined by direct computation.
It seems worthwhile however to give a more abstract
approach which will apply to more general problems. Let
be the generator of a semigroup
C
wt
II T (t) II < Me . I A I ~ exp(wt) . group
U(t) .
values of e
At
A
Hence the spectrum of A E L(Z).
Let
Q=
in
consists only of points
e
,
A E
be compact.
such that
Proof:
A
Then T (t)
U(t)
II T (t) II
with
T (t)
From Theorem 3, A
+
Then i f
C
+
{/.:
A E
A
I AI
Q,
is comwt } > e
Q.
Theorem 5 (Vidav [67], Shizuta [62] ) . tor of a semigroup
isolated eigen
in the set
U(t) At
generates a semi 
We prove that i f
U(t) .
pact then the spectrum,of
U(t)
C
Re (A) > w}.
{A:
is an eigenvalue of
A E L(Z)
+
Suppose that there are only + C
lies in the disc
T (t) A
Then
with
T(t)
Let
C
wt < Me
be the generaand let
generates a semigroup
is compact. C
generates a semigroup
U(t)
which is a solution of the functional equation U(t)
T(t)
We prove that the map in norm on in norm.
[0, t]. Since
s
+
+ {
to T(s)AU(ts)ds.
f(s)
= T(s)AU(ts)
(6.2.17) 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.
l1Z
6.
INFINITE DIMENSIONAL PROBLEMS
To prove the claim on the continuity of prove that t
0
> S >
s
AU(ts)
+
and
hI
is continuous in norm.
0
>
f
with
hI
we first Let
sufficiently small.
Then
the set {A(U(tsh)  U(ts))w: Ilwll has compact closure.
For
£
>
0
=
Let
g(s)
Ihl < hI}
it can be covered with a
finite number of balls with radius wl,wZ, ... ,w n .
1,
and centres at
£
= AU(ts).
Then for
Ilwll
= 1,
Ilg(s+h)w  g(s)wll ~ IIA(U(tsh)  U(ts)) (ww k ) II + IIA(U(tsh)  U(ts))wkll . Hence if
0
is chosen such that
IIA(U(tsh)  U(ts))wkll < for
Ihl < 0
(constant)
£.
and
k
=
1,Z, ... ,n
The continuity of
£
then f
Ilg(s+h)  g(s)11
~
fol1ows 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)
and
U(t) T(t)
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 spectrum of U(t) lying outside the circle IAI = e wt consists of isolated eigenvalues with finite algebraic multiplicity.
6.2.
Semigroup Theory
113
The above result is also useful for decomposing the space
Z.
Suppose that the assumptions of Theorem 5 hold.
Since there are only isolated points in the spectrum of U(t) wt outside the circle IAI = e , there is only a finite number of points in the spectrum of ~
Re(A)
w +
£
for each
£
sum
such that
£
>
in the halfplane
Let 0
AI, ... ,A n denote these and let P be the corres
Then we can decompose
PU(t) + (IP)U(t)
M(a)
O.
>
eigenvalues for some fixed ponding projection.
A + C
and for any
a >
£
II (IP)U(t) II ~ M(a)e(w+a)t.
U(t)
into the
there exists Finally, PZ
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. product on
H
pact.
, ). such that
Let
Let
H
be a Hilbert space with inner
A be a positive selfadjoint operator
AI
is defined on all of
H
and is com
Consider the equation
v
+ 2av + (A+B)v
0
where B € L(H) and a > O. Let Z be the Hilbert space D(A I / 2 ) x H with inner product < > as defined in Example
s.
We can recast the above equation in the form
w
Cw
C
As in Example 5, C
is a bounded perturbation of a skew
selfadjoint operator and so
C
generates a group
Set).
Let
114
6.
u (t) Then
U(t)
o
[ a:
I
INFINITE DIMENSIONAL PROBLEMS
J
S (t) [
;l
I aI
is a group with generator
[ aI
+
and the compactness of the injection C2 : Z + Z is compact. Since C1 generates a group that
the asymptotic behavior of
:J
[ a 2 I:B
a: ]
A
D(A 1 / 2 )
+
H
implies
Thus the above theory applies. U1 (t) with IIU 1 (t)ll.::.. e at , U(t) (and hence Set)) depends
on a finite number of eigenvalues.
H
=
D(A)
The above theory applies, for example, to the case 2 L (n), where n is a bounded domain in mn, A = ~, {v E H:
=
~v
E H and 1
elliptic theory, A Example 7. U
tt
v
0
=
an}
on
since by standard
is compact.
Consider the coupled set of wave equations +
2au t  u xx
+
Bf:
g(x,s)u(s,t)ds 
Bv
=
0
(6.2.18)
for
0 < x <
~
with
u
=
v
0
at
I
g(x,s)
x
=
O,~,
where
ns.
n=l
As in Example 5 we can write (6.2.18) as a first order system w on
1
2
Cw
=
Z = (H 0 (0, 'If) x L (0, ~))
2
•
Since
C
is a bounded pertur
bation of a skewselfadjoint operator it Rcnerates a
6.Z.
Semigroup Theory
group
are a
U5
T(t). An easy computation shows that the eigenvalues of C Z a ± (a  an ) l/Z ' n = 1,Z, ... , where a n = n Z or An n Z + (Brr) / (ZnZ) . For a > 0 and B small enough the
n real parts of aU the eigenvalues are negative so we expect that liT (t) II < Me wt , t > 0, for some w > O. This formal
argument can be rigorized by applying the above theory. a = O.
We now suppose that
In this case, for
sufficiently small all the eigenvalues of ginary and they are simple.
M.
are purely ima
By analogy with the finite di
mensional situation we expect that some constant
C
B
IIT(t) II.::. M, t ~ 0, for
We show that this is false.
Consider the following solution of (6.Z.18): 1
u(x, t)
Zrr
v(x,t)
m 1 sin mx cos mt
where
m(cos mt  cos amt)sin mx
m is an integer and
(6.Z.19)
a~
the initial data corresponding to (6.Z.19) is bounded independently of m. Let tm Zm 3rr. Then for large m, Z
1  cos(rrzB) so that
Z lux(x,t m) I ~ (constant)m . Z
+
Thus
O(m 4 )
>
IIT(t m)
constant,
112.
(constant)m . The above instability mechanism is associated with the fact that for
a
= 0, B f 0, the eigenfunctions of C do not
form a Riesz Basis.
For further examples of the relationship
between the asymptotic behavior of solutions of the spectrum of
C see [15,16].
We now consider the nonlinear problem
w = Cw
and
116
6.
w where
C
= Cw
+
INFINITE DIMENSIONAL PROBLEMS
N(w),
= Wo
w(O)
is the generator of a semigroup
N: Z + Z.
(6.2.20)
E Z, T(t)
on
Z and
Formally, w satisfies the variation of constants
formula wet) = T(t)w O Definition.
A function
of (6.2.20) on and if for each
[O,T]
w E C([O,T];Z) if
C*
ing between
is a weak solution 1
w(O)
wOI f(w(')) E L ([O,T];Z)
the function
[O,T]
is the adjoint of
<wet) ,v>
is ab
and satisfies
d dt <wet) ,v> = <wet) ,C*v>
where
(6.2.21)
to
v E D(C*)
solute1y continuous on
ft T(ts)N(w(s))ds.
+
+
C and
< ,>
a.e.
denotes the pair
Z and its dual space.
As in the linear case, weak solutions of (6.2.20) are given by (6.2.21).
More precisely:
Theorem 6 [9].
A function
of (6.2.20) on
[O,T]
and
w
w: [0, T] + Z
if and only if
is a weak solution 1 f(w(·)) E L ([O,T];Z)
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 [61].
Let
N: Z + Z
be locally Lipschitz.
Then
there exists a unique maximally defined weak solution wE C([O,T);Z)
of (6.2.20). Ilw(t)II+ex>
Furthermore, if as
T < ex>
then (6.2.22)
t+T
As in the finite dimensional case, (6.2.22) is used as a continuation technique.
Thus if for some
Wo
E Z, the
6.3.
Centre Manifolds
solution
wet)
of (6.2.20) remains in a bounded set then
wet)
exists for all
6.3.
Centre Manifolds Let
Z
117
t > O.
11'11.
be a Banach space with norm
We consider
ordinary differential equations of the form
w = Cw where
C
semigroup
N(w),
+
w(O) E Z,
(6.3.1)
is the generator of a strongly continuous linear Set)
N: Z
and
second derivative with
Z
+
0, N' (0) = 0 [N'
N(O)
Frechet derivative of
has a uniformly continuous is the
NJ.
We recall from the previous section that there is a unique weak solution of (6.3.1) defined on some maximal interval
[O,T)
and that if
T <
then (6.2.22) holds.
00
As in the finite dimensional case we make some spectral assumptions about (i)
Z = X
C.
We assume from now on that:
Y where
~
X
is finite dimensional and
Y
is closed. (ii)
X
is Cinvariant and that if
tion of
C
to
eigenvalues of (iii)
If
Y
U(t) is
B = (IP)C
X, then the real parts of the A
are all zero.
is the restriction of
S (t)
to
Y, then
a,b,
II U(t) II P
is the restric
U(t)invariant and for some positive con
stants
Let
A
~ ae
bt
be the projection on
lind for
t > O.
X along
x E X, Y E Y, let
(6.3.2)
Y.
Let
118
6.
= PN(x+y),
f(x,y)
INFINITE DIMENSIONAL PROBLEMS
g(x,y)
= (IP)N(x+y).
(6.3.3)
Equation (6.3.1) can be written
X
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
=
There exists a centre manifold for (6.3.4), 2 hex), Ixl < 0, where h is C . 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 In general if
yeO)
is not in the domain of
will not be differentiable. yet) = h(x(t)), and since and consequently Theorem 9.
+ f(u,h(u)).
(6.3.5) B
then
yet)
However, on the centre manifold X is finite dimensional
x(t),
yet), 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))
be a solution of (6.3.4) with
\I (x(O) ,y(O))
\I
tion
of (6.3.5) such that as
u(t)
sufficiently small.
