Theory and Applications of Hopf Bifurcation 0521231582, 9780521231589

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London Mathematical Society Lecture Note Series. 41

Theory and Applications of Hopf Bifurcation

B.D.HASSARD,N.D.KAZARINOFF and Y.-H.WAN Department of Mathematics State University of New York at Buffalo

CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE LONDON NEW YORK NEW ROCHELLE MELBOURNE SYDNEY

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Uf"'htGr~ity

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i

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trurnpington Street, Cambridge CB2 lRP 32 East 57th Street, New York, NY 10022, USA 296 Beaconsfield Parade, Middle Park, Melbourne 3206, Australia

© Cambridge University Press 1981 First published 1981 Printed in the United States of America Printed and bound by BookCrafters, Inc., Chelsea, Michigan

British Library cataloguing

~n

publication data

Hassard, B D Theory and applications of Hopf bifurcation. -(London Mathematical Society Lecture Note Series;41 ISSN 0076-0552). 1. Differential equations, Nonlinear 2. Bifurcation theory I. Title II. Kazarinoff, N D III. Wan, Y H IV. Series 515' .35 QA371 80-49691 ISBN 0 521 23158 2

CONTENTS

CHAPTER 1.

The Hopf Bifurcation Theorem

1

1. Introduction

CHAPTER 2.

1

2. The Hopf Bifurcation Theorem

14

3. Existence of Periodic Solutions and Poincare Normal Form

25

4. Stability Criteria

36

5. Reduction of Two-Dimensional Systems to Poincar~ Normal Form

45

6. Restriction to the Center Manifold

52

7. Exercises

71

Applications: (by Hand)

Ordinary Differential Equations 86

1. Introduction and a Recipe-Summary

86

2. Examples

92

Example 1.

The Mass-Spring-Belt System

92

Example 2.

van der Pol's Equation

95

Example 3.

Bulk Oscillations of the LefeverPrigogine System (the Brusselator) 101

Example 4.

Langford's System (Including an Elementary Bifurcation to Tori)

117

3. Exercises CHAPTER 3.

106

Numerical Evaluation of Hopf Bifurcation Formulae

129

1. Introduction

129

2. Location of the Critical Value of the Bifurcation Parameter and the Corresponding Equilibrium Solution

131

3. Evaluation of the Coefficient

c 1 (0) of

137

Poincare Normal Form 3.1

Right and Left Eigenvectors of

the

A

138

3.2. 3.3. 3.4.

Normalization of

v 1 and

u

1 Numerical Differencing for Second Partial Derivatives

Evaluation of

139 140

g 20 , g 11 , g 02 , h 20 , and 141

hll 3.5.

Solution for the Coefficient Vectors w11 , w20 in the Expansion of the Slice v = v of the Center Manifold 142

3.6.

Numerical Differencing for

3.7.

Computation of

c

4. Evaluation of

c21

c 1 (0)

a' (0), w' (0),

144 ~2 ,

r 2 , and ~ 2

144

5. Error Estimation

145

6. Sample Applications

146

Example 1.

The Mass-Spring-Belt System

148

Example 2.

The Centrifugal Governor

149

Example 3.

The Lorenz System

156

Example 4.

The Hodgkin-Huxley Current Clamped System 159

Example 5.

The Brusselator with Fixed Boundary Conditions

166

A Panel Flutter Problem

169

Example 6.

7. Exercises CHAPTER 4.

142

174

Applications: Differential-Difference and Integra-differential Equations (by Hand)

181

1. Introduction

181

2. Theory and Algorithm for DifferentialDifference Equations

182

3. The Hutchinson-Wright Equation and Related Examples

191

4. An Example with Two

Distin~t

Lags

200

5. Unbounded Delays

205

6. A Three-Trophic-Level Model

210

7. Exercises

219

CHAPTER 5.

Applications: Partial Differential Equations (by Hand)

224

1. Introduction

224

2. Semiflows

226

3. Hopf Bifurcation and Associated Stability Computations for Local Semiflows

233

4. The Brusselator with Diffusion and No Flux Boundary Condition on an Interval or a Disk

239

5. The Brusselator with Diffusion and Fixed Boundary Conditions 6. Exercises

247 263

APPENDICES

266

A. The Center Manifold Theorem

266

B. Summary of Results on Continuous Semigroups

277

C. A Regularity Theorem for Maps of Sobolev Spaces

282

D. Truncation Error, Roundoff Error and Numerical Differencing

284

E. The Code BIFOR2 for Numerical Evaluation of Hopf Bifurcation Formulae

288

F. A Sample Program Using BIFOR2

296

G. Contents of Microfiche

298

REFERENCES

300

INDEX

310

PREFACE We were motivated to write these Notes by our joint belief that it would be useful to scientists in many fields to have a mature and effective version of the Hopf bifurcation algorithm available together with examples of how to apply it, both by hand and by machine. Our capacities for writing these Notes were much enhanced by colleagues who both stimulated our efforts and gave generously of their knowledge.

We thank them all:

Jim Boa, Steve

Bernfeld, Shui-Nee Chow, Don Cohen, Paul Fife, Jim Greenberg, Jack Hale, Alan Hastings, Philip Holmes, In-Ding Hsu, Bryce McLeod, Milos Marek, Jerry Marsden, Alistair Mees, Piero de Mottoni, Aubrey Poore, John Rinzel, Reinhard Ruppelt, David Sattinger, Luigi

Salvador~Agnes

Schneider, Joel Smaller, and

Pauline van den Driessche. We also thank the National Science Foundation for support under Grants MCS-7905790, MCS-7819647, and MCS-770108. We owe our typists special thanks. reading was typed by Gail Berti.

The copy you are

Earlier versions were typed by

Marie Daniel, Sue Szydlowski, and Lynda Tomasikiewicz.

Their

work enabled us to improve these Notes significantly.

Buffalo, N. Y., March, 1980

Brian Hassard Nicholas Kazarinoff Yieh-Hei Wan

CHAPTER 1.

1.

THE HOPF BIFURCATION THEOREM

INTRODUCTION Periodic phenomena or oscillations are observed in many

naturally occurring nonconservative systems.

The purpose of

these Notes is to describe some of the recent developments in the study of such phenomena; namely, those related to what is called Hopf Bifurcation.

In this Introduction we shall limit our

attention to Hopf Bifurcation for an autonomous system of ordinary differential equations

-dx = dt

x E JRn

where ~9

and

\!

(1 .1)

f (x \!)

'

is a real valued parameter on an interval

• The crucial hypotheses made are that (1.1) has an isolated

stationary point, say at

x

=

x*(\!) , and that the Jacobian

matrix

i,j

has a pair of complex conjugate eigenvalues

\ 1

=

l, ...

and

,n)

(1. 2)

\ 2 , (1.3)

such that for some number

1

= The number

0, and

a 1 ( \! c )

is called a critical value of

\)

c of eigenvalues

\)

(1.4)

=f. 0 •

.

If the

other than :- iw0 ) all have strictly ) c ) negative real parts, the assumption (1.4) means that there is a A(\!

loss of linear stability of the stationary point

x*(\!)

as

\!

Under certain additional technical assumptions, c we will prove in this Chapter that the system has a family of crosses

\!

periodic solutions

=

X

p

E:

e

(t)

measures

the amplitude

and

e0

is sufficiently small.

This appearance of periodic

solutions out of an equilibrium state is called Hopf bifurcation [55].

Most usually, the periodic solutions exist in exactly one

of the cases

\!

>

(i.e. indexed by

\!

c

,

\!


y c

It was shown in [45]

(ii)

y

=

y

c

(iii)

y




(3H(e)

(2. 6)

is orbitally asymptotically stable

(3H(e)


o, (4)

the remaining

n- 2

o, a'

a(O) =

(O) =/:

eigenvalues of

A(O)

o, have strictly

negative real parts, then the system (2.1) exist an

ep

~

p

>

0

and

has~

~

family of periodic solutions:

L+l

C

.

-funct1on

(e)

such that for each

~

P

there

(e) ,

(2. 7a)

e E (O,ep)

there exists~ periodic solution

16

p

e of

(t) , occurring for

X

any

1-1

EJ

There is

h

h

neighborhood

0

containing

are members of the family

is a

(t)

~

such that for

the only nonconstant periodic solutions of (2.1) that

satisfying

e

p 1-1 (e)

= 0 and an open interval

lie in

p

=

1-1

p

= 1-1

(e)

1-1

L+l

C

e

e E (O,ep) •

,

e

for values of

p (t)

The period

of

.

-functl.on

(2. 7b)

Exactly two of the Floguet exponents of

e

~

0 •

One is

0

(t) approach e e E (O,ep) , and the other is ~

for

p

0

as L+l

c

function

L

[-2] p =

"" LJ

21..

R 1 fJ2i e

+ 0 (,1+1) "'

(2. 7 c)

The periodic solution

p (t) is orbitally asymptotically stable e with asymptotic phase if {3p(e) < 0 but is unstable if p

f3 (e) > 0 • ,,P

~""2K

If there exists a first nonvanishing coefficient

(1 ::::;: K ::::;: [L/2 ]),

then there is an

such that ---

the open interval

(2 .8)

has the following properties. unique

e

E (O,e 1 )

periodic solutions p(t;!J.) exponent

(!J. E J 1 ) • f3(1J.)

For any

1-1

=

for which

!J.p(e)

pe(t)

< e < e 1)

For

are

vanishing coefficient

(0 1-1

{32 K

of

J1

there is a

Hence the family of

1-1 •

may be parameterized as

E J 1 , the period

c1 -functions p

in

I~J.Il/K

T(!J.)

and Floquet

The first non-

is given £y (2. 9a)

17

and p

sgn ~(~)

= sgn ~2 K

p(t;~)

Thus the members

(~

(2. 9b)

E J 1)

of the family of periodic

solutions are orbitally asymptotically stable with asymptotic

~~K


and are unstable if

0 •

Theorem III. Suppose that the hypotheses of Theorem I

are satisfied. p

~2 K

Also assume there exists a first nonvanishing coefficient in the expansion (2.7a).

X

=

J1

0

and

~

e: 1 > 0

number

p(t;~)

periodic solution

T(~)

E J1

h

neighborhood

E J1

~

such that for each

1:11/K:.

is an analytic function of

0

One is

~

for

~(~)

valued function

-h.

of (2.1) which lies in

p(t;~)

of the Floguet exponents of ~

exist~

of

, where

Qy (2.8), there exists exactly one nonconstant

is given

period

Then there

approach

0

Exactly two ~ ~

as

E J 1 , and the other is

, analytic in

The

~

0 ,

real

For each

in

J1 ,

the interval

I

p

sgn ~(~) = sgn ~ 2 K , p

~2K

where p(t;~)

Qy (2.9a).

is described

The periodic solutions

are orbitally asymptotically stable with asymptotic

phase if

~ 2:


0 •

We amplify these theorems in the following Remarks. Remark 1.

If, for all sufficiently small positive

periodic solutions ~

> 0

(resp.

~

defined to be +1

p (t)

e:

0,

~

in Theorems I, II or III exist for

< 0),

the direction of bifurcation is

(resp. 0, -1) .

vanishing coefficient

p

~2 K

e: , the

(or

the direction of bifurcation.

If there exists a first nonH

~ 21 )

, then its sign determines

In applications, it is more

18

difficult to find the direction of bifurcation than it is to determine that a Hopf bifurcation occurs.

In the remaining

sections of this Chapter we shall derive and describe Bifurca-tion Formulae that enable one to evaluate p

p

p

~2' ~4'

p

~2 I 0

If

'

or if

p ~2

p

T 2'

= 0

p

(3~

and

,.4' (32' p

~4 I 0

and

0

then the direction of

)

bifurcation is explicitly determined from these Bifurcation Formulae. The terms supercritical and subcritical also appear in the literature.

>

~

0

ex' (0) > 0

If

)

periodic solutions which exist for

< 0) are termed supercritical (resp. If ex' (0) < 0 '- however, there is disagreement

(resp.

~

subcritical).

about the definitions of these terms, and we suggest that the reader proceed cautiously when encountering this situation. Remark 2.

Theorem I is a restatement of E. Hopf's original

version [81, pp. 163-205 and 55]. remains in Theorem I: powers of

e

A defect of the original

namely, the possible presence of odd

~H(e),

in the expansions of

tee), and

TH(e) •

In-consequence, Hopf's Theorem (Theorem I) does not provide the full analyticity results possible for the functions (3(~)

~

If the Hopf expansion of

•

H

~2 J

vanishing coefficient ~ Te lL)

H

and

Q(11.) ~ ~

1'n

,

(e)

T(~)

and

has a first non-

Theorem I only proves analyticity of

\~~.\ ~ 1 /2J

In this same case, we claim that

the first nonvanishing coefficient in the expansion (2. ?a) of ~

p

(e)

2K = 2J

occurs for the same index

Thus whenever

(The proof

e in the derivations of

involves a comparison of the role of Theorems I and II.)

0

~

H

(e)

does not vanish

identically, i.e. has a first nonvanishing coefficient Theorem III shows that ~ 111.11/J

0

T(~)

In particular, 1"f

and ~2

I 0

19

(3(~)

H

~2J ' are actually analytic in

(it has been shown that

p H [46]) both T(\-1) and 8(\-L) are analytic in 1-1 • 1-12 1-12 Another consequence of the odd powers of € in Theorem I

is the appearance in the literature of approximations of the form

T3 , ~ 3 are nonzero. When encountering such approximations, perform the following substitution. Let 1-1 3 ,

in which some of

€

=

~

€

+

~2

Y€ , and form ~2

1-L

=

\-12€

T

=

2TI

8

wo

(1

vanish simultaneously. 'iy

and

~3

~ ~3

\-13€

+

~2 'i 2 €

+

0 (€4)

+

82€~2 + ~3€3 +

Then, for some choice of

\-1 3 ,

+

)

~ ~3) T3€

0 (€4)

y , all of

+

0 (€4)

'

. 1-13 , ~ 3 , and

~3

will

This demonstrates that the appearance of

was caused by an unfortunate choice of algorithm

for computing the bifurcating solutions. Remark 3.

The case where some (or all) of the remaining

eigenvalues of

A(O)

n - 2

(cf. Hypothesis (4) above) have strictly

positive real parts for all

1-1 E c:9

is ,easily handled.

The

periodic solutions that arise are, however, unstable because the eigenvalues with positive real parts give rise to characteristic (Floquet) exponents with positive real parts. A variety of interesting phenomena, including bifurcation to tori, may occur if some (or all) of the remaining

20

n - 2

eigenvalues are on the imaginary axis when

0

see [62,

pp. 67-69; 81, pp. 206-218], for example. Remark 4.

The system

.=

r

-

~r

. e=1

r

2£

sin(.!.) r

'

'

(in polar coordinates) or xl

=

~xl

- x2

-

x 1r

2£-1

sin(.!.) r (r

x2

xl

+

~x2

-

x 2r

2£-1

2

=

2 xl

2

+ x2)

sin(.!.) r

(in Cartesian coordinates), where

J.,

'

is a positive integer, has ~

finitely many periodic solutions for each hood of 0 E R 1 and infinitely many for

~ =

in a full neighbor0 •

In this

example,

=

~ (8)

1 , and

8

2£-1

1 sin(-) 8

The orbits of the periodic solutions of the system are circles with radii determined from the equation . (1) r 2£-1 s~n­ r Theorem I does not apply to this example; but, if Theorem II does apply. 8

,

There is one periodic solution for each

but there are different numbers of periodic solutions for

different values of

~

• CXl

Remark 5. p

~2K

J., ~ 5 ,

=

0

In Theorem II, if for

k

=

1,2, ••. , and

F E C

0 ERn

jointly in its arguments, is attracting for

one cannot conclude stability; see Chafee Salvadori [88].

21

[15]

~

and Negrini-

=

0,

Remark 6.

The difference between Theorems I and II lies with

the conclusions involving uniqueness and analyticity of and

8 .

~'

T,

We shall prove Theorem II in Sections 3-6 of this

Chapter using the Center Manifold Theorem and a reduction of (2.1) to Poincare normal form.

This approach yields a stronger

uniqueness result than that of Theorem I, bvt this is balanced by a loss of analyticity of

~'

~

T, and

e

in

in Theorem II.

Clearly, Theorem II can be used with analytic hypotheses to sharpen the uniqueness result of Theorem I.

What is not so

obvious is that Theorem II also sharpens the analyticity results, with respect to

, of Theorem I.

~

We do this in the following:

Proof of Theorem III. Theorem I establishes the existence of analytic functions ~

H

(e)

)

TH (e)

assumption for H

~2 J

~

)

~H (e)

and

p =!= 0 ~2K

)

(superscript

for Hopf).

H

By the

the bifurcating periodic solutions occur

t 0 ; hence, there exists a first nonvanishing coefficient

in the power series H

~

co

(e)

= r;

H j

~.e

J

j=2

Therefore, by Lagrange's theorem on reversion of series [23, pp. 123-125], there exists an inverse function EH(~) analytic in the variable l ll.ll/2J ~ and is such that

EH(~H(e))

=e

(0

~

e

~H (EH (~)) = ~

(0

~

~~~2J

for some sufficiently small positive

~

eo)

H

eo

that is

' H H ~ Ceo)/~2J)

~

Theorem II also

0

applies under the hypotheses of Theorem I and establishes the p

existence of functions

~

for Poincare) that are

c1

p

(e) , T (e) , in

e

~

p

(e)

(superscript

P

for any positive integer

L

and which have expansions in even powers of

22

e •

For any integer

I > K , the function

p 1-1 (e)

has an expansion

p

1-1 (e)

and since the form

where

1-L~K #

0, there exists an inverse function

M= I + 1 - K •

This function

for some sufficiently small positive

EP(!J.)

of

has the properties

e1 .

Now, since the periodic solutions are unique,

and, further,

for all sufficiently small Since

TH(e)

and

1-1

SHCe)

having the same sign as are analytic in

e

and

p

1-LzK •

EH(!J.)

analytic in 1~-Lll/ZJ, T(!J.) and $(1-J.) are analytic in l/2J p p p p 1-1 • ·However T (E (1-J.)) and $ (E (1-J.)) may be expanded

I l

• ~ 1 ,~11/K

to arbitrary order in powers of of

~ l ,ll/ZJ

Thus the on 1y powers

· th e power ser1es · f or t h at appear 1n

23

T(11) ~

and

is

B(~)

~ 1/K III,\

are powers of

,

This completes the proof of

Theorem III. Remark 1.

In the usual case

Floquet exponent sufficiently small

~(~) ~

p

~2

=f 0 , the period

are analytic functions of

and the ~

, for all

•

H The explicit formula for ~ 2 derived in [57] is the p same as the formula for ~ 2 derived in [46]. Thus we may set

Remark 2.

p ~2

H

~2

~2

In these Notes we shall follow the Center Manifold approach [ 46,

81L and for convenience we adopt the notations p

~·J

= ~· J

~·J = ~~J

)

·r. = J

24

p 'i.

J

(j

=

1,2, •.. )

.

3.

,

EXISTENCE OF PERIODIC SOLUTIONS AND POINCARE NORMAL FORM Our goal in the remainder of this Chapter is to give a proof

of Theorem II. Part I.

Our proof is divided into three main parts:

2X2 systems in Poincare normal form; Part II.

The

reduction of general 2X2 systems to Poincare normal form; Part III.

Application of the Center Manifold Theorem to reduce

general nXn systems to the 2X2 case on the center manifold. There are more efficient techniques available for proving Theorem II [25, 94] than those we use to produce the apparently roundabout proof given in these Notes.

We have proceeded as we

have both for pedagogical reasons and because we desired to exhibit efficient algorithms for computing the form of the bifurcating periodic solutions, their periods, and their stability.

The formulae which are the end results are composed

of constituent parts, the different parts corresponding to segments in our proof, and each part being significantly simpler than the whole. Part I(A).

Existence.

We assume we are given a 2X2 system in the following Poincare normal form

[6,

Chapter 5]:

(hJ 2 X= A(ft)X

+

(3 .1)

~

j=l

where

(3. 2)

25

Re c. (f1)

- Im c. (f1) J

J

(1 ::;; j ::;;

B. (f1) = J

and

F(X,f1)

is jointly

c1 +2

in

X and

[~]) '(3.3)

1-1 •

Equation (3.1) is equivalent to

(3 .4)

S

where Remark.

=

x1

+ iX 2 •

Some material on the uses and history of Poincare's

normal form is given by Arnol'd in [6, Chapters 5 and 6]. Although we have not seen an instance in the work of Poincare in which he makes explicit use of this normal form, all the ideas necessary for derivation of the form (3.1) or (3.4) are present in Poincare's work, and we feel that it is appropriate to attach his name to this form; see, for example [92].

We note that

Poincare does introduce what is now often called Birkhoff normal form for Hamiltonian systems [10; pp. 82-85]. Following E. Hopf's method, we let

X = ey , and we consider

the system

with initial condition (3 • 1) •

y (0)

(The superscript

jointly in CL+l

X and

J·o;ntly ;n ~ ~

denotes

T

(3. 5) makes sense even for

(1, 0) T

=

€

=

0

.

which is

' "trans pose.

Since

(e, 0) 11 )

F (X, f1)

T

for

Note that is CL+2

1-1 , the right-hand side of equation (3.5) is

y , 1-1, a n d

Le t

e •

26

y

= y (t ,e,l-1 )

d eno t e th e

solution of (3.5) that satisfies the given initial condition. By standard theory

[43,

enough, y(t, O, ~) positive

for

e

=0

and

small

exists sufficiently long to cross the

y 1 -axis for

see Figure

5],

Chapter

t

=

T 0 (~)

, where

1.3.

Figure 1.3.

Also, y

is

cL+l

jointly in

t, e, and

~

•

Now,

and from (3 .5)

Since

w(~)

> 0

~

for

in a neighborhood of

0 , we may apply

the Implicit Function Theorem to solve the equation

= 0 for a function T(O,~) = T 0 (~) and

t

=

T(e,~)

T E CL+l

y 2 (t, e,

~)

with the properties jointly in

e

and

~

•

Let (3. 6)

27

Then, for all sufficiently small ~

and

in

•

Since

I

~

(0,0)

~ , = 2n a' (0) I w 0 and

€

I E CL+l

jointly

, we may again

apply the Implicit Function Theorem to conclude that for some L+l in €p > 0 there exists a function ~ = ~(€) , C I(e,~(e))

such that

for

=1

•

We have there-

fore proved the existence of a family of periodic solutions, one for each

e E (O,ep)

of

. = F (X,~)

X

,

X (0)

=

(e,O)

T

,

in the 2x2 case, provided our system is in Poincare normal form. Part I(B).

Bifurcation Formulae

We next derive formulae for the initial coefficients in the MacLaurin expansions of

~

= ~(e)

We begin with some motivation.

and

T = T(e) (=

T(e,~(e)))

Consider the differential

equation (3. 7)

where

z

and the

is a complex variable, A. (0) cj(~)

are complex valued.

in Poincar~ normal form. rotationally invariant: for any real number

=

iw 0 , M

::?:

1

is arbitrary,

This canonical equation is

Observe further that (3.7) is if

z

is a solution, then so is

zei¢

¢,and the trajectories of (3.7) are

circles with centers at

z

=

0 .

This simple geometry is

reflected in efficient computation of the MacLaurin expansions of ~

( e)

and

T (e) •

Forming

z

z +

z

z

from (3.7), we obtain

(3. 8) .

The right-hand side of (3.8) is zero if and only if

28

z

0

or

(3. 9)

But if (3.9) holds, then (3.8) implies that 8 ~ 0 .

some

Setting

zz = 8

2

zz

~ = ~(8)

and

= 82 ~

0 , for

in (3.9) now

gives

(3. 10)

This equation determines the coefficients in the expansion

a'

(In the analysis below of the case ~l'

~3 ,

~S'

... are all shown to vanish, which is a priori

obvious from (3.10).

~

However, we shall later consider the case

~ 0 ,

a' (0) = 0 , a" (0) Expanding

(0) ~ 0 , the coefficients

in which case

in powers of

8

~l

is in general nonzero.)

in (3.10), we find that

(3. 11)

At

O(e) ,

a'

(3.11) implies that

(O)~l

0 ' since

a'(O) ~ 0

find that at

by hypothesis.

=0

•

Thus

(3.12)

Using this result in (3.11),

we

2

O(e )

~2

=-

Re c 1 (0) a' (O)

29

(3.13)

At

O(e 3 ) , (3.11) implies that

a' (O)f.L 3 = 0 •

Therefore (3. 14)

0 (€

Using (3.12) and (3.14) in (3.11), we obtain at

4

) :

or \-14

where

= - a'

1

(O)

[Re c 2 (0)

+ 1-Lz

Re

c{ (0) +

a" (O) 2

2

1-1 2 J

,

(3.15)

I-L2 is given by (3.13). Given that (3.10) holds, we may rewrite (3.7) as (3.16)

Thus

z

= ee 2nit/T(e)

,

where (3. 17)

From this equation the coefficients in the expansion

may be found.

Explicitly at

0(1), (3.17) yields

,o =

1

and, to higher order,

30

(3.18)

(3 .19)

Hence

=0 or

(3.20)

since

w0 > 0

by hypothesis.

Then at

-1 ,2 = -w [Im c 1 (0) 0

At

0 (e:3)

O(e?)

(3.19) becomes

+ ~ 2 w' (O)] •

(3. 21)

(3.19) yields

-wo , 3

=

0 '

or

'1'3 = 0 •

Finally, at

O(e:4)

(3.22)

(3 .19) becomes

or

(3.23)

31

~4

where Remark.

is given by (3.15), ~ 2 If

a'(O)

=

0

but

a"(O)

~. 1 s and J

above for computing the

~2

by (3.13) and

by (3.21).

f. 0 , the algorithm given r. 's can still be used.

In

J

this case (3.11) becomes

r.._-M,~. ,..j} 2 + ... + Re c 1 (0)" 2 + 1..~ 1 ~-"j

a"(O) 2 Th us l· f

u

Re c 1 (0)

and ~l

O(e 3 )

-- 0 •

u

"' (0)

u. 11

=

have opposite signs, k

[-2Re c 1 (0)/a"(0)}

2

;

and there exist bifurcating periodic solutions in a full neigh-

=

=

=

borhood of

~

but

f. 0 , the above algorithm fails, and fractional

a"'(O)

e

powers of T (e)

(see [46]).

0

If, however,

a'

a"(O)

must be introduced in the expansions of

~(e)

0

and

•

The computation of exact formulae for the (i

(0)

~

5)

~­l

and

~.

l

can easily be carried further, but we do not compute

more of these coefficients as they are not needed in the applications which follow in subsequent chapters. We shall next show that the bifurcation formulae derived above for systems of the specific form (3.7) actually hold for general

2X2

Lemma.

systems in the Poincare normal form (3.24) below. If the Poincare normal form of (2.1) is

(3. 24)

where of

C(s,s,~)

is

L+2

C

jointly in

s, s,

~

in~ neighborhood

0 E C XC X R1 , then the periodic solution of period

such that

of (3.24) has the form

32

T (e)

S = € expi2n it/T(e)] + O(eL+2 ) ,

(3.25)

2TT [l T( "') "' = w 0

(3.26)

where '"'L

+"-~l'T"i€

i] +O(€) L+l

and L

i

L+l + 0 (€ ) '

~ ( €) = ~ 1 ~i € ~i

and the

and

'T"i

(3.27)

are again given £y (3.12) - (3.15), (3.20)

- (3.23). Proof.

Let T = t/T(e)

Then in the variables

s

and

(T,f1)

ee

2ni1'

Y1

(3.24) becomes L

[2]

d

2n iT]+~= T(e)T][A.(~) + ~ l

. 2.

cj (~) (f1Y;)J e J] + O(eL+l). (3. 28)

The assumed smoothness of solution

Y1 , with

C(S,,S,,~)

f1(0) = 1

permits us to write the

in the form

(3. 2 9) (1 : :; ; i :::;;; L)

•

We shall show that Ylo - 1 '

(1 : :; ; i

Yl·~ - 0

: :; ; L)

We substitute the right-hand side of (3.29) for Then at

0

0 (€ )

(3. 28) yields

33

•

Y1

in (3.28).

2rr

i71 0 or

d'i

0 .

Hence - 1 .

At

(3.28) yields

O(e) ,

or

where

d1

is a constant independent of

where

d2

is a constant.

1-periodic. d2

=0

.

But

71 2

O(e )

d3

71 dl = 0

)

and hence and since

Thus

is 711 711 (0) = 0 )

Therefore

2

At

where

Consequently

But

e .

,

'

(3.28) yields

is a constant •. Then

is 1-periodic since

71 2 (0) = 0, d 4 = 0.

71

is.

Therefore

Continuing in this way, we show that

34

Hence

d3

=

0

and since

T).

~

=0

(1

s:

i

S:

L) .

If we now substitute the right-hand side of (3.25) for

s

in (3.24), then we obtain the already computed values (3.12) (3.15) and (3.20) -

(3.23) for

and

just

as in the first part of this Section 3. This completes the proof of the lemma. Remark 1.

The condition

that guarantees the

n1

T(e)-periodicity of

is the first of the

orthogonality conditions that arise in Hopf's original paper [55], wherein the Poincare-Lindstedt method is followed. used this condition to evaluate

~1

and

We have not

T1 , nor have we used

subsequent of the orthogonality conditions. Remark 2.

By the statement

s,s,~" we mean that

"C(s,s,~)

C(s,s,~)

is

c1 +2

jointly in

is a function of the real

and that the partial sl = Re s, s2 = Im s and ~ derivatives of c with respect to sl' s2 and ~ of combined order ::;; L + 2 are all continuous. The variable s is

variables

)

included in order to indicate that the right-hand side of (3. 24)

is a function of combination

sl

function theory.

sl

+ is2

and

s2

independently, not just in the

as commonly understood in analytic

C(s,~,~)

need not even be defined for

n# S.

Our use of complex variables is merely for the computational convenience which the complex arithmetic provides.

35

4.

STABILITY CRITERIA In applications of the Hopf Bifurcation Theorem to systems

modeling natural phenomena it is important to determine if the bifurcating periodic solutions are stable.

We next present two

approaches to the question of stability, one based on the Poincare-Bendixson Theorem and the other a calculation of the Floquet exponent that determines stability.

The approach using

Floquet theory provides slightly more information. We begin with the first mentioned approach. our 1-L 2

2X2

f. 0 ,

We assume that

system has the Poincare normal form (3.24) and that i. e . , Re c 1 ( 0)

dr dt

f. 0 •

s = rei 8 •

Set

d8

= 2r Re f(r,e,I-L) ,

ss = /

Then

Im f ( r, 8 , 1-L) ,

dt

and

(4 .1)

where L

f(r,e,I-L) = A(I-L)

[2 J 2j + ~ 1 cj(I-L)r +

The periodic solution indexed by

Let that for

61 E (0,1) 0

< e < e2 c1

. )

e

I

0( (r,I-L)

IL+l )

•

(4. 2)

is just

Then there exists an

e2

< ep!J2 such

the circle

2 2 = [Cr,e) I r = e (1 + 61 )}

lies outside the orbit of the periodic solution indexed by and for any 62 E (0) ~) the circle c2 = [ (r, e)

I

r

2

=

e

e 2 6~}

lies inside the orbit of the periodic solution indexed by

e •

There is no other periodic solution within the annulus determined by

c1

and

c 2 ; see Fig. 1.4.

36

Let

P

be any point

Figure 1.4. on

c1

, and consider the trajectory of (4.1) passing through

On this trajectory at dr de

=

2

P

Re f(r,8,bL(e)) rIm f(r,e,f.L(e))

k

r=e (1+6 1 )

37

2

P.

and

by (3.27) and (3.12) dr/de




0

at

P E

c1 c2

c1

and

determined by

(3.15).

Hence, if

Re c 1 (0)


0 , then

is instable, for we may apply the Poincare-Bendixson

Theorem as above to show that, in the limit

t-+ -co,

p(t,~(e))

is asymptotically orbitally stable; and hence, for increasing p(t,~(e))

t

is instable.

Our second approach to stability is to use Floquet theory, which we now briefly review for the benefit of those readers unfamiliar with it; see G. Floquet [33], J. Hale [39, p. 118], or

P. Hartman [43, p. 60].

(p(t p

We begin with a definition.

Definition.

Let

p(t)

+ T) = p(t)

for all

be a T-periodic solution of t E JR1 ) , where

f E

c1 (Rn,lRn)

x

= f(x)

.

Then

is asymptotically, orbitally stable with asymptotic phase if

and only if there exists an solution of time

~

=

f (x)

e

> 0

for which

such that if

liP (t 0 )

t 0 , then there exists a constant

- p (t 0 )

¢ ,

- p ct

38

+

¢) 1

I
0)

y = A(t)y where of

A

t

(A (t

+

T)

= A (t)

for all

t

is a continuous real or complex

1

E 1R ) ,

n Xn

(4. 3)

matrix function

Floquet's theory gives the structure of solutions of

(4.3). Definition.

A fundamental matrix solution of (4.3) is a time-

dependent matrix

Y

such that any solution vector

y of (4.3)

may be expressed as y(t) for some constant vector Theorem (Floquet).

Y(t)c

c •

Every fundamental matrix solution of (4.3)

has the form

-----Y(t) for some

=

P(t)e

Bt

(-co

2 ) calculate the following.

Let

(k = 3) ••• , n) ,

Solve the linear systems

for the

n - 2

dimensional vectors

89

The

matrix

is from step 5.

D

2 1 1 [ 0 F

k-2 GllO

Let

+

= 2 oy 1oyk

02 2 [ 02F2 F + i oy 2oyk oyloyk

02F1

oy 2oyk

J]

o2F2 2 2 + i [ iF1 + o F )] oy 1oyk - oy 2oyk oy 2oyk oy 1oyk '

2 1 1 [ 0 F

k-2 GlOl

=2

and let n-2 g21

8.

~

k=l

k k k k (2 GllOwll + G101w20)

Let

~2

= -Re c 1 (0)/a' (0) ,

,2

=

-(Im c 1 (0) + ~ 2 w'(O))!w 0 , and

~2

=

2 Re c 1 (0) ,

where 9.

= G21 +

a' (0) = Re·

A~ (\!c) , w' (0) = Im

>-i

(\!c) •

The period and characteristic exponent are:

2

4

8 = $2 e +

O(e ) , where

(provided

~2

2 €

= _____c~ + \)

\)

~2

O(\! _ \! )2

c

'

# 0) ; and the periodic solutions them-

selves are (except for an arbitrary phase angle)

90

where y 1 = Re

z ,

Im z ,

y2

and

z

=

ee

2nit/,T + ie 2 [

6wo g02e

-4n it/T

] ( 3) 4n it/T + 6 - 3 g20e gll + 0 8 10.

•

Have a friend check the calculations independently, or use numerical techniques (Chapter 3) to verify the results.

Remark 1.

Exercise 8 at the end of this Chapter will guide the

interested (or dubious) reader through the derivation of this Recipe-Summary from the results of Chapter 1. Remark 2.

If one or more of

then one \-12' 1'2' and (32 is 0 However, the may be interested in calculating \-14' 1'4' and {34 hand calculation of \-14' 'i4 and (34 tends to be a tedious )

procedure, which we do not in general recommend. 8 must be performed for arbitrary ~

=

~c, so that

c 2 (0)

c 1 (\-1)

~

and hence

near

~

c{(O)

c

Steps 4 through

, not just at

may be obtained.

Also,

must be found and this calculation involves 5th order

partial derivatives of

F

at

~

::::

~

c ) y

=0 .

Only for

certain simple systems (such as the van der Pol example below) and

should hand calculation of Remark 3.

(34

be attempted.

The procedure described above is an effective

technique for analyzing the Hop£ bifurcation when the system of

91

ordinary differential equations is simple enough to treat by hand.

The question, what is the 'best' technique, has no single

answer because different classes of problems can have different properties which bestow selective advantage upon one technique or another.

For example, if the linearized system simplifies

greatly under Laplace transformation, then direct application of harmonic

balancing

[3 J

has advantages.

As another example,

note that although we advocate preliminary coordinate transformation in the present Chapter, in subsequent Chapters we generally avoid such transformations. In support of the use of bifurcation formulae as opposed to direct application of a general theory, we note that the use of formulae takes advantage of the simplification performed in derivation of the formulae, which in many problems helps considerably.

Also, the various general theories available

(Center manifold-Poincare Normal Form, Lyapunov-Schmidt, Lyapunov Functions-Poincare Normal Form,

Integral Averaging,

Harmonic Balancing, Describing Functions, etc.) all necessarily produce exactly the same formulae, once the formulae have been appropriately compared.

The use of bifurcation formulae may thus

be considered as application of one's favorite theory, in simplified form. In support of our particular recipe-summary for hand computations, we note that if the preliminary coordinate transformation is not employed, the analytic effort involved in the change of coordinates may not be eliminated but rather postponed to a later stage in the computation where it may further complicate matters.

Similarly, our complex arithmetic may be

replaced with real arithmetic, but with complex arithmetic the task is generally simpler. Example 1.

The mass-spring belt system.

The system

92

was partially analyzed in the Introduction to Chapter l. critical value

~

c

' where

system linearized about

0'

c~

c

) = -c

x ... (~ ) " c

'

At the

the eigenvalues of the

(0(~ ) /k,O)

c

are

where

and the eigenvector corresponding to The matrix

P

is

v1

is therefore

y

and the variables

In terms of the

A. 1

y

l

and

are given by

variables) the system becomes

or

93

=

(l,iw 0 ) T.

The only nonvanishing second and third order derivatives of

w

w

- __Q_t)"(\i ) m c

are

- __Qt) 111 (\i ) • m c

and

~2 ,

the summary of the evaluation of

~2 ,

82

Referring to

, we compute

iw 0 4m- """ u (\! c )

w

'

2

__Q_ 0 , and

Since

~2

> 0 , the periodic solutions exist for

since

~2

> 0 , they are unstable.

solutions is the expression

X'7(

(\! c )

+[

\)

-

1/2 \)

c]

Re

~2

where

94

F

2

v

>

\ic

and

An approximation to the

or

+

0 (\! - \) ) '

c

as given in Chapter 1.

The interested reader may follow step 9

of the Recipe-Summary to evaluate the approximation for Example 2.

0(\! -

\!

c

)-term in the

x(t) •

van der Pol's equation.

The equations for the RLC electric circuit illustrated below

R

I L

can be written as dvc

ic

=

c

iR

=

iL

dt =

VL

)

-i

c

=

d'~L L

dt

VR + VL

95

' =

= 0 (iR)

VR

vc

)

'

where the i's are the currents in the branches indicated by the subscripts and where

= ¢(iR)

vR

is a generalized Ohm's law,

characteristic of the "resistor" device.

If we set

= x ,

i1

R , which is actually an active

= -(L/C)

vC

1/2

y

and

t

= (LC)

1/2

'T,

then the equations take the form

=

x y

where

=

f(x)

-y - f (x) X

(L/c) 112 ¢(x)

I

and

I

denotes differentiation

with respect to the scaled time variable

'T

for a full

;

derivation and discussion of this system see Hirsch-Smale [51, Ch. 10].

Further, if the resistance is described by the function f(x)

-~x

+

x

3

'

then the system is a form of van der Pol's equation. ~

meter

The para-

controls the amount of "negative resistance" or "gain"

of the device

R • ~

For all values of

,

(x,y)

=

(0, 0)

is a stationary point.

Now

' so the linear stability of this stationary solution is determined by the eigenvalues

Al For -2


0 , the periodic solutions exist for

3/4 ,

are stable.

r2

=

0 ,

and

~2

=

c 1 (0)

=

-3/8 ,

-3/4 •

~

> 0

and

This information is consistent with the global

analysis of Hirsch-Smale [51, Ch. 10] which shows that for each 0




':>':>

i

32

-3

S +

5 O(e )

'

' and e

2

4

=3~ +

3

0 (~ )

•

The following table compares these approximate results with the "exact" results obtained by a simple shooting scheme. (numerical integration) began on the positive initial position

x0

x

axis, and the

was varied until a.trajectory was

generated that returned to the initial position.

100

Each shot

TABLE 2.1

___lL_

T ~1.12

2n{l+~ 2 /162

.025 .05 .1 .2 .4

6.28343 6.28417 6.28711 6.29888 6.34574

6.28343 6.28417 6.28711 6.29889 6.34602

The column

x(y=O)

{y=02

{4wJ3 21/2

.18257 .25817 .36496 .51537 • 72462

.18257 .25820 .36515 .51640 .73030

X

contains the intercepts with the positive

x-axis of the periodic solutions so computed. The agreement between the numerical results and the approximate analytical results is of the sort expected; namely, it is ~ •

better for smaller values of

The numerical results support

the correctness of our calculated values for the bifurcation parameters. Example 3.

Bulk oscillations of the Brusselator.

The pair of nonlinear diffusion equations

was posed by Lefever and Prigogine [75] in 1968 as a model system for an autocatalytic chemical reaction with diffusion. Here

A

and

B

are concentrations of certain "initial"

substances and are assumed to be constant. Y

The unknowns

X

n1

D2

are concentrations of two intermediates, and

their respective diffusion coefficients.

The symbol

and

and are

6

represents the Laplacian in the appropriate number of space variables, and the term

x2 Y represents the autocatalytic step.

We shall return to this system in Chapters 3 and 5. shall assume, however, that consequentl~

X

and

Y

Here we

are space independent;

the reaction is governed by the ordinary differen-

tial system

101

.

-(B + l)X + x 2Y +A

X

.

2

y

BX - X Y •

The only stationary point of this system is

X

=A

,

= B/A

Y

(A, B

>

0) ,

and the Jacobian matrix of this system at this equilibrium is

B - 1 [

-B

A obey the characteristic equation

The eigenvalues

2

2

2

A - (B - 1 - A )A + A = 0 • Let

a= l [B- (l + A2 )] • 2

Then if

a 2 < A2 , the roots

A form a complex conjugate pair

where

We now choose as

B

(A,B/A)

B

as the bifurcation parameter, and we note that

is increased past

B0

=

1

+

A2 , the stationary point

loses linear stability since the complex conjugate pair

of eigenvalues then has positive real part. The eigenvector

v1

corresponding to

102

Al

is

We define

~0~ l and

Then the system for

y2

y 1 ,y 2

= w y 1 +a y 2 + y

is

2

h (y 1 , [ (a + 1 - B) y 1 - w y 2 ] I A ) ,

where h(x,y)

= BA-l x 2 + (x + 2A) xy

and y

= a+

1 - B

+

A2

w

The bifurcation formulae we have previously derived apply immediately to this system. ~2

, and

~2

Since we shall evaluate only

, which are obtained from

103

c 1 (0)

~2

,

alone, we may set

so that in the above

o

ex=

and

W = W

0

=A ...

Thus the system becomes 1 . yl = -Ay 2 + F (y 1 'y 2 ' 0)

.

2 y2 = Ay 1 + F (y 1 'y 2 ; 0)

)

where

and

Substituting in our formulae, we obtain 1 1 - A) gll = 2 (A

)

1 g21 = 4 [-3+i/A]

g02 = gll

-

i

)

and g21 2

+--

= -

i

1 - A) 2 + - 1 - 1 + -1 + ~. [ - 1 (8 6A A 2 4A 4 A2

104

J}

•

Now

a' 1-1 2

(O) = 1/2

=

w'

and

(O) = 0 , so that

-Re c 1 (0)/a'(O)

=-1-+.!.>o 2A2

4

'

= - [Im

c 1 (0)

and

1" 2

+ 1-1 2 w' (0) ] I w(O)

[(A1- 2 41]

=

A)

+

>

0

Thus the bifurcation is always supercritical.

(3 2 = -2a' (0)!-1 2
0

(For any fixed

.

0

applies and shows that

\)


•

1/2 , are

asymptotically orbitally stable, and are approximated by

108

1 p(t;\)) = x;'/2) + (\) - 1)2 Re (eit 2 1 1

a·!

=

0

1 ; -I 2 )'

1cos

1 + (\) - 1)2 2

I sin \.. _

0

[i1)

:1.J

+

+

0(\) - 1)

2

'

0 (\) - 1)

2

In the case of this example we can check the results analytically. x3

=

x3

In cylindrical coordinates

x1

=

r cose , x 2

=

r sine ,

the system becomes

r

.

= (\) -

e=

l)r

+

rx 3

1

and by inspection, we find the (exact) solution

r

=

e=

R(\))

t

where

R(\))

=

[(1- \))(2\)- 1)] 1 / 2

= (\)- 1/2) 112 + 0[(\)- 1/2)] 312 In Cartesian coordinates, this solution is

109

+

R(\!)

and is the one whose approximation we derived above. R(\!)

about

period

T

= 1/2

\!

J

=1

~2

we see that

is a constant) namely) 2n ; so

Expanding

is correct. 'T'

2

=0

The

is also

correct. The periodic solution corresponds to the stationary solution r

=

R(\!)

J

x3

=

1 -

of the 2-dimensional autonomous system

\!

(*)

The Jacobian of this system at

r

=R

J

x3

=

1 -

\!

is

with characteristic polynomial

and eigenvalues

These eigenvalues are also the nonvanishing characteristic (Floquet) exponents associated with

p(t;\!)

•

In particular)

the Floquet exponent whose expansion is

=

~2(\!- 1/2)

is exactly

110

+

0(\!-

1/2)

2

as given above.

Expanding

Al (v)

about

v

=

1/2 , we confirm

that our previous evaluation

f32 =

-4

is correct. We leave it as an exercise for the reader to show that at

v

=

1

there is a subcritical bifurcation from the stationary 0 x* to the same periodic solution p(t;v) , except that

point for

v s:::: 1 , v

< 1, p (t ;v)

is unstable by virtue of two

positive characteristic exponents. There is even more bifurcational behavior in Langford's system, which we shall now pursue.

The eigenvalues

Al 2

above

'

may be rewritten as

where

v

1

- 2o/6 = 18 25

'

v2

= 18 + 2~/6 25

Since

1/2 < v 1 < 2/3 < v 2 < 1 , we see that the stationary

point

r

x

=

1 - v

' 3 is stable for

(equivalently, the periodic solution

< v < 2/3 , with the eigenvalues being real and negative for 1/2 < v ~ v 1 but forming a

p(t;v)) Al 2

=R

1/2

' complex conjugate pair with negative real part for < v < 2/3 •

= 2/3 the stationary point loses linear stability. For 2/3 < v < v 2 , the eigenvalues form a complex conjugate pair with positive real part; and for v 2 ~ v < 1 the

v1

At

v

eigenvalues are both real and positive. for the 2-dimensional system occurs at family of periodic solutions in the

111

Thus a Hopf bifurcation v

=

(r,x 3 )

2/3 •

The resulting

variables occurs

"on top of" the basic periodic solution, and so represents a family of bifurcating tori in the original coordinate space. The general theory of bifurcation to tori is beyond the scope of these Notes [60, 74, 97].

However, because of special symmetries

in Langford's system, Hop£ Bifurcation Theory is adequate to describe this phenomenon in the present case. ~

For eigenvalue

near Al

2/3

the eigenvector

associated with

v1

corresponding to the

(*) is

l[a(~) (

=

+

'

where

a(~) = (Note:

(3v - 2)/2

a

These

and

w(~) = i[36~

and

a

are not the

W

bifurcation calculation.)

-

25~ 2

- 12] 1 / 2 •

w of the previous

and

We next define

and we define {

I =

I I

I Y2 '-

(

'r '

yl

~

j

p~ 1~

R

I

I

'

-1 pl 1

Lx3j '-

Again, since we shall only compute enough to set

~

= 2/3

'I

-

I

~~

I

.)

~2 ,

T2

and

~2

, it is

throughout the calculation, in which case

112

a =o ,

1

= 1/3 ,

R

-

wo = 3 J2 '

1

Y1 =

r

3 '

-

In the new variables

the system

(~' ITMAX.

The value for

ITMAX is defined within BIFOR2.

~ = ~k

At each iterate

computed by Newton's method. of/ox(x,~)

continuity of

Jacobian matrix for all

is

(The Hopf hypotheses, together with

' guarantee the invertibility of the (x,~)

sufficiently close to x .... c~

The user-supplied estimate of initial point in the Newton iteration for 1

x*(~k)

, the stationary point

"

x(~ 1 )

c>

is the

, and the result

of this iteration is used as the initial approximation in the

x .k

x(~ 0 )

Newton iteration for x*(~ 0 )

iteration for

•

For

=

0

Let k

=

denote the result of the

1,2, .•• , the point

k X )"(

(2. 9)

-

is used as the initial approximation to k

from (2 .5), and

x.,./~k+l)

yk

is

denotes the result of the Newton iteration

x,._.(~k)

for

For fixed

k

~

0 , let

k

X.

J

(j

=

0,1,2, •.• )

sequence generated in the Newton iteration for k (x.). J

, where

1

(1 s: i

N)

S:

x ;\- ("vk )

'· and let

denote the components of the vector

x*(~k)

iteration for

denote the

The

is then

k J

x. -

(j = 0,1, .•• ) . (2.10)

The iteration is normally stopped at the first either

134

j

such that

N

2)

~

k

((X.

J

i=l where

and

2 · ; ~ ) 2 < 10 - Ns~g X. l ) . I X f ) J- ~ re k

-

N .

are as above and

components of

xref

being scales for the corresponding

components of

x •

€

s~g

The components of

(2.ll)

is anN-vector, the xref

are computed within

BIFOR2 from the corresponding components of the user-supplied estimate for above.

x,..(\J c)

)

using the function

The current iterate

k x.

described

when the iteration is stopped

J

k

is taken as the approximation

ref(•)

to the solution

x,,~

X7,~ (\Jk)

.

The Newton iteration itself is performed within subroutine NWTN.

LINPACK [29] routines SGEFA and SGESL are used to solve

the linear systems

is defined by

indicated by (2.10), and then k

xj - sk • The Newton iteration terminates abnormally and an error return from NWTN occurs when

j

> ITMAX , where ITMAX is as

above.

After Ak

x*(\Jk)

k = of/ox(x*;

\Jk)

is found as above, the Jacobian matrix is evaluated, and then the double-step QR

algorithm [29, llO] (EISPACK [100] subroutine

HQR, called by sub-

routine EIGR) is used to compute all the eigenvalues of this matrix.

The eigenvalue

Al (\Jk)

is then selected according (2.3).

In Method 1, the computation of

Al (\Jk)

subroutine EVALS, which calls NWTN to find

135

is organized by X 7,~

(\Jk) , then EIGR to

>.. 1 .

find

~

Method 2 for location of

c

This method differs from the method described above only in the technique used to solve the eigenvalue problems. method is again performed by subroutine ANUCRT.

The secant

Just as above,

the QR algorithm (subroutine EIGR, called by EVALS) is used to evaluate

Al c~l) •

Then, however, inverse iteration [27, 91]

rather than the QR algorithm is used to evaluate addition

v 1 (~ 0 ) , the corresponding eigenvector.

close to

~1

, >.. 1 c~ 0 )

is in general close to >., 1 c~ 0 )

inverse iteration for

>.. 1 c~ 0 )

z

>..1

>.. 1 (~ 0 ) , and in Since c~ 1 )

~O

is

; and the

, based upon the approximation

>.. 1 c~ 1 ), tends to converge rapidly even though an arbi-

trary initial guess is used for

vl c~o) •

The inverse iteration

is performed by subroutine INITER which is called by EVALl. Al c~k+l)

Inverse iteration is also used for the evaluation of for

k ~ 1, and the iteration is ba~ed upon the (extrapolated)

approximation (k

1,2, .•• ) ) (2. 12)

where

is as in (2.5).

The extrapolation

(2 .13) provides the initial approximation for the eigenvector v 1 (~k+l) , k ~ 3 . employed except that

For

k = 1

vl c~o)

and

2 , the same formula is

is used instead of

vl c~l) ) which

was not calculated. The difference in execution times between Methods 1 and 2 will depend both upon the machine and upon the problem.

In the

Hodgkin-Huxley system the expense in evaluating the Jacobian matrix dominates the expense of solving the algebraic eigenvalue problem, so use of the second method rather than the first makes

136

little difference in execution time.

For the panel flutter

problem, we have observed Method 2 to be twice as fast as Method 1; and so, for this problem, a large fraction of the computational effort is evidently spent on the algebraic eigenvalue problem. At one stage in the development of BIFOR2, we investigated an algorithm for the location of

v

closely related to one

c

recently proposed by Kubi~ek [73].

See Appendix E for a brief

description of this algorithm, and the reasons we have not included it in BIFOR2.

3.

,

EVALUATION OF THE COEFFICIENT

c 1 (0)

OF THE POINCARE NORMAL

FORM Once

vc, x,./vc) , and

w 0 = w(vc)

have been determined,

BIFOR2 calls subroutine ClPNF to evaluate the coefficient

c 1 (0)

of the Poincare normal form. The stages in evaluating 1)

c 1 (0)

are as follows. and

Find the right and left eigenvectors

the Jacobian matrix

A = (lf/(lx(x,.((vc); vc)

of

which correspond to

>.. 1 (vc) = iw 0 • so that the first nonvanishing element is Normalize

the eigenvalue 2)

identically 1, and normalize T

u 1v 1

u1

relative to

v1

so that

=1 • 3)

Perform numerical differencing to approximate the second

partial derivatives

I

f 20 = ('d 2 /oz 2 )f(x,.,+ Re(v 1z); vc) z=O

(3 .1)

fll = (o 2 /oz'dz)f(x*+ Re(v 1z); vc)\z=O

4)

Calculate T-::" T T 2u 1 f 11 , g02 = 2ulf20 g20 = 2ulf20' gll =

(3. 2)

= P/u h20 = p J. f20' hll

137

'

where

5)

Solve the linear systems

(3 .3)

for the coefficient vectors expansion of the slice 6)

~

w11 , w20

= ~c

of the quadratic terms in

of the center manifold.

Perform numerical differencing to evaluate the third

partial derivative

(3 .4) 7)

Evaluate

(3 .5)

Remark.

The derivation of this particular algorithm from the

results of Chapter 1 is assigned as Exercise 1 at the end of the present Chapter. 3.1

Right and left eigenvectors of

A

Inverse iteration is used to compute the right eigenvector of the Jacobian matrix to the eigenvalue

Al (~c)

=

=

of/ox (x_,_( ~ ) ; ~ ) , corresponding '" c c iw 0 . An arbttrary initial guess is

A

made for the eigenvector, then ClPNF calls subroutine INITER to perform inverse iteration.

In INITER, the initial estimate of

the eigenvalue is slightly degraded before being used.

(In our

original version of INITER, the estimate of the eigenvalue was not degraded.

When applied to the mass-spring-belt problem,

138

however, this version failed: reader to explain why.)

we leave it as an exercise for the

The members of the sequence of vectors

generated by the inverse iteration are normalized by subroutine ENRML to have Euclidean norm 1, and so that the component of largest complex magnitude which has the lowest index, is real and positive.

The iteration is stopped the first time that the

Euclidean norm of the difference between successive approxima-N .

s~g

tions to the eigenvector is less than 10

Inverse iteration is similarly used to compute the left eigenvector

u1 •

Just one LU factorization is performed in LINPACK routines CGEFA and CGESL

and

computing both are used by !NITER.

CGEFA performs the LU factorization; then

CGESL solves for the successive iterates. 3.2

Normalization of

and

vl

ul

The approximation to the periodic solution is of the form x(t; 'J)

where

=

X~,_.(\))

+ Re(v 1z + w20z

z = e:exp(2nit/T('J))

2

+ O(e: 2 )

the first component is 1, 2e:

+ w11 zZ) + oclzl3)

)

(3. 6)

is normalized so

If

represents (to within

the

peak to peak (max minus min) amplitude of the first component of the periodic solution.

It may not be possible to normalize

in this fashion, however. therefore normalizes is identically 1.

v1

v1

The scheme used (subroutine BFNRML), so that the first nonvanishing component

(Here, "normalize 11 means only multiplication

by a nonzero scalar.) Under the Hopf hypotheses, Al ('Jc) value of the Jacobian matrix eigenvectors

and

A('J )

c

•

=

iw 0

is a simple eigen-

Thus the right and left

are uniquely defined, up to multipli-

cation by nonzero scalars, and

uT 1v 1 # 0 •

purposes, the desired normalization of

u1

For subsequent is such that

This normalization is performed by subroutine RLNRML.

139

3.3.

Numerical differencing for second partial derivatives The second partial derivatives

I

f 20 = (a 2 /az 2 )f(x* + Re(v 1 z); vc) z=O , fll = ca 2 /azaz) f

(x'),~ +

I

Re (vl z); v c) z=O

are computed by differencing the first partial derivatives as evaluated by the user-supplied subroutine.

Let

z = y 1 + iy 2 .

Then, symbolically, a/az = ca/ay 1 - ia/ay 2 )/2, a/az = ca/ay 1 + ia/ay 2 )/2 , 2 2 2 2 2 2. 2 a /az = ca /ay 1 - a /ay 2 - 2~ a /ay 1 ay 2 )/4 a 21 azaz- = (a 2/ ayl2 + a 2; ay22)/4 ; and the derivatives to evaluate are,

where

Now

)

-[a f ( ax X;,~ +

where

af/ax

denotes the Jacobian matrix.

are formed, and the

approxin~tions

140

V

r

Y1

The matrices

2

2

r

o f /oy 1 (x~.) ~ [ (A+ - A_) v ] 1 (2 6y) , - [(A

are employed.

+

A )vi] (26y)

Then, the matrices =

i

of/ox(x* ~ 6y(-v ))

are formed, and the approximation 2 2 i o f/oy 2 (x,.) ~ - [(A+ - A )v ]/ (26y) is employed.

The computation of

f 20

and

f 11

as just

described is performed within subroutine DIF2. The increment used is

I

6y = u 1/3 ref (jjx~·~ll !\\v 1 \) • The factor

1/3

arises because the truncation error in the difference quotients is 0((6y) 2 ) , and the roundoff error is 0 (u/ 6y):

u

see Appendix D.

priate scale for

The ratio

6y , provided

jjx*\1 =/= 0 ; if

scale 1 is assigned by the function

The functions terms of

f

g(z,

z,

w)

and

by means of g (z, z, w)

where

P

\\x,\-\\1 \\v 1 11 gives an appro-

is the real matrix J.

The partial derivatives

141

1\x~\-1\ = 0 , the

ref(•).

h(z, z, w)

are defined in

2

2 2 h 20 = o h/oz , at

w

z

hll = 0 2 h I ozoz

= 0 are all computed in terms of f 20 =

at

= 0 2 g I oz_2

= 0 2 g I ozoz,

2

o g/oz ,

~ u

2 f/~z 2 . u

f ll =

,

~ u

2; ~

-

~ uzuz

z = w = 0 , which were approximated above.

Specifically,

and This straightforward computation is performed by subroutine PRJCT2.

3.5.

Solution for the coefficient vectors

\) = \) c

the expansion of the slice The vectors

and

and

in

of the center manifold.

are solutions of the N-

dimensional linear systems

where

A

system for for

is the Jacobian matrix

A= of/ox (x*(vc); vc) •

The

w11 has a real coefficient matrix, while the system involves a complex matrix. The Hopf hypotheses imply

that both coefficient matrices are invertible, and so solution for

and

is a straightforward task.

Subroutine CMAN2

sets up the linear systems and calls LINPACK subroutines SGEFA, SGESL, CGEFA and CGESL to solve the systems.

3.6.

Numerical Differencing for

c21

Let G(z;z) = g(z;z,w(z,z)) , 142

where the real

N -dimensional vector-valued function

coefficient

c 1 (0)

= ~c

~

represents the slice

of the center manifold.

w (z, z) The

in the Poincare normal form involves the 3rd

partial derivative

c21 at

z

=

0

w = 0

)

.

Since

= a 3GI oz 2oz

aglow

=

0

at the origin, G21

may be

computed as G21

=

03

oz 2oz

g(z,z,w 2 (z,z))

l

)

z=O

where

is the quadratic approximation to

w(z,z)

Approximations to

the coefficient vectors since

W

Let

w20 , w11 were obtained above; and is a real vector-valued function, w02 = 20 •

w

z

= y1 +

(a 2 1ayi + a 2 lay;)l4 and

iy 2 •

= ~14

Then, symbolically, a 2 lazoz = , where

~

denotes the Laplacian in

Now

Subroutine DIF3 approximates

c21

by applying a finite

difference operator, the 9-point Laplacian variables

y1

= Re(z)

and

~ 9 (h)

in the

y 2 = Im(z) , to the function

(oloz)g(z,~,w 2 (z,~)) , which is evaluated as above.

The 9-point

Laplacian is constructed as the Richardson extrapolation

of the customary 5-point Laplacian

143

6 5 (h)I/)(O,O)

= (ap(h,O) + 1/J(-h,O) + ljJ(O,h) + 1/)(0,-h) - 41/J(O,O))/h 2 ,

and the increment

The factor

h

u 1/6

employed is

arises because the truncation error in the O(h 4 ) , and the roundoff error is

9-point Laplacian is see Appendix D.

O(u/h 2 ):

Subroutine DIF3 calls DGFUN to evaluate

expressions

y 1 = Re (z)

as functions of the variables

and

y

2

= Im (z) • The

user-supplied subroutine FNAME is called by DGFUN to evaluate the Jacobian matrix. 3.7.

Computation of

c 1 (0)

The coefficient

c 1 (0)

of the cubic term in the Poincare

normal form is given by c 1 (0)

=

The computation of

c 1 (0) , once the critical value

the corresponding stationary point organized by subroutine ClPNF.

x(~

c

)

~

c

and

are known, is

g 20 , g 11 , g 02 and have been obtained as described in Sections 3.1 through 3.6

G21 above, ClPNF evaluates

4.

EVALUATION OF Once

c 1 (0)

needed to evaluate

ex'

c 1 (0)

c~ ) )

c

Once all of

using the given formula.

w' c~c)) \.12' 'i2 AND (32

has been found, only \.12' 'i2

A.i c~c)

and

82

=ex'c~c)

144

.

and w' c~ ) c c The derivative

ex'

+ iw'

c~ )

c~

c

)

are

is approximated by the symmetric difference quotient,

where l::::v

=

u

1/3 \)ref

1/3 arises \)ref is as in Section 2. The factor u because the truncation error in the difference quotient is 0((6\J) 2 ) and the roundoff error is O(u/t:N): see Appendix D. The eigenvalue Al (\Jc + 6\J) is evaluated by solving for

and

x~,( (\J c

+ 6\J)

by means of Newton 1 s method, then using inverse

iteration for the eigenvalue. evaluated similarly.

Al (\Jc

The eigenvalue

Al (\Jc

~

6\J) •

Subroutine BIFOR2 organizes the computation of .

is

Subroutine DEVALl computes the difference

quotient, and calls EVALl to compute ~2

- 6\J)

~2 ,

r2

and

BIFOR2 calls ANUCRT to determine

calls ClPNF to evaluate

\), x(\) ), and W(\J) , c c c c 1 (0) , calls DEVALl to evaluate

A{(\Jc) , and then calculates

5.

ERROR ESTIMATION There are errors in the computed values of

~2 ,

T2

and

~2

due to the use of finite difference approximations to the f 20 , f 11 , c21 , and A{(\Jc) • (The use of a difference approximation to start the iteration to locate derivatives

does not in general affect the computed value of ignored.)

c 21

and

\)

c

\Jc ,

and may be

The schemes employed in approximating for A{(\Jc)

are all such that the total error due to

differencing (truncation and roundoff) is relative machine precision.

O(u 213 ), u

being the

Thus the error due to the use of

145

differencing in the computed values of

~2 , ~ 2

~2

and

is also

O(u2/3) • In order to provide information about the actual error involved, BIFOR2 will (when

JJOB

=

1

is specified) estimate

the error due to the use of differencing. c 1 (0)

A{C~c)

and of

are performed both with increments based

upon the user-supplied value u

The computations of

u , and with increments based upon

increased by a factor of 1000

from the value input.

Let

err 4 , errS' err 6 and err 7 denote (respectively) the absolute values of the changes in the computed values of Re c 1 (0), Im c 1 (0), a'(~c) and w'(~c) • Since ~ 2 = 2 Re c 1 (0), err 3 = 2 err 4 is an estimate of the error due to differencing in ~2

•

The quantities

w'(~c)

~

~2

are then recomputed on the

~2

~2

and

(~c) ~ err 6

are used as estimates

of the error due to differencing in

and

a'

err 7 , and the absolute values of the changes in

the computed values of

6.

and

Re c 1 (0) ~ err 4 , Im c 1 (0) ~ errS'

basis of values and

~2

~2

and

~2

err 1

.

SAMPLE APPLICATIONS In this section, we give six applications of the code for

evaluating bifurcation formulae.

The last two applications are

to partial differential systems. The first example, the mass-spring-belt problem, was discussed both in Chapters 1 and 2.

Here, the example serves as

a simple introduction to use of the code. The second example, Watt's centrifugal governor, is a third order system representing one of the oldest control systems known

([93], pp. 213-220).

of a function

p

=

p0 (K))

We present a criterion (in the form for deciding whether the bifurcation

that occurs when Vyshnegradskii's stability condition is violated is to stable periodic solutions or to unstable periodic solutions.

Our study has design implications.

In a control

system in which there is a possibility of violating the stability condition (an elevator starting from rest, say) it would be good

146

practice to ensure that should the equilibrium state lose stability, the loss of stability is to stable, small amplitude periodic solutions. The third example is Lorenz' model for dynamic turbulence. We find that the bifurcation is to unstable periodic solutions for all values of the parameters considered.

This agrees with

Marsden and McCracken's corrected version of their original analysis [81; pp. 141-148], and with [104]. The fourth example is drawn from the study [44] of periodic solutions of the Hodgkin-Huxley model nerve equations.

The

present technique, however, is simpler in that numerical differencing rather than symbolic manipulation is used to evaluate the second and third partial derivatives. find unstable periodic solutions for periodic solutions for

I Z I2 , I


1, our computations indicate that

for all values of

-1

p , 0 < p < 1 •

Table 3.3 provides a simple criterion for deciding the direction of bifurcation • p

yc


p0 (K) , f.L 2 (p,K) > 0; and there is an unstable family of oscilla-

is still relatively well behaved.

tions for

y

slightly larger than

If y

•

and

For these values of

y ' even though the stationary point is asymptotically stable, it is

155

c

unstable with respect to certain perturbations of amplitude

If

p

=

~ 2 (p,K)

p 0 (K) , then

= 0

determine the direction of bifurcation.

and our results do not See Exercise 8 at the

end of this Chapter. An interesting feature of Table 3.3 is the lower bound for p0 (K) , or equivalently, the upper bound for

'Po(K) .

We

have

This bound implies the following:

If the parameters

p, K

are varied in any manner such that either K > 1 or -1 0 cos p > ~ ~ 39.3 , then any loss of stability of the m stationary point that may occur is a Hopf bifurcation to stable

and

y

periodic solutions. Given a control system that may at times operate near a limit of stability of an equilibrium state, it is clearly good practice to adjust the control parameters so that any loss of stability that may occur is to stable rather than unstable periodic solutions.

In the present example, we have shown how

this may be accomplished for Pontryagin's model. Example 3.

The Lorenz system.

The system of equations

yl

I

=

-ayl

y' 2

=

-ye3

y' 3

=

yly2 - by 3

+ ay2 +

yyl- y2

(6. 9)

was studied by Lorenz (see [81], pp.l41-148)asamodelfor fluid dynamic turbulence.

The system remains interesting, even though

156

it has little to do with some current theories of turbulence. In the Lorenz system, bifurcation with respect to the para-

y

meter

a# 0

If

a

is considered for fixed values of and

b(y- 1)

and

b

> 0 , the stationary points of

(6.9) are yl = 0 '

~[b(y-

yl =

1)]1/2 (6.10)

2

and

2

y3 = yl = Y/b

The point corresponding to the positive square root is of interest. The Jacobian of the system at this stationary point is

a

-a

where

(6. 11)

-1

1

y 1 = [b(y- 1)] 112 , and has characteristic polynomial A3

+

(a

+ b + 1 ) A2 + b ( y +

a) A

+ 2 b cr Cy - 1) •

The general real cubic with real root imaginary roots

Al 2

'

x?

= +- iW 0 -

A >...2 3

A3 =

=

and pure

is

2

2

+ w 0 A - >...3w0 •

therefore have y ' we must

At the critical value of

Al 2

A3

-(cr +

b

~ iw o = "±" i

+

1) '

[b (y c + cr) J1/2

' and 2bcr(~

.c

- 1)

= b(y c + cr)(cr + b + 1) ,

157

or

(6.12)

- 0' (CJ + b + 3) Yc - CJ - b - 1

(6.13)

Thus

+

CJ)

which is positive if

CJ

b (y

(J

c

2 bcr Ccr + 1) I Ccr - b - 1 ) ,

> b + 1 .

(We only consider

b

> O,

> 0 .) Differentiating (6.12) with respect to

y

gives

;x,'(y) = -b(:\ + 2cr)/ [3:\ 2 + 2(cr + b + 1):\ + b(y +cr)], so -~b~-~1~)____ -~b~~~CJ__ 2 [ w ~ + (cr + b + 1) 2 J

The loss of stability at

;x. 1 (yc)

= iw 0 )

0 .

given by (6.13) is thus a c The eigenvector v 1 corresponding

y=y

classical Hopf bifurcation. to

>

is

Again, it is possible, in principle, to write out the expression

1-12 explicitly. The task would make another nice exercise in symbolic manipulation. for

Subroutines LR and LRFUN were written to evaluate and

~2

numerically.

The following table of values of

is taken from program output. ~ 2 (cr,b)

and

~ 2 (cr,b)

The corresponding tables for

may be found on the microfiche.

158

Table 3.4.

1-1 2 (cr, b)

for the Lorenz system.

cr 100

b

20

40

60

10.0

-.0399

-.0262

-.0238

-.0219

-.0202

20.0

-. 0175

-.0134

-. 0125

-.0121

30.0

-.0285

-.0111

-.0091

-.0085

40.0

-.0136

-.0082

-.0069

50.0

-.0276

-.0091

-.0064

60.0

-.0130

-.0069

7 0.0

-. 02 74

-.0086

For all values of

cr

and

b

80

that we consider, we find

< 0, i.e. the bifurcation is to unstable (~ 2 = -2a'(O)~-L 2 > 0) periodic solutions for yz yc ) y < y c .

1-1 2 Ccr,b)

Remark 1.

An analytical study of Hopf bifurcation in the

Lorenz system is given in [81; pp. 141-148], and it was Jerry Marsden who suggested we consider the system. revealed a mistake in Figure 4B .1 of [81].

Our study

See [104] for more

recent analytical work on Hopf bifurcation in the Lorenz system. Re'mark 2.

In the Lorenz system, perhaps more interesting than

the Hopf bifurcation is the presence of a strange attractor; see [81, pp. 368-381]. Example 4.

The Hodgkin-Huxley current clamped system.

In the early 1950's a series of experiments was performed on the giant axon of the squid ''Loligo" by Hodgkin, Huxley, and coworkers.

One result was the famous model of nerve conduction

which continues to play a central role in the theory of the nerve [52, 96].

The Hodgkin-Huxley model, in the general case,

is represented by a partial differential system which governs the propagation of nerve impulses.

There are two independent

variables, time, and distance along the axon.

159

Solutions of the partial differential system which are functions of a single traveling wave variable may be obtained as solutions of a 5th order ordinary differential system.

One of

the successes of the Hodgkin-Huxley model is the prediction of the experimentally observed wavespeed from numerical solutions of this 5th order system. Solutions of the partial differential system which are functions of time alone, independent of space, may be obtained as solutions of a 4th order ordinary differential system.

Such

solutions correspond to "space clamped" experiments in which a thin platinized-platinum wire (electrode) is inserted along the length of the axon.

Because the platinized-platinum wire is a

good conductor, the electric potential (voltage) at every point along the wire is essentially the same at any one time; this is assumed to eliminate all spatial dependence.

In voltage clamped

experiments, an external constant voltage source is connected to the electrodes (the other electrode being in the bath), and the resulting current is measured as a function of time.

In

current clamped experiments, an external constant current source is connected to the electrodes and the resulting voltage is measured as a function of time. In the current clamped experiments, an interesting threshold phenomenon is observed.

For small values of the external current

stimulus, the voltage simply decays to a rest value.

For

slightly larger values of the current I, the nerve axon responds repetitively, with a number of large amplitude "action potentials", before the voltage decays to rest.

Numerical inte-

grations of the Hodgkin-Huxley model current clamped system reveal much the same phenomenon, except that the succession of action potentials continues forever as a periodic solution.

The

fact that the Hodgkin-Huxley model does not predict that the axon will eventually become "tired out" in the current clamped experiments is an inadequacy of the model.

It does seem reasonable,

however, to suppose that periodic solutions of the model

160

correspond to repetitive firing of the axon.

We note that in a

modified experimental setting with a low calcium solution, the axon can be made to produce long sequences of action potentials. In what follows, we shall use bifurcation theory to describe two different families of small amplitude periodic solutions of the model current clamped system.

The system is

given by CM

~~

= G(v,m,n,h)

+ I

dm = [ (1 - m)a (v) m dt

m[3 (v) ]ci>

dn = [(1- n)cx (v) n dt

n[3 (v) ] ci>

m

n

dh dt where

v

msec, m

is the voltage in

mv , the time

is the sodium activation, n

activation, h

t

is measured in

is the potassium

is the sodium inactivation, and

is a temperature compensation factor.

ci>

The temperature

°C

"constants"

cxm (v), [3m (v), cxn (v), {3n (v) , ah (v)

and the factor

T

and

~1 (v) •

Explicitly, Hodgkin and Huxley give

am (v)

= 1/ expc

an (v)

= .1/expc

(-.lv

+ 2.5),

(-.lv

+

ah(v) = .07expc(-.05v),

1))

{3 (v) = 4exp (-v/18) m

{3 (v) n

= exp

~ (v) = 1/(1

(-v/80)/8

+ exp(-.lv + 3))

where

-- lJ

(ex - 1) /x

expc (x)

if

X

f: 0 ,

if

X

=

1

161

is

ci> alters the reaction rate

measured on

)

= 3(T- 6 · 3 )/lO

0 •

The function G(v,m,n,h) is G = 120 m3h(ll5 - v) - 36n 4 (v + 12) + .3(c 10 - v) , where the constant c 10 is adjusted so that G(O, m~(O), n (0), h (0)) = 0 . ~

~

The subscript

indicates rest values: m~ (v) =

Ci.

n (v) =

Ci.

~

m

n

(v) I

+ {3m (v)) ,

(Ci. (v) m

(v)/(a (v) + {3 (v)) , n n

h~ (v) = Ci.h (v) I (Ci.h (v)

One finds

c 10

~

10.599 .

CM

+

~ (v)) •

is the membrane capacitance per

unit area, ~ = 1.0 ~f/cm

2

stimulus.

G are measured in

Both

~

I

and

, and

I

For each value of the parameter stationary point

(v.,.~(I),

x.k(I) =

reasonable temperatures

is the external current ~

•

I , there is a unique

m,'=

·r-l

0

:>

0.. 0..

•

co

Figure 3.2.

~A/em

in

I

2

Bifurcation diagram for the current-clamped Hodgkin0

Huxley system, T = 6.3 C. Each point

(I, v

) on the curve represents a periodic solution PP of the current-clamped Hodgkin-Huxley system for a current

stimulus I) and of amplitude measured by the peak-to-peak voltage v

PP

- max v(t) - min v(t) t

t

Figure 3.2 was constructed by purely numerical techniques,

163

I ~ I1

and includes work of Rinzel and Miller [96]. For I~ I 2 ,

and

the numerical results agree with the bifurcation

analytic predictions in precisely the way expected; see [44] for the comparison. At

I

= Ia

)

I3

and

I4

Figure 3.2 is for

"knees".

the curve has limit points or

)

=

T

at higher temperatures,

6. 3°C:

at T = 18.S°C ) I3 and I4 straighten out, and there is no "switchback" portion of the curve. At still higher the knees at

temperatures, the stationary state becomes stable for all current stimuli For

I, and the bifurcations disappear. I = I 0 ~ 6. 26S

T = 6. 3°C , the ''knee" at

represents

the first occurence of nontrivial periodic solutions. slightly larger than

For

I

Io , there is a pair of large amplitude

solutions, one stable and one unstable.

As

I

~

I0

from above,

these solutions coalesce into a single large amplitude solution for which two characteristic exponents vanish and two are negative. Figure 3.2 indicates that for

I3

< I < I 4 , there are (at

least) 3 unstable periodic solutions, in addition to the stable rest state and the stable large amplitude periodic solution.

In

fact, recent work by Rinzel and Miller [96] shows the situation to be still more complicated. range

IS

< I < I 6 , where

It appears that for

IS

and

I6

obey

I3

I

in the

< IS < I 6 < I 4 ,

there is a branch of unstable, period-doubled periodic solutions. The period-doubled solutions have voltage peaks which are alternately larger and smaller. of secondary bifurcations at

These solutions arise by means

IS

and

I6

in which a character-

istic multiplier associated with the basic periodic solution passes through the point -1

At present· writing, the full

branch of period-doubled solutions has not been computed and the possibility of tertiary bifurcations has not been excluded.

It

seems to be a fairly safe conjecture, however, that no additional stable periodic

solutions will be found.

Of the periodic solutions of the current-clamped

164

Hodgkin-Huxley system discussed for

I0

< I < I 1 , only the rest

state and the largest amplitude periodic solution can be expected as long-time behavior of the system because the periodic solutions of intermediate amplitudes are unstable by computations of the associated characteristic

as indicated

' multipliers.

Physically, the most relevant unstable periodic solutions are the small amplitude solutions for

I~

I1 , I

< I1 •

These

solutions are analogous to the unstable limit cycles in the masssprin~belt

system (Example 1) in that they represent a

mechanism by which perturbations from the rest state either decay back to rest or grow as

t ~

oo

presumably to the largest

,

amplitude solution. For

I

I~

fixed,

I1 , I

< I 1 , the unstable small ampli-

tude periodic solution is an unstable limit cycle in the two dimensional slice

I

=

constant of the center manifold.

If

trajectories are begun with initial conditions on this twodimensional invariant manifold and in the interior of the set bounded by the limit cycle, the trajectories will decay in an oscillatory manner to the rest state.

If the initial conditions

are on the invariant manifold but in the exterior of this same set, the trajectories oscillate with increasing amplitude.

The

unstable periodic solution also lies on a 3-dimensional hypersurface in

R4

, which defines a boundary for the stable manifold

associated with the stable rest state

x*(I) ,

I~

I1 , I

< I1 .

The set

6

where

is small but independent of

mation to the stable manifold.

For

I , is a local approxiu 1 , see Section 3.2.

Remarks. 1.

It was the Hodgkin-Huxley current-clamped system which

inspired the derivation of bifurcation formulae [46] upon which Chapters 1 and 2 are based.

The study [44] initiated the

165

development of the computer program described in the present Chapter. 2.

In Hassard [44]J the evaluation of 2nd and 3rd order

partial derivatives was by means of symbolic manipulation and equations (4.3) and (4.4).

The present computation using the

program for evaluation of bifurcation formulae is simpler) in that explicit expressions for the 2nd and 3rd order partials are not required. 3. (3.2)J

Simple shooting was employed to compute most of Figure with the variable order) variable step size Gear stiff

system solver used for the individual shots; see [44]. As described in [44]) however) the simple shooting scheme was unable to compute the full branch of unstable periodic solutions because of the extreme sensitivity of the trajectories with respect to initial conditions.

The primary branch of

unstable periodic solutions was completed by Rinzel and Miller [96] with a finite difference scheme using Gear's 5th order stiffly stable formula to discretize the time derivatives on a uniform mesh.

The resulting large systems of nonlinear algebraic

equations were solved by Newton's method) with special techniques to take advantage of the band structure of the Jacobian matrices. Example 5.

The Brusselator with fixed boundary conditions

Consider the system

Qy

at

a2 x or 2

=

d -

=

8d

2

+ (B - l)x +A y + h (xJy)

a 2y

or

2 2 - Bx -A y - h(xJy) J

where h (xJy) and

x

=

y

=

0

at

r

=

=

BA

-1 2 x + (x + 2A)xy J

OJl •

The present version is obtained by 166

D = d and D = 8d 1 2 the version discussed in Chapter 2 (Example 3, p. 101).

setting

X

=x

+A ,

=

Y

y + B/A ,

in

To convert this system to an approximating, ordinary differential system we use a simple finite difference scheme. n ~ 1

After choosing an integer and

r

y(ri)

i

=

i6r , for

by

xi

and

we let

i=O, .•. ,n+l yi

N

=

2n , 6r = 1/(n + 1)

Approximating

respectively

and

x (r.) ~

(i = O, ... ,n + 1)

and

using 3-point centered difference approximations for o 2x/or 2 ~2y/~r2 o o , we o b tain the N-dimensional ordinary differential system

h (x., y.) ~

=

(i

where

D

=

~

l, ... ,n) ,

d/(6r) 2

and

for all

t

Subprograms BD and BDFUN were written to evaluate bifurcation formulae for the approximating system. To investigate the effect of the finite difference scheme, we set for

e

A = 1, d = .1,

N = 2, 6, 14, 30 •

= .5

and computed

Be'

~2 ,

~2

and

~2

Before one compares these results,

however, it is necessary to standardize the normalizations. Suppose and

t

for

and satisfy

u(r,t) 0

~

r

~

and 1

v(r,t)

and

~

=

u(v,t) = u(l,t)

convenient norm for the pair IICu,v)ll

2

1

= :r

< t < v(O,t)

(u,v)

JT rJ l 0

are smooth functions of

u(r,t)

~

, are

T periodic in

=

v(l,t)

=

0 •

r t ,

A

is 2

2 + vCr,t) dr]dt .

0

Now by the trapezoidal rule,

II ( u, v) II

2

1

= -T

J0T [6r

n

~ [ u ( r . , t) . 1 ~ ~=

2

16 7

2 2 + v ( r i , t) ] d t + 0 ( ( 6r) ) .

In the present application such functions y(r,t))

(u,v)

are approximated using a vector function

which, from Hopf theory, expands as

Therefore

-1T JT0

n

6r 1:; [ x . ( t) i=l ~

2

+y .

~

( t)

2

] dt

and it is evident that

should be compared for Similarly, define

increa~ing

N

r2 = 2~ 2 /(6r\v 1 \ 2 )

rather than

,

~2

~2

•

= 28 2 /(6r\v 1 \ 2 )

~ 2 , T 2 , ~ 2 are the values for ~ 2 , ~ 2 and which arise when the eigenvector v 1 is normalized such that

These quantities ~2

~r

\vl\2 With

and

~2

=

1 A

for

1 , d = .1 and N = 2, 6, 14

Table 3.6.

N

6r

2

B

8 = .5, we computed

and 30.

Be'

~2 ,

~2

The results form Table 3.6.

Effect of increasing N (A = 1, d = .1, e = .5).

{12

-c

1:!!.2

!.2

.50000

3.200000

1.046083

.033241

-1.046083

6

.25000

3.405887

• 768911

.017429

-.768911

14

.12500

3.461513

• 763703

.015498

-.763703

30

.06250

3.475690

.762614

.015254

-.762614

168

We extrapolated these results using Richardson extrapolation, based upon assumed functional dependence of the form f(6r) = 2 3 4 f(O) + c 2 (6r) + c 3 (6r) + 0((6r)) This procedure resulted in B

c

~

3.4804 ,

i:L2 ~ • 7623 '

.0152 '

which may be compared with the exact values

~2

=

.76226 •• , ~ 2

=

.01518 •• , ~2

=

-.76226 ••

The effect of differing values of

B

c

=

3.48044 •• ,

from Chapter 5.

A, d and

e

is investi-

gated in Chapter 5. Remarks. 1.

Both the centered difference approximation to the 2nd

partial derivatives and the trapezoidal rule approximation to the normalization condition have truncation errors of the form 2 3 canst (6r) + 0(6r ) . If a higher order difference scheme is adopted, a corresponding higher order quadrature scheme should also be adopted. 2.

Since BIFOR2 is designed for o.d.e. applications, it

takes little advantage of the sparseness of the matrices involved in the present bifurcation calculation.

To take full advantage

of this sparseness, one could write a code similar to BIFOR2 but intended for one space-dimensional partial differential systems: such a code would use a nonlinear two-point boundary value problem solver and an ordinary differential eigenvalue problem solver. 3.

See Chapter 5, Exercise 7 (p. 265).

A simple way to 'get the most' out of BIFOR2 in appli-

cations to p.d.e. is to use very high order discretization schemes, since BIFOR2 was designed to handle dense matrices. Example 6.

A panel flutter problem.

This problem, which also involves a partial differential system, is included here to illustrate another way the program for evaluation of bifurcation formulae for systems of o.d.e. 's

169

may actually be applied to p.d.e. 's. The original problem is that of 'one-dimensional' panel flutter [30, 53].

We thank P. Holmes for suggesting the problem

and for much of the following discussion.

dynamic p pressure -----=>:;;.

z- 0

z = 1

Figure 3.3.

Panel in wind tunnel

If spanwise bending is negligible,then one can reduce the von Karman equations to an Euler type beam equation with flow effects derived from simple ''piston theory":

'V +

J'Po;,

+ pv, + a~ 1111 + v 1111 1

=

S

2

[r + x 0 v' (z) dz +a

1

S0

;_,, (z)v' (z)dz}v" ,

where pv' = energy from flow,

JP

6~ = 'flow' damping,

r= X,

ex, a

=

axial tension, (membrane) stiffness,

= damping

coefficients of panel,

170

(6.14)

with 'hinged' boundary conditions

=

v

=

v"

at

0

=

z

0,1 •

The finite dimensional Galerkin approximation is obtained, for

m modes of vibration, by setting

=

v(z,t)

m I; X. ( t) Sin (jTT Z) j=l J

(6.15)

in (6.14) and carrying out the standard Galerkin method of taking the product with

sin iTT z and integrating from

z

0

=

to

z = 1 This procedure produces the system

X

d dt

where

c,

D

I

0

X

m

=

m

--------I

xl X

0

xl

xl

c

m,

l

and

H are

0

+

(6 .16)

xl

Hl

X

H m

D

J

m

'

1

D ..

lJ

rr +

-6 .. (irr) 2 lJ

clJ .. =

-6 .. [(irr) 4a + (p)l/26] lJ m

H. l

(irr) 2 ] - 2 p jrr

[ ~ j=l

(jrr) 2 (ux.

J

With the definitions .

.

+a~.)x.] J

N

sinirrz cosjrrz dz

so =

(i,j

l, •.• ,m)

and

(6.17)

(irr) 2x. (i = l, ••• ,m).

l

2

J

= 2m and

t

y ~ (x 1 , .•• ,xm,x 1 , ••• ,xm) , the original p.d.e. is then approximated by an N dimensional o.d.e. for the vector y • The parameters

u , a , a , 6

and

bifurcation with respect to the parameter

171

r

are fixed, and p

is considered.

For the values ~

Ci.

= .01 , cr = .0001

=

.005 )

6

= .1

Holmes has computed that, for of

p

~

(6.18) and

N

r

= -2.4n

2

'

= 8 , there is a critical value

112.75 , with stationary point y~._z

(2.4,-l.04,.094,-.0l7,0,0,0,0).

Since the number of modes of vibration is to be increased for a better approximation of the p.d.e., the code was written for variable (even) dimension

N .

Subroutines

For successively higher dimensions

values of

p

C

and

y ~·c (p C )

were obtained as Table 3.7.

N , the converged

for the previous dimension

The following set of values of

used as estimates.

for

and of

were written to use Holmes' estimates of N = 8 •

PF and PFFUN

N

were

1-12' ,. 2) 82

N was increased. and

1-12' ,.2

82

for the panel flutter problem.

N

PC

1-12

,.2

(32

8

112.778

-.058007

.089784

.019916

12

112.841

-.057878

.091871

.019806

16

112.846

-.057865

.092087

.019794

20

112.846

-.057862

.092130

.019791

In this example, the normalization provided by BIFOR2 for the eigenvector

v1

happens to be appropriate for the approxima-

tion scheme, so the values of may be compared immediately.

~'

T2

and

~2

for increasing

N

Similar convergence of the station-

ary point and of the eigenvector

v1

increased,

172

was observed as

N was

Y~.,-+ (2.2976, -.9995, .0906, -.0161, .0041, -.0018, .0007, •.•

v 1 -+ (1, -.4145+.0084i, .0370- .0012i, -.0069+.0001i, .0017 -.OOOli, .••. Since

