The Hopf Bifurcation and Its Applications

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J. E. Marsden M. McCracken Applied Mathematical Sciences 19

The Hopf Bifurcation and Its Applications

Springer-Verlag New York Heidelberg Berlin

Applied Mathematical Sciences EDITORS Fritz John

Lawrence Sirovich

Courant Institute of Mathematical Sciences New York University New York, N.Y. 10012

Division of Applied Mathematics Brown University Providence, R.J. 02912

Joseph P. LaSalle

Gerald B. Whitham

Division of Applied Mathematics Brown University Providence, R.J. 02912

Applied Mathematics Firestone Laboratory California Institute of Technology Pasadena,CA.91125

EDITORIAL STATEMENT The mathematization of all sciences, the fading of traditional scientific boundaries, the impact of computer technology, the growing importance of mathematicalcomputer modelling and the necessity of scientific planning all create the need both in education and research for books that are introductory to and abreast of these developments. The purpose of this series is to provide such books, suitable for the user of mathematics, the mathematician interested in applications, and the student scientist. In particular, this series will proVide an outlet for material less formally presented and more anticipatory of needs than finished texts or monographs, yet of immediate interest because of the novelty of its treatment of an application or of mathematics being applied or lying close to applications. The aim of the series is, through rapid publication in an attractive but inexpensive format, to make material of current interest widely accessible. This implies the absence of excessive generality and abstraction, and unrealistic idealization, but with quality of exposition as a goal. Many of the books will originate out of and will stimulate the development of new undergraduate and graduate courses in the applications of mathematics. Some of the books will present introductions to new areas of research, new applications and act as signposts for new directions in the mathematical sciences. This series will often serve as an intermediate stage of the publication of material which, through exposure here, will be further developed and refined. These will appear in conventional format and in hard cover.

MANUSCRIPTS The Editors welcome all inquiries regarding the submission of manuscripts for the series. Final preparation of all manuscripts will take place in the editorial offices of the series in the Division of Applied Mathematics, Brown University, Providence, Rhode Island. SPRINGER-VERLAG NEW YORK INC., 175 Fifth Avenue, New York, N. Y. 10010 Printed in U.S.A.

Applied Mathematical Sciences I Volume 19

J. E. Marsden M. McCracken

The Hopf Bifurcation and Its Applications with contributions by

P. Chernoff, G. Childs, S. Chow, J. R. Dorroh, J. Guckenheimer, L. Howard, N. Kopell, O. Lanford, J. Mallet-Paret, G. Oster, O. Ruiz, S. Schecter, D. Schmidt, and S. Smale

I] Springer-Verlag New York 1976

J. E. Marsden

M. McCracken

Department of Mathematics

Department of Mathematics

University of California

University of California at Santa Cruz

at Berkeley

AMS Classifications: 34C15, 58F1 0, 35G25, 76E30

Library of Congress Cataloging in Publication Data Marsden, Jerrold E. The Hopf bifurcation and its applications. (Applied mathematical sciences; v. 19) Bibliography Includes index. 1. Differential equations. 2. Differential equations, Partial: 3. Differentiable dynamical systems. 4. Stability. I. McCracken, Marjorie, 1949- joint author. II. Title. III. Series. QA1.A647 vol. 19 [QA372] 510'.8s [515'.35] 76-21727

All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. © 1976 by Springer-Verlag New York Inc. Printed in the United States of America

ISBN 0-387-90200-7 ISBN 3-540-90200-7

Springer-Verlag Springer-Verlag

New York' Heidelberg· Berlin Berlin· Heidelberg' New York

To the courage of G. Oyarzun

PREFACE The goal of these notes is to give a reasonahly complete, although not exhaustive, discussion of what is commonly referred to as the Hopf bifurcation with applications to specific problems, including stability calculations.

Historical-

ly, the subject had its origins in the works of Poincare [1] around 1892 and was extensively discussed by Andronov and Witt [1] and their co-workers starting around 1930. paper [1] appeared in 1942.

Hopf's basic

Although the term "Poincare-

Andronov-Hopf bifurcation" is more accurate (sometimes Friedrichs is also included), the name "Hopf Bifurcation" seems more common, so we have used it.

Hopf's crucial contribution

was the extension from two dimensions to higher dimensions. The principal technique employed in the body of the text is that of invariant manifolds.

The method of Ruelle-

Takens [1] is followed, with details, examples and proofs added. Several parts of the exposition in the main text corne from papers of P. Chernoff, J. Dorroh, O. Lanford and F. Weissler to whom we are grateful. The general method of invariant manifolds is common in dynamical systems and in ordinary differential equations; see for example, Hale [1,2] and Hartman [1]. methods are also available.

Of course, other

In an attempt to keep the picture

balanced, we have included samples of alternative approaches. Specifically, we have included a translation (by L. Howard and N. Kope11) of Hopf's original (and generally unavailable) paper. These original methods, using power series and scaling are used in fluid mechanics by, amongst many others, Joseph and Sattinger [1]; two sections on these ideas from papers of Iooss [1-6] and

PREFACE

viii

Kirchgassner

o.

and Kielhoffer [1]

(contributed by G. Childs and

Ruiz) are given. The contributions of S. Smale, J. Guckenheimer and G.

Oster indicate applications to the biological sciences and that of D. Schmidt to Hamiltonian systems.

For other applica-

tions and related topics, we refer to the monographs of Andronov and Chaiken

[1], Minorsky

[1]

and Thom [1].

The Hopf bifurcation refers to the development of periodic orbits ("self-oscillations") from a stable fixed point, as a parameter crosses a critical value.

In Hopf's

original approach, the determination of the stability of the resulting periodic orbits is, in concrete problems, an unpleasant calculation.

We have given explicit algorithms for

this calculation which are easy to apply in examples.

(See

Section 4, and Section SA for comparison with Hopf's formulae). The method of averaging, exposed here by S. Chow and J. MalletParet in Section 4C gives another method of determining this stability, and seems to be especially useful for the next bifurcation to invariant tori where the only recourse may be to numerical methods, since the periodic orbit is not normally known explicitly. In applications to partial differential equations, the key assumption is that the semi-flow defined by the equations be smooth in all variables for

t > O.

This enables the in-

variant manifold machinery, and hence the bifurcation theorems to go through (Marsden [2]).

To aid in determining smoothness

in examples we have presented parts of the results of DorrohMarsden. [1].

Similar ideas for utilizing smoothness have been

introduced independently by other authors, such as D. Henry [1].

PREFACE

ix

Some further directions of research and generalization are given in papers of Jost and Zehnder [1], Takens [1, 2], Crandall-Rabinowitz [1, 2], Arnold [2], and Kopell-Howard [1-6] to mention just a few that are noted but are not discussed in any detail here. Ruelle [3]

We have selected results of Chafee [1] and

(the latter is exposed here by S. Schecter) to

indicate some generalizations that are possible. The subject is by no means closed.

Applications to

instabilities in biology (see, e.g. Zeeman [2], Gurel [1-12] and Section 10, 11); engineering (for example, spontaneous "flutter" or oscillations in structural, electrical, nuclear or other engineering systems; cf. Aronson [1], Ziegler [1] and Knops and Wilkes [1]), and oscillations in the atmosphere and the earth's magnetic field at a rapid rate.

(cf. Durand [1]) a~e appearing

Also, the qualitative theory proposed by

Ruelle-Takens [1] to describe turbulence is not yet well understood (see Section 9).

In this direction, the papers of

Newhouse and Peixoto [1] and Alexander and Yorke [1] seem to be important.

Stable oscillations in nonlinear waves may be

another fruitful area for application; cf.Whitham [1].

We hope

these notes provide some guidance to the field and will be useful to those who wish to study or apply these fascinating methods. After we completed our stability calculations we were happy to learn that others had found similar difficultv in applying Hopf's result as it had existed in the literature to concrete examples in dimension

~

3.

They have developed similar

formulae to deal with the problem; cf. Hsu and Kazarinoff [1, 2] and Poore [1].

PREFACE

x

The other main new result here is our proof of the validity of the Hopf bifurcation theory for nonlinear partial differential equations of parabolic type.

The new proof,

relying on invariant manifold theory, is considerably simpler than existing proofs and should be useful in a variety of situations involving bifurcation theory for evolution equations. These notes originated in a seminar given at Berkeley in 1973-4.

We wish to thank those who contributed to this

volume and wish to apologize in advance for the many important contributions to the field which are not discussed here; those we are aware of are listed in the bibliography which is, admittedly, not exhaustive.

Many other references are contained

in the lengthy bibliography in Cesari [1].

We also thank those

who have taken an interest in the notes and have contributed valuable comments.

These include R. Abraham, D. Aronson,

A. Chorin, M. Crandall., R. Cushman, C. Desoer, A. Fischer, L. Glass, J. M. Greenberg, O. Gurel, J. Hale, B. Hassard, S. Hastings, M. Hirsch, E. Hopf,

N~

D. Kazarinoff, J. P. LaSalle,

A. Mees, C. Pugh, D. Ruelle, F. Takens, Y. Wan and A. Weinstein. Special thanks go to J. A. Yorke for informing us of the material in Section 3C and to both he and D. Ruelle for pointing out the example of the Lorentz equations (See Example 4B.8). Finally, we thank Barbara Komatsu and Jody Anderson for the beautiful job they did in typing the manuscript.

Jerrold Marsden Marjorie McCracken

TABLE OF CONTENTS SECTION 1 INTRODUCTION TO STABILITY AND BIFURCATION IN DYNAMICAL SYSTEMS AND FLUID DYNAMICS .•.•.•••.••.•

1

SECTION 2 THE CENTER t1ANIFOLD THEOREM

27

SECTION 2A

•

SOME SPECTRAL THEORY

50 SECTION 2B

THE POINCARE MAP

56 SECTION 3

THE HOPF BIFURCATION THEOREM IN R2 AND IN Rn

63

SECTION 3A OTHER BIFURCATION THEOREMS

85

SECTION 3B MORE GENERAL CONDITIONS FOR STABILITY

91

SECTION 3C HOPF'SBIFURCATION THEOREM AND THE CENTER THEOREM OF LIAPUNOV by Dieter S. Schmidt .•..•..•.••....•.

95

SECTION 4 COMPUTATION OF THE STABILITY CONDITION

••••.•....• 104

SECTION 4A HOW TO USE THE STABILITY FORMULA; AN ALGORITHM

.•• 131

SECTION 4B EXAMPLES

136 SECTION 4C

HOPF BIFURCATION AND THE METHOD OF AVERAGING .•••..••.•...•.•.•• 151

by S. Chow and J. Mallet-Paret

xii

TABLE OF CONTENTS

SECTION 5 A TRANSLATION OF HOPF'S ORIGINAL PAPER by L. N. Howard and N. Kopell ••.•.....•.•...••.•

163

SECTION 5A EDITORIAL COMMENTS by L. N. Howard and N. Kopell .•••.•••••.•.•.•.......•.........•.

194

SECTION 6 THE HOPF BIFURCATION THEOREM FOR .••.• , •....•.••.•.•.•. " . . • . •. . . • DIFFEOMORPHISMS

206

SECTION 6A THE CANONICAL FORM

219 SECTION 7

BIFURCATIONS WITH SYMMETRY by Steve Schecter

224

SECTION 8 BIFURCATION THEOREMS FOR PARTIAL DIFFERENTIAL EQUATIONS ..•••...••.•..••....... " .••..•. " . . . • .

250

SECTION 8A NOTES ON NONLINEAR SEMIGROUPS

258

SECTION 9 BIFURCATION IN FLUID DYNAMICS AND THE PROBLEM OF TURBULENCE

285

SECTION 9A ON A PAPER OF G. IOOSS by G. Childs

304

SECTION 9B ON A PAPER OF KIRCHGASSNER AND KIELHOFFER by O. Ruiz ••••••••••••••••••••••••••. . •••. . . •.••

315

SECTION 10 BIFURCATION PHENOMENA IN POPULATION MODELS by G. Oster and J. Guckenheimer •.••••••••••••.••

327

TABLE OF CONTENTS

xiii

SECTION 11 A MATHEMATICAL MODEL OF TWO CELLS by S. Smale

354

SECTION 12 A STRANGE, STRANGE ATTRACTOR by J. Guckenheimer

368

REFERENCES

382

INDEX

405

THE HOPF BIFURCATION AND ITS APPLICATIONS

1

SECTION 1 INTRODUCTION TO STABILITY AND BIFURCATION IN DYNAMICAL SYSTEMS AND FLUID MECHANICS Suppose we are studying a physical system whose state is governed by an evolution equation unique integral curves. of

X; i.e., X(X )

O

=

O.

upon the system at time in state

xo.

will remain at answer to this

dx dt

=

X(x)

x

which has

Let

X be a fixed point of the flow o Imagine that we perform an experiment

t

=

0

and conclude that it is then

Are we justified in predicting that the system X o

for all future time?

qu~stion

The mathematical

is obviously yes, but unfortunately

it is probably not the question we really wished to ask. Experiments in real life seldom yield exact answers to our idealized models, so in most cases we will have to ask whether the system will remain near

X if it started near xo. The o answer to the revised question is not always yes, but even so, by examining the evolution equation at hand more minutely, one can sometimes make predictions about the future behavior of a system starting near

xo.

A trivial example will illustrate

some of the problems involved.

Consider the following two

2


di~~erential

equations on the real line: X'

(t)

=

-x (t)

(1.1)

X

(t)

=

x

(1. 2)

and I

(t) •

The solutions are respectively: (1.1')

and (1.2')

Note that

0

is a

~ixed

point

both

o~

~lows.

In the first

E R, lim x(xo,t)

case, for all

X

= O. The whole real line t+ oo moves toward the origin, and the prediction that if X is o

near

x(xo,t)

0, then

o

is near

0

is obviously justified.

On the other hand, suppose we are observing a system whose state

x

at time

is governed by (1.2). t

=

0, x'( 0)

An experiment telling us that

is approximately zero will certainly not

permit us to conclude that

x(t)

all time, since all points except

stays near the origin for 0

move rapidly away from O.

Furthermore, our experiment is unlikely to allow us to make an accurate prediction about

x(t)

because if

moves rapidly away from the origin toward x(O) > 0, x(t)

moves toward

+00.

all

t

but if

Thus, an observer watching

such a system would expect sometimes to observe and sometimes

x(O) < 0, x(t)

x(t)

t~

x(t)

~ +00. The solution x(t) o for t+ oo would probably never be observed to occur because a

slight perturbation of the system would destroy this solution. This sort of behavior is frequently observed in nature. is not due to any nonuniqueness in the solution to the

It dif~er-

ential equation involved, but to the instability of that solution under small perturbations in initial data.


3

Indeed, it is only stable mathematical models, or features of models that can be relevant in "describing" nature.+ Consider the following example.*

A rigid hoop hangs

from the ceiling and a small ball rests in the bottom of the hoop.

The hoop rotates with frequency

w

about a vertical

axis through its center (Figure l.la).

Figure l.la For small values of

Figure l.lb w, the ball stays at the bottom of the

hoop and that position is stable. some critical value

However, when

w

reaches

w ' the ball rolls up the side of the

hoop to a new position

o

x(w), which is stable.

The ball may

roll to the left or to the right, depending to which side of the vertical axis it was initially leaning (Figure l.lb). The position at the bottom of the hoop is still a fixed point, but it has become unstable, and, in practice, is never observed to occur.

The solutions to the differential equations

governing the ball's motion are unique for all values of

+For further discussion, see the conclusion of AbrahamMarsden [1].

* This

example was first pointed out to us by E. Calabi.

w,

4


but for

o'

w > w

this uniqueness is irrelevant to us, for we

cannot predict which way the ball will roll.

Mathematically,

we say that the original stable fixed point has become unstable and has split into two stable fixed points.