Then there exists a solut
+ .. ,
6.3.
Centre Manifolds
119
x(t) yet) where
y
>
= u(t) + O(e yt )
(6.3.6)
h(u(t)) + O(e yt )
O.
The proof of the above theorem is exactly the same as the proof given for the corresponding finite dimensional result. Using the invariance of
h
and proceeding formally
we have that h' (x) [Ax + f (x, h (x) ) 1
= 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 Let
E
Xo
the domain of
B. be small.
X
Let
(6.3.4) with
To prove that
h (x O)
is in
it is sufficient to prove that
B
lim+ t+O exists.
hex)
= xO.
h(x O)
h(x(t))
x(t), yet) x(O)

U(t)h(x O) t
be the solution of
As we remarked earlier, yet)
differentiable.
From (6.3.4)
yet)
= U(t)h(x o) +
J:
is
U(tT)g(X(T) ,Y(T))dT,
so it is sufficient to prove that lim+ t1 t+O exists.
Jt
0
U(tT)g(X(T) ,y(T))dT
This easily follows from the fact that
strongly continuous semigroup and h(xO)
is in tlie domain of
B.
g
is smooth.
U(t) Hence
is a
120
6.
Theorem 10. origin in Hx) q
>
X into
D(B).
E
~
Let
be a
INFINITE DIMENSIONAL PROBLEMS
1 C
map from a neighborhood of the
Y such that
~(O)
= 0,
~'(O)
x'" 0, (M~)(x)
Suppose that as
= 0 and
= O(lxlq),
1, where (M~)
Then as
=
(x)
~'(x)
[Ax
+ f(x,~)l
 BHx)  g(x,Hx)).
= O(lxlq).
x'" O,llh(x)  Hx)11
The proof of Theorem 10 is the same as that given for the finite dimensional case except that the extension
s: X ... Y
of
domain of 6.4.
~
must be defined so that
is in the
B.
Examples
Example 8. V
tt
+
Consider the semi1inear wave equation v t  vxx  v v
where as
Sex)
f
v'" O.
C3
is a
+
fey)
=
0
0, (x,t)
=
at
E
(O,rr)
(0,"') (6.4.1)
x
= O,rr
x
function satisfying
fey)
=
v3
+
O(v 4)
We first formulate (6.4.1) as an equation on a
Hilbert space. Let
Q
=
(d/dx)
2
+
I, D(Q)
Q is a selfadjoint operator.
=
2
1
H (O,rr) n HO(O,rr).
Let
1
Then
2
Z = HO(O,rr) x L (O,rr) ,
then (6.4.1) can be rewritten as
w = Cw
+
N(w)
(6.4.2)
where N(w)
Since
C
is the sum of a skewselfadjoint operator and a
6.4.
Examples
121
bounded operator, C generates a strongly continuous group. 3 Clearly N is a C map from Z into Z. The eigenvalues of +
A1
and all the other eigenvalues have real part less
0
o.
than
Care
The eigenspace corresponding to the zero eigen
value is spanned by
q1
where
~ ] s in x.
[
To apply the theory of Section 3, we must put (6.4.2) into canonical form.
We first note that
Cq2
= q2
where
and that all the other eigenspaces are spanned by elements of the form
an sin nx, n
2
~
2, an E1R.
other eigenvectors are orthogonal to X
span(q1) , V
Z
X III Y.
= span(q1,q2) ,
The projection
In particular, all q1
and
q2·
Let
span(Q2) III V.l , then
Y
P: Z ... X
is given by
(6.4.3) where w.
J
Let
w
= 2 TT
JTT
0
w.(e)sin e de. J
= sQ1 + y, s E 1R, Y E Y and B
(IP)C.
Then we can
write (6.4.2) in the form
(6.4.4) y
= By + (IP)N(sQ1+Y).
122
6.
INFINITE DIMENSIONAL PROBLEMS
By Theorem 8, (6.4.4) has a centre manifold h(O)
=
=
0, h' (0)
0, h:
y
=
h(s),
By Theorem 9, the equa
(eS,eS) + Y.
tion which determines the asymptotic behavior of solutions of (6.4.4) is the onedimensional equation sq1
PN( sq 1 + h(s)).
=
(6.4.5)
Since the non1inearities in (6.3.4) are cubic, h(s) so that
or (6.4.6)
s
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)
Hence, if vt(x,O)
v(x,t)
> 0
then as
set)
=
t
(4 t) 1/2
+
00,
+ oCt 1/2).
is a solution of (6.4.1) with
small, then either
v(x,t)
(6.4.7) v(x,O),
tends to zero exponen
tially fast or v(x,t) where
set)
=
±s(t)sin x + 0(s3)
(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. Suppose that fey) = v 3 + av 5 + 0(v 7 ) as v + O. In order to calculate an approximation to (M4» (s)
=
h(s)
set
4>' (s)PN(sq1+4>Cs))  B4>Cs)  (IP)N(sQ1+4>Cs)) (6.4.9)
6.4.
Examples
where
4>: IR
(M4» (s)
+
123
Y.
= 0 (s 5)
•
To apply Theorem 10 we choose 4>(s)
If
= 0(s3)
4>(s)
so that
then
(6.4.10)
where
sin 3x.
q(x)
If
4>(s)
(6.4.11)
then substituting (6.4.11) into (6.4.10) we obtain
(M4» (s)
Hence, i f
a
= 3/4,
B1
1/32, BZ
0, then
M4> (s)
so by Theorem 10 h(s)
=
43 qzs 3
+
1[1] 32 0 qs 3
+
O(s 5 ).
(6.4.12)
Substituting (6.4.1Z) into (6.4.5) we obtain
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 
[B +
4 (Z/rr)
f1 (v o
s
(s,t)) Zds]v
xx
= 0, (6.4.13)
• 0 at x = 0,1 and given initial conditions xx V(x,O), vt(x,O), 0 < x < 1. Equation (6.4.13) is a model for
with
v • v
124
INFINITE DIMENSIONAL PROBLEMS
6.
the transverse motion of an elastic rod with hinged ends, v
B a constant.
being the transverse deflection and
The
above equation has been studied by Ball [7,81 and in particu
B = _~2
lar he showed that when
the zero solution of
(6.4.13) is asymptotically stable.
However, in this case the
linearized equations have a zero eigenvalue and so the rate of decay of solutions depends on the nonlinear terms.
In
[131 the rate of decay of solutions of (6.4.13) was found using centre manifold theory.
Here we discuss the behavior
B + ~2
of small solutions of (6.4.13) when
is small.
As in the previous example we formulate (6.4.13) as an ordinary differential equation. vatives and
z
We write
for time derivatives.
2
1
2
H (0,1) n HO (0,1) x L (0,1),
w
[
Ww
1
2
1'
Cw
=
[
Let
v''''
Qv
2 w ] ' Qw w l 2
for space deri
+
Bv",
N(w)
Then we can write (6.4.13) as w
It is easy to check that group on If
Z and that
N
Cw
+
N(w).
C
generates a strongly continuous
is a
COO
A is an eigenvalue of
trivial solution
u(x)
(6.4.14)
map from
Z
u""  Bu" + (A+A 2)u
0
u(O)
u" (1)
u" (0)
=
Z.
C then we must have a non
of
=
into
u(1)
o.
6.4.
Examples
lZ5
An easy computation then shows that C
A is an eigenvalue of
if and only if
Let
£
=
~ZB + ~4.
Then the eigenvalues of
Care
An(£)'
where
2A (£) = 1 + [1_4£]1/Z, AZ(£) = 1  A (£), and all l l the rest of the An(£) have real parts less than zero for £
sufficiently small.
The eigenspaces corresponding to
are spanned by
and
and
.[
~x,
then
Z= X
(jj
Y
~x
] sin
Z
span(ql)' y = span(ql ,qZ)' Y = span (qZ)
(jj
are spanned by elements orthogonal to
X
1 A (£) Z
>
while the eigenspaces corresponding to
Let
where
and the projection
and
ql
n
for
An(£)
qZ'
P: Z ... X
yol,
is given by
where 1
w. = Z J J
Let
w
=
0
w. (e)sin J
sql + y, where
s E ffi
~e
de.
and
y E Y.
Then we
can write (6.4.14) in the form Al(£)sql + PN(sql+Y) (6.4.15)
By + (IP)N(sql+Y) £ =
O.
By Theorem 8, (6.4.15) has a centre manifold lsi
<
0, lEI
is the second
<
EO'
Using
~omponent
of
h(s,£) N(w)
then
y
=
h(s,£),
126
6. 2 ;r
INFINITE DIMENSIONAL PROBLEMS 1
[I 0
s 2~ 4 cos 2~ede ] ~ 2 s sin
~x
+ 0(s4 + le:s 3 1) s3 sin ~x + 0(s4 + le:s 3 1). Hence
= (s 3 +O(s 4 +Ie:s 3 1))q1' so that by
PN(sq1+h(s,e:))
Theorem 9, the asymptotic behavior of small solutions of (6.4.15) is determined by the equation (6.4.16) We can now determine the asymptotic behavior of small solutions of (6.4.14).
For
asymptotically stable.
0 < e: < 6, solutions of (6.4.14) are For
6 < e:<0, the unstable manifold
of the origin consists of two stable orbits connecting two fixed points to the origin. Exercise 17.
For
e:
(See Figure 1.)
= 0, show that equation (6.4.16) can be
written as s
s 3
Phase Portrait for Small Negative Figure 1
e:
6.4.
127
Examples
Example 10.
Consider the equation U
tt
+
V
tt
+
2u t  u xx + a 2v + f(u,v) 2v t  vxx  u + g(u,v) = 0
(x,t) E (O,rr) x (O,rr)
for where For
f(u,v), g(u,v)
u = v = 0
with
at
o
(6.4.17)
x = O,rr,
have a second order zero at
u
= v = o.
= 2, we show that the linearized problem has two purely
a
imaginary eigenvalues while all the rest have negative real parts.
We then use centre manifold theory to reduce the prob
lem of bifurcation of periodic solutions to a twodimensional problem. Let
• • T , t h en we can write ( 7 as ( u,v,u,v) 6.4.1)
w
w = Cw on
Z
=
+
N(w)
1 2 (HO(O,rr)) x (L 2 (O,rr)) 2 •
(6.4.18)
Let
is an eigenvalue of A then A is an eigenvalue of 2 C where 1. + 2A + IJ = 0 and all the eigenvalues of C
If
IJ
arise in this way. values of
An easy calculation shows that the eigen
are given by
C
n = 1,2, ....
For
a
= 2, the eigenvalues of
C
are
all the rest have negative real parts.
erad so that
Al (a)
zero speed.
and
Al = i, Xl
i, while
Also
(Re 1. 1 (2)) > 0
II (a)
cross the imaginary axis with non
It is now trivial to apply centre manifold theory
128
6.
to conclude that for
a  2
INFINITE DIMENSIONAL PROBLEMS
small, the behavior of small
solutions is determined by an equation of the form
5
where
s E m2 , a
=
[a 1
1 ] s + J (s, a) a
is a real parameter and
(6.4.19)
J(s,a)
=
0(s2).
To apply the theory in Section 2, Chapter 3, we need to calculate the quadratic and cubic terms in
J(s,O).
To do this we
need to put (6.4.18) into canonical form and to calculate the centre manifold when On the subspace
a
2.
From now on we let
{r sin nx: r E m4}
can be represented by the matrix
o o 4 n
Note that the eigenvalues of
2
Cn
a
= 2.
the operator
C
where 1
o
o
1
2
o
o
2
C are given by the eigenvalues
Cn for n = 1,2,... . To put (6.4.18) into canonical form we first find a basis which puts C1 into canonical
of
form.
then Let
Calculations show that if
C1 q 1 = q2' C1q 2 = q1' C1 q 3 =  2q 3 + Q4' C1Q4 • Q3 2Q 4' w· Qz where Q. [Q1,Q2,Q3,Q41, then we can rewrite
6.4.
Examples
129
(6.4.18) as .
= Q1 CQz
z Let
= {[sl,s2'O,O] TljI: E lR}, Y = V (jj [X (jj
(jj
Y
5
and the projection w1 w2
Pw
(6.4.20)
E lR} , V = {[O,O,r ,r ] TljI: 1 ,5 2 1 2 V].l where ljI (x) = sin x. Then
X
r ,r 1 2 Z =X
1
Q N(Qz).
+
on
P
X along
is given by
Y
ljI
0 0
Let
z
f:
2
w. J
:rr
w.(e)sin e de. J
T
1 ,5 2 ,0,0] ljI write (6.4.20) in the form [5
=
10 ]
y, si
+
5
+
E
lR, Y
Y, then we can
E
By
Y
where
+
B = (I_P)Q1 CQ ,
(I_p)Q1 N(Qz) 4>
= [l,l,O,O]TljI and
By Theorem 8, (6.4.21) has a centre manifold From
r
w
F (z) ]
L
=
Qz
=
_[
we have that
G(z)
Suppose that F (z) 4 O(lzI ) where F3
=
t
13
t
23 F3 (z)
and
w1
y
[5
G(z)
,5
2
]
T
= h(s).
2z 1  2z 4
and
1
and
G (z) + 3 are homogeneous cubics. Then if
+ 0
(I z 14)
=
=
5
G3 denote the first two components of
=
130
6.
INFINITE DIMENSIONAL PROBLEMS
on the centre manifold,
with a similar expression for
J
2 (s).
Hence, on the centre
manifold (6.4.22)
and we can apply the theory given in Section 2 of Chapter 3. If the constant
K associated with (6.4.22) is zero (see
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. F(z) = F2 (Z) + F 3 (Z) 4 0(lzI ) where F2 and
If
+
4 0(lzI )
and
G(z) = G2 (z) are homogeneous quad
G3 (z) + G2 ratics, then the calculation of the nonlinear terms is much 2 more complicated. On the centre manifold, zl sll/J + 0 (s ), 2 z2 = s21/J + 0(s2), z3=0(s), and z4 = 0(s2). The terms of order s 2 make a contribution to the cubic terms in PQlN(QZ). to
h(s).
Hence, we need to find a quadratic approximation This is straightforward but rather complicated so
we omit the details. Example 11.
Consider the equations Du
xx
+
(Bl)u
+
A2v
+
2Auv
+
2 u v
+
l BA u 2
eDv xx  Bu  A2v  2Auv  u 2v  BA l u 2 u = v = 0 where
A,B,e,D
at
(6 . 4 . 23)
x = 0,1,
are positive.
The above equations come from
a simplified model of a chemical reaction with
u
+
A and
+
6.4.
v
+
131
Examples
BA 1
as the chemical concentrations [5,12]. 1 2 We study (6.4.23) on Z = (HO(O,l)) . Set
w
J.
= [:
C
=
[D d' . dx
2
A2
(B1)
aD d
B
2
dx 2
_ A2
]
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,a,D, C has two zero eigenvalues such that
the restriction of block.
C
to the zero eigenspace has a Jordan
The bifurcation of static solutions where
zero eigenvalues and the restriction of
C
C
has two
to the zero eigen
space is zero, has been studied in [25,26). On the subspace
{r sin nnx, r
€
~2}
the operator
C
can be represented by the matrix 22 D
n n
+
B1
C
n
[
The eigenvalues of for C
n
= 1,2, ••.
B
C are given by the eigenvalues of
Cn We suppose that two of the eigenvalues of
are zero while a11 the rest have negative real parts.
For simplicity we assume that the eigenvalues of If
Cl
Cl is to have two zero eigenvalues then
are zero.
13Z
6.
INFINITE DIMENSIONAL PROBLEMS
(6.4.Z5)
We make the following hypotheses: (HI)
There exists
AO,BO,eO,D O such that (6.4.Z5) is satis
fied and the real parts of the rest of the eigenvalues of (HZ)
C
For
are negative.
A,B,e,D
parametrize
in a neighborhood of
AO,BO,eO,D O' we can
trace(C I ) (6.4.Z6)
The first hypothesis is satisfied, for example, if DO > 0
and (6.4.Z7)
If we vary
B
and
and keep
A = AO' D = DO' then the mapping (B, e) + (det(C I ), trace(C )) has a nonzero Jacobian I at B = BO' e eO i f AO,BO,e O are given by (6.4.Z7), so by the implicit function theorem, (HZ) is satisfied. In e
order to simplify calculations we assume (6.4.Z7) from now on. Let X = {sljJ: s € lR Z}, Y X.L where ljJ(x) = sin 1TX, then
Z
=X
~
Y.
By Theorem 8, the system
w
Cw
E
0
+
N(w)
has a centre manifold
h: (neighborhood of
where we have written
E
= (EI,E Z)'
Z XxlR)+Y,
On the centre manifold,
the equation reduces to (6.4.Z8)
6.4.
where
Examples
s
133
and
=
iii. (z) 1
=
Z Jl N. (z (e) ) sin 0
IT
1
e de.
We treat the linear and nonlinear parts of (6.4.Z8) separately. [l,p] T , qz = [_p,l]T where p = _AlB l / Z I f ql 00' then Cl(O)ql 0 and CZ(O)qz = (A Z+B) ql • Let Q = [ql ,qzl • then
Let
and let
T
z0
M( E)
A +B Then for
E
sufficiently small, M(E)
1
by (6.4.Z6).
Let
s
= QMl(E)r, + M(E)Q
1
J
+
O(d.
is nonsingular and
J [ :, :, J
then (6.4.Z8) becomes
N(QM
1
(E)rljl + h(s,E)). (6.4.Z9)
To check the hypotheses of Section 9 of Chapter 4 we only need calculate the nonlinear terms when the fact that
h(s,O)
E =
O.
Using
= O(SZ), routine calculations show that
M(O)QlN(QMl(O)rljl + h(s,O)) • [Pl'PZ1TR(rl,rZ) 3
3
+ O(lrll + Irzi )
6.
l34
INFINITE DIMENSIONAL PROBLEMS
.. here
3~(1+p2) l(l_p),
PI
R(r l ,r 2)
P2
= 3!(1+p2) l(l+p) (A 2+B)
= alr l2 + a 2r l r 2 + a 3 r 22 a2
Using (6.4.27) , a l 2 zero i f IT DO 1.
=
= (lT2Do)1(1lT4D~)
so that
al
is non
We assume that IT 2DO + 1 from now on. Note also that since 1 + P =  (IT 2DO) 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) Substituting (6.4.30) into (6.4.29) and using the above calculations we obtain
p
0 [ El
=
a
(6.4.31)
1 ] p + F(p,E)
E2
B
o. We have already checked that of
DO,B
is nonzero [B
a
+ O.
is only zero when
tion of a certain algebraic equation).
If
For most values DO B
is a solu
+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
a
in (6.4.23) the theory given in Sec
tion 3 does not apply since the map even defined on the whole space.
(a,v)
~
av xx
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]
R. Abraham and J. Marsden, Foundations of Mechanics, 2nd ed., Benjamin, New York (1978).
:2]
R. A. Adams, Sobolev Spaces, Academic Press, New York (1975) .
[3]
V. I. Arnold, Bifurcations in versal families, Russian Math. Surveys ~, 54123 (1972).
[4]
V. 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).
[5]
J. F. G. Auchmuty and G. Nicolis, Bifurcation analysis of nonlinear reactiondiffusion equations, BUll. Math. Biology lZ., 323365 (1975).
[6]
A. V. Balakrishnan, AP~lied Functional Analysis, Applications of Mathematics, 01. 3, SpringerVerlag, Berlin and New York (1976).
[7]
J. M. Ball, Stability theory for an extensible beam, J. Diff. Eqns. li, 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 (1974).
[9]
J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc . .§l, 370373 (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. ~, 4757 (1976). 136
References
137
[11]
M. Berger, Nonlinearity and Functional Analysis, Academic Press, New York (1977).
[12]
J. Boa and D. S. Cohen, Bifurcation of localized dis
[13]
turbances in a model biochemical reaction, SIAM J. Appl. Math. 1Q., 123135 (1976).
J. Carr and NajiAlAmood, Rate of decay in critical cases, J. Math. Anal. and Appl. li, 242250; II, Ibid., 293 300 (1980).
[14]
[15]
J. Carr and R. G. Muncaster, The application of centre manifolds to amplitude expansions, submitted to SIAM J. Appl. Math. J. Carr and M. Z. M. Malhardeen, Beck's Problem, SIAM
J. Appl. Math.
[16]
lZ.,
261262 (1979).

J. Carr and M. Z. M. Malhardeen, Stability of nonconservative linear systems, Springer Lecture Notes in Math., Vol. 799, SpringerVerlag, Berlin and New York TI9'80) .
[17]
N. Chafee, The bifurcation of one or more closed orbits from an equilibrium point of an autonomous differential system, J. Diff. Eqns. ,!, 661679 (1968).
[18]
N. Chafee, A bifurcation problem for a functional differential equation of finitely retarded type, J. Math. Anal. Appl. 1i, 312348 (1971).
[19]
S. N. Chow and J. MalletParet, Integral averaging and bifurcation, J. Diff. Eqns. ~, 112159 (1977).
[20]
S. N. Chow and J. MalletParet, Fuller index and Hopf bifurcation, J. Diff. Eqns. ~, 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, 209235 (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. ll, 193226
r
(1971) .
[24]
N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Diff. Eqns. ~, 5398 (1979).
[25]
M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory, Commun. Pure Appl. Math. g, 2198 (1979).
[26]
M. Golubitsky and D. Schaeffer, Bifurcation analysis near a double eigenvalue of a model chemical reaction, NRC Rep. 1859, Univ. of Wisconsin (1978).
38
REFERENCES
27)
J. K. Hale, Ordinary Differential Equations, Wiley
28)
J. K. Hale, Critical cases for neutral functional dif
Interscience, New York (1969).
ferential equations, J. Diff. Eqns. !Q, 5982 (1970).
29)
30)
P. Hartman, Ordinary Differential Equations, Second Edition, P. Hartman (1973).
31)
P. Hassard and Y. H. Wan, Bifurcation formulae derived from center manifold theory, J. Math. Anal. Appl. ~(l), 297312 (1978).
32)
A. R. Hausrath, Stability in the critical case of pure imaginary roots for neutral functional differential equations, J. Diff. Eqns. !l, 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) . D. Henry, Geometric Theor~ of Parabolic Equations, to be published by Springer erlag. M. Hirsch, C. Pugh and M. Schub, Invariant Manifolds, Springer Lecture Notes in Math., Vol. 583, 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. g, 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. Math. Phys., Kl. Sachs Akad. Wiss. Leipzig 94,1722 (1942) . 
[40)
G. Iooss, Bifurcation of Maps and A~plications, NorthHolland MathematIcs Studies, Vol. 3 , NorthHolland Publishing Company, Amsterdam, New York and Oxford (1979) .
References
[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. ~,
[42)
139
2, 1 3 5  17 2 ( 1 9 7 7) •
M. C. Irwin, On the stable manifold theorem, Bull. London Math. Soc. ~, 196198 (1970).
[43) [44)
A. Kelley, The stable, centerstable, center, centerunstable, and unstable manifolds, J. Diff. Eqns. 1, 546570 (1967).
[45)
A. Kelley, Stability of'the centerstable manifold, J. Math. Anal. Appl. l!, 336344 (1967).
[46)
W. Klingenberg, Lectures on Closed Geodesics, SpringerVerlag, Berlin and New York (1978).
[47)
N. Kopell and L. N. Howard, Bifurcations and trajectories joining critical points, Adv. Math. 18, 306358 (1976). 
[48)
solutions into Takens, (1972) .
[49)
J. E. Marsden, Qualitative methods in bifurcation theory, Bull. Amer. Math. Soc. ~, 11251147 (1978).
[50)
J. E. Marsden and T. J. R. Hughes, Topics in the mathematical foundations of elasticity, Nonlinear Analrsis and Mechanics: HeriotWatt Symposium (ed. R. J. nops) Vol. 2, PItman, London (1978).
[51)
J. E. Marsden and M. F. McCracken, The Hopf Bifurcation and Its AvPlications, Applied Math. SCIences, Vol. 19, SprInger er1ag, Berlin and New York (1976).
[52)
s.
[53)
J. Moser, Proof of a generalized form of a fixed point theorem, Springer Lecture Notes in Math., Vol. 597, SpringerVerlag, Berlin and New York (1977).
[54)
A. I. Neishtadt, Bifurcation of the phase pattern of an equation system arising in the problem of stability loss of selfoscillations close to 1 : 4 resonance. P.M.M. ~, 896907 (1978).
[55)
A. Pazy, Semigroups of linear operators and applications to partial differential equations, University of Maryland Lecture Notes, Vol. 10, (1974).
J. Merrill, A model of the stimulation of Bcells by replicating antigen, I., Math. Biosci. 41, 125141 (1978); II. Ibid., ~, 143155 (1978). 
40
REFERENCES
56)
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Se1fadjointness, Academic Press, New York (1975).
,57)
D. Ruelle and F. Takens, On the nature of turbulence, Comm. Math. Phys. ~, 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 Theery, Springer Lecture Notes In Mith., Vol. 309, SprIngerVerlag, Berlin and New York (1973).
[60)
l!, 339364
[61)
I. Segal, Nonlinear semigroups, Ann. Math. (1963) .
[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, l43l45 (1979).
[64)
F. Takens, Forced Oscillations and Bifurcations, Communication 3, Math. Institute, Ri]ksuniversIteit, Utrecht (1974).
[65)
F. Takens, Singularities of vector fields, Publ. Inst. Hautes Etudes Sci. il, 47100 (1974).
[66)
D. A. Vaganov, N. G. Samoilenko and V. G. Abramov,
Coo, Math. Z.
Periodic regimes of continuous stirred tank reactors, Chern. Engrg. Sci. ll, 11331140 (1978). [67)
I. Vidav, Spectra of perturbed semigroups with applications to transport theory, J. Math. Anal. Appl., ~, 264 279 (1970).
[ 68)
Y. H. Wan, Bifurcation into invariant tori at points of resonance, Arch. Rat. Mech. Anal. ~, 343357 (1978).
[69)
J. Zabczyk, A note on
Co semigroups, Bulletin de L'Academie Polonaise des Sciences, Serie des SCIences math. astr. et phys., 23, 895898 (1975).
INDEX Analytic semigroup, 105, 106, 110
Contraction mapping, 14, 17, 19, 22, 2426, 102104, 108109
Asymptotic behavior, 4, 812, 19, 29, 32, 3739, 46, 89, 114115, 118, 122123, 126, 128
Coupled equations, 1
Asymptotic stability, 1, 2, 4, 68, 21, 3536,118, 122,124,126
Dissipative operator, 108109 Eigenvalues, 3, 7, 1113, 16, 18, 20, 25, 34, 40, 44, 4647, 50, 5355, 8991, 98, 109, 111112, 114115, 117, 120, 124125, 127128, 131132
Averaging, 42 Autonomous equations, 13 Bifurcation (see also Hopf bifurcation), 1112, 4041,43, 47, 50, 5253, 5559,82, 8485, 94,127, 131, 134135
Equilibrium point, 8, 13, 28, 55, 57, 59, 6567, 69, 7375, 82, 84, 126, 131 Evolution equation, 13, 97 First integral, 65 Generator, 97, 99134
Cauchy problem, 101 Geodesics, 53 Center manifold approximation of, 2, 57, 9,13, 25, 32, 35, 38, 49, 120, 122123
Gronwall's inequality, 18, 20, 2324, 98 Group, 106107, 113114, 124
defini t ion, 3
Hamiltonian system, 53
existence of, 14, 67, 9, 1217, 2934, 38, 46, 89, 93, 97, 117118, 122, 124125, 127129, 132
Hi11eYosida theorem, 102104, 107, 109
flow on, 26, 8, 10, 19, 2122, 29, 32, 38, 4647, 89, 92, 119, 126, 128129,
12, 35, 118132
Homoc1inic orbit, 57, 6768, 7475, 80, 82, 84 Hopf bifurcation, 3950
Imp1i cit function theorem, 4041,43, 48, 52, 56, 6768, properties of, 1920, 2830, 70,73,94,132 95, 118 Infinitesimal generator (see generator) Chemical reactions, 8, 12, 130131 Instability, 2, 4, 67, 21, 35, 4344, 115, 118 Closed graph theorem, 106 Compact operator, 111112, 114 Continuation, 116
Invariance of domain theorem, 24 Invariant manifold, 23, 13, 16, 30  34
.42
INDEX
[nvariant set, 13, 3334, 5153, 117
Separatrix, 8
Jacobian, 3, 9394, 132
Skewselfadjoint operator, 107, 113, 120
Laplace transform, 102
Spectral radius, 110
Liapunov function, 37
Spectrum, 109113, 115, 117
LiapunovSchmidt procedure, 12
Stability, 2, 4, 6, 8, 19, 2122, 35, 4344, SO, 57, 118
Linearization, 7, 1113, 15, 34, 37, 40, 44, 4647, SO, 5354, 57, 71, 8182, 88, 9294, 127
Stable manifold, 3, 8, 13, 15, 67
LumerPhillips theorem, 108
Symplectic mapping, 53
Neutral functional differential equations, 13
Uncoupled equations, 1
Nilpotent, 20
Uniform boundedness theorem, 100
Normal form, 52, 56, 89
Unstable manifold, 8, 13, 67, 82, 126
Strong solution, 104105
Parabolic equations, 105 Weak solution, 105, 116 Partial differential equations, 13, 98, 102 Periodic solutions, 28, 33, 4045,47, SO51, 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