Applications of Centre Manifold Theory (Applied Mathematical Sciences)

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)(y-w)

(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

-,,(u-U ) +

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 p-dimensional 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 non-zero 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 Liapunov-Schmidt procedure in the analysis of static problems. Our use of centre manifold theory in bifurcation problems follows that of Ruelle and Takens [57) and of Marsden and McCracken [51).

For the relationship between centre

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

In

general, given a parameter dependent differential equation it is difficult to classify all the possible phase portraits. For an example of how complicated such an analysis can be, see [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 1-3 are the simplest such results in centre

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

The assumption that the eigenvalues of the lin-

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

The equations need not be autonomous.

(3)

In certain circumstances we can replace 'equilibrium

point' by 'invariant set'. (4)

Similar results can be obtained for certain classes

of infinite-dimensional evolution equations, such as partial differential equations. There is a vast literature on invariant manifold theory [1,8,22,23,27,28,30,32,34,35,42,44,45,48,51).

For applications

of invariant manifold theory to bifurcation theory see [1,14,

17,18,19,24,31,34,36,37,38,47,48,49,51,57,65).

For a simple

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

Our proofs of

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

Theorem 3 is

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

The method of approximating the centre manifold in

Theorem 3 was essentially used by Hausrath [32) in his work on stability in critical cases for neutral functional 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 cut-off 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:)[lx-x'l + ly-y'll, IG(x,y) - G(x',y')1 < k(e:)[lx-x'l + ly-y'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)e-rtlxo-xil + (l+Pl)M(r)k(e:)

Ito

er(s-t) 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

n-dimensional system u

Au + f(u,h(u)).

(2.4.1)

In this section we prove a theorem which enables us to relate the asymptotic behavior of small solutions of (2.3.1) to solutions of (2.4.1). We first prove a lemma which describes the stability properties of the centre manifold.

20

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 (t-s) (T~)

I~(t)eatl ~ K

sup{ I ~ (t) eat I: t ~ O}, then

be the solution of (2.4.6) wit-h 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

~

Clzole-at

+

Ck(e:)( e-a(t-s)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 one-to-one, 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 one-to-one 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 one-to-one 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

(2i-2)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

-he-x) .

•

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)

~

-he-x)

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(x-y) (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(x-cp(x)) + cp(x) + f(x-cp(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 one-parameter 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 co-ordinates 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 non-zero, it is straightforward to prove the existence of periodic solutions by means of the implicit function theorem. The specific type of bifurcation depends on the sign of so it is necessary to obtain a formula for

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

+

£ 2-3 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 non-zero

speed and that A(O)

[0

-OlO] 0

Olo

with 3.3.

non-zero. 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 + ( a-Zx+ -x ~

EX

where a

and

E,Ii'Yl'Y2 b

The equations are

1

a

2" 1i(1-x) - a - ylab

b

-ylab + Y2b

are positive parameters.

(3.3.1)

In the above model

represent certain concentrations so they must be

non-negative.

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(l-F(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 (l-x ) - - 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 (x-x O) - xoY - z 2 1 3x O + a O - 2· Then assuming 1jJ is non-zero, 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(1-x ) [6y l x O O

if

e:

is sufficiently small.

Hence, the eigenvalues of

cross the imaginary axis with non-zero 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 non-zero also.

K(O)

To calculate

is non-zero 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)

- ljI-lYCP2(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 non-zero 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 two-dimensional manifold so we can assume

m

2

without loss of generality.

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

~

A(~)

is a one-dimencrosses 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 two-dimensional 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+le-in8

+ 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(~)e-ine)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 one-to-one 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(~~n-2)/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 two-dimensional 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 two-dimensional 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. 333-348), 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 co-ordinates 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 2-5.

Sections

3-8 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 non-zero 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 3-5 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. 337-338] 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 + (Yl-l) (Yl+l) Thus, for

c

close to

1

z

=

z

z

(c-l) (c+l) . (4.4.3Z)

the closed curves are approximately

Z Z Z YZ + Z(Yl-l) = Z(c-l) .

(4.4.33)

So, instead of equation (4.4.Z9) we consider the equation

=

G(\l,a,c) If we prove that that

(C-l)-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)

= -fe-x), 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

=

(8b-4)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(8b-4-Sb+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(h-l.o) = _yhZQ(h-l)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 one-to-one 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 6-7.

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 4-6 use the transformations in Exercise 8.) Figure 7

4.9.

Quadratic Nonlinearities

87

ON

L

REGION 3

Figure 7 (cont.)

CHAPTER 5 APPLICATION TO A PANEL FLUTTER PROBLEM

5.1.

Introduction In this chapter we apply the results of Chapter 4 to a

particular two parameter problem.

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. !r-rO!

~

By numerical

Then for

!p-PO!

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

!p-PO!

and

!r-rO!

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 non-zero 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 one-dimensional 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(z-z),

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

= S-lx.

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

(A-AO)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

Ir-rol

and

Ip-pol

(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(lr-rol

(Yl'Y2) ~ (Yl'Y2) +

0, £2 = O.

Ip-pol)

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 non-zero, 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 non-zero 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

S-l(A-AO)S, it is easily seen that

J

4 x 4

matrix

is a non-zero

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

J

can use

is non-zero, 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 --c-Yl - c: Y2 + °3' x 4

Xl

where

Yl

(5.4.1)

3 03 = 0(IY l 1 + 1Y213). Let S-l = [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((I-P)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(t-s))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

e-wtT(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(t-h)h

T(t~h)lw

-1

[T(h)w - wl

shows that (6.2.6) holds for the two-sided 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

e-AtT(t)w dt,

IIR(A) II < A-I

(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 Hille-Yosida Theorem.

Theorem 1 (Hille-Yosida 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

(AI-C)-lll ~ A-I

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)

h-l(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

e-AtT(t)wdt

J: e-AtT(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 Hille-Yosida 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 Hille-Yosida 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 < A-I

for

A> 0,

so by the Hille-Yosida 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(t-s)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(t-s)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(t-s)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 Hi1le-Yosida 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 skew-selfadFor

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 (Lumer-Phillips 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(AI-C)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 non-empty set is

H}

is open and closed and use

the Hille-Yosida 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(AI-C)-lwll < (2a

+

bllwll,

0 < 2a < 1. +

Use the inequality

bA-l)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(t-s)ds.

f(s)

= T(s)AU(t-s)

(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(t-s)

+

and

hI

is continuous in norm.

0

>

f

with

hI

we first Let

sufficiently small.

Then

the set {A(U(t-s-h) - U(t-s))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(t-s).

Then for

Ilwll

= 1,

Ilg(s+h)w - g(s)wll ~ IIA(U(t-s-h) - U(t-s)) (w-w k ) II + IIA(U(t-s-h) - U(t-s))wkll . Hence if

0

is chosen such that

IIA(U(t-s-h) - U(t-s))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(t-s).

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) + (I-P)U(t)

M(a)

O.

>

eigenvalues for some fixed ponding projection.

A + C

and for any

a >

£

II (I-P)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

A-I

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 skew-selfadjoint 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(t-s)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 C-invariant and that if

tion of

C

to

eigenvalues of (iii)

If

Y

U(t) is

B = (I-P)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)

= (I-P)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(t-T)g(X(T) ,Y(T))dT,

so it is sufficient to prove that lim+ t1 t+O exists.

Jt

0

U(t-T)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 self-adjoint 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 skew-selfadjoint 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

(I-P)C.

Then we can

write (6.4.2) in the form

(6.4.4) y

= By + (I-P)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 one-dimensional 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) - (I-P)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 non-linear 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 + (I-P)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 two-dimensional 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 .

= Q-1 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)Q-1 CQ ,

(I_p)Q-1 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 PQ-lN(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

+

(B-l)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

(B-1)

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

+

B-1

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 non-zero 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 = _A-lB 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

= QM-l(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)Q-lN(QM-l(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(1-lT4D~)

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 non-zero. 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 1-2 in Section 3.4 of

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P. Hartman, Ordinary Differential Equations, Second Edition, P. Hartman (1973).

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32)

A. R. Hausrath, Stability in the critical case of pure imaginary roots for neutral functional differential equations, J. Diff. Eqns. !l, 329-357 (1973).

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F. G. Heinenken, H. M. Tsuchiya and R. Aris, On the mathematical status of the pseudo-steady state hYpothesis of biochemical kinetics, Math. Biosci. 1, 95-113 (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).

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P. J. Holmes, Bifurcations to divergence and flutter in flow induced oscillations - a finite dimensional analysis, J. Sound and Vib. g, 471-503 (1977).

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P. J. Holmes, Coupled flutter and divergence in autonomous nonlinear systems, to appear in A.S.M.E. Symposium on Nonlinear Dynamics.

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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). -

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INDEX Analytic semigroup, 105, 106, 110

Contraction mapping, 14, 17, 19, 22, 24-26, 102-104, 108-109

Asymptotic behavior, 4, 8-12, 19, 29, 32, 37-39, 46, 89, 114-115, 118, 122-123, 126, 128

Coupled equations, 1

Asymptotic stability, 1, 2, 4, 6-8, 21, 35-36,118, 122,124,126

Dissipative operator, 108-109 Eigenvalues, 3, 7, 11-13, 16, 18, 20, 25, 34, 40, 44, 46-47, 50, 53-55, 89-91, 98, 109, 111-112, 114-115, 117, 120, 124-125, 127-128, 131-132

Averaging, 42 Autonomous equations, 13 Bifurcation (see also Hopf bifurcation), 11-12, 40-41,43, 47, 50, 52-53, 55-59,82, 84-85, 94,127, 131, 134-135

Equilibrium point, 8, 13, 28, 55, 57, 59, 65-67, 69, 73-75, 82, 84, 126, 131 Evolution equation, 13, 97 First integral, 65 Generator, 97, 99-134

Cauchy problem, 101 Geodesics, 53 Center manifold approximation of, 2, 5-7, 9,13, 25, 32, 35, 38, 49, 120, 122-123

Gronwall's inequality, 18, 20, 23-24, 98 Group, 106-107, 113-114, 124

defini t ion, 3

Hamiltonian system, 53

existence of, 1-4, 6-7, 9, 12-17, 29-34, 38, 46, 89, 93, 97, 117-118, 122, 124-125, 127-129, 132

Hi11e-Yosida theorem, 102-104, 107, 109

flow on, 2-6, 8, 10, 19, 21-22, 29, 32, 38, 46-47, 89, 92, 119, 126, 128-129,

12, 35, 118132

Homoc1inic orbit, 57, 67-68, 7475, 80, 82, 84 Hopf bifurcation, 39-50

Imp1i cit function theorem, 4041,43, 48, 52, 56, 67-68, properties of, 19-20, 28-30, 70,73,94,132 95, 118 Infinitesimal generator (see generator) Chemical reactions, 8, 12, 130-131 Instability, 2, 4, 6-7, 21, 35, 43-44, 115, 118 Closed graph theorem, 106 Compact operator, 111-112, 114 Continuation, 116

Invariance of domain theorem, 24 Invariant manifold, 2-3, 13, 16, 30 - 34

.42

INDEX

[nvariant set, 13, 33-34, 51-53, 117

Separatrix, 8

Jacobian, 3, 93-94, 132

Skew-selfadjoint operator, 107, 113, 120

Laplace transform, 102

Spectral radius, 110

Liapunov function, 37

Spectrum, 109-113, 115, 117

Liapunov-Schmidt procedure, 12

Stability, 2, 4, 6, 8, 19, 2122, 35, 43-44, SO, 57, 118

Linearization, 7, 11-13, 15, 34, 37, 40, 44, 46-47, SO, 53-54, 57, 71, 81-82, 88, 92-94, 127

Stable manifold, 3, 8, 13, 15, 67

Lumer-Phillips 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, 104-105

Parabolic equations, 105 Weak solution, 105, 116 Partial differential equations, 13, 98, 102 Periodic solutions, 28, 33, 40-45,47, SO-51, 55, 57, 65-67, 69-71, 73-80, 82, 84, 127 Perturbation, 9, 12, 30, 33, 44-45, 113-114 Phase portraits, 11-13, 54, 60-63, 80, 82, 84-87, 126 Poincare map, 33 Rate of decay, 2, 4, 10, 37, 124 Reduction of dimension, 12, 19, 89 Resolvent, 102-103 Saddle point, 67 Selfadjoint operator, 104105, 107, 113, 120 Semigroup, 97-134