~2

< 0

and

~2

> 0 , the bifurcation is subcritical

and to unstable periodic solutions. of

Now the first

m components

Re ( e 2nit/Tv ) + O(p - pc) 1 provide approximations to the Fourier sine coefficients of v(z,t;p).

Thus

2nt C (z) _ T

'

where m

:E (x?,, (p)) . sin jnz ,

j=l

J

m

~ (Re v 1 )J.sin jnz,

C (z)

j=l S (z) =

m ~

j=l and the approximation to

(Im v 1 ) .sin jnz, J

v(z,t;p)

contains error due to

truncation of the Fourier series in addition to the usual error O(p- pc) •

The functions

for

are shown in Figure 3.4.

m = 10

v,.,Cz;pc)' C(z)

and

S(z)

The function

obtained v*

is the

base profile, buckled due to the combined effects of compression (negative tension) and dynamic pressure. linear stability as

p increases past

The base profile loses Pc •

For

P~

Pc'

P
(3 0 ,

~O > a>

0 ,

nB > 2

and

=

For the system (3.18) -

=

8[B~o I a]

there is a unique steady state

xe

about this equilibrium (3.18) -

(3.19) is

du (t) = - au(t) - bu(t--r) dt

(u

= x-x e'

The characteristic equation for (3.20) is 'f

8 ,

vklere

This is consistent with the data.

Let

~O'

are positive constants adjusted to fit experimental We shall assume that

B

In (3.19)

be the bifurcation parameter.

199

1/n

Linearized

b = y(nB-1)) •

X. +

a

(3.19)

(3. 20)

+ be -x.~ = 0 •

This analysis continues

as Exercises 1 - 3 at the end of this Chapter.

4.

AN EXAMPLE WITH TWO DISTINCT LAGS The field vole (Microtus agrestis) is a small grass-eating

rodent.

It is a stable food for hawks, owls, crows, weasels,

stoats, foxes, etc.

In simple habitats such as plantations and

open grasslands the concentration

x(t)

of field voles is

oscillatory with peaks at four-year intervals. habitats less periodicity is observed.

In mixed

Stirzaker [103] has

argued that there are two principal contributions to the growth rate

~(t)

•

The field vole mates after the age of seven weeks.

It has a life expectancy of 60-70 weeks.

But the reproductive

productivity of voles in a simple habitat is affected by crowding and thus will be a complicated average of the population at previous times. Voles are easy prey for their predators.

Hence one can

assume that the consumption of voles by predators depends only on the capacity of the predators and their number, i.e., is some linear function of at time

t

•

p(t) , the predators' concentrations

But the number of mature predators depends on the

number of infants reaching maturity in previous generations and involves additional time lags, as well as the availability of prey to feed the young.

Hence

p(t)

is a complicated function

of the vole population at previous times. Thus

x(t)

is a complicated function of

x

evaluated at

several (or even infinitely many) previous times. the above discussion is an oversimplified one.

Of course,

But it gives an

explanation for the presence of two or more delays in even a simple model of vole populations in homogeneous habitats. Having given this discussion, we leave it to the reader to study Stirzaker's paper if she or he desires to learn more about field vole populations; and instead we study an idealized model equation having nothing to do with voles but which has been studied previously by different methods.

200

This model

equation is

-

rr,J3

~ (t)

2 + 1-1) [x (t-1) + x (t-2)] (1-x (t))

( 9

(4 .1)

1 (x E R ) •

Kaplan and Yorke [68] proved that for periodic solution of period 6.

1-1

> 0 , (4.1) has

Jones [65] also proved

existence of periodic solutions of (4.1) for

1-1

> 0 , and he

computed them numerically, obtaining a period of 6. compute

~2

1-1 2 ,

and

82

for (4.1) and give the form of the

periodic solutions. period

=

T(\-1)

We shall find that 6(1+0(!-1 2 )) •

Linearized about

q (t)

at

1-1

=

=0 •

e

We shall

x

0

=

~2

=0

so that the

equation (4.2) has the solution

irrt/3

(4. 2)

A

We identify the

of Section 2 by choosing

be that characteristic value with

A(O) = irr/3 •

A

to

The

characteristic equation for

• x(t)

= -( rrll 9

+ 1-1)

(4. 3)

[x(t-1) + x(t-2)]

is

(6 = rrD)

(4.4)

9

We shall prove that at pure imaginary roots

1-1

=0

+irr/3

this equation has exactly two

and no root with positive real

The straight line with equation u = -v/6 does not -v -2v intersect the curve u = e + e for v ~ 0 • Thus, at part.

1-L

=0 ,

(4.4) has no nonnegative real roots.

For (4.4) to have a pure imaginary root it must be that

201

A=

iw

at

1-1

=

0

cos w + cos 2w

=2

cos

2w

3w 2 =o,

cos

(4. 5)

and

=

w

=

cos w/2

If

(4. 6)

6(sin w +sin 2w).

Thus the pairs

0 , the second equation is not satisfied.

(+ W , 6 ) , where - n n

2n+l

w

n

= c--)7T

3

(n=O,l,2, ••. , and

n

f= 1 mod 3)

and

6

w = ---'n;,.;___ 2 sin w n

n

yield all solutions of the simultaneous equations (4.5) and

> 6 = 7T/J/9 if n > 0 • Hence ~i7T/3 n the only pure imaginary roots of (4.4) at ~ = 0 . (4.6).

But

6

are

It remains to prove that (4.4) has no complex root with positive real part for Then

~

=

0

A is a root of (4.4) at

a.=

Suppose ~

=

0

A = a+ iw ,

a> 0

if and only if

-6 (e -a cos w + e - 2a cos 2w)

(4. 7)

6 (e -a sin w + e - 2a sin 2w) .

(4. 8)

and

w If

=

a> 0, (4.8) implies that

\w\

~ 26 •

Therefore we only

need to look for solutions of (4.7) - (4.8) for

wE (0,26] •

Now (4.7) implies

6 cos

w

= e -a

+ cos 2w e -2a cos w

Since 202

(4. 9)

cos 2t cos t if

t E

:2

-1

1T

(0,3] ,

the right-hand side of (4.9) is positive there.

Hence we need only look for solutions of (4.7) -

(4.8) for

But w-Oe -Ct sin w - Oe - 2 a sin 2W

=

G (w)

has a positive derivative for such

w

and

G(%)

(4.4) has no roots with positive real parts for

> 0 • ~

=

Hence

0

Finally, the characteristic equation (4.4) may be differen~

tial with respect to

A.' (O)

= iu/3

[1

to yield

+oct-

3

f3) J ,

where

\)

-1

2

= 1I.._ + 4

2

+ .Q.)

(1

2

Hence

w(O)

= ~ , a'

(O)

1T

u./312 > o ,

and

In particular, the transversality condition is satisfied, and we may apply the Hop£ bifurcation theory to (4.1) at Recall that ~~

q (8)

=

De

q

iTTe/3

is defined by (4.2).

,

where

203

We define

~ = 0 •

D = u[1 + O~i 1T] •

(4 .10)

Then if

(¢,~) = ¢co)~Co) -

o

8

S

8=-2

S

;cs-8)~Cs)dsdnC8) ,

s=o

where dn(8) =

-o

[6(8+2) + 6(8+1)] ,

Also if A¢ C8)

-2

9


0

(bifurcation occurs for -1

{32

=

lOT < O

(the bifurcating periodic solutions are asymptotically, orbitally stable), and '1'2

= 1/12 > 0

0

Furthermore, the asymptotic form of the bifurcating periodic solution is

7o

EXERCISES

Exercise lo

Show that for (3ol8) - (3ol9) the Hop£ spectral 219

hypotheses are met for

'i = 'i

where

J

c

-a

Arc cos (b) "!'

c

=

(b2 _ a2) 1/2

is stable for e increases through "!'

The steady state as

'i


0

such that the resolvent and

¢

(O

< ¢ < TI/2)

(AI - A)-l

there exists

exists for all

larg (A- a)l < ¢ + TI/2}

for s orne

C> 0

and

A E S"" ...,,a .

These results also hold for more general second order elliptic 227

operators [85]. that

generates an analytic semigroup Tt • In particular, for any t > 0 , u E 1 2 (0) , and IITtll , IIATtll are

A

Given any A. in the resolvent A, (A,I - A) -l (L 2 (0)) C H2 (0) • By Rellich 1 s

continuous in set

Thus, it follows by Theorem 4 in Appendix B

p (A)

t

of

t > 0

for

compactness theorem, (A,I - A)-l

is compact.

5 in Appendix B, one concludes that

A

Hence, by Theorem

generates a compact

s emigroup. The domain

l

norm

.

l~

DA

of

becomes a Banach space under the

A

For future use we let D 2 =

to the subspace

A

be the restriction of

[u E DA\Au E DA}

Since

E DA

Ttu

A

if

A

u E DA , we may also let subspace on

DA

DA

of

be the restriction of 0

L (0) , and we obtain a

semigroup

C

Tt

Tt

to the

semigroup

\u\~ ~ C(\6u\~ + \u\~)

(recall that

C0

The

Tt

2

Tt

u E DA) •

for

is compact and has infinitesimal

~

generator

A •

(b) on

oO .

Consider the Neumann problem: Here

6u

=

f

with

ou/on

=0

o/on

denotes the directional derivative of u in the outward normal direction on oO • For u E c2 (0) with

= 0 on oO , one can establish the a priori estimate: \u\~ ~ C(\6u\~ + \u\~) as in (a).

ou/on

Denote by

L2 (o) , which consists of

DB , the dense subset of

functions belonging to the closure of the set

[u

E

c2 (o)l

ou/on

=0

on

oOt

in

H2 (0) •

Again, the Laplacian

can be extended to a closed, selfadjoint operator

B:

DB~ L2 (0)

[85, pp. 237-240] that generates an analytic contraction semiTt

on

L2 (0) , where each

The space

DB

becomes a Banach space under the norm

group

Tt

is compact for

t > 0 .

I

~~ ~

As

above, we restrict

B

on this space

~

namely,

DB

and call it

B

~

B: D 2

~

DB .

Let

Tt

be the restriction of

Tt

on

B

It is not hard to show that semigroup with infinitesimal generator

228

is again a B

compact

Next, we want to show how local semiflows (nonlinear) arise from nonlinear partial differential equation of evolution in a special situation.

We follow the presentation in Holmes and Marsden [53]. Let L be the generator of a co semigroup ut on X and f: X -+ X be a map of class Ck(k ~ 1) Now, ' consider the abstract evolution system du/dt = Lu + f(u)

.

Clearly, any solution

u (t)

for this system with

.

u (0) = u

0

E DL

satisfies the integral equation (Duhamel s formula) 1

t

=

u(t) Since

f

Utuo +

J0 Ut-sf(u(s))ds

is locally Lipschitz and

M,f3 •

constants

llutll

~

(t)

Me tf3

for some

Picard iteration, as for ordinary differential

equations, shows that the solutions of the integral equation (t) F: l9 c JR+ X X -+ X •

de fine a unique loca 1 semi flow defined as follows.

For each

u

maximal interval of the form integral equation with J(u )

u(O)

as

U (J(u ),u ) 0 0 u EX

on

J (u )

in

u

and

in

X , let

J(u )

be the

0

[O,a) , where the solution of the

=

u

0

exists.

u(O) = u

with initial condition

0

0

is

l9

t-+ F(t,u ) 0

0

This solution on is unique.

is given

l9

is the unique solution

0

0

•

Furthermore, Ft (Ft(u) = F(t,u))

for each

t > 0 •

Let us call

F

is of class

the local semiflow

associated to the abstract differential equation du/dt Ut

=

=

Lu + f(u) •

DxFt(O)

for all

If

=

f(O)

0

and

du/dt

=

=

L

and

0 , then

t ~ 0 •

In the bifurcation problem setting, equation

Df(O)

Lu + f(u)

f

depend on some parameter

in the ~

lying

Thus, Define f.L = ~ - ~ c c Ff.L the Denote by uf.L one has du/dt = L u + f (u) t ' t f.L f.L and L + f corresponding semigroup and local semi flow of L f.L f.L f.L respectively. Now, assume

in an open interval containing

~

.

229

(1)

ui-L

is jointly continuous in

(2)

ui-L

is of class

ck

(3)

f

is of class

c

t t

k

in

u,\-L

in

u,\-L

t, u,

(t

1-L

0)

:?:

t > 0

for each

.

1-L

Under the hypotheses (1)' (2) and (3) on

L

and

follows by the same arguments as above that in

1-L

F\-L t

'

it

is of class

ck

Consider a reaction-diffusion system:

(c) du.

~

dt

rn 6u. + ~. 1 c .. u. + f. (u) ~ ~ q J J=

= d.

~

on a bounded domain condition

ul, ... ,urn

0

(n

~

with Dirichlet boundary

3)

.

~

with

f. (0) ~

for real du/dt

lRn

in

(i = l, ... ,rn)

on the smooth boundary oo Here, d.' c .. ~ ~J ern -+ 0 d. > f.: c are smooth functions in

u = 0

are real numbers

form:

.

t > 0

for each

u,\-L

f

tL

=

'

0

~

'

D. f (0) ~

=

and

0

is real

f. (u) ~

We write this system in the more compact

=

D 6u + Cu + f(u)

in· 0

with

u

=0

on

oO .

Here

u

D

=(~1·:·~) . . . •

• •

0 ••• d

'

c

(c .. ) , and ~J

f

=

(ff···rnl)

rn

Denote by

the operator which is the operator

d.6 ~

ass.ociated with the Dirichlet problem for

as in a).

Recall that

230

is the closure

2 -

[u.~ E C (O)\u.~ = 0 on and A =A X XA X=DA X ••• XDA 1 m

of the set

m a compact analytic semigroup on

Set Clearly

A generates

1

A

perturbation of

~

n to

L

A+ c

L

C •

be a

By Theorem 6

again generates a compact, analytic

From the remark made just before Example (a)

3), it follows that X •

Let

by a bounded operator

stated in Appendix B, semigroup.

X •

f(u)

(here

defines a smooth function from

Therefore, the Dirichlet problem for the reaction-

diffusion system above defines a smooth local semiflow on D = D

Now, we assume

and

C = C

1-1 ~

a real parameter L

= D 6 1-1

X

+ C

1-1

X •

depend analytically on

1-1

O. Clearly, the family

near

of closed linear operators on

X

can be extended

1-1

naturally into a holomorphic family of closed linear operators of type (A) near 0 • semiflow

ul-1

is jointly continuous in

t

smooth in

u,l-1

f = f(u,l-1) smooth in

for each

t

u,l-1 •

and

r--

ul-1

is

t

Let us assume that 1-1

and

f(u,IJ.)

aO

defines a

is

Consequently, the equation

+ C u + f(u,IJ.)

1-1

with

u = 0

on

1-1

local smooth semiflow

Fl-1

on

t

X

enjoying the same smoothness of

u~ .

In many situations, Y =

> 0 •

t,u,ll

also depends on the parameter

du/dt = D 6u

that of

Thus, by Theorem 7 in Appendix B, the

[u E Xlu

u

only takes real values.

is real}, the real subspace of

X •

Set

The local

Fl-1 defined through Picard iterations, leaves y t In other words, invariant; each Picard iterate maps y to y and it carries Fl-1 can be regarded as a local semi flow on y t semi flow

'

.

Fl-1 t

the same smoothness as that of (d)

or

Y •

X

.

Similarly, the reaction-diffusion equation

du/dt = D 6u au/on = 0

on

'

+ Cu + f(u)

on Here,

in

0

with Neumann condition

a 0 , also generates a local smooth semiflow on U

D

C

' ' '

f

X

h ave the same meaning as that in (c),

231

~

~. ~

X = D X ••• X DB B. is the infinitesimal generator in Bl m ~ introduced for the Neumann problem in (b), and y = [u

E X\u

near semi flow

0

If

D, C, f

depend on a real parameter

in the same way as in (c), then the associated local F~

t

on

B=

(c). Now (e)

is real} •

X

Y has the same smoothness as that in

or

BX ••• X 1

B

m

and

L

= B+

C•

Suppose the bifurcation of interest is fran a non-

trivial stationary solution du dt

u ..k

D6u

of the system (x E 0 c IR n)

+ Cu + f (u)

with either Dirichlet or Neumann conditions as in (c) or (d). Set

v = u - u* •

Then dv dt

= D6v + C*(x)v + g(x,v) ,

where c~_.(x) '

g(x,v) = Provided

u*

is smooth in

f(u~ ...,

=c +

f X (u~) r~

J

+ v) - f(uJ) - f u (u~)v • 7\ '"

belongs to the appropriate Banach ~

and

, the analysis of (c) or (d) carries through

with minor modifications.

Consequently, the system for

either Dirichlet or Neumann conditions on smooth semiflow

space

() 0

v

with

defines a loca 1

with the same smoothness properties as in

(c) or (d).

For further examples of nonlinear semiflows see HolmesMarsden [53] and Marsden-McCracken [81] . . Indeed bifurcation in reaction-diffusion systems has been studied by a large number of authors; see Paul Fife's recent surveys [31; 32, pp. 152-154]. The Proceedings of the October 1979 Madison, Wisconsin Advanced Symposium on Dynamics and Modelling of Reactive Systems [102] deal with reaction-diffusion in general.

232

3.

HOPF BIFURCATION AND ASSOCIATED STABILITY COMPUTATIONS FOR LOCAL SEMIFLOWS Let

0 .

Y

c0

Now, consider a family of

in a neighborhood of in

JR

C00

be a Banach space admitting a

F~ (0)

with

=

t

in

0

0 •

Y

norm away from

for

0

~

t

These semiflows

~

r

F~ t

Y XJR -+ Y X JR

(y,~)-+ (F~(y),~).

F~ t

of

defined

t

~

and

near

0

may come from

some partial differential equation of evolution. local semi flow

F~

local semiflows

The suspended

is defined by F~

To study Hopf bifurcation for

we need

t

some hypotheses: Smoothness hypothesis.

t,y,~ y,~

(t

~

0)

is jointly continuous in

t > 0 ,

and to each

F~(y)

is of class

Ck+l

in

(k~S).

~

near

0 •

(2)

A

t > 0 •

(3)

~

A

for all

t

DYF~(O)

The semigroup

infinitesimal generator for all

F~(O) = 0

(1)

Spectral hypothesis. and

F~(y)

t

in

[O,r]

has

exp(ta(A )) = cr(DYF~(0))\[0}

and

~

t

has a pair of simple complex

~

conjugate eigenvalues and

dReA,(~)/d~\u=O

A.(~), A.(~)

> 0 •

Re[cr(A )\[A.(~), A.(~)}]~ j.1

We assume

(4)

-0
0)

iw

6> 0

such that

0 •

satisfies the smoothness and the spectral

hypotheses as described above.

By the center manifold theorem

for semiflows (see Appendix A), there exists a locally invariant, locally attractive, three-dimensional submanifold suspended local semiflow (0,0) • to

(y,~)-+ (F~(y),~)

in

~

of the

YXJR

through

This manifold (called the center manifold) is tangent

Y X R , where c

Y

c

is the eigenspace associated with the

eigenvalues

iw 0 , -iw 0 • The existence of center manifolds for the partial differential equations studied in this Chapter (i.e.

reaction-diffusion equations) also follows readily from

233

=

~

the slice

~

For all sufficiently small

Appendix A.

denote

, let

11\

Then

constant of the center manifold

is

locally of the form

for some small

>

E:

0

6 > 0 •

and

Next we need a result of Chernoff and Marsden [12; 17; 81, p. 265] which is a generalization of the Bochner-Montgomery theorem: Let

Theorem.

G

be a local

dimensional smooth manifold reversible, is jointly ck k-1 vector field on \n c

11\ in

.

Ck

t,y

By this result, the restriction to

field

.

R

(yc,ys'~)

and

' ~

(y,~) ~ (F~(y),~)

~

=

~y

y

c (y c'f.L)

is generated by a vector y

of s Clearly, the projection

R

on

Y

local recurrence of

G

bifurcation problem for

R

~

•

the semiflow

G

~

reduces to that for

~

near

R

FI-L t

~

or

has a family of periodic solutions

, which can be parametrized by

234

imply that

By Theorem 2 in 1-L

T

The

(0,0) •

Chapter 1, we have a Hopf Bifurcation of. R

period

11\ , the

can be regarded as the "essential model" for

obeys the same kind of hypotheses.

Therefore, F~

In this

1-L

The smoothness and spectral hypotheses on R

.

'

Since all

t

R

\n

happens in the center manifold F~

y

in

c

is actually a family of

parameterized by

c

~

vector fields

y

s defines a coordinate system on

coordinate system, the vector field vector fields

is generated £.y -a

of the suspended

Take any complementary space y

and write

G:

is locally

Then, G

.

local semiflow

semiflow on a finite

E:

near

F~ t

at

p(t,~) 0 :

1-L

= 0 . with

T

=

=

T(e)

[1

+

~2€

2

4

+ 0(€ )] •

Any periodic solutions sufficiently close to the origin in \~\

with

y

small must appear in this family.

It will be shown below that the differential equation describing

R0

in

can be written in the form:

Y

c

dz -=

with

dt

when one identifies

Y c

with

g .. E C ,

lJ

C through a wise choice of

coordinates. For the convenience of the reader, the bifurcation formulae for

~2' ~ 2

in terms of

g ..

lJ

and

A' (0)

, which have been

obtained in Chapter 1, Section 3, are collected here:

where

~2

=

~2

=

Re c 1 (0) C/1

1

wo =

a(~)

(0)

[Imc 1 (0) + ~ 2 w' (O)] ,

+ iw (~)

and

Re c 1 (0) < 0 and unstable periodic solutions bifurcate if Re c 1 (0) > 0 • Here, Stable periodic solutions bifurcate if

one must use the local attractivity of the center manifold and note that for partial differential equations this implies global existence of orbits near any stable periodic solution in a center manifold; see Appendix A.

235

For the rest of this section, we show how the differential equation describing

R0 on the manifold M0 can be obtained in explicit form for a local semiflow defined by a partial differential system of a specific type.

The process will be

formally the same as that used in previous Chapters. At form

~

= 0 , our partial differential system has the abstract

du/dt =Au + f(u)

YEE>iY = (yl + iy2 \Yl'Y2 group and (k

f

E Y}

is of class

5), and

-:2:

on a Banach space

ck+l

Ey

Au, f (u)

where

'

A

u

E Y • Since Hopf bifur-

cations are being considered, we know that simple eigenvalues Suppose now: y

-±- iw 0 ,

wo>

0

'

A

has a pair of

with eigenvectors

(1) There is a real inner product

and it is extended to

Y$iY

so that




1,2, ... ) .

B , there must 0

.

Thus for

2 2 K - n TI D , the trace must vanish and the determinant must be positive, i.e. for some n = m B = Be and m ' 2 2 2 r If m > l , then at W = m TI ed (B 0 m one of the matrices

2 2 trace (K - n TID) is positive for all least

l

~

=

(m

2

2 2 - n )TI d(l +e)

< m and the operator £ has at

n

m - l eigenvalues with positive real parts.

value of

B

The only

at which the Hopf hypotheses may be satisfied is

therefore c

B = Bl

=

l +A

2

2 + TI d(l +e) ,

and we must further assume that

w~

== -disc 1 (B~)

Assume finally that at 2 2 2 2 r c det(K- n TID) = n TIed (Bn - B1 ) for

n

~

2 •

> 0

With these assumptions, it is straightforward

(see Exercises 1, 2 and 3) to construct an interval

249

I

and having the properties

containing i)

has a complex conjugate pair of eigen1 c 2 (B - Bl)' w = -disc 1 (B) a~ iw, where a=2

for B E I, S., values

the remaining eigenvalues

ii)

B E I, where

E:

A

obey

ReA :::;; -e

> 0 is independent of

B

.

Thus the Hopf hypotheses on the eigenvalues are satisfied at

c Bl •

It is computationally advantageous to perform a change of variables at this stage. Note that for B near B~ , the eigenvector of K - n 2 D corresponding to A , is v 1 = (l,(a + iw+ dn2 - B + l)/A 2 )T. We let

~ ~ ~) ~

P1

where

[(Rev 1 )

(

y = (a+ dn

2

( • Imv 1 ) ] ,

2 2 - B + 1)/A , 6 = -w/A , and change variables

according to

The system becomes

1(~)

(i)

=

2 +

.L)

+

(1)

h(u,v) ,

where

a+ d(n

2

-w

or 2

L =

w+ (8 - 1)

~

2

(n2 +

.L) or

2

ot + 8d(n

2

02 +-- ) or 2

with the domain

D1

= ~u,v)

\u,v

E H2 [0,l]

and

250

u = v

'

for all

= 0 at

0,1}

In the above

{3

=

(a

+

d1l

2 B+l+A)/w,

a = B/A + 2Ay , and

Since we shall only compute

it suffices \.12' ,.2 and {32 ' to find the restriction of this system to the slice B = Be of 1

the center manifold, and so we set B Denote by

L*

Be

in the following.

1

the operator

wo + (e L*

=

-

l) rl (TT2

o

2

+ _£__) 'Or2

= -w 0 L

with the same domain as

.

Choose q

=

I t is easy to check that

b

in

(q*,~)

n1 , =0

and that

•

Lq

q .k

=

[_~)

sin nr

= (a,Lb)

(L·ka,b)

= iw 0 q ,

L;'(q·k

1

=S

-T a b dr

0

denotes the usual inner product in Write

(vu)

=

zq + zq + w; z

251

for all

a

in

= -iw 0 q*, (qi(, q) =

Here,

(a, b)

. D1 *, 1 ,

Thus u = (z + z) sin 11 r + w1 = ull + w1 , v = i (Z - z ) sin 11 r + w2 = vii + w2 • Therefore

2

h(u,v) = CT(ull + 2u\\ w1 ) + 2AO(ull Vjj + ull w2 +vii w1 ) 3 2 4 + JIU\1 + bull vii + O( z \ ) '

I

where we have assumed that

w = O(\z\ 2 )

the terms necessary to compute

c 1 (0)

and have retained only

•

The integrals

1

1

S . 2 11 r

2 sin11r dr = 11 ' 0

S

s~n

=

0

1

S .

3 11 r dr = 4s~n 311 ' 0

are used below.

dr

1

S0 s . 4 11 r ~n

d

r

1 2 '

= 83 ,

One computes

(q * ,f)= (q~"',

(h)

f3h) = (1 + if3)

S01h

sin11r dr,

where

1 soh sin11r dr = cr((z + z) 2

3~

+ 2(z +

z)~l}

J",.f -2 2 4 " + 2Avti(z - z ) 311 + (z + z)w 2 + i(z- z)w 1 }

252

. 2

Here

w s1n n r dr . The system in

zJ w coordinates is then dz dt

dw dt

= iw 0 z +

-=

(q"~) f)

Lw+f- 2Re((q*Jf)q)

where h

.L

=h =

- 2 sin n r

+

[a (z

.2 ) ( s1n nr

.L

z) 2

sin nrdr

+ 2A6i cz-2

.2 nr - 8 =s1n 3TT sinnr =

sin nn r J

and

1 I

n

= 2

so

0

if

is even

sin nr sin n n rdr = 8

-

. . nn(4

Now) if

n

2

w20 2 WQ2 -2 w = - 2-z +w 11 zz + - - z + 2

if n2)

...

center manifold) WQ2 = w20 and w20 wll linear) two-point boundary value problems )

(2iw 0 - L) w20 = 2 (rr - 2A6i) (

-Lwll = 2a

is odd.

represents the must solve the

~)

(sin 2 nr)

1 ) 2 f3 (sin n r) .L

253

n

.L

,

w20 = w11 = 0 be solved for

with

at

r = 0,1 •

Although these problems may

in terms of closed form expressions,

the solutions are more easily described in terms of Fourier series

. -1n

w20

=

wll

= - 2cr !; 3

-2 Ccr - 2AOi) !;; In sinnnr (Ln - hw 0 )

eX)

{3

,

-1 (1)

In sin nn r Ln

{3

,

where 2

dn (1 -

2

-w 0

n )

Ln =

2

2

8dTT (1 - n )

Now

"= s . 1

w

0

w

s~n2

nr d r

= "w11 zz +

Re ("w20 z 2) + 0 ( \ z 13 ) ,

where 1 "

wll =

so 1

"

w20 =

s0

wll

. 2 nr d r =-cr ~co I2L -1 3 n n

s~n

2 w20 sin nr dr = - (cr - 2AOi)

(~) ' ~~

I n2 (L n

2iw 0 )

-1

l~l

Restricted to the center manifold, the system therefore has the form dz -=

dt

where the coefficients

g. . ~J

arise in the expansion of

254

(q * 'f)

=

(1 + ifj)

I

1

h sinrrr dr 0

=

thus

(1 + i,g)h. . , 2 ::;; i + j ::;; 3 , where l.J

h..

~

._.1:j__

L.J

• , • ,

2::;;i+j::;;3 l..J.

. . 1-J z z

1 I = h sin rr r dr

0

+ 0

(I z I4 ) ,

and

hll

= fu. 3rr

h21 --2--

= 2(cr - Aoi) w11 + 2AOw~ 1

'

1

+

,a

I

~ ) w ~1 + Auw ~~2 + l (3y - iu) ~ • + Aui 20 20 8

Finally,

h

+ (1 + i,g) where the

h .. 's are given above. l.J

The parameters T2

=-[Im

;1 '

c 1 (0) +

~2

~ 2 , T2 , ~ 2

w' (0) ]!w 0

are then

, {3 2

~2

=

-Re c 1 (0)/a'(O) ,

= 2 Re c 1 (0) , where

a' (O) = 1/2 and w' (O) = -derr2 /2w 0 • The periodic solutions themselves are approximated as

255

x(t,r)

= u(t,r)

y(t,r)

= yu(t,r) + 6v(t,r)

(u) v

zq + zq +o 0

, the equation

=0 .

Dh(O)

is said to be locally invariant,

0 EM

,. = 1'(x)

and

for some interval

The tangent space

is invariant under the linear operator If

T0M

of

M

L •

(*) is an ordinary differential

equation as in Chapters 2 and 3.

If one takes

=L

as in equation (2 .1) of Chapter \.1 represents a delay equation. If H = y

L

(*)

Here, L Lt

such that

0 0 (x)

~

e

+ h(x)

U in a real Hilbert space

in some open set

containing the origin.

f(x) - Lx

4,

H

= C [ -r, 0]

the form L =A

and

and

(*) as

(*) reduces to a

in examples c) or d) of Chapter 5, the form reaction-diffusion equation. Assume now that

L

satisfies the following spectral

conditions: a)

vc

'

H

where

is a direct sum of two closed invariant spaces ReO' (LI Vs)

dimension with b)

O'(eLt)

< a
0

.

vc

vs ' has finite

For ordinary differential equations with

cr(L) ~ 0 , the

spectral conditions are always met. Definition.

A locally invariant manifold

M

is called a center

manifold if Note that for simplicity, we merely study the center manifold in

L ,

the absence of an unstable part of

The following example shows that there may be more than one center manifold locally for an equation (*). Example.

Consider the system

For any real number

U[

(x,y)lx ~

O,

y

= 0}

CL ,

2

x'

X

y'

-y

the set

M

CL

= ( (x,y) IY = CL e 1/x

x

< o}

is a center manifold for the system.

The Center Manifold Theorem (a)

Under the hypotheses above on r-1 possesses ~ (C ) center manifold (b)

M

U

of

Remark 1.

= IR

h

), a

l2l

the equation

M .

0

There

such that whenever

approaches

for all

(H

and

is locally attractive in the following sense.

exists an open neighborhood

n

L

t

M as

_.

co

•

In the case of ordinary differential equations Cr

· ( x·) d oes possess a equat1.on

Cr

center rnan1.' fold

[43]. Remark 2.

If the equation

manifold may fail to exist.

(*) is of class

co

C

, a

co

C

center

Van Strien [105] shows that the

system

267

co

c

x' = -x y' I

J..L

has no

co

C

Remark 3.

-y

2

+

-

2 J..L

(x

2

-

J..L2)

= 0

center manifold. If the system ()'~) is analytic, an analytic center

manifold may fail to exist. analytic center manifold.

However, there is at most one For instance, the analytic system

x' = -x y' has no analytic center manifold.

2

-y +

X

2

For if this system had an

analytic center manifold it would be represented by the co

divergent series Remark 4.

y =

~ (n- l)!xn . n=2

The existence of a center manifold for ordinary

differential equations in a more general setting can be found in Kelley [72]. We shall now sketch a proof of the center manifold theorem stated above.

We use a combination of the methods used by

Hartman [43] and Marsden-McCracken [81].

Thus, the center

manifold will be obtained as an invariant manifold of the time one map of the associated semiflow. Outline of proof Let

(x ,x ) c s decomposition H Then the equation

denote the coordinates on

= V/~ Vs

H

defined by the

, and denote the norm of

(*) becomes

268

H

by

I •I .

Let Re [cr (A) ] on

6

=

Cx c + Y(x c ,x s )

x's

=

Ax s +

-a >

obeying

0) •

(t

y (y ,z)

e: 1 ; and for any

e:

and with

z (y' z) are 0 < e: < e: 1

let

y (y' z) =

i

lE:

c

1P I Cy, z )

I)

y (e:y' e:z)

0

if

l (y,z)l

if

ICy,z)l ~ l/(2k),

if

I (y,z) l

if

lCy,z)l ~l/(2k).


e >

0

'¥

A(Y,Z)

defined on is small if

H

and

Cr

e is small.

the system y

I

z

1

Cy + y (y' z) = Az + Z (y, z)

has the same properties local to

(0,0)

as does the system(*),

For 0 < e < e 1 the system (***) s is defined globally and generates a cr semiflow ¢t(y 0 ,z 0 )

with

=

ey = xc

ez = x

and

(yt' 2 t) · Define

and

Lemma.

~

following statements hold

a)

y

b)

A(Y t ,e

t ,e

=

0

'

zt

,€

,z t ,e ) ~ o

The time one map is

=

0

as

IIY0 11

if €

uniformly~

~

>

~

0 :::;; t :::;; 1

for small

•

€

0 •

¢ 1 (y 0 ,z 0 ) = (y 1 )z 1 ) .

The existence

of an invariant manifold for this map is guaranteed by the following: Lemma.

(Lemma (2 .4) in Marsden-McCracken [81, p. 32].

close enough to

0 , then for sufficiently small

exists an invariant manifold of

¢1

270

If

6

is

e , there

that is defined

£y

~

r-1 C

function

z

= g 8 (y)

\\Dg e (y) \\ < 1 • where

with

= 0, Dg 8 (0) = 0, and

g 8 (0)

Furthermore, \\zn - g 8 (y n) \\ -+ 0

= ¢~(y 0 ,z 0 ) = ¢n(y 0 ,z 0 )

(yn,zn)

Fix any such small c - a > 4A. , where

=

e

1\e -c\\

= \\eA\\ , 1/ c

a

n -+

co

•

6 , say

and

e

as

.

e0 , 6

= o0 with

One can show,

following Hartman's proof of existence of invariant manifolds

[43, Chapter IX], that (a)

0 = g€ (y 0) , then 0 m = 1' 2' •••

({3)

llvm\\ s: (a + 2A.)m \\vall

if

\\Ymll ~ (c - 2A,)miiY 0 \\

2

vt = zt

g€ (y t) 0

for

for

m = 1, 2, ••• , where

.

Furthermore, one can show that (y)

For

c > 2A.

manifold

z

the restriction of

= g 8 (y)

¢1

on the invariant

is a diffeomorphism (onto).

0

This last result follows from: .

Proposition.

Let

~:

10

~

n _.

10

~ ~

~(y)

n

be a

= By + G(y) ,

where B is a nonsingular constant matrix. If \\B- 1[[ sup \\DG(y)\\ < 1 , then ~ is ~diffeomorphism onto. y

To establish (y) one applies this Proposition to the time one map

c = e y + yl € (y, g€ (y)) ' ' 0

which has the form ¢ 1 (y)

= By + G(y) • 271

0

To outline a proof of part (a) of the Center Manifold z = g

Theorem, it suffices to show the manifold invariant under the maps z 0 = ge 0 (y 0 ) . such that

z_n

for t 0 between to By (y), one can find (y_n,z-n)

=

g~

~

0

¢

(y_n)

ling (y) II < 1 , g (0) = 0

and

is also

(y) €0

and

0

1 •

n = 1,2, •••

for

= (y 0 ,z 0 ) .

¢ (y ,z ) n -n -n

By

and ({3), one has

llz_n+t II+ IIY_n+t II:::: llz_n+t II+ 1\gCy_n+t )II 0 0 0 0 :?:

Let

llv _ + II n to

:?:

(a + 2A.) -n\\v

to

(i)

\I •

and

::;; 21\Y -n II [e -a + 2A.] . By (i), (ii), and (a), 2 (e -a + 2A.) (c - 2A.) -n IIYoll :::: (a + 2A.) -n \\v

to

II ,

or 2 (e Let

n

~ ~

z = g 8 (y) 0 0 and 1

.

-a

n

+ 2A.) (a + 2A.) (c - 2A.)

Then since

-n

\IYoll .

:?:

\\v

to

c - a > 4A. , one obtains

is invariant under the map

¢t 0

Let

272

II • v

to for any t 0

=

0 ; or

between

8 =

sup rllnyY(y)z)l\) llnyZ(y)z)ll) lin Y(y)z)ll) lin Z(y,z)\11 • (y)z)EH z z

Clearly

9 __.. o

as

€

--t

0 •

Thus) we may assume that

To prove part (b)) writing we will establish

v (t)

llvCt) II ~ llvCO)IIe(a+28)t

can assume

for

t:e:O. ("ob''1.1.11 ::::: k 1 > 0 for some k1 > 0 •

K > 0 ,

Furthermore, for some II y ( X

Now choose ~=~(5 1 ) •

c ~

' X

s

)

-

y (X

= n(O) = Hence,

c

'

g (X 1-1

c

) )

II

:s: Kll X s

c(0)/(2K) •

Then

g II( X

-

~

c

)

II

~ K17 •

¢'(t) < 0

on

B 01 ,~(0 1)

BO

1 '~

x(t)

1

x(t)

does not enter

275

B~

A

v 2'~(v 2

of

BO,~(O)

(O ) but that does not enter each

But if

( t) •

is positively invariant for

Lastly, suppose that there exists a trajectory that enters

if

S

1.1.

)

,

(t)

for

say, with

0
6 2 , for more than a finite time since

where

exponentially attracting, and of

D

B

61,n

are dissipative.

contraction semigroup.

c0

an analytic semigroup is relevant.

= [s

Let

is dissipative and the range

To deal with t-smoothness of

68

(x,Ax) ~ 0

is dissipative.

Suppose

whose space.

+

(Ax,x)

with dense

DA •

Theorem 3 [8; pp. 174-175]. semigroup

A

and

\argument of

semigroups the concept of

Given

s\

e

in

< e} •

(0,

¥)

set

A semigroup

Tt

is said to be analytic if it can be extended to a family 278

T

s some

of bounded linear transformations with

e

(i)

TT

in

T

= Ts • Ts

sl+s2

(ii)

Ts (x)

(iii)

lrs(x) 68

)

1

xl _. 0

as

Is X

in

X

Proposition 2.

for

2

is analytic in the sector

for any

68

such that

2) '

(0'

lying in

s

Suppose

infinitesimal generator

A •

1)

I

_. 0

68

for each

X

in

X

in any closed subsector of

.

Tt

is an analytic semigroup with

Then, to each for

=

n

t > 0 ,

1, 2, •••

dnT 2)

t dtn

= AnTt

in the operator topology on bounded transfor-

mations for t _. \IATtll Let

n

=

In particular, t _. \\Tt\\ ,

1,2, ...

are continuous in

¢ E (0,

Theorem 4.

TT 2)

and

t

a > 0

Suppose that

---- ----

on

(O,oo)

be given.

A

is a closed operator with dense

domain such that

~~~

1)

The resolvent set of

\\R(\,A)\\

Then, A

contains the sector

= [\ E Cl\ I= a, larg (\-a) I 0

(0 ,oo)

279

\

.

is compact is and only i f

Tt

for~

is said

in

p (A)

.

(Hence, R(A,A)

is compact for any

A

in

p(A)) .

It follows from this theorem that any compact semigroup has the following properties: 1)

The infinitesimal generator

A

has a pure point spectrum

consisting at most a countable sequence of points lf~ ""k }

corresponding eigenvectors

) and

[Ak}

with

cannot have an

f\ } l/\k

accumulation point in the finite part of the plane. 2)

T

The spectrum of

3)

for

t

> 0 ;

=

t

the closure of

[e

Akt

I

k = 1,2, ... }

= exp (tcr(A)) ).

(i.e., cr(Tt)

In conclusion, we quote some perturbation results of semigroups from Kato [69].

Banach space

X

and

)

B

semigroup on a

generates a

A

Suppose

Theorem 6.

is a bounded transformation on

X

.

Then, co

+ B also generates a

semigroup on

X

1)

A

2)

If

A

generates an analytic semigroup, so does

3)

If

A

so does generates a- compact semigroup, --

0

A+ B A + B

.

Here, of course, A + B

denotes the linear transformation from

the domain of

X

A

into

which is defined by

=

(A+ B)(x)

Ax + Bx • A family

of closed operators on

L

X , defined for

in

1-L

a domain ~

and

(A)

(2)

Do

of the complex plane, is said to be holomorphic of

if LX

(1)

the domain

D(L ) 1-L

is holomorphic in

~

=

D

is independent of

for every

x

in

1-L

D

1-L

Theorem 7.

(Kato [ 69, Chapter IX, Th. 2, 6]).

holomorphic family of

~

(A)

defined near

1-L

Let

=0 .

L

be a

1-L

If

10

is the generator of an analytic semigroup, the same is true for tL L with 11-LI sufficiently small. In this case, U(t,I-L) = e 1-L 1-L

is holomorphic in

1-L

and

t

when

280

t

is in some open sector

containing in

t

~

t

to

> 0 • t

=

Moreover, all

0 •

281

are strongly continuous

Appendic

Let

c.

A Regularity Theorem

. be a bounded domain with smooth boundary 1n

0

H~(O,R)

Recall that

Rn •

denotes the real Sobolev space intro-

duced at the beginning of Section 2 in Chapter 5. Proposition

.!.·

Assume that

. f unct1on, and suppose

from

$~£(0,lR)

-+

Nn

f: 0 X lRm -+ lR

> !! 2

H~(O,lR)

is a smooth F: s-+ f(·,s(•))

is well-defined and smooth.

To verify this well-known proposition [ 1 ], the following lemma from Palais [90] at page 31 is needed: Lemma.

Assume that

~>I

suppose

~~~(O,lR)

-+

g: OXJR

Then the map

H~(O,lR)

m

-+ lR

is a smooth function, and

s-+ g(·,s(·))

from

is well-defined and continuous.

Proof of Proposition 1.

By the Taylor Theorem, for any

r

f(x,s +h)==

with

~ '{J

a'

R

Ci

smooth in their arguments and

R (x,s,O) == 0 • a

From the above Lemma, the maps

a h-+ h (·), s-+ ¢ (·,s(·)), a are continuous in the norms

\\

s,h-+ R (·,s(·),h(·)) a

\\ ~

induced by

\\

\\ 0~ .

Th us'

¢k (s)

the multilinear map, defined by

k!

(h) ==

is bounded and from

H'e, (O,JR)

into

.

.

LkfH~(" ,JR) ' H~(" ,JR)) s \

282

s -+ k (s)

.

Since,

is continuous

,

Ra(·,s(·),h(·))-+ Ra(·,s(·),O)

= 0 as

(s,h)-+ (s,O).

1\R(s,h)\\_e --------~-+

llhll~

0

as

(s,h) -+ (s,O) , where

~ R(·,s(·),h(·))ha(·)

•

aa

F(s +h)

One gets

=

R(s,h)

Clearly,

=

"'r k(s) u

k!

h

k

+ R(s,h)h

r

.

k=O

By a converse of Taylor's Theorem in Banach space (see Theorem 2.1 at page 6 in Abraham and Robbin [ 1 ]) •

One obtains that

F: E9 ~.e, (O,IR) -+ H£ (O,IR)

Now

positive integer, F

is of class

Cr

is, therefore, smooth.

283

r

can be any

Appendix D.

Truncation Error, Roundoff Error and Numerical Differencing

Our object in this Appendix is to explain the various choices of increments made in Chapter 3.

The concepts are standard, see

for example [27, 112l. Suppose that a number where t (h)

g (0)

go

go t(h)

by

is then

g(h)

= g0 -

= lim g(h)

' h-+ 0 The truncation error

may or may not be defined.

in approximating

g0

is defined by

g(h) .

(The expression "truncation error" is more natural in the context of series, where the partial sum of "truncated" series.

terms is the

In the present context, the limit

"truncated" by stopping at a nonzero Now suppose that

N

g(h), h

rather than exactly, and let

h -+ 0

is

h.)

# O, is evaluated numerically G(h)

The roundoff error in evaluating

denote the machine result. g(h) is

r(h) = g(h) - G(h) • (A machine only carries numbers to a certain finite precision, and so during computations must approximate those numbers which cannot be represented exactly.

The cumulative effect of these

errors is termed "roundoff error" because rounding is a common technique, although not the only one, for approximating numbers by machine representable numbers.)

Note that if

G(h)

is not

identically constant, it has jump discontinuities since its values are machine representable numbers. The total error in approximating

g0

by

G(h)

sum of the truncation and roundoff errors: e(h)

= g0 -

G(h)

284

t (h)

+ r

(h) •

is then the

The question arises:

how best to choose the value of

minimize the total error?

h

to

Often there is not enough information

available to answer this question, and so one looks instead for the value of

h

that minimizes the sum

+

T (h)

where

T(h)

and

R(h)

R (h) ,

are bounds for

t(h)

and

r(h) ,

respectively. Suppose T(h) where

p > O, q > O,

u

is the machine precision (the smallest

number such that the machine distinguishes between

of

c1

(1.0)), and

and h

and

u •

c2

and

The sum h =

T(h)

(1.0 + u)

are positive constants independent

+

R(h)

is minimized for

(.9. ~ u)l/ (p+ q) • p

c1

For the one-sided difference approximation

p

=

1

and

q

as an unknown.

=

1 •

However, the ratio

For

c1

involves

must be treated

f"(x 0 ) , and since the result

of the computation is an approximation to unreasonable to assume that

Cic 1

f"(x 0 )

f

1

(x 0 ) , it would be

is known.

Moreover,

c2

involves the actual mechanism of accumulation of roundoff error, which is often inadequately known.

In the absence of this

additional information, the choice

h = X

can be made, where

xref

ref

U

1/2

is a scale for the variable

this choice, the total error is

O(u 112 ) 285

The "O"

x •

For

symbol here

refers to the limit as one performs the same computation using floating point arithmetic of successively greater precision. Although a somewhat unrealistic limit, the power

u

1/2

indicates

that under "normal" circumstances, one-sided numerical differencing can produce a value for

f

1

(x 0 )

with roughly 1/2

the number of significant figures carried by the machine. For the two-sided difference approximation

=

g(h) p

=

2

and

q

(f(x 0 +h) - f(x 0 - h))/2h ~ g 0

=

1

as unknown since h

=

xref u 113

and the ratio cl

involves

c 1 /c 2 f"'(xo)

=

f'(x 0 ) ,

must again be treated An increment

is then appropriate, for which choice the total O(u 2 / 3 )

error in the approximation to the derivative is For the three point approximation

= (f(x 0 +h) + f(x 0 -h) - 2f(x 0 ))/h 2

g(h)

the exponents are

p

=

q

=

2 , and the choice

appropriate, for which the total error is

~

h

g0

=

=

X

f"(x 0 ) ,

ref

U

1/4

is

O(u 112 )

Let

This is the "5-point Laplacian".

the exponents are again is appropriate, where x

and

p s ref

=

q

=

For the approximation

2 , and the choice

1/4

u ref is a common scale for the variables

y •

Let

286

h = s

This is the "9-point Laplacian", and may be thought of as having been constructed from the 5-point Laplacian by the process of Richardson extrapolation [27], which has the effect of increasing

p

from 2 to 4.

so the choice error is

h

=

O(u 213 )

s

ref

u 116

The exponents are

p

=

4

and

q

=

2

is appropriate, for which the total

•

The first step in the secant iteration for location for location of

v

(Chapter 3, Section 2) may be thought of as a c Newton step, in which a one-sided difference approximation is used for the derivative

a'(v 1 )

and the size of the increment

employed is an informed guess of the value required to minimize the total error in the approximation. value of

Note that the computed

v

will be (essentially) independent of the scheme c used to start the secant iteration. A decision was made that the error due to numerical differencing in all of should be

O(u 213 ) •

c 1 (0),

a'

(0),

w'

(0), ~ 2 , ~ 2

and

~2 ,

Therefore two-sided (rather than one-sided)

differencing is used in computing

f 20

and

f 11 (Chapter 3,

Section 3.3), the 9-point (rather than 5-point) Laplacian is used in computing

c21

(Chapter 3, Section 3.6), and two-sided

(rather than one-sided) differencing is used in computing (Chapter 3, Section 4). accept an

O(u 1 / 2 )

A.{ (vc)

Note that if the decision were to

error, the number of Jacobian evaluations in

this phase of the computation could be reduced, roughly by a factor of 2.

287

Appendix E.

1.

BIFOR2

The Code

Introduction. The code BIFOR2 is the product of several years of evolution.

A brief history of the code will perhaps help others avoid some of the pitfalls we encountered. Our first idea was to use a language for symbolic manipulation in order to perform the tedious algebraic manipulations which made hand calculations impossible.

This approach was

employed both in the derivation of bifurcation formulae [46] and in the application [44].

With symbolic manipulation, however,

there were numerous technical problems.

The number of

potentially distinct analytic expressions involved in the term g 21

is at least

N2 (N 2

+ 3N + 2)/6 , the n~m~er of potentially

distinct third order partial derivatives i,j,k,l = l, ••• ,N.

o f~/oxjoxkox 1 ,

The program employed in [44] was therefore

limited to relatively low order systems.

Also, because of the

hybrid symbolic/numeric nature of the program, it was tied to the particular machine used (a CDC 6400) and to the peculiar combination of languages employed (SYMBAL, SNOBAL, FORTRAN). A new version, was therefore written, entirely in FORTRAN [45].

The symbolic manipulation phase was eliminated, the

partial derivatives being calculated instead by numerical differentiation.

Once symbolic manipulation had been eliminated,

a striking economy became apparent. formed into real canonical form, only partial derivatives and

0(1)

When the system is transO(N)

distinct second

distinct third partial derivatives

are needed to evaluate the bifurcation formulae.

The new version

therefore performed the numerical differentiation in the canonical coordinate system rather than in the original coordinates.

Program storage requirements were reduced to

O(N 2 ) , and the program became applicable to higher order

288

systems.

Because there is a certain loss of accuracy associated

with numerical differentiation, procedures for estimation of this error were built into the code.

Also, preliminary checking of

the user-supplied subroutine for consistency between the Jacobian matrix and the function values was incorporated because such checks had proven useful in the study [44].

BIFORl

followed the "recipe" given in Chapter 2, and was used to verify the analytical results presented there.

Every disagreement

between BIFORl and the hand calculations was traced to mistake(s) in the latter. Although able to analyze systems much more difficult than could be treated by hand or with the aid of symbolic manipulation, the code BIFORl was soon pressed to the limit of its capabilities in the panel flutter

problem,

efficiency became apparent.

and

the

need for increased

A new version) BIFOR2, was therefore

written. In BIFOR2, the explicit construction of the matrices P-l

P

and

of the transformation to real canonical coordinates was

eliminated in favor of the technique employing right and left eigenvectors as described in Chapter 3.

Also, the numerical

differentiation procedure was reorganized into two distinct stages:

the reorganization reduced the number of Jacobian

evaluations required during the numerical differencing from to

0(1)

value of

O(N)

Different techniques for location of the critical

• ~

were introduced.

The two techniques

(MTH

=

1 and 2)

retained in the current version are not as fast as a third technique which we developed, based upon solving the

N

+ 2 (real)

dimensional system f(x; ~)

det

simultaneously for

c