See

Figure 1.2 and Exercise 1.16 below.

w

fixed

points

(0 )

(b)

Figure 1.2

Since questions of stability are of overwhelming practical importance, we will want to define the concept of stability precisely and develop criteria for determining it. (1.1)

Definition.

Let

F

t

semiflow)* on a topological space variant set; i. e. , Ft (A) C A

cO

be a

flow (or

M and let

for all

t.

A

be an in-

We say

A

is

stable (resp. asymptotically stable or an attractor) i f for any neighborhood

U

of

A

there is a neighborhood

V

of

A

such

*i.e.,

F t : M ; M, F O = identity, and. F t + s = FsoFt for all t, s ER. C means Ft(x) is continuous in (t,x). A semiflow is one defined only for

t

~

O.

Consult, e.g.,

Lang [1], Hartman [1], or Abraham-Marsden [1] for a discussion of flows of vector fields.

See section BA, or Chernoff-

Marsden [lJ for the infinite dimensional case.


x(xo,t) - Ft(x ) O

that the flow lines (integral curves) long to

U

Thus

if A

X

o

E V

(resp.

n

t>O

(resp. tends towards A

be-

F (V) = A). t

is stable (resp. attracting) when an initial

condition slightly perturbed from

If

5

A).

A

remains near

A

(See Figure 1.3).

is not stable it is called unstable.

stable fixed point

as ymptot ieo Ily sto bl e fixed point

stable closed

or bit

Figure 1.3 (1.2)

Exercise.

Show that in the ball in the hoop

example, the bottom of the hoop is an attracting fixed point w < W = Ig/R and that for W > o ing fixed points determined by cos e for

W

o

=

there are attract2

g/w R, where

the angle with the negative vertical axis, R of the hoop and

g

e

is

is the radius

is the acceleration due to gravity.

The simplest case for which we can determine the stability of a fixed point X o is the finite dimensional, n linear case. Let X: R + Rn be a linear map. The flow of


6

x

is

=

X(Xo,t)

e

tX

(x ). Clearly, the origin is a fixed O Ajt point. Let {A } be the eigenvalues of X. Then {e } j tX are the eigenvalues of e Suppose Re A. < 0 for all j • J Re A.t J Then ->- 0 as t ->- 00. One can check, using I/jtl = e the Jordan canonical form, that in this case

A.

totica11y stable and that if there is a real part, 0 (1.3)

is unstable. Theorem.

Let

map on a Banach space

E.

fixed point of the flow of

z E o(X)

with

J

pos~tive

More generally, we have: X: E

->-

E

be a continuous, linear

The origin is a stable attracting X

if the spectrum

is in the open left-half plane. there exists

is asymp-

0

o(X)

of

X

The origin is unstable if

such that

Re(z) > O.

This will be proved in Section 2A, along with a review of some relevant spectral theory. Consider now the nonlinear case. Let P be a Banach 1 manifo1d* and let X be a c vector field on P. Let Then

dX(PO): T (P) PO

linear map on a Banach space.

->-

is a continuous

T (P) Po

Also in Section 2A we shall

demonstrate the following basic theorem of Liapunov [1]. (1.4)

Theorem.

Banach manifold X(PO) = O.

P

Let

X(F t (x», FO(X)

F =

If the spectrum of O(dX(Po»

*We

t x.

Let

and let

X

be a

Po

vector field on a

be a fixed point of

be the flow of (Note that dX(PO)

c1 X

Ft(P O)

i. e. , =

Po

a at

X, i. e. ,

Ft(X)

for all

t. )

lies in the left-half plane; i.e.,

C {z EQ::IRe z < O}, then

Po

is asymptotically

shall use only the most elementary facts about manifold theory, mostly because of the convenient geometrical language. See Lang [1] or Marsden [4] for the basic ideas.


7

stable. If there exists an isolated Re z > a, Pa there is a

is unstable.

If

z E o(dX(Pa))

z Eo(dX(Pa))

O(dX(Pa))

such that

c

Re z

such that

{zlRe z S a}

and

a, then stability

=

cannot be determined from the linearized equation. (1.5) ~

2

Exercise. 2

: X(x,y) =

(y,~(l-x

Consider the following vector field on )y-x).

Decide whether the origin is un-

stable, stable, or attracting for

< a,

~

~

=

a, and

~

>

a.

Many interesting physical problems are governed by differential equations depending on a parameter such as the

w

angular velocity X : P ~

TP

+

in the ball in the hoop example.

be a (smooth) vector field on a Banach manifold

Assume that there is a continuous curve that of

X

~

X~.

(p(~))

a, i.e.,

=

Suppose that ~

unstable for

> ~a.

p(~)

X

in

p(~)

P

such

~ < ~a

is attracting for (p(~a)'~a)

The point

X~.

For

and

is then called ~

< ~a

the flow

can be described (at least in a neighborhood of

~

by saying that points tend toward true for

~

> ~a' ~a.

abruptly at ~

> ~a'

~

> ~a.

P.

is a fixed point of the flow

p(~)

a bifurcation point of the flow of of

Let

p(~).

p(~))

However, this is not

and so the character of the flow may change Since the fixed point is unstable for

we will be interested in finding stable behavior for That is, we are interested in finding bifurcation

above criticality to stable behavior. For example, several curves of fixed points may corne to(A curve of fixed points is a

gether at a bifurcation point. curve

a: I

+

P

such that

X

~

(a(~))

such curve is obviously

~ ~ p(~).)

stable fixed points for

~

>

~a.

=

a

for all

~.

One

There may be curves of

In the case of the ball in


8

the hoop, there are two curves of stable fixed points for W

> w ' one moving up the left side of the hoop and one moving

O

up the right side (Figure 1.2). Another type of behavior that may occur is bifurcation to periodic orbits. form

a: I .... P

closed orbit

This means that there are curves of the

such that Y]1

a(]10)

of the flow of

=

p(]10)

X]1.

Hopf bifurcation is of this type.

and

a(]1)

is on a

(See Figure 1.4).

The

Physical examples in fluid

mechanics will be given shortly.

-----t=====-f--unstable fixed point -

sta b Ie closed orbit

y -stable fixed point

x unstable fixed point - - stable fixed point

x

(a) Supercritica I Bifurcation

(b) Subcritical Bifurcation

(Stable Closed Orbits)

(Unstable Closed Orbits)

Figure 1.4 The General Nature of the Hopf Bifurcation


9

The appearance of the stable closed orbits (= periodic solutions) is interpreted as a "shift of stability" from the original stationary solution to the periodic one, i.e., a point near the original fixed point now is attracted to and becomes indistinguishable from the closed orbit.

{See

Figures 1.4 and 1.5).

further bifurco~

tions

stable point

appearance of

the closed orbit

a closed orbit

grows in amplitude

Figure 1.5 The Hopf Bifurcation Other kinds of bifurcation can occur; for example, as we shall see later, the stable closed orbit in Figure 1.4 may bifurcate to a stable 2-torus.

In the presence of symmetries,

the situation is also more complicated.

This will be treated

in some detail in Section 7, but for now we illustrate what can happen via an example. (1.6)

Example:

The Ball in the Sphere.

A rigid,

hollow sphere with a small ball inside it hangs from the

10


ceiling and rotates with frequency w through its center

about a vertical axis

(Figure 1.6).

w<w o

Figure 1. 6 For small but for

w >

W

o

w, the bottom of the sphere is a stable point, the ball moves up the side of the sphere to

a new fixed point.

For each

w

>

wo ' there is a stable, in-

variant circle of fixed points (Figure 1.7).

We get a circle

of fixed points rather than isolated ones because of the symmetries present in the problem.

stable circle

Figure 1. 7 Before we discuss methods of determining what kind of bifurcation will take place and associated stability questions,


11

we shall briefly describe the general basin bifurcation picture of R. Abraham [1,2]. In this picture one imagines a rolling landscape on which water is flowing.

We picture an attractor as a basin

into which water flows.

Precisely, if

and

A

is an attractor, the basin of

x E M which tend to

A

as

t

+

+00.

F

t

A

is a flow on

M

is the set of all

(The less picturesque

phrase "stable manifold" is more commonly used.) As parameters are tuned, the landscape, undulates and the flow changes.

Basins may merge, new ones may form, old

ones may disappear, complicated attractors may develop, etc. The Hopf bifurcation may be pictured as follows.

We

begin with a simple basin of parabolic shape; Le., a point attractor.

As our parameter is tuned, a small hillock forms

and grows at the center of the basin.

The new attractor is,

therefore, circular (viz the periodic orbit in the Hopf theorem) and its basin is the original one minus the top point of the hillock. Notice that complicated attractors can spontaneously appear or dissappear as mesas are lowered to basins or basins are raised into mesas. Many examples of bifurcations occur in nature, as a glance at the rest of the text and the bibliography shows. The Hopf bifurcation is behind oscillations in chemical and I

biological systems (see e.g. Kopell-Howard [1-6], Abraham [1,2] and Sections 10, 11), including such 'things as "heart flutter". One of the most studied examples comes from fluid mechanics, so we now pause briefly to consider the basic ideas of

* That "heart flutter" is a Hopf bifurcation is a conjecture told to us by A. Fischer;

cf. Zeeman [2].

*

12


the subject. The Navier-Stokes Equations Let

D

C R3

be an open, bounded set with smooth boundary.

We will consider

D

to be filled with an incompressible

homogeneous (constant density) fluid.

Let

u

and

velocity and pressure of the fluid, respectively.

p

,

be the

If the

fluid is viscous and if changes in temperature can be neglected, the equations governing its motion are: au + (u.V)u - v~u = -grad p

at

(+ external forces)

(L3)

div u = 0 The boundary condition is if the boundary of that u(x,t)

u(x,O) and

D

ul

o

aD

(or

ul

aD

prescribed,

is moving) and the initial condition is

is some given p(x,t)

(1. 4)

for

t

uO(x). > O.

The problem is to find

The first equation (1.3) is

analogous to Newton's Second Law

F

rna; the second (1.4)

is

equivalent to the incompressibility of the fluid.* Think of the evolution equation (1.3) as a vector field and so defines a flow, on the space vector fields on

D.

I

of all divergence free

(There are major technical difficulties

here, but we ignore them for now

-

see Section 8. )

The Reynolds number of the flow is defined by where

U

and

L

with the flow, and

R = UL

v

,

are a typical speed and a length associated

v

is the fluid's viscosity.

For example,

if we are considering the flow near a sphere

.... toward which fluid is projected with constant velocity

U00 i

*see any fluid mechanics text for a discussion of these points. For example, see Serrin [1], Shinbrot [1] or HughesMarsden [3].


(Figure 1.8), then sphere and

U

L

13

may be taken to be the radius of the

Uoo •

-Figure 1.8 If the fluid is not viscous

(v

0), then

R

00, and

the fluid satisfies Euler's equations: au at + (u·V)u divu The boundary condition becomes: ul laD from

=

0

on

(1.5)

o.

(1.6)

is parallel to aD, or aD This sudden change of boundary condition

for short. u

-grad p

aD

to

ul laD

ul

is of fundamental significance

and is responsible for many of the difficulties in fluid mechanics for

R

very large (see footnote below).

The Reynolds number of the flow has the property that, if we rescale as follows: u* x*

t* P*

U*

UL*

L T*

u x

T t u *) 2p

rlU

14


then if R

=

and

T

=

L/U, T*

UL/v, u* t*

=

L*/U*

and provided

R*

=

U*L*/V*

=

satisfies the same equations with respect to

that

u

satisfies with respect to

* + (u*·V*)u* ~ dt* div u* with the same boundary condition

x

and

x*

t; i.e.,

-grad p*

(1. 7)

a

(1. 8)

= a

u*1

as before.

(This

dD

is easy to check and is called Reynolds' law of similarity.) Thus, the nature of these two solutions of the Navier-Stokes equations is the same.

The fact that this rescaling can be

done is essential in practical problems.

For example, it

allows engineers to testa scale model of an airplane at low speeds to determine whether the real airplane will be able to fly at high speeds. (1.7)

Example.

Consider the flow in Figure 1.8.

If

the fluid is not viscous, the boundary condition is that the velocity at the surface of the sphere is parallel to the sphere, and the fluid slips smoothly past the sphere (Figure 1.9).

Figure 1. 9 Now consider the same situation, but in the viscous case. Assume that

R

starts off small and is

gradual~y

increased.

(In the laboratory this is usually accomplished by increasing


the velocity

-+

Uooi, but we may wish to think of it as

i.e., molasses changing to water.)

15

v

-+

0,

Because of the no-slip

condition at the surface of the sphere, as the velocity gradient increases there.

Uoo

gets larger,

This causes the flow

to become more and more complicated (Figure 1.10).* For small values of the Reynolds number, the velocity field behind the sphere is observed to be stationary, or approximately so, but when a critical value of the Reynolds number is reached, it becomes periodic.

For even higher

values of the Reynolds number, the periodic solution loses stability and further bifurcations take place.

The further

bifurcation illustrated in Figure 1.10 is believed to represent a bifurcation from an attracting periodic orbit to a periodic orbit on an attracting 2-torus in

I.

bifurcations may eventually lead to turbulence.

These further See Remark

1.15 and Section 9 below.

--. 0::

R = 50 (a periodic solution)

II further bifurcation as R increases

~

t

~-ct~~~ ~~ "-....Y

"---./

R = 75 (a slightly altered periodic solution) Figure 1.10

*These

large velocity gradients mean that in numerical studies, finite difference techniques become useless for interesting flows. Recently A. Chorin [1] has introduced a brilliant technique for overcoming these difficulties and is able to simulate numerically for the first time, the "Karmen vortex sheet", illustrated in Figure 1.10. See also Marsden [5] and Marsden-McCracken [2].

16


(1.8)

Example.

Couette Flow.

A viscous,

*

incompressi-

ble, homogeneous fluid fills the space between two long, coaxial cylinders which are rotating.

For example, they may

rotate in opposite directions with frequency For small values of

w

(Figure 1.11).

w, the flow is horizontal, laminar and

fluid

stationary.

I

I I

wfH-

W

I

Figure 1.11 If the frequency is increased beyond some value

wo' the

fluid breaks up into what are called Taylor cells (Figure 1.12).

top view

Figure 1.12

*couette

flow is studied extensively in the literature (see Serrin [1], Coles [1]) and is a stationary flow of the Euler equations as well as of the Navier-Stokes equations (see the following exercise).


17

Taylor cells are also a stationary solution of the NavierStokes equations.

For larger values of

w, bifurcations to

periodic, doubly periodic and more complicated solutions may take place (Figure 1.13).

doubly periodic structure

he Iicc I stru cture

}o'igure 1.13 For still larger values of

w, the structure of the Taylor

cells becomes more complex and eventually breaks down completely and the flow becomes turbulent.

For more informa-

tion, see Coles [1] and Section 7. (1. 9)

Exercise.

Find a stationary solution

..,.

to the

u

Navier-Stokes equations in cylindrical coordinates such that

U

depends only on

f

0

and

U

r, u

r

= U

z and the angular velocity

wlr=A 2

= + P2

= 0, the external force w

satisfies

(i.e., find Couette flow).

Wlr=A

1

= -PI'

Show that

is also a solution to Euler equations. (Answer:

where

and

s=

Another important place in fluid mechanics where an instability of this sort occurs is in flow in a pipe.

The


18

flow is steady and laminar (Poiseuille flow) up to Reynolds numbers around 4,000, at which point it becomes unstable and transition to chaotic or turbulent flow occurs.

Actually if

the experiment is done carefully, turbulence can be delayed until rather large

R.

It is analogous to balancing a ball

on the tip of a rod whose diameter is shrinking. Statement of the Principal Bifurcation Theorems k X: P + T{P) be a c vector field on a manifold lJ depending smoothly on a real parameter lJ. Let F~ be the Let

P

flow of

Let

be a fixed point for all

attracting fixed point for point for

lJ > lJO'

lJ, an

lJ < lJO' and an unstable fixed

Recall (Theorem 1.4) that the condition

0{dX {PO)) C {zlRe z < O}. At lJ lJ = lJO' some part of the spectrum of dXlJ{po) crosses the for stability of

imaginary axis.

PO

is that

The nature of the bifurcation that takes

(PO,lJ ) depends on how that crossing O occurs (it depends, for example, on the dimension of the place at the point

generalized eigenspace* of

dX

(PO) lJO the spectrum that crosses the axis).

belonging to the part of If

P

is a finite

dimensional space, there are bifurcation theorems giving necessary conditions for certain kinds of bifurcation to occur. If

P

is not finite dimensional, we may be able, nevertheless,

to reduce the problem to a finite dimensional one via the center manifold theorem by means of the following simple but crucial suspension trick. flow

_ lJ F t - (Ft,lJ)

0{d~{Po,lJn)) =

* The

e

on

P

o (dX{PO,lJ o ) )

Let x IR.

~

be the time 1 map of the

As we shall show in Section 2A,

That is,

definition and basic properties are reviewed in Section 2A.


19

~

The following theorem is now applicable to

(see

Sections 2-4 for details). (1.10)

Center Manifold Theorem (Kelley [1], Hirsch-

Pugh-Shub [1],

Hartman [1], Takens [2], etc.).

a mapping from a neighborhood of to

P.

~

We assume that

has

a k

~

Let

be

in a Banach manifold P O continuous derivatives and has

We further assume that

splits in-

spectral radius 1 and that the spectrum of

to a part on the unit circle and the remainder, which is at a non-zero distance from the unit circle. d~(aO)

generalized eigenspace of

Let

Then there exists a neighborhood

and a

ck- l

of

of dimension

Y

V at

submanifold

d, passing through

If

in

P

a O and tangent to

x EM

(Local Attractivity):

~(x)

and

0,1,2, ••• , then as Remark.

n ~

00,

~n(x) E V

If

E V, then

It will be a

corolla~y

so that properties (a) and (b) apply to

E~

for all

00

C

F

M can be chosen t

for all

t > O.

The Center Manifold Theorem is not

Remark.

always true for

to the proof of

is the time 1 map of

~

defined above then the center manifold

(1.12)

for all

~n(x) ~ M.

the Center Manifold Theorem that if

~

aO

M, called a center manifold for ~,

(Local Invariance):

(1.11)

Ft

of

has dimension

E M. (b)

n

V

Y

aO' such that:

(a) ~(x)

denote the

belonging to the part of

the spectrum on the unit circle; assume that d
~o),

X

as

~

0(dX~(pO»

has two isolated nonzero, simple complex conjugate eigenvalues A(~)

and

A(~)

that

d(RedA(~» ~

0(dX~(po»

such that

I

~=~

Re

O.

>

A(~)

0

(resp. > 0)

and such

Assume further that the rest of

o

remains in the left-half plane at a nonzero

distance from the imaginary axis.

Using the Center Manifold M C P, tangent to the eigen-

Theorem, we obtain a 3-manifold space of

=

A(~O),A(~O)

and to the

invariant under the flow of

~-axis

at

~

=

~O'

locally

X, and containing all the local

recurrence.

The problem is now reduced to one of a vector 2 field in two dimensions R2 + R • The Hopf Bifurcation

X: ~

Theorem in two dimensions then applies (see Section 3 for details and Figures 1.4, 1.5): (1.13)

Hopf Bifurcation Theorem for Vector Fields

(Poincar~ [1], Andronov and Witt [1], Hopf [1], Ruelle-

k Takens [1], Chafee [1], etc.). Let X be a c (k ~ 4) ~ 2 vector field on R such that X (0) = 0 for all ~ and ~ k X = (X~,O) is also C • Let dX~(O,O) have two distinct, simple* complex conjugate eigenvalues that for for

~

~

O.

> 0, Re

Then there is a

ck - 2

O.

such that

* Simple means that the generalized eigenspace (see Section 2A) of the eigenvalue is one

dimens~onal.


(xl,O,~(xl»

~

is on a closed orbit of period

2rr 1),,(0)

radius growing like such that in

~3

~(O)

O.

the flow of

> 0

(c) if

0

U

and

xl I 0

for U

of

and

(0,0,0)

is one of the above.

is a "vague attractor"*

xl I 0

for all

X

There is a neighborhood

such that any closed orbit in

Furthermore, ~(xl)

=

~,of

I

21

for

XO' then

and the orbits are attracting

(see Figures 1.4, 1.5). If, instead of a pair of conjugate eigenvalues crossing the imaginary axis, a real eigenvalue crosses the imaginary axis, two stable fixed points will branch off instead of a closed orbit, as in the ball in the hoop example.

See

Exercise 1.16. After a bifurcation to stable closed orbits has occurred, one might ask what the next bifurcation will look like.

One

can visualize an invariant 2-torus blossoming out of the closed orbit (Figure 1.14).

In fact, this phenomenon can occur.

In order to see how, we assume we have a stable closed orbit for

F~.

Associated with this orbit is a Poincar~ map.

To

define the Poincar~ map, let let

N

X be a point on the orbit, o be a codimension one manifold through transverse

to the orbit.

The Poincar~ map

P

~

takes each point

x E U,

a small neighborhood of

F~(X)

intersects

diffeomorphism from

*This

X in N, to the next point at which o (Figure 1.15). The poincar~ map is a

N U

to

V -

P~(U)

eN, with

P~(xo)

= Xo

condition is spelled out below, and is reduced to a specific hypothesis on X in Section 4A. See also Section 4C. The case in which d Re )"(~)/d~ = 0 is discussed in Section 3A. In Section 3B it is shown that "vague attractor" can be replaced by "asymptotically stable". For a discussion of what to expect generically, see Ruelle-Takens [1], Sotomayer [1], Newhouse and Palis [1] and Section 7.

22


Figure 1.14

x ----+-+........-

N

Figure 1.15 (see Section 2B for a summary of properties of the Poincar~ map).

The orbit is attracting i f

0

»

and is not attracting if there is some that

Iz I

z E

{z

I

I z I < I}

o(dP 1I (x »

o

such

k C

vector

> 1.

X : P

We assume, as above, that field on a Banach manifold We assume that

closed orbit

Y(lI)

map associated with assume that at

P

1I

with

is stable for

comes unstable at

TP

+

XlI (PO) 1I < lIO'

is a

=

0

for all

1I.

be-

and that

lIO' at which point bifurcation to a stable,

takes place. Y(lI)

Let

and let

X

P

be the Poincar~

1I

o (1I)

E Y (1I).

We further

1I = 1I1' two isolated, simple, complex con-

jugate eigenvalues

A(ll)

and

the unit circle such that rest of

c

(dP 1I (x o

o(dPll(xO(lI»)

dIA(ll)j dll

of

I

1I=1l

dPlI(xO(ll» > 0

cross

and such that the

1

remains inside the unit circle, at a

nonzero distance from it. Theorem to the map

A(ll)

We then apply the Center Manifold

P = (Pll,ll)

locally invariant 3-manifold for

to obtain, as before, a P.

The Hopf Bifurcation


23

Theorem for diffeomorphisms (in (1.14) below) then applies to yield a one parameter family of invariant, stable circles for P~

for

~

> ~l.

x~,

Under the flow of

come stable invariant 2-tori for

F~

t

these circles be-

(Figure 1.16).

-

/

I

/ N

I \

\

/

~

.

.......

- --

Figure 1.16

Hopf Bifurcatio; Theorem for Diffeomorphisms

(1.14)

(Sacker [1], Naimark [2], Ruelle-Takens [1]). be a one-parameter family of

ck

(k > 5)

Let

P~: R2

+

diffeomorphisms

satisfying:

=a

(a)

P (0)

(b)

For

~

(c)

For

~

~

< 0,

a

for all

~

a(dP (O»C {zl ~

(Il

> 0), a(dP (0» ~

simple, complex conjugate eigenvalues

I A (~) I =

that

a(dP (0» ~

1

(I A (~) I

I z'

> 1)

;\(~)

0 ~=O

•

Then (under two more "vague attractor" hypotheses which will be explained during the proof of the theorem), there is a

R2

24


continuous one parameter family of invariant attracting circles of

].l E (0,

P, one for each ].l

(1.15)

Remark.

E:)

for small

> O.

E:

In Sections 8 and 9 we will discuss how

these bifurcation theorems yielding closed orbits and invariant tori can actually be applied to the Navier-Stokes equations.

One of the principal difficulties is the smooth-

ness of the flow, which we overcome by using general smoothness results (Section 8A).

Judovich [1-11], Iooss [1-6], and

Joseph and Sattinger have used Hopf's original method for these results.

Ruelle and Takens [1] have speculated that

further bifurcations produce higher dimensional stable, invariant tori, and that the flow becomes turbulent when, as an integral curve in the space of all vector fields, it becomes trapped by a "strange attractor"

(stran';Je attractors are

shown to be abundant on k-tori for

k

~

4); see Section 9.

They can also arise spontaneously (see 4B.8 and Section 12). The question of how one can explicitly follow a fixed point through to a strange attractor is complicated and requires more research.

Important papers in this direction are

Takens [1,2], Ne'i"house [1] and Newhouse and Peixoto [1]. (1.16) Theorem. and

H

Let

].l

that the map

Define

R x H L].l

be a Hilbert space (or manifold)

a map defined for each

: H -r H

from

(a) Prove the following:

Exercise o

(].l,x) to

~ ,(x) ].l

is a

H, and for all

D'].l(O)

c

k

].l E R

map, k > 1,

].l E IR, '].l(0)

].l < O.

A (0)

=0

1,

(dA/d].l) (0)

L].l

Assume further

there is a real, simple, isolated eigenvalue such that

O.

and suppose the spectrum of

lies inside the unit circle for

of

such

A(].l)

> 0, and


25

has the eigenvalue 1 (Figure 1.17); then there is a Ck - l

curve

near

(0,0) E IH x

(0,0)

in

of fixed points of

1

iH x iR

points of (0,1.1)

iR.

(x,l.1) r>- (l.1 (x),l.1)

The curve is tangent to

(Figure L 18).

at

IH

These points and the

are the only fixed points of

in

(0,0).

a neighborhood of (b)

:

Show that the hypotheses apply to the ball

in the hoop example (see Exercise 1.2). Hint: iH x iR

Pick an eigenvector

with eigenvalue

1.

(z,O)

for

(LO'O)

in

Use the center manifold theorem

Figure 1.17

fL

unstable,

H Figure 1.18 to obtain an invariant 2-manifold the (a,l.1)

1.1

axis for on

eigenvector

C

(x,l.1) =

where

a for

C

(l.1(x),l.1).

tangent to

(z,O)

and

Choose coordinates

is the projection to the normalized Set

in


26

these coordinates.

Let

g (O:,I.l)

== f (O:,I.l)

_

1

0:

and we use the

implicit function theorem to get a curve of zeros of C.

g

in

(See Ruelle-Takens [1, p. 190]). (1.17)

Remark.

The closed orbits which appear in the

Hopf theorem need not be globally attracting, nor need they persist for large values of the parameter

I.l.

See remarks

(3A. 3) •

(1.18)

Remark.

The reduction to finite dimensions

using the center manifold theorem is analogous to the reduction to finite dimensions for stationary bifurcation theory of elliptic type equations which goes under the name "LyapunovSchmidt" theory. (1.19)

See Nirenberg [1] and Vainberg-Trenogin [1,2]. Remark.

Bifurcation to closed orhits can occur

by other mechanisms than the Hopf bifurcation. is shown an example of S. Wan.

Au

~

0

0

Fixed pOints_O

come together

Fi xed points move off axis of symme~ry and a closed orbit forms

Figure 1.19

In Figure 1.19


27

SECTION 2 THE CENTER MANIFOLD THEOREM In this section we will start to carry out the program outlined in Section 1 by proving the center manifold theorem. The general invariant manifold theorem is given in HirschPugh-Shub [1].

Most of the essential ideas are also in

Kelley [1] and a treatment with additional references is contained in Hartman [1].

However, we shall follow a proof

given by Lanford [1] which is adapted to the case at hand, and is direct and complete.

We thank Professor Lanford for

allowing us to reproduce his proof. The key job of the center manifold theorem is to enable one to reduce to a finite dimensional problem.

In the

case of the Hopf theorem, it enables a reduction to two dimensions without losing any information concerning stability. The outline of how this is done was presented in Section 1 and the details are given in Sections 3 and 4. In order to begin, the reader should recall some results about basic spectral theory of bounded linear operators by consulting Section 2A.

The proofs of Theorems 1.3


28

and 1.4 are also found there. Statement and Proof of the Center Manifold Theorem We are now ready for a proof of the center manifold theorem.

It will be given in terms of an invariant manifold ~,

for a map

not necessarily a local diffeomorphism.

Later

we shall use it to get an invariant manifold theorem for flows. Remarks on generalizations are given at the end of the proof. (2.1)

Theorem.

Center Manifold Theorem.

Let

a mapping of a neighborhood of zero in a Banach space

Z.

~

We assume that

is D~(O)

further assume that

D~(O)

the spectrum of

ck + l

, k > 1

~

be

Z

into

~(O)= O.

and that

has spectral radius

1

We

and that

splits into a part on the unit circle

and the remainder which is at a non-zero distance from the unit circle.* D~(O)

Y

Let

denote the generalized eigenspace of

belonging to the part of the spectrum on the unit circle;

assume that

Y

has dimension

d
1.

Perhaps surprisingly, this

assumption is actually rather restrictive. if

Y

real-valued

is finite-dimensional or if

Y

It holds trivially

is a Hilbert space; for

a more detailed discussion of when it holds, see Bonic and Frampton [1]. We can now state the precise theorem we are going to prove. (2.3)

Theorem.

Let the notation and assumptions be as

above.

Then there exist

{y E Y:

Ilyll

< d

e: > 0

into

X,

ck-mapping

and a

u*

from

with a second-order zero at zero,

such that a) IIYII ~

0,

Ilxll

then

.·IIB-lll·llu2-ulII0'

~

>.. yo

I I u 2 - u 1 1I •

I

J

(2.16)

Now insert estimates (2.15) and (2.16) in

to get

If we now require a

3t

=

II All (l+y>.) + >. (l+2y>.)

< 1,

will be a contraction in the supremum norm.

(2.17)*

39

THE HOPF BIFURCATION ANO ITS APPLICATIONS

ii)

The assertions we want all follow directly from

the following general result. (2.5)

Lemma.

a Banach space

Y

that, for all

Let

(un)

be a sequence of functions on

with values on a Banach space

nand

Assume

y E Y, j

O,1,2, ..• ,k,

is Lipschitz continuous with Lipschitz

and that each constant one. (un(y))

X.

Assume also that for each

y,

the sequence

converges weakly (i.e., in the weak topology on

to a unit vector a)

u

u(y).

X)

Then k th derivative

has a Lipschitz continuous

with Lipschitz constant one. b) y

and

ojun(y)

converges weakly to

oju(y)*

for all

j = 1, 2 , ••• , k • If

X, Yare finite dimensional, all the Banach space

technicalities in the statement of the proposition disappear, and the proposition becomes a straightforward consequence of the Arzela-Ascoli Theorem.

We postpone the proof for a moment,

and instead turn to step IV). IV) Ilxll

< 1

We shall prove the following:

and let

y E Y

be arbitrary.

Let

Let

(xl'Yl)

* This statement may require some interpretation. n,y,ojun(y)

x E X with 'I'(x,y) .

For each j y

is a bounded symmetric j-linear map from

to X. What we are asserting is that, for each Y'Yl' ..• 'Yj' the sequence (ojun(y) (Yl' •.• 'Yj)) of elements of X converges in the weak topology on X to oju(y) (Yl' ... 'Yj).


40

Then II xIII

~ 1

and

~ CY.·llx-u*(y) II,

Ilxl-u*(Yl) II where

CY.

is as defined in (2.17).

[ I x n -u* (Y n ) II

< cy'nll x-u* (Y)

(2.18)

By induction,

II

as

0

-+

n

-+

00,

as asserted. To prove

I Ixll

I < 1,

we first write

Xl = Ax + X(x,y),

/lxlll

~

IIAlj·llxll

+ It ~

so that

"All

+ It

< 1

by

(2.6)

To prove (2.18), we essentially have to repeat the estimates made in proving that

j7

is a contraction.

Let

Yl

the solution of

On the other hand, by the definition of Y l

=

Y l

we have

By + Y(x,y).

Subtracting these equations and proceeding exactly as in the derivation of (2.16), we get

Next, we write

xl = Ax + X(x,y). Subtracting and making the same estimates as before, we get

be

41


as desired.

This completes step IV).

Let us finish the argument by supplying the details for Lemma (2.5). Proof of Lemma (2.5) We shall give the argument only for ization to arbitrary

k

k = 1; the general-

is a straightforward induction argu-

ment. Yl'Y2 E Y

We start by choosing

and

¢ E X*

and con-

sider the sequence of real-valued functions of a real variable

From the assumptions we have made about the sequence

(un)'

it follows that lim ~n(t) = ¢(u(Yl+tY2)) n->-QO for all

t,

~n(t)

that

I~~ (t)1

- 00,

we get

~'

n j (t)

which con-

We shall temporaX(t).

We have


42

1jJ(t) =1jJ(O) + which implies that

1jJ (t)

f:

X(T) dT,

is continuously differentiable and

that 1jJ

I

(t)

X(t) •

To see that lim1jJ~(t)

= 1jJ'(t)

nT OO

(i.e., that it is not necessary to pass to a subsequence), we note that the argument we have just given shows that any subsequence of

(1jJ' (t» n

has a subsequence converging to

1jJ

I

(t) ;

this implies that the original sequence must converge to this limit. Since

we conclude that the sequence

converges in the weak topology on shall denote by only suggestive.

DU(Yl) (Y2);

we have

to a limit, which we

this notation is at this point

By passage to a limit from the correspond-

is a bounded linear mapping of norm for each

X**

< 1

from

We denote this linear operator by

yto

X **


i.e., the mapping to

~

y

Du(y)

43

is Lipschitz continuous from

Y

L(Y,X ** ). The next step is to prove that

this equation together with the norm-continuity of will imply that

u

is

(Frechet)-differentiable.

Du(y)

y~

The integral

in (2.19) may be understood as a vector-valued Riemann integral. By the first part of our argument,

for all

¢

E X**

and taking Riemann integrals commutes with

continuous linear mappings, so that

for all

¢

* EX.

Therefore (2.19) is proved.

The situation is now as follows: if we regard

u

as a mapping into

X**

We have shown that, which contains

then it is Frechet differentiable with derivative other hand, we know that want to conclude that

x. X

u

actually takes values in

t+O

DU(Yl) (Y2)

and

belongs to

u(Yl+ tY 2)-u(Yl) t

the difference quotients on the right all belong to is norm closed in

X

But ~~l~

X

On the

it is differentiable as a mapping into

This is equivalent to proving that for all

Du.

X,

X**

the proof is complete. CJ

X,

and

44


(2.6)

Remarks on the Center Manifold Theorem It may be noted that we seem to have lost some

1.

differentiability in passing from that

~

is

ck + l

to

and only concluded that

fact, however, the k th

~

u*

u* ,

since we assumed

u*

ck .

is

In

we obtain has a Lipschitz continuous

derivative, and our argument works just as well if we

~

only assume that

k th

has a Lipschitz continuous

deriva-

tive, so in this class of maps, no loss of differentiability occurs. k th

Moreover, if we make the weaker assumption that the

derivative of

~

is uniformly continuous on some neigh-

borhood of zero, we can show that the same is true of (Of course, if

X

and

u*.

Yare finite dimensional, continuity

on a neighborhood of zero implies uniform continuity on a neighborhood of zero, but this is no longer true if

X

or

Y

is infinite dimensional). 2.

As

C. Pugh has pointed out, if

~

is infinite-

ly differentiable, the center manifold cannot, in general, be taken to be infinitely differentiable. that, if

~

It is also not true

is analytic there is an analytic center manifold.

We shall give a counterexample in the context of equilibrium points of differential equations rather than fixed points of maps; cf. Theorem

2.7 below.

This example, due to Lanford,

also shows that the center manifold is not unique; cf. Exercise 2.8. Consider the system of equations: ~

0,

= -

.where zero.

h

(2.20 )

is analytic near zero and has a second-order zero at

We claim that, if

h

is not analytic in the whole com-


p1ex plane, there is no function neighborhood of

(0,0)

u(Y1'Y2)'

45

analytic in a

and vanishing to second order at

(0,0),

such that the manifold

is locally invariant under the flow induced by the differential equation near

(0,0).

To see this, we assume that we have an

invariant manifold with

Straightforward computation shows that the expansion coefficients

c,

,

are uniquely determined by the requirement of

J 1 ,J2

invariance and that (j1+ j 2) : (j1): h j1 + j2 , where h(Y1) If the series for series for

h

u(o'Y2)

= L

j.:. 2

j h 'Y1 J

has a finite radius of convergence, the diverges for all non-zero

Y2'

The system of differential equations has nevertheless many infinitely differentiable center manifolds. one, let

h(Y1)

agreeing with

To construct

be a bounded infinitely differentiable function h

on a neighborhood of zero.

Then the manifold

defined by (2.21)

is easily verified to be globally invariant for the system


46

0,

dx dt

(2.22)

and hence locally invariant at zero for the original system. (To make the expression for sketch its derivation. volve

x

u

less mysterious, we

The equations for

Yl'Y2

and are trivial to solve explicitly.

do not inA function

u

defining an invariant manifold for the modified system (2.22) must satisfy

for all

t, Yl' Y2.

The formula (2.21) for

u

is obtained

by solving this ordinary differential equation with a suitable boundary condition at 3. ~

t

=

_00.)

Often one wishes to replace the fixed point

by an invariant manifold

on a normal bundle of 9.

V.

V

0

of

and make spectral hypotheses

We shall need to do this in section

This general case follows the same procedure; details are

found in Hirsch-Pugh-Shub [lJ. The Center Manifold Theorem for Flows The center manifold theorem for maps can be used to prove a center manifold theorem for flows. time

t

We work with the

maps of the flow rather than with the vector fields

themselves because, in preparation for the Navier Stokes equations, we want to allow the vector field generating the flow to be only densely defined, but since we can often prove that the time

t-maps are

00

C

this is a reasonable hypothesis for

many partial differential equations (see tails) •

Section 8A for de-

47


(2.7) Let

z

Theorem.

Center Manifold Theorem for Flows.

and let

Ft

o < t
O.

Assume that the

spectrum of the linear semi group DF (0): Z -+ Z is of the t tal form e t (01U0 2 ) where lies on the unit circle (Le. e ta 2 lies inside the °1 lies on the imaginary axis) and e unit circle a nonzero distance from it, for is in the left half plane.

Let

Y

t> 0; i.e.

02

be the generalized eigen-

space corresponding to the part of the spectrum on the unit circle.

a

o

Assume

dim Y

d
0

(b)

If

t > 0

and

then

Ft(x) E V,

Ft(X) EM

V

for all

n

0,1,2, .•• ,

and

F~(X)

then

remains defined and in

F~(X)

-+

M as

n

-+

This way of formulating the result is the most convenient for it applies to ordinary as well as to partial differential equations, the reason is that we do not need to worry about "unboundedness" of the generator of the flow.

Instead

we have used a smoothness assumption on the flow. The center manifold theorem for maps, Theorem 2.1, applies to V

and

M

F , t > O. However, we are claiming that t can be chosen independent of t. The basic reason e~ch

48


for this is that the maps~t}

commute:

Fs

F t -- F t+s-

0

F

0 F ' where defined. However, this is somewhat overs t simplified. In the proof of the center manifold theorem we

would require the

F

to remain globally commuting after they

t

$.

have been cut off by the function

That is, we need to

ensure that in the course of proving lemma 2.4,

A can be

chosen small (independent of

are globally

t)

and the

Ft'S

defined and commute. The way to ensure this is to first cut off the Z

outside a ball

B

in such a way that the

turbed in a small ball about tity outside of of

F

t

0

00

function

C

and is

0

outside

where

T =

it is easy to see that Z

G t

which coincides with

f B.

remains a

Ft , 0

semiflow.

t

and

and are the iden-

x

which is one on a neighThen defining

(2.23)

extends to a global semiflow t on ~

t

~ T

on a neighborhood of B.

Moreover,

Gt

(For this to be true we required

the smoothness of the norm on smoothness in

T,

Z

and that for

jointly * )

t > 0

.

F

t

Now we can rescale and chop off simultaneously the outside F

t

in

are not dis-

It f(F (x))ds, o s

zero, and which is the identity outside Ck + l

-
-

-00.

Show

that the curve which is the union of the positive y-axis with any integral curve in the lower half plane is a center manifold for the flow of

X.

(see Kelley [1], p. 149).

Exercise.

(2.9)

(Assumes a knowledge of linear semi-

group theory). Consider a Hilbert space space) and let

A

Let

be a

K: H

H

->-

H

(or a "smooth" Banach

be the generator of an analytic semigroup. Ck + l

mapping and consider the evolution

equations dx dt (a) on

H

Ax + K(x),

x(O)

= X

(2.24)

o

Show that these define a local semiflow

c k +l

which is

in (t,x)

Duhamel integral equation

for

t > O.

x(t) = eAtx

o

+

Ft(x)

(Hint: Solve the

t-

f

0 e (

t-s ) AK(x(s))ds

by iteration or the implicit function theorem on a suitable space of maps (references: Segal [1], Marsden [1], Robbin [1]). (b) of

i~variant

Assume

K(O)

=

0, DK(O)

=

O.

Show the existence

manifolds for (2.24) under suitable spectral

hypotheses on

A

by using Theorem 2.7.

50


SECTION 2A SOME SPECTRAL THEORY In this section we recall quickly some relevant results in spectral theory. Schwartz [1,2].

For details, see Rudin [1] or Dunford-

Then we go on and use these to prove Theorems

1. 3 and 1.4. Let space

E

T: E

and let

{A E ~[T - AI Then I ITI

a(T)

I·

+

Let

E

be a bounded linear operator on a Banach

a(T)

denote its spectrum,

a(T) =

is not invertible on the complexification of

is non-empty, is compact, and for r(T)

r(T) = SUp{IAI

A E a(T),IAI

El
0, E

l

and

o (T ) i

E2 o .• 1.

f(T)

Let

= La

L a n zn

f(z) Tn

then

On'

Suppose

otT)

0 (f (T» = f (0 (T» •

E

i

E

=

El

e

E2

0 1 U 02

=

then there are unique such that

where

T-invariant subspaces and if

=

Ti

TIE.'

then

1.

is called the generalized eigenspace of Let

Lemma 2A.2 is done as follows: curve with

be an entire func-

o

in its interior and

O

2

0i'

be a closed

in its exterior,

then

A is not al-

Note that the eigenspace of an eigenvalue ways the same as the generalized eigenspace of

A.

In the

finite dimensional case, the generalized eigenspace of

A is

the subspace corresponding to all the Jordan blocks containing

A in the Jordan canonical form. (2A. 3)

Lemma.

Let T, 0 1 , and O 2 be as in Lemma °1 °2 2A.2 and assume that d(e , e ) > O. Then for the operator tT the generalized eigenspace of e i e tT is Ei • Proof.

Write, according to Lemma 2A.2,

E

Thus,

From this the result follows easily. (2A.4)

Lemma.

Let

T: E

+

E

[] be a bounded, linear


52

operator on a Banach space

E.

Let

r

be any number greater

than

r(T),

the spectral radius of

I

on

equivalent to the original norm such that ITI

I

E

We know that

Proof.

II Tnll

sup

Therefore,

n~a

we see that

(~~~ r sup n~a

I

E

r

n

I

:s

and let

Let

r > cr(A)

I = sUb

A: E

+

Ie

of

I tAl

I

E

i.e.

n~a

r

II Tn(x) II n r

n

be a bounded operator

A E

cr(A) , Re

on

E

A
a,

< e rt.

Note that (see Lemma 2A.l) e

etA;

sup

I}/n.

II Tn+l(x) II

n~

(i.e. if

there is an equivalent norm

Proof.

=

r.

0

rl xl .

Lemma.

I xl

a}),

then the origin is an attracting (resp. repelling) fixed point for the flow

¢t = e tT

of

T.

'!'HE POPF BIFURCA,TION AND ITS APPLICATIONS

Proof. {Z

This is immediate from Lemma 2A.5 for if

IRe z < O},

is compact.

53

there is an r < 0 with 0 (T) < r, tA Thus le I.:5 e rt .... 0 as t .... +00. D

0(T)~

as

0 (T)

Next we prove the first part of Theorem 1.4 from Section 1.

(1.4)

Theorem.

Banach manifold

P

Let

X

cl

be a

and be such that

o (dX (p 0)) C {z E 0

and that

such that

dX (0) •

=

From the local existence theory of ordinary differen-

. * , there is a tial equat10ns

r-ball about

for which the

0

time of existence is uniform if the initial condition lies in this ball.

such that

where

o

Let R(x)

Find

X

X(x) - DX(O) ·x.

=

Ilx II

2 r 2 implies

IIR(x)11

[J

(0,0) = e 2n (Re A(O))/Im A(O) _ 1

av (0 0) a)1 , av aX

av

V(O,O) = O.

is

1) I )1=0

= I

V(O ,11) - \7(0,0) )1 (0,0) = 1

m

~70)

ItO.

d(Re/()1)) )1 )1=0

(Note that this is where the hypothesis that the eigenvalues cross the imaginary axis with nonzero speed is used.)

The

rest of the lemma follows from the implicit function theorem. Step 5.

Conditions for Stability

Now let us assemble results on the derivatives of and

V

~

11

at zero. (3 • 1 0)

Lemma.

Proof.

By the way that the domain of

know that if

)1 I (0) = O. V

was'chosen, we

V(x ,ll(x )) = 0, then the orbit through l l

78


that

and

have opposite sign.

(In polar coordinates, (xl,O,~(xl»

this corresponds to the fact that the orbit of crosses the line X

n

O.

f

=

~(Yn)·

Choose a sequence of points

-TI).

Then for each

~(xn)

and

e =

x ' there is a n

neighborhood of

(0,0)

Proof.

V (0,

~'(O)

=

=

0

and

o. 0 O.

l

We already know that

av 3x

V(O,O) =

(0,0)

=

O.

l

3 ~ (0,0) = 0, we differentiate the equation 3x l

+ 3V I 3~

o

then

~(O)

(0,0)

3V

3x

(To show

is uniformly

~

2

To see that

O.

Yn < 0

is bounded in a

, we have

Yn

0)

~

Yn

Therefore, since ~(Yn)

has opposite sign to Lemma.

T(xl'~)

such that

and the fact that

continuous on bounded sets.)

( 3 • 11)

~,

By continuity of

this, one uses the fact that

Y n

(xl,~(xl

))

~"(x

1

2

and we get the equation

3

) = O.

VI

3 2

0,

If

O. 0

xl (0,0) (3.12) if

Definition.

(0,0)

is a vague attractor for

3 3 V (0,0) O. Im A (0) d~ ~=O

for small

> 0

~(O)

Since

has a local minimum at

the equation

is a vagueattractorfor

xl ~ 0.*

for small

Proof.

79

Recall that

Therefore,~" (O)

(xl,O,~(xl})

> O.

To show

is attracting, we must

show that the eigenvalues of the derivative of the Poincare map associated with this orbit are less than value (see Section 2B).

P(Xi,~(xl».

The derivative of

~I ax l .

.

(xl,l-L(x l

hood of

»

(O,O)

I

av aX l

I

P~(xl}

(xi) =

at the point

Xl

< 0.

(O,O) Th us, we nee d on 1 y s h ow

We show that the function

f(x l )

(xl'~ (Xl))

experiences a local maximum at

Xl

O.

(Xl'~ (Xl))

*This

is

1, there is a neighbor-

for

that for

av aX l

=

ap > -1.. --aX l

is

P~(xl}

~I ax 1

in which

in absolute

Clearly, the Poincare map associated

with the orbit through . (xl,O,~(xl})

Because

1

is not the most general possible statement of the theorem; see Section 3B below for a generalization.

We


80

already know thatf(O}

= ~~

I

= O.

1 (O,O) Thus,

fl (O) = O.

Therefore, fll (O) 2 d V

dj.Jdx

[,3~

(O,O)

dX

l

Thus, f (xl)

2

d V ( 0 , 0 0 dxldj.J

3 2 d V

(0,0 ) < O.

=30 Xl

(0,0) ]

l

experiences a local maximum at

0

xl

and the

orbits are attracting. 0 Step 6.

The Uniqueness of the Closed Orbits

(3.14)

Lemma.


such that any closed orbit in through one of the points Proof. that i f

N

N

of the flow of

(0,0,0)

of X

passes

(xl'O').!(x }). l


N

of

E

(x l ,x 2 ').!) ENE' then the orbit of

(O,O,O)

such

¢t through

(x ,x 2 ').!) l

crosses the xl-axis at a point

I~ll

This is true by the same argument that was used to

< E.

show the existence of E Domain (P)

).! ).!

and

dP dX l

P (xl ,).!).

I

> 0

Assume that the

nothing to prove.

N = {(xl'x2 ,).!) I (xl').!)

We choose and

such that

T(X ,).!} l

> E > 0

and

(xl').!)

is small enough so that j.J(X }. l

(xl,O,).!)

Suppose

V(x ,).!} = 0 l

for

).! E N

iff

(xl,O,).!) E y, is a closed orbit 0, then

).!

V(x ,).!} f l

O.

Then

P(x ,).!} l

> xl'


? O.

n

(P

n-l

(x

l

81

,)1)

has the same sign as P

P

n-l

n-l

(xl,jl) - P

(xl ,jl) ,

n-2

n n-l (P (xl,jl) < P (xl,jl»)

there is a nonzero lower bound on this shows that

(x ,O,)1) l

n P (Xl,jl) >

By induction

(Xl,jl).

for all T(Xl,jl)

n.

Because

for

(xl,O,jl) E N,

is not on a closed orbit of

~t.

CJ

The Hopf Theorem in Rn Now let us consider the n-dimensional case.

Reference

is made to Theorem 3.1, p. (3.15) field on

Theorem.

X

Let

)1

be a

ck+l

, k ~ 4, vector

n R , with all the assumptions of Theorem 3.1 holding

except that we assume that the rest of the spectrum is distinct from the two assumed simple eigenvalues conclusion (A) is true.

A()1) ,A()1).

Conclusion (B) is true if the rest of

the spectrum remains in the left half plane as zero.

Then

Conclusion (C) is true if, relative to

)1

crosses

A()1),A()1),

0

is a "vague attractor" in the same sense as in Theorem 3.1 and if when coordinates are chosen so that

I A (0) I o

o

-I A (0) I o (3.16)

Remarks.

o (1)

is independent of the way sponding to the

A(O),XTOf

d Xl (0) 3

2 d X (0) 3 3 d X (0) 3

The condition n R

3 A(O) ~ O(d X (O» 3

is split into a space corre-

space and a complementary one

since choosing a different complementing subspace will only

82


3 d X (Ol 3

replace

by a conjugate operator:

easily seen. The condition

(2 l n

=

3

since the matrix (3)

is automatic if dXO(O)

is real.

Further details concerning this theorem are given

in Section 4. The proof of Theorem 3.15 is obtained by combining the center manifold theorem with Theorem 3.1; i.e., we find a A(~l

center manifold tangent to the eigenspace of and apply Theorem 3.1 to this.

and

A(~)

One important point is that

in (B) of Theorem 3.1 we concluded stability of the orbit Here we are, in (B), claiming it

within the center manifold. n R

in a whole

neighborhood of the orbit.

The reason for

this is that we will be able to reduCe our problem to one in which the center manifold is the is invariant under the flow. with period

If

x ,x -plane and that plane 2 l (x,~) is on a closed orbit

T,

d¢

T,~

(xl

all

a l2

a

a

21 0

I

22

d 3 ¢T(x) 2 d 3 ¢T(x)

0

d3¢~(x)

The two-dimensional theorem will imply that the spectrum of the upper block transverse to the closed orbit is in

{z! Izi

O.

Consider the differential system of

put into a normal form as outlined above.

= ••• =

u' (0)

let ~

D

D

=

u(n-l) (0)

=

0 u(n) (0) ~ 0

u(n) (0) Re{So+i aOb }. o

Then if

n

is

there exists at least locally a one parameter

family of periodic orbits which collapse into the origin as the parameter tends to zero and the period tends to even then there are two such families in case in case

2n.

D < 0

If

n

and none

D > O. The result is very close to that of Chafee [1), dis-

cussed in Section 3A.

See also Takens [1].

is

104


SECTION 4 COMPUTATION OF THE STABILITY CONDITION Seeing if the Hopf theorem applies in any given situation is a matter of analysis of the spectrum of the linearized equations; i.e. an eigenvalue problem. normally straightforward.

This procedure is

(For partial differential equations,

consult Section 8.) It is less obvious how to determine the stability of the resulting periodic orbits.

We would now like to develop a

method which is applicable to concrete examples.

In fact we

give a specific computational algorithm which is summarized in Section 4A

below.

(Compare with similar formulas based on

Hopf's method discussed in Section SA.)

The results here are

derived from McCracken [1]. Reduction to Two Dimensions We begin by examining the reduction to two dimensions in detail. Suppose smoothly on

X: N ~,

+

T(N)

is a

ck

on a Banach manifold

vector field, depending N

such that

X

~

(a(~»

= 0

105


~,

for all zeros of

a(~)

where

x~.

O(dX~(a(~»)

is a smooth one-parameter family of

Suppose that for

C {z! Re z

~

< ~O'

a(~)

< Ol, so that X~.

fixed point of the flow of

the spectrum is an attracting

To decide whether the Hopf

Bifurcation Theorem applies, we compute

dX~(a(~».

simple, complex conjugate nonzero eigenvalues cross the imaginary axis with nonzero speed at if the rest of

o(dX~(a(~»)

A(~)

~

=

If two A(~)

and ~O

and

remains in the left-half plane

bounded away from the imaginary axis, then bifurcation to periodic orbits occurs. However, since unstable periodic orbits are observed in nature only under special conditions (see Section 7), we will be interested in knowing how to decide whether or not the resultant periodic orbits are stable.

In order to apply

Theorem 3.1, we must reduce the problem to a two dimensional one.

We assume that we are working in a chart, i.e., that

N= E, a Banach space. assume that (Xl x 2 X3 ) ~' ~' ~

~O

0

where

For notational convenience we also

and xl ~

a

and

(~)

x

2 ~

= 0

for all

Let

~.

X

~

are coordinates in theeigen-

space of

corresponding to the eigenvalues A (0) and dXO(O) 3 A (0) , and x is a coordinate in a subspace F complementary ~

to this eigenspace.

We assume that coordinates in the eigen-

space have been chosen so that

dXO(O,O,O)

o

I A (0) I

o

-I A(O) I

o

o

o

o

This can always be arranged by splitting

(4.1)

E

into the

subspaces corresponding to the splitting of the spectrum of

106


{zl Re z < O} C {A(O) ,A(O)}, as in 2A.2.

into

dXO(O)

By

the center manifold theorem there is a center manifold for the flow of

XTOf

X

(X ,0) tangent to the eigenspace of A(O) and II and to the ll-axis at the point (0,0,0,0). The center

manifold may be represented locally as the graph of a function, that is, as

{(x ,x ,f(x l ,X ,ll),ll) l 2 2

neighborhood of

(O,O,O)}.

for

(x ,X ,ll) l 2

in some

Also, f(O,O,O) = df(O,O,O) = 0

P(x ,x ,f(x l ,X 2 ,ll) ,ll) = (x ,X ,ll) is 2 2 l l a local chart for the center manifold. In a neighborhood of and the projection map

the origin

X

is tangent to the center manifold because the

center manifold is locally invariant under the flow of We consider the push forward of

X.

X: X(x ,X ,ll) l 2

1

TPoX(x l ,x 2 ,f(xl'X 2 ,ll),ll) = (Xll(xl,x2,f(xl,x2,ll», 2

Xll(xl,x2,f(xl,x2,ll»,0)

by linearity of

P.

If we let

" 1 2 X (x l ,x 2 ) = (X (x l ,x 2 ,f(x l ,X 2 ,ll», Xll(xl,x2,f(xl,x2,ll»)), U

" X is a smooth one-parameter family of vector fields on II " such that (0,0) = 0 for all ll. We will show that X II II satisfies the conditions (except, of course, the stability

x

condition) of the Hopf Bifurcation Theorem. are the flows of

X

and

" X

respectively, then

for points on the center manifold. sultant closed orbits of will not be either.

¢t

¢t PO¢t

~t

and

=

" ¢tOP

Therefore, if the re-

are not attracting, those of

¢t

We will also show that if the origin is

a vague attractor for of

" ¢t

If

" ¢t

at

II

=

0, then the closed orbits

are attracting. Since the center manifold has the property that it

contains all the local recurrence of

¢t' the points

(O,O,O'll)

are on it for small II and so f(O,O,ll) = 0 for small ll. " 1 2 Thus, X (0,0) = (X (O,O,f(O,O'll»' X (O,O,f(O,O'll») = II II (X1 (0,0,0), X 2 (0,0,0» = o. " poX Xop on the center manifold,

107


so

" PodX = dXoP

for vectors tangent to the center manifold.

A

typical tangent vector to the center manifold has the form (u,v,dlf(xl,x2'~)u + d2f(xl,x2'~)v (xl,x2,f(xl,x2'~)'~)

+ d3f(xl,x2'~)w,w), where

is the base point of the vector.

Because

we wish to calculate

a(dX (0,0», we will be interested in

the case

podX(O,O,O,~) (u,v,dlf(O,O,~)u

=

O.

dl(O,O,~)v,O)

=

dX(O,O,~} (u,v,O).

d2f(0,0,~)v)

Since

=

dX (0,0) . ~

Now

~

w

i

dX~(O,O)

(u,v)

That is, for

i

+

~(O,O,O) (u,v,dlf(O,O,~)u +

= 1,2.

A

A Ea(dX~(O,O».

Let

is a two-by-two matrix, A is an eigenvalue A

and there is a complex vector

(AU,AV). and

We will show that

(u,v,dlf(O,O,~)u

cause

X

+

(u,v)

such that

dX~

A is a eigenvalue of

d2f(0,0,~)v)

(0,0) (u,v) = dX~(O,O,O)

is an eigenvector.

Be-

is tangent to the center manifold,

Therefore,

dldlf(xl,x2'~)oX

+

dlf(xl,x2,~)odlX

+

dlf(xl,x2,~)od3X

1

1 1

(xl'x2,f(xl'x2'~) ,~)u (xl,x2,f(xl,x2'~)'~}u

(xl,x2,f(xl,x2,~},~}odlf(xl,x2'~)u

2

+ dld2f(xl,x2'~}X (xl,x2,f(xl,x2'~)'~)u

and

2

+

d2f(xl'x2'~)

odlX (xl'x2,f(xl,x2'~),~)u

+

. 2 d2f(xl,x2,~)od3X (xl,x2,f(xl,x2,~),~)odlf(xl,x2'~)u

108


3 + d 3 X (Xl ,x 2 ,f (xl ,x 2 ,\l)

,\l)

°d 2 f (xl ,x 2 ,\l)v

1

d d f(x l ,X ,\l)oX (x ,x ,f(x ,X ,\l),\l)v l 2 l 2 l 2 2

+ d l f(x l ,X 2 ,\l)od 2 X1 (x l ,x 2 ,f(x ,X 2 ,\l),\l)v + l + dlf(Xl,X2,\l)Od3Xl(Xl,X2,f(Xl,X2,\l) ,\l)od 2 f(x l ,X 2 ,\l)v + d 2 d 2 f(x l ,X 2 ,\l)X 2 (x l ,x 2 ,f(x ,X ,\l),\l)v l 2

+ d 2 f(x l ,X 2 ,\l)od 2 X2 (x l ,x 2 ,f(x l ,X ,\l),\l)v 2 2

+ d 2 f (xl ,X 2 ,\l) od 3 X (xl ,x 2 ' f (xl ,X 2 ,\l) At the point

(O,O,O,\l),

X

1

= X

2

°

,\l)

od 2 f (xl ,X 2 ,\l) v.

and so we get

3

dX (O,O,O,\l) (u,v,dlf(O,O,\l)u + d f(O,0,\l)v) 2 3 3 · 3 dlX (O,O,O,\l)u + d 2 X (O,O,O,\l)v + d 3 X (O,O,O,\l)odlf(O,O,\l)u

+ d 3 X3 (O,0,0,\l)od 2 f(O,0,\l)v 1

1

dlf(O,O,\l)o(dlX (O,O,O,\l)u + d 2 X (O,O,O,\l)v 1

1

+ d 3 X (O,O,O,\l)odlf(O,O,\l)u + d 3 X (O,0,0,\l)od 2 f(O,0,V)v) 2 2 + d f(O,0,\l)o(d X (O,O,O,\l)u + d X (O,O,O,\l)v 2 2 l 2 2 + d 3 X (O,O,O,\l)Odlf(O,O,\l)u + d 3 X (O,0,0,\l)Od 2 f(O,0,\l)v)

by the assumption that

with eigenvalue When

\l

=

(U,V)

is an eigenvector of

\. 0, df

°

and we have that

dX \l (0,0)


"-

The eigenvalues of

dX (0,0) \1

are continuous in

they are roots of a quadratic polynomial.

a

a 2 (\1)

and

(\1)

l

a

Because and

a 2 (\1)

l

(\1)

t \ (\1)

2

then

,

(0) = \ (0)

l

Re a

and

\(\1)

V'"

~t

~

hood

of

(\1) t \(\1)

l

(\1) = \(\1) \(\1)

and

are simple

{\(\1),\(\1)}

at the

< 0

The map

X, then the closed

Q(x ,x ,x .\1) = 1 2 3

onto a neighborhood

~

to the center manifold

V

for

M

QI M = P, xl{x=o} = X, and 3

where

"-

¢tl

{x =O} 3

is invariant under

Y Y

\1 \1

fies the conditions for Hopf Bifurcation with eigenspace of manifold for Y

Y

\(\1) y

{x =0}

~

and

is

at

and let the point

3

d~T(X ) (x l ,O,\1(x l

1

»

~t

of

au

a

a

a

2l 0

is tangent

Y.

~.

E

Therefore,

a vector field and

R

2

Y\1

satis-

being the

The center

V'" (0)

Assume that

< 0

(x ,O,O,\1(X » be on a l l 2 Because R is invariant, a

12

a

22 0

= ~t·

(0,0,0,\1).

{(x ,x ,0,\1)}. 1 2

closed orbit of the flow

X

(0,0,0,0)

(x ,x ,f(x ,X ,\1) ,\1) l 2 2 l

"-

Ell F

of

small enough so that

we are immediately reduced to the case of 2

"-

for

is a diffeomorphism from a neighbor-

(0,0,0,0)

where we have chosen

Clearly

(0)

are attracting.

(x ,x ,x -f(xl'x ,\1),\1) 2 3 2 l

for

a

XTOf.

(0,0,0,\1).

orbits of

R

(0) =

2

O(dX\1(O,O,O», we must have that the center

We show now that if

on

l

and

manifold is tangent to the eigenspace of point

a

would be bounded away from

(\1)

i

Furthermore, since

eigenvalues of

Let these roots be

Since this is not true, a

\1.

(\1) = \(\1).

a

because

\1

a 2 (\1) E o(dX\1(O,O,O», i f

and

zero for small a

so that

109

3

13 23

d3~T(X ) (xl'0,0,\1(x 1

1

»

110


By assumption,

3 (d 3 4>T(0) (0,0,0,0»

3

e

a{T(0)d X (0,0,0» 3

is inside the unit circle.

V'" (0)

Since

so is

2 R

eigenvalues of the Poincare map in less than

By continuity,

db t ]

I

T (xl) + dX

T' (Xl)

+

I

:::it~

113

114


In the case equation.

xl = 0, we can considerably simplify this

We know the following about the point

=

at(O,O)

bt(O,O)

aa

d~t(O,O)

e

at

for all

(0,0»

tdX (0,0)

T(O)

=

(0,0)

" dX(O,O)

A

t

0

ab t

t at (0,0)

dX(at(O,O), b

=

(0,0): (4.5)

t

0

(4.6)

I'~o)

[-1':0)1

I]

[ cos I A(0) It

sinIA(O) It]

-sinIA(O) It

COSIA(O) It

(4.7)

(4.8)

(4.9)

21T/IA(0) I

and (4.10)

T' (0) = O.

Proof of (4.10) • S' (0)

=

S(x)

=

S (y) •

have opposite signs.

= oX ~T

S (Xl)

because given small

0

such that

S' (0)

Let

Thus,

Choosing

(0,0) + )1' (0) l

=

T(x l ,)1(x l

»·

Then

x > 0, there is a small

-

S(x)

S(O)

x x

n

+

~T)1 (0,0).

and

S(y)

-

y < 0 S(O)

y

0, we get the result. But

)1' (0)

=

0, as was

shown in the Proof of Theorem 3.1 (see p. 65).

Therefore, V'"

d 3X

We now evaluate

21T / IA (0)

(0)

fo

I a 3 x" 1

1 --3-

dX

l

3 dX l

(at (xl,O) ,b (xl,O) )dt. t

and get: at (0,0) ,b (0,0) t


21T/IA(0)

v' "

Io

(0)

-

3

a3

I ~3"" ax aa

x2 l

3 3 (O,O)eos IA(O) It -

aab3

115

x3 (O,O)sin 3 IA(O) It

2 (O,O)eos IA(O) It sinIA(O) It

aa ab

a3 xl

2

+ 3 aaab 2 (O,O)eosIA(O) It sin IA(O) It

a2

xl

2

---- (0,0) o~a2

a at aX l

---2- (O,O)eosIA(O) It

a at aXl

2 - -2sinIA(O) It]

+

I A (0) I

In order to get a formula for derivatives of

....

X

V'" (0)

depending only on the

at the origin, we must evaluate the

derivatives of the flow

(e.g. ,

(0,0) )

from those of

.... X

116

at


This can be done because the origin is a fixed

(0,0).

" X.

point of the flow of

Because this idea is important, we

state it in a more general case. (4.1) such that

Theorem.

=

X(O)

Let (~

0

X X(p)

c k vector field on

be a

=

~t

Let

0).

be the time

map of the flow of

X.

The first three (or, the first

derivatives of

at

0

~t

three (or, the first

Proof.

Rn t

j)

can be calculated from the first

j) derivatives of

X

at

O.

Consider

Hi

because

~t(O)

=

O.

for all equation

aat

Furthermore, x.

(d~t(O)) =

Thus, d~t(O) = etdX(O). a2~i

t

So

d~t(O)

dX(O).

ax~

(0)

=

0ij

because

J

satisfies the differential

d~t(O)

and

d~O(O)

= I.


Furthermore,

We get the differential equation:

The solution is:

J:e-

2 i Cl X ClxpClxR,

2 i Cl X + ClxpClXR,

0

o

sdX

2 (0)d X(0) [::; (0),

:~

(0) )dS.

Cl 3 <jli ."....-~_t""""'"""_ (0): ClxjClXkClX h

Finally consider

+

117

<jl

[ ,',Pt ,,'

~+

t ClXjClX h ClX k

Cl<jlP t ClX j

ClP Cl 2 <jlR, i t ClX t + -<jlt ClX ClxR, h ClxjClx k

,',' ]

ClXkCl~h o

<jlt

a'o'J

t (0) ClxjClxkClX h

118


t dcj>q dcj>P H 3 i d X t t ( 0) dX t (0) dX. (0) dX (0) dX dXtdX h k P q J +

+

a2xi dXpdX t

[ ,2.p (0)

dXjd~h

d 2 cj>t H P t t (0) dX dXjdX k h

t d 2 cj>t H HP t t (0) + t (0) (0) dX (0) dX dXkdX j h k

(0)]

i + dX dX

o. equation:

and

The solution is: d 3 cj>

t (0) dXjdXkdX h

t

(0)

d 3 cj>t t (0) • dXjdXkdX h

We get the differential


In the case we are considering, dX(O,O)

A

and so

d¢t(O,O)

d

2

=

cos/A(O) It

(

-sinl A(0) It

¢

~ (0,0)

calculate

dX

=

l

2a [d---2t dX

d~S

A

l

Consequently,

(0)]

Also,

2

=

[ddxiX1 (O,O)cos 2 IA(O) Is

l

2 2

+

e-sdX(O)

-SinIA(O)ls]. cosIA(O) Is

d 2X(O) [d¢S dX (0), dX

(0,0),

d X (O,O)sin2 IA(O) Is ] • ---2-dX 2

/A. (0)

0 [

-I A (0)

/

We wish to

cosIA(O) It

l

First note that

_ [COSIA(O) /s - sin IA(0) I s

sin/A(O) It].

119

and

0

I}

120


and thus

2 2 2 1 d 2 ;{2 d 3 X ]. 3 2 - + 2 dX dx + - 2 - cos [\(0) + [- dX 2 dX 1 1 2

X

2 1 d [2 dX 2

X

-2- -

d2 2

X

2 dX dx

1

It,

2 l 32 X-] + -

2

dX 1


(where all derivatives are evaluated at the origin) . Putting this all together,

]

(0) ds =

2

d at

-y-

[dXl

2]

d bt

(0), -2- (0)

dX l

1

31 A (0) I

x

,'Xl]

2 d2 X d2 l 3 + [- -dX2 - + 2 dX dx + -dX2 - coslA(O) It sin lA(O) It l 2 2 l

x

,'Xl]

2 l 2 2 d d X + 2 dX dX - - 2 - sinlA(O) It cos 3 11..(0) It + [ dX l 2 dX l 2 --2-

121

122


"']

2 2 2 3 3 + 2 2 3x 3x + 3 ; + [- 3x 3x 1 2 2 l

Xl

X

4 cos IA(0)

2 2 2 l 3 3 - 2 3x 3x + --2- sin IA(O) --2+ [- 3x 2 3x 1 l 2

X

[,

X

3 2 }(l -2- 3x 2

"X'] , 2-1]

COSIA(O)

It

x

"']

sinIA(O)

It

2 l

X

3 d + [, --2- + 2 3x 3x + 3 ; 3x 3x 1 2 2 l

+

[3

It,

X

2 2 3 2 3x 3x + 3 ; 1 2 3X l

2 2

It

2 2 3 --2- - 3 - 2 - sinIA(O) It cos I A (0) It 3x 3x 2 l

X

"X']

2 l 2 l 2 3 2 3 - 3 - 2 - sin IA(0) It - 3 -2-cOS IA(O)lt 3x l 3x 2

X

X

3 cos IA(O) It

+

[32~.1 3x

_

l

3 3 b

Before computing

t

--3- (0), which is a lengthy calcula-

3x

l

tion, we will use the information above to simplify our expression for

V'II

(0).

To do this, we must compute


t1T/IA(O) I

0

I: I:

2

a at :-z aX 1

(O,O)COSIA(O) It dt,

2

1T /

I A (0) I a at :-z aX 1

(O,O)sinIA(O) It dt,

1T /

a2b t I A (0) I --2aX l

(O,O)COSIA(O) It dt,

a2b t -2aX l

(O,O)sinIA(O) It dt.

and

f:1T/1A(O)

I

The results are:

21T/IA(O)

Io

21T/IA(O)

Io

I a2at -2aXl I

1T (O,O)eosIA(O) It dt =

2

d at -2-- (O,O)sinIA(O) It dt

aXl

21T/\A(O)

I. o

2 3/A(O) I

I a2bt

=

.

1T 2 3IA(O) / 1T

'_.-2- (O,O)eos/A(O) It dt =

aX1

21T/'A(O)

1o

I a2bt

---2-- (O,O)SinIA(O) Itdt =

aX1

2 3/A(O) /

2

IT

31A(Oli

Therefore,

V'"

(0)

(O,O)dt

2 a at - 2 - (O,O)cosIA(O) It

dX

l

123

124


Le.

(where all derivatives are taken at the origin).

()3 b

To compute

~ (0,0), we use the equation: ()x

l


125

The calculation involved is quite long,* but is straightforward, so we will merely indicate how it is done and then state the results.

I:

'IT/I

A(0) 1e tdX ( 0 )

+

+

It

["

x x

2 'IT

x

x x

41

a2 <j>

s e- sdX (0)d 2 X(0) ~ (0,0) , -2(0, 0) ] d, dt o aX l aX l

a2 _1 _ a2 _2 + a2 1 _ 5 'IT -241 A(0) /3 aX l I A(0) 13 aX l2 aX 22 a2 _1 _ a2 _2 + a2 1 a2 1 _ 7 'IT 5 'IT -22 aX 2 3 A (0) 1 aX l 41 A(0) 1 ax l ax 2 aX 2 l

[,orne thing ,

+

The lengthy computation alluded to is:

3 'IT

41 A(0)

1

3

a2x2 a25{ 2 ax l ax 2 -=-r aX l

x

'IT

3

1ft. (0) 1

a25{_1 _ a2 _2 + _ aX 22 aX l2

x

41 A(0) 3 'IT

41 A(0)

1

x

a2x1 a2 5{ 2

5 'IT

+

a2x1 ax l ax 2

3

1

3

-=-r aX -=-r aX 2

2

,'i 2 a'i 2 J ax l ax 2 ax~ ;

(at

(0,0) )

::~

::~

(0,0) ]dS dt

and one easily sees that

(0,0),

(0,0),

The final result of the computations is, therefore,

*We

are not joking! One has to be prepared to shack up with the previous calculations for several days. Details will be sent only on serious request.

126


+

2,,2 [2 _a_ 2X1 (0,0) a x (0,0) + a2xI (0,0) a2~ (0,0) 3 2 ;Z aa IA(0)1 1f

aT

alT

x

2 l 2"1 2"2 a2}{2 a + ~ a; (0,0) aaab (0,0) + ~~ (0,0) aaab (0,0) 4 ab2 aa 2"2 2"2 2}{1 2}{1 (0,0) + f~ (0,0) a; (0,0) + ~~ (0,0) _a_ 4 ab2 ab2 aa aa

x

x

2 2"1 5 a l 3 a2 2 a2~ } + '4 aaab (0,0) a x (0,0) + '4 aaab (0,0) -2- (0,0) • aa

;z

Thus we get our formula:

(4.2) V'"

Formula (0) =

37T 41 A (0)

I

x

3 l [ 3,,1 (0,0) a ~ (0,0) + a 2 aa aaab

x

x

3 2 3 2 a (0,0) + - - (0,0) } +a 2 3 aa ab ab

+

31f 2

4IA(0) 1

(-

x

2 l a 2 aa

x

x

x

x

(0,0)

x

2 l a (0,0) aaab

2 2 2 2 a (0,0) +a-2- (0,0) aaab ab 2 2 2 l 2 2 a a a + -2- (0,0) aaab (0,0) - .ab 2 aa

x

2 l a2~2 +a-2- (0,0) 2 (0,0) aa aa

x

- aba x2

x

(0,0)

l a2 (0,0) aaab

(0,0)

2 2 a 2 (0,0) ] • ab

2 1

x


In two dimensions, where directly to test stability if

X

127

" this can be used X,

=

dXO(O,O)

is in the form on

108.

p.

Expressing the Stability Condition in Terms of

X.

We now use the equation

~(Xl,X2,f(Xl,X2'~)) = dlf(xl,x2'~) +

d2f(xl,x2'~)

near

respect to

X

o ~

at

0

Xl(Xl,X2,f(Xl'X2'~») ~(Xl'X2,f(Xl,X2'~))

(0,0,0,0)

to compute the derivatives of those of

0

(0,0,0).

" X

o

at

in terms of

Since no differentiation with

occurs, we drop all reference to

differentiating this with respect to

and

(0,0)

xl

and to

~.

Upon

x ' we get: 2

128


We now differentiate the first equation with respect to and the second with respect to

and

three expressions.

df

o

and

dX

xl

=

x

2

=

0, where

o

1\ (0) I

-I \ (0) I o

o

o

This procedure yields

3 3 dldlX (0,0,0) + d 3 X (0,0,0) 3 3 d d X (0,0,0) + d X (0,0,0) l 2 3

i.e. 3 d X (0,0,0) 3 -1\(0) 0

I

We get

These expressions are easy to write down

and to evaluate at the point

f

only.

xl

21\(0)

I

3 d X (0,0,0) 3 -21\(0)

0 1\(0)

dldlf(O,O)

I

I d3~ (0,0,0)

dldl(O,O) d d f(O,O) 2 2

o o d

3

x3

129


3 -dldlX (0,0,0) 3

-d l d X (0,0,0) 2

-d2d2~(0,0,0) See formula (4A.6) on p. 134 tained by inverting the

for the expression for

3 x 3

matrix on the left. 3

that since the determinant is

{zlRe z < 3

d 3 X (0,0,0) -211.(0) Ii)

a(d 3 X (0,0,0»

O} U {A(O),A(O)} 2

+ 411.(0)

1

2

=

(d X (0,0,0)

3

c

2

3

+

a(dX(O,O,O» C

implies both

d 3 X3 (0,0,0)

3

and 3

(d 3 X (0,0,0) + 211.(0) Ii) (d X (0,0,0) 3

are invertible, the matrix is invertible.

Finally we must compute the first three derivatives of at

(0,0)

ob-

Note

3

d X (0,0,0)

3

411.(0) ,2), and since

didjf

in terms of those of

X

o at

(0,0,0).

Remember

that 1,2. Therefore, i

djX (x ,x 2 ,f(x l ,x 2 ) l

+ d Xi (x ,x ,f(x l ,x 2 » 3 l 2 for

i,j = 1,2.

0

11. (0)1]

So

[

-I A (0) I

Differentiating again, we get:

0

0

d j f(x ,x 2 ) l

Xo

130


~djxi(Xl,X2)

=

~dj~(Xl,X2,f(Xl,X2))

+

d3dj~(Xl'X2,f(Xl,X2))

+

~d3X

+

d3d3~(Xl,X2,f(Xl,X2))

i

(Xl ,x2 ,f(Xl ,X2 ))

i

+ d 3X (X l ,x2 ,f(Xl ,X 2)) Evaluating at

t

=

0

0

0

0

~f(Xl,X2) d j f(X l ,X2 ))

~f(Xl,X2)

0

d j f(X l ,X2 )

~djf(Xl,X2)'

i,j,k

0, we get: i, j,k

We differentiate once more at evaluate at

d~~djXi(o,o)

= 1,2.

=

d~~dj~(O,O,O) + d3dj~(0,0,0)

+

~d3~(0,0,0)

0

1,2.

0:

0

d~~f(O,O)

d~djf(O,O)

This can be inserted into the previous results to give an explicit expression for

V'"

(0)

on p.

Below in Section 4A, we shall summarize the results algorithmically so that this proof need not be traced through, and in Section 4B examples and exercises illustrating the method are given. (4.3)

Exercise.

In Exercise 1.16 make a stability

analysis for the pair of bifurcated fixed points. proof, write

f(a,O) = a + Aa 3 + •••

stability if

A < 0

and show that we have

and supercritical bifurcation and in-

stability with subcritical bifurcation if explicit formula for example.

(Reference:

In that

A

A > O.

Develop an

and apply it to the ball in the hoop

Ruelle-Takens [1], p. 189-191.)

131


SECTION 4A HOW TO USE THE STABILITY FORMULA; AN ALGORITHM The above calculations are admittedly a little long, but they are not difficult.

Here we shall summarize the re-

suIts of the calculation in the form of a specific algorithm that can be followed for any given vector field.

In the two

dimensional case the algorithm ends rather quickly. longer~

aI, it is much

In gener-

Examples will be given in Section 4B

following. Stability is determined by the sign of object is to calculate this number.

V"

I

(0), so our

We assume there is no

difficulty in calculating the spectrum of the linearized problem. Before stating the procedure for calculation of V'"

(0), let us recall the set up and the overall operation. Let

Banach space

X: E

+

.E

(if

~

E

be a X~

ck

(k > 5)

-

vector field on a

is a vector field on a manifold, one

must work in a chart to compute the stability condition). Assume

x~(a(~»

=

0

for all

~

and let the spectrum of

132


dX~(a(~))

For

satisfy:

~ 0, we look at and

A(0) = i.

dXO(O,O) =

(-~ ~J

Recall that to use the stability formula

developed in the above section we must choose coordinates so that a I m AO(O)] [ -1m A(O)

dXO(O,O)

Thus, the original coordinates are appropriate to the calculation; an example where this is not true will be given below. We calculate the partials of

anx

~

axjayn-j since

X

o at

(0,0)

a

(0,0)

for all

up to order three:

n > 1

Xl (x,y) = y. 2 a x 2 -2- (0,0) = 0, aX l

a3x 2 -3- (0,0) aXl Thus, V"'(O)

a2X:2 ax ax l 2

(0,0) = 0,

2 a x 2 -2- (0,0) aX 2

0,

3 3 3 a x a x a x 2 2 2 (0,0) = -6. 0, - 2 - (0,0) = 0, - - 2 (0,0) = 0, -3aX ax ax ax ax l 2 l 2 2

=~

(-6) < 0, so the periodic orbits are attract-

ing and bifurcation takes place above criticality.

138


(4B.2)

Example.

On

consider the vector field

3 2 (x+y,-x -x Y+(1l-2)x+(1l-1)y).

XIl(x,y)

=

x (0,0) 11

0

and dXIl(O,O) = (1 11-2 11 ±

The eigenvalues are

1 ) 11-1

~

-1 < 11 < 1,

Let

2

then the conditions for Hopf Bifurcation to occur at are fulfilled with

A(O)

3 2 (x+y,-x _x y_2x_y). required form.

e2

vectors

1.

e

l for finding

=

(_~

XO(x,y)

-i),

=

0

=

which is not in the

We must make a change of coordinates so

so that A

Consider

dXO(O,O) =

becomes

dXO(O,O) and

=

11

th~t

That is, we must find vectors

A

e

= -e 2 and dX o (0,0)e 2 = €l. The e = (0,1) will do. (A procedure 2

dXo(O,O)e l

(1,-1)

and

and

A

is to find

eigenvectors; we may then take

e

l

=

a

and a +

a

the complex

and

e

2

=

i(a-a).

See Section 4, Step 1 for details.) X (xe +ye ) = XO(x,y-x) l 2 O 3 2 2" 2 (y,-x -x (y-x)-2x-(y-x» = (y,-x y-x-y) = ye l + (-x y-x)e • 2 Therefore in the new coordinate system, Xo(x,y) = (y,-x 2 y-x). A

o 2 a x al a3x ax3

2

(0,0)

2 (0,0)

a2x 0,

l

for all

2 (0,0) = 0, axay

2 a x

n > 1.

2 (0,0)

0,

al

3 3 a x a x a3x 2 2 (0,0) 2 0, - 2 - (0,0) = -2, - - 2 (0,0) = 0, ay3 ax ay axay

O.

139


Therefore, V"

I

37f (0) = 41 A(0)

I

(-2) < 0.

The orbits are stable and

bifurcation takes place above criticality. (4B.3)

Example.

The van der Pol Equation 2 2 dx The van der Pol equation d x2 + )leX -l)crr + x = dt is important in the theory of the vacuum tube. (See Minorsky

°

[1], LaSalle-Lefschetz [1] for details.) for all

)l

As is well known,

> 0, there is a stable oscillation for the solution

of this equation.

It is easy to check that the eigenvalue

conditions for the Hopf Bifurcation theorem are met so that bifurcation occurs at the right for )l

°

=

all

=

)l

0.

However, if

the equation is a linear rotation, so

n.

For

)l

=

°

v(n) (0)

for

0, all circles centered at the origin are

closed orbits of the flow.

By uniqueness, these are the

closed orbits given by the Hopf Theorem.

Thus, we cannot use

the Hopf Theorem on the problem as stated here to get the existence of stable oscillations for

)l

> 0.

In fact, the

closed orbits bifurcate off the circle of radius two (see LaSalle-Lefschetz [1], p. 190 for a picture).

In order to

obtain them from the Hopf Theorem one needs to make a change of coordinates bringing the circle of radius 2 into the origin. u" + f(u)u ' + g(u)

In fact the general van der Pol equation

=

can be transformed into the general Lienard equation x'

=

y - F(x), y'

=

-g(x)

by means of

x

=

u, y

=

u ' + F(u).

This change of coordinates reduces the present example to 4B.l.

See

Brauer~Nohel

[1, p. 219 ff.] for general information

on these matters.) (4B.4)

2

Example.

2

On

3 R , let

2 ()lx+y+6x ,-x+)lY+Yz, ()l -l)y-x-z+x).

X)l(x,y,z) = Then

X)l(O,O.O)

°

and

°

THE HOPF BIFU~CATION AND ITS APPLICATIONS

140

l.l -1

dXl.l(O,O,O)

For

0

l.l

0

2

-1 l.l ± i.

1

l.l -1

which has eigenvalues

-1

0, the eigenspace of

l.l

and

-1

is spanned by

{(1,0,-1), (0,1,0)}.

is spanned by

(0,0,1).

+" -~

The

for

dXO(O,O,O)

complementa~y

subspace

With respect to this basis X (x,y,z) II 2 (llX+y+6x ,-X+lly+yz,llx+ll Y-z+x ). We now compute the stability 2

2

condition.

I

\A(O)

= 1

d3X~(0,0,0) =

and

2

3

dldlf(O,O)

2

d d f(0,0) l 2

-1

-1

1

d d f(0,0) 2 2

2

-2

3

Al

d~djdkXO(O,O)

0

A2 dldldlXO(O,O)

0

A2 d d d X (O,0) l l 2 O A2 d d d X (O,0) l 2 2 O Therefore,

V'" (0)

31T

~

1

5

-1.

2

6/5

0

-2/5

0

4/5

1· (6/5)

6/5

3'1·4/5

12/5.

(6/5 + 12/5) > 0, so the orbits are un-

stable. The next two exercises discuss some easy two dimensional examples. (4B.5) B(X,y) llO = 0

=

Exercise.

Let

222 2 (ax +cy ,dx +fy )

X(x,y) = A (x) + B(x,y) II y and

A

1)

II •

where

Show that

is a bifurcation point and that a stable periodic

orbit develops for

l.l > 0

provided that

cf > ad.

(This

example is a two dimensional prototype of the Navier-Stokes equations; note that

X

is linear plus quadratic.)


(4B.6) in

2 R

Exercise (see Arnold [2]).

using complex notation.

141

z = z(iw+~+czz)

Let

Show that bifurcation to

periodic orbits takes place at

z

=

~

=

O.

Show that these

c < O.

orbits are stable if

For some additional easy two dimensional examples, see Minorsky [1], p. 173-177.

These include an oscillator in-

stability of an amplifier in electric circuit theory and oscillations of ships.

x+

sin x + ex

=M

The reader can also study the example

for a pendulum with small friction and

being acted on by a torque

M.

See Arnold [1], p. 94 and

Andronov-Chailkin [1]. The following is a fairly simple three-dimensional exercise to warm the reader up for the following example. (4B.7) 3

Exercise.

X (x,y,z,w) = ~

(~x+y+z-w,-x+~y,

Show that bifurcation to attracting closed orbits'

-z,-w+y ).

takes place at (Answer:

Let

V'" (0)

(x,y,z,w) =

=

(0,0,0,0)

and

~

=

O.

-91T/4).

The following example, the most intricate one we shall discuss, has many interesting features.

For example, change

in the physical parameters can alter the bifurcation from sub to supercritical; in the first case, complicated "Lorenz attractors" (see Section 12) appear in plcace of closed orbits. (4B.8)

Example (suggested by J.A. Yorke and D. Ruelle).

The Lorenz Equations (see Lorenz [1]). The Lorenz equations are an idealization of the equations of motion of the fluid in a layer of uniform depth when the temperature difference between the top and the bottom is maintained at a constant value.

The equations are

142


dx at

-ax + ay

dE ~t

-xz + rx - y

dz dt

xy - bz.

Lorenz (1] says that"

x is proportional to the intensity

of the convective motion, while

y

is proportional to the

temperature difference between the ascending and descending currents, similar signs of

x

and

y

denoting that warm fluid

is rising and cold fluid is descending.

The variable

z

is

proportional to the distortion of the vertical temperature profile from linearity, a positive value indicating that the

a

strongest gradients occur near the boundaries".

K

the Prandtl number, where expansion and

v

=

K-IV

is

is the coefficient of thermal

is the viscosity; r, the Rayleigh number, is

the bifurcation parameter. r > 1, the system has a pair of fixed points at

For x

=y =

±/b(r-l), z

=

r - 1.

field at the fixed point

a

x

The linearization of the vector

=

y

=

+/b(r-l), z

-1

-/b(:-lll·

Ib (r-l)

-b

=r

- 1

is

dX (/b(r-l) ,/b(r-l) ,r-l). r

The characteristic polynomial of this matrix is x

3

+ (a+b+l)x

2

+ (r+a)bx + 2ab(r-l)

0,

which has one negative and two complex conjugate roots.

For

a > b + 1, a Hopf bifurcation occurs at

We

r - er(a+b+3) - (er-b-l) •

shall now prove this and determine the stability.


143

Let the characteristic polynomial be written (X-A) (x-X) (x-a) = 0, where

A

Le. Clearly, this has two pure imaginary roots iff the product of the coefficients of

x

2

and

x

equals the constant term.

That is, iff

or

Thus, we have the bifurcation value. Ai(r

o).

r

o(o+b+3) (a-b-l)

o=

We now wish to calculate

Equating coefficients of like powers of

x, we get

(o+b+l) (r+a}b and

2 Thus, a = -(a+b+l+2A ) and (r+a}ba 2A l a - 2Ab(r-l), so l that -(a+b+l+2A ) (r+a)b = -2ab(r-l) + 2A (O+b+l+2A )2 l l l Differentiating with respect to calling that

r, setting

r= r

' and reO

Al(r O) = 0, we obtain b(a-b-l} > 0 2[b(r +o} + (a+b+l)2]

for

a > b + 1.

O

Thus, the eigenvalues cross the imaginary axis with nonzero speed, so a Hopf bifurcation occurs at

V"'(O)

We will compute

for arbitrary

it at the physically significant values

is minus the coefficient of is the coefficient of

at Following for

R3

in which

I(A)

a,b

_ a(a+b+3) rOa-b-l

and will evaluate

a = 10, b = 8/3.

x 2 , so x, so

At

a = -(a+b+l)i and

I AI2

=

2ab(a+l) a-b-l

of Section 4A, must compute a basis

•

144


o

-0

1

M

-~ o

-1

-b

becomes

o

/

2cr b (cr+l) cr-b-l

o

_/2crb (0'+1) cr-b-l

o

o

o

The basis vectors will be Mw

[

=

aw.

u,v,w

An eigenvector of

A,I

where

Mu

=

-IAlv, Mv

M with eigenvalue

-0' b+l !.(cr+b+1) (O-b-l)b} " 0+1 •

to the eigenvalues

- (cr+b+l)

The eigenspace of

a

is

M corresponding

is the orthogonal complement of the

eigenvector of

t M

eigenvector is

fo+b-l,- (o-b-l) ,_jb (cr+b+;~io-b-l)J.

choose

u

=

corresponding to the eigenvalue

(-(a-b-l),-(a+b-l),O).

on the eigenspace of

-IAlu,

Because

A,I, we may choose

2crb(a-b-l) _!2b(0-b-l) (/ 0+1 ' 0(0+1)'

v

2 M

=-

(cr_l)!2(0+b+.l»).

=

a.

This

We will 2 [-IAI ] 0 2

_1:1

rfr

Mu

We now have

0

our new basis and, as in Example 4B.4, after writing the differential equations in this basis to get ready to compute

V"

'(D).

This is a very lengthy computation

so we give only the results.*

Following (II) of Section 4A,

*Details will be sent on request.


the second derivative terms of 311

~

[-

V'" (0)

145

are

2 2 2 2 l 2 2 2 l d 2 X2 d2 X d X d X d X d X + - 2 - dX dx + - 2 - dX dX - 2 - dX dX 2 dX 2 dX l l dX 2 l 2 l l

2 l 2 l d X d X + ,2 X1 ,'Xl ,2 X'] - - 2 - dX dx 2 2 2 dX dX dX dX 2 l l 2 2 1

•

2 311 (o-b-l) / 2 b (o-b-l) 40b (0+1) 3 w2 0 (p+l)

{

2 2 [20 b (o+b..,.l) -

2 20b

(o-b-l)

+ 20b(0-1) (o-b-l) (o+b+l) + 20(0-1)2(0+1) (o+b+l)] [b(o+1) (o+b-l) (o-b-l) - 2b 2 (0-b-l) + 2b(0-1) (o-b-l) (o+b+l) + 2(0-1)2(0+1) (o+b+l)] + [b(o+l) (a-b-l)

2

2 2 + (0 -1) (a-b-1) (a+b+1) - a(a -1) (a+b-1) (0+b+1)

- ab(o+l) (a+b-l) (a-b-l) -

2 2 2 (0 -1) (o+b+l) (o-b-l) ] [(a -1) (a+b-l) (o+b+l)

2 + b(o+l) (a+b-l) (o-b-l) - 2b(a-l) (o+b+l) - 2B (a-b-l) + 2B(0-1) (o-b-l) (a+b+l)] + 2OO(a-l) (o+l) 2 (a-b-l) (a+b+l) (cr+b-l) 2 2 2 2 + 2b a(o+l) (a-b-l) (o+b-l) -

2

2 [8b a (a-I) (a-b-l) (a+b+l)

+ 8ba(a-l) 2 (0+1) (o+b+l) - 8b3a (o-b-l)] [(a-I) (a-b-l) (o+b+l) - b(a-b-l) -

(a-I) (a+b+l)]j =:

~~.

The third derivative terms are

311(0-1) (o-b-l) 20b(o+1)3 w2

2

!2b(a-b-l)

a(a+l)

1 [(a+b+l)2(a-b-l) + 8ab(o+1)]

146


2 2 2 2 {_4cr b (cr _1) (cr-b+1) + crb(0+b-1) (0-b-1) (0+b+1) 2 + 2crb (cr+1) (0-b-1) (cr+b+1) (0 +b-1) - 20 2b 2 (cr+1) (cr+b-1) (cr+b+1)] [30b(cr+1) (cr+b-1) - b(cr+1) (2b+1) (cr-b-1) 2 - 2b(0-b-1) 2 (cr-iliH) _4(0 _1) (cr-b-1) (cr+b+1) + (i-I) (b+2) (cr+b+1)] 2 2 + [4ib (cr+1)2(0+b-1) - 2crb(cr+1) (cr+b+1) (cr-b-1) (cr +b-1) +

2cr~2 (cr+1)

2 2 2 (cr-tb+1) (cr+b-1) - 8cr b (cr+1) (cr +b-l)

2 2 2 2 - 2crb (cr+b+1) (0-b-1) (cr +b-1)] [2b(cr+b+1) (cr-b-1) 2 - 4(0 -1) (cr-b-1)0+b+1) + crb(cr+1) (cr+b-1) 2 + b(a+1) (2b+5) (cr-b-1) + 3 (b+2) (cr _1) {cr+b+1)] 2 - [2ab (cr+1) (cr+b-1) (cr+b+1) + 4crb(cr+b+1) (i+b-1) -

2 2 2 (cr+b+l) (cr-b-1) (cr +b-1) + crb(cr+b+1) (cr+b-1)] 2 2 [2crb (cr+1) (b+2) (cr-b-1) + 2crb (o-b-1) 2 (cr+b+1)

- cr(cr-l) (cr+1)

2

(cr+b+1) (cr-b-l) + crb(cr+1) 2 (cr-b-l) (cr+b-l)

2 2 - (cr-1) (cr+1) (b+1) (cr-b-1) (cr+b+1) - b(b+1) (cr+1) (cr-b-1/ - b(a+1) (cr+b+1) (cr-b-1)3] } :::

~E,

where _ (cr-1) w - (cr+1) /

12- (cr+b+1) cr

[(b+1) (cr+1) (cr-b-1) + a(cr+1) (cr+b-1) + b(cr+b+1) (cr-b-1)]

and 2 E, = 37T(cr-b-1) 2crb(cr+l)3 w2

Since

V'"

(0) =

2b(cr-b-1) cr(cr+1)

(A +A )E, 1 2

and

E, > 0, the periodic

orbits resulting from the Hopf bifurcation are stable if A + A < 0, and unstable if A + A > O. For cr = 10, b 1 2 1 2 9 9 9 1.63 x 10 , A = 0.361 x 10 , A + A = 1.99 x 10 ; A 2 l 2 l

8/3,

147


therefcre, the orbits are unstable, i.e. the bifurcation is subcritical. Our calculations thus prove the conjecture of Lorenz [1], who believed the orbits to be unstable because of numerical work he had done.

For different

a

or

b

however, the sign

may change, so one cannot conclude that the closed orbits are always unstable.

A simple computer program determines the

regions of stability and instability in the

b-a

plane.

See Figure 4B.l Lorenz' value u= 10, b= 8/3 (0,0)

/

25

~~ STABLE

STABLE PERIODIC ORBITS

100

100

50

b

V"'IOJ = 0

PERIODIC

LORENZ ATTRACTOR

ORBITS --(SUPER CRITICAL (SUB-CRITICAL HOPF BIFURCATION) HOPF BIFURCATION)

V,I/(O»O

----

V"'(O) 0

as

a

and McLaughlin [lJ.

+

00.

This has also been done by Martin

Our result agrees with theirs.

We

148


proceed as follows: {(0+b+l)2(0-b-l) + 80b(0+1) }A l + A = p(b,o) 2 is a polynomial of degree highest order term is

in

11

For

o.

(8b 2 +12b) all.

(0) > 0

this co-

b > 0

If

efficient is positive, so for large positive fixed), V'"

fixed, the

b

a

(with

b

and the bifurcation is subcritical.

This example may be important for understanding eventual theorems of turbulence (see discussion in Section 9). The idea is shown in Figure 4B.2.

For further information on

the behavior of solutions above criticality, see Lorenz [1] and Section 12.

(L. Howard has built a device to simulate

the dynamics of these equations.)

as

b

(4B.9) Exercise. Analyze the behavior of V" I (0) ro for fixed a and as b + ro for a = Sb, for

+

various

B

>

o.

!

turbulent?

-+-

stable I e qu iii brium I unstable I

r

r =bifurcation parameter

-stable cycles

Lore

z ott octor unstable fixed pOint

stable ~--t......,,""'-

_unstable cycles ~

periodic soLutions

_ _...l-_ _- - - ,

supercritical

f Figure 4B.2

149


(4B.IO)

Exercise.

By a change of variable, analyze the

stability of the fixed point

x = y = - b(r-l), z

Show that Hopf bifurcation occurs at

r o -- o 0-6-1 (0+b+3)

r - 1. and that

the orbits obtained are attracting iff those obtained from the analysis above are. (4B.11)

Exercise.

Prove that for

r > 1, the matrix

(:_l)j

-0

~

[

Ib

-1

-1

has one negative and two complex conjugate roots. (4B.12) 3 R

Exercise.

Let

F

denote the vector field on

defined by the right hand side of the Lorentz equations. (a)

Note that

div F

=

-0 - 1 - b, a constant.

Use this to estimate the order of magnitude of the contractions in the principal directions at the fixed points. (b)

If

V

=

(x,y,z), show that the inner product

is a quadratic function of

x,y,z.

By considering

~t
and

and that the remainder of the spectrum of

A(a)

positive distance away from the imaginary axis.

z ERn

stays uniform Decompose

as z

=

(x,y)

according to the spectrum of space corresponding to

A(O), so that

A(O) ,A(O)

and

With this decomposition we may assume A p (a)

0 (a)

y (a)

-w

A(a)

where

r

l w (a)

(a)]

y(a)

Q

P

is the eigen-

is its complement.

153


and the spectrum of imaginary axis.

AQ(a)

stays uniformly away from the

We may also represent x

x

in polar coordinates

(r cos 8, r sin 8).

Consider a periodic solution bifurcating from (0,0,0).

The differential equation for

r

(x,y,a)

is

2 y(a)r + Orr )

r

and it follows that when

r

attains its maximum on the solution,

r = 0, and hence

=

a

(4C.3)

Orr).

Now the periodic solution also lies on the center manifold

~,

described by ~:

The fixed point over,

~

(r,y)

=

is tangent to

y

(4C.4)

¢(r,8,a).

=

(0,0)

lies on

px(-aO,a O)

at

~

for all

(r,a)

=

(0,0)

a; morewhich

implies (4C.4) has the form ~:

=

y

r~(r,8,a)

~(0,8,0)

=

0.

Thus, we have on the periodic solution (4C.5)

y

Choosing

£ >

°

of the same order as the amplitude of the

solution, we may scale the equation by making the replacements

r

~

The estimates (4C.3),

£r,

a

~

£a,

y

~

(4C.5) imply then

£y.

154


r

0(1) } 0(1) 0(£)

y

The precise relation between later when

a

(4C.6)

in scaled coordinates.

£

and

a

will be determined

will be chosen as a particular function of

£.

We will in fact show then that a

=

in scaled coordinates.

0

Expand the differential equation (4C.2) in a Taylor series, in scaled coordinates.

It is not difficult to show

the estimates (4C.6) imply the equation takes the form

(4C.7) AQY + £Jx

y

2

2

+ O(£a) + 0(£ )

where homogeneous polynomial of degree j

B,

J

A

A

Q

J

Q

R

2

x

E R2 , ta k'1ng va 1 ues in

->-

R

2

2 R

bilinear

(0)

x R

2

->-

2 R

symmetric, bilinear.

In polar coordinates (4C.7) becomes r

. y where

=

2

2 3

£avr + £r C (6) + £ r c (e) + £rG (6)y 2 4 3 + 0(£2 a ) + 0(£3) 0(£2) 6)2 + O(£a) +

in

155


I 2 (cos 6) B, I (cos 6, sin 6) + (sin 6) B, I (cos 6, sin 6)

c, (6) J

J-

,

J-

2 (cos 6) B, I (cos 6, sin 6 ) J-

D (6) j

-

I (sin 6) B, I (cos 6, sin 6 ) J-

homogeneous trigonometric polynomials of degree j homogeneous trigonometric polynomial of degree 2

G (6) 2

Q*

taking values in

Q.

the dual of

The goal of the method of averaging is to "average out"

.

the dependence of

r

on

radial coordinate

-r

6

and

y, that is, to find a new

in which the equation for

is

If this were done, then all periodic solutions would simply be circles

=

r

r(e)

satisfying

F(r(e) ,c)

=

O.

Actually it is

not necessary to entirely eliminate dependence on generally all that is required is the absence of

e e

and and

y; y

from a finite number of terms in the Taylor series expression in

e

and

y.

For example, in (4C.8), generically it is

enough to average the

e,ey

and

e

2

terms.

More precisely, consider any differential equation r·

I' l

~l e j R'k(r,6,a)y k

j=l k=O

e

The series for

.

r

J

w + O(e)

may be a finite Taylor series with remainder.

In order to average a given term, say ordinate

r

Rpq ' define a new co-

by

In the new coordinates, the coefficient of

ePyq

becomes

Rpq

156


where

Two cases are considered. Case I, q

O.

In this case we choose

- !

u(r,8,o:)

Observe that of

8

f8 R O(r,~,o:)d~

wOp

u

is 2'IT-periodic in

u

to be 2'IT

+

8 2'ITW 8

foRpO

and

(r,~,o:)ds.

R pO

is independent

and, in fact, is the mean value

Therefore, we have averaged the coefficient of Case II, q > O.

Here we wish to choose

identically zero thus eliminating the we seek a 2'IT-periodic function

£PyO

u

so that

£Pyq

term.

u(r,8,o:)

~

pq

is

Therefore,

satisfying (4C.9)

By considering

R pq

that such a unique

as a forcing term in (4C.9), it follows u

always exists if and only if the homo-

geneous equation

o has no nontrivial solution 2'IT-periodic solution. shown that this is the case provided that n-2

I n./... 'I- integer j=l J J

It can be

157


n-2

In.

n. > 0,

for all integers

J -

q

j=l J

In particular this can always be done if either

q

is stable. We return to the bifurcation problem (4C.8) and now average the

E,Ey

and

E

2

in the above fashion by means of

the transformation r

=

2

r + Eu(r,8,a) + Ew(r,8,a)y + E v(r,8,a).

In fact, the transformation has the form 2

2 3

r = r + Er u(8) + Erw(8)y + E r v(8), and this yields the equation for

-r

r

-2 -2 davr + r C (8) + r u ' (8)w] 3

+ Er[G (8) + w(8)AQ + w' (8)w]y 2

+ E2-3 r [C (8) + u ' (8)D (8) + w(8)J(cos 8, sin 8)2 4

(4C.10)

3

- 2u(8)u ' (8)w + v' (8)w]

Since

C

3

has mean value zero, we choose u(8) =

so that the coefficient of

E

in (10) is

avr.

equal to the unique solution of G2 (8) + w ( 8 ) AQ + w(8)

W

of period

I

(8) w 2n.

0

Set

w(8)

158


so the

Ey

term vanishes.

Finally, we may choose 2-3

make the coefficient of

E r

v(6)

to

the constant

mean[C 4 (6) + u' (6)D 3 (6) + w(6)J(cos 6, sin 6)

K

-2u(6)u' (6)w] 1 J2'IT 1 2TI 0 C4 (6) - W ~3(6)D3(6) + w(6)J(cos 6, sin 6)

Thus in the new coordinates

r

(r,6,y),

e

w + O(E)

1

y

AQY + O(E).

J

y

the center manifold

•

(4C.8) becomes

3 2 - + E2-3 ECi.Vr r K + O(E Ci.) + O(E )

The equation for

2

(4C.ll)

may be neglected now as we restrict to y

r~(Er,6,ECi.).

Moreover, it is not

difficult to show that the unique branch of periodic solutions bifurcating from the origin has the form

r

=

IRI

I 2 / + O(E)} in scaled, averaged coordinates

Ci.= -E sgn(vK) that is, r

= in original coordinates.

y

(4C.12)

2

Ci. = -E sgn (vK) The amplitude of the bifurcating solution is therefore approximately In case

(_ Ci.KV) 1/2 K

and one sees the period is near

2 'IT W

0, one simply averages higher order terms

159


in

and

E

y

in the same manner.

The possible normal forms

one arrives at in this case are r

w

for integers

+ O(E) ~

p

2

K' i

and

O.

The bifurcating solution in

this case has the form

r

=

K' 1~l

l/2P E

2 + O(E ) in original coordinates

y a

J

=

so has amplituded near

[_ aKv,) 1/2p

and period near

271 W

Observe that in all these cases bifurcation takes place only on one side of

a

solutions occur at

=

O.

For cases in which all bifurcating

a = 0

(for example in the proof of the

Lyapunov center theorem - see Section 3C), the method of averaging gives no information. For more details of the above method, as

~ell

several applications, see Chow and- Mallet-Paret [1].

as We

mention here two examples treated in this paper using averaging.

(1)

Delay Differential Equations (Wright's Equation).

The equation

z(t)

-az (t-l) [Hz (t) 1

arises in such diverse areas as population models and number theory, and is one of the most deeply studied delay equations.


160

For

21T '

a >

topological fixed point techniques prove the

existence of a periodic solution.

Using averaging techniques,

one can analyze the behavior of this solution near In particular, for from

z

=

;

0; the corresponding statement

)1 2 < 0 . ttt

The Characteristic Exponents of the Periodic Solution. In the following we shall sometimes make use of deter-

minants; this, however, can be avoided.

In the linearization

about the periodic solutions of (2.3), L

(u)

-t,E: -

ttt see editorial comments in §5A below.

(5.1)


185

we have, by (2. 3), (5.2 )

A fundamental system

formed with fixed initial con-

u. (t,S)

-1

ditions depends analytically on u. (T,e;) = 'a. (e;)u (0) L 1 \! -v

~he

(t,s).

are analytic at

-1

coefficients in

s = O.

The deter-

minantal equation

I Ia ik (s)

II

AT(e;)

I;:

e

determines the characteristic exponents

A k

~,

-

I;: 0 ik

= 0,

(5.3 )

and the solutions

of (1. 3), where

e

~

At

v

Since (5.1) is solved by ( 5. 3).

'The exponent

u

y,

1

I;: =

is a root of

13, which was spoken of in the intro-

duction, corresponds to a simple root of the equation obtained by dividing out s = 0, 13 = S2 s

2

-

I;:

+

same reasons as

13 (s)

l.

...

is also equal to zero for the

(131

and

)11

is thus real and analytic at

Now i f

T 1)'

there is some minor of order

n - 1

follows that (1.3), A = 13, has a solution

order

s

n - 2

=

O.

Even if

13

which is not zero.

= 0,

tv + ~

with periodic

v

0

w

u

y. *

and

That

0, then

O.

From this it v

$

0

which is

there is a minor of

As we know, in this case,

there is a solution of (5.1), analytic at ~

-

in the determinant (5.3)

(with the corresponding 1;:) which is not

analytic at

is not

13

s

~,~, where either

=

0, of the form ~

$

0, or

is linearly independent of the solution tv + w

is a solution implies that

* Cf. e.g. F. R. Moulton, Periodic Orbits, Washington, 1920, p.

26.

186


v=

v + w = L

(v),

L

-

-t,e:-

-

(w).

(5· 4)

-t,e:-

After these preliminary observations we shall calculate

62 ,

~2

We assume here that

f O.

possible as will subsequently be proved. introduce

s

as a new

t

=0

6

is then im-

If we use (4.7) to

into (1.3) we get L

(v).

-t,e: -

Also, we have (with the new

~i

where all the

t)

have the same period

the power series for

~,

6,

~,

TO'

If we introduce

y, it follows (dropping the

subscript zero on the operators as before) that (5.5)

~o

.

(5.6)

~l

(5.7)

These equations have the trivial solution

6.l

(i

0,1, .•• ) •

(5.8)

Thus, one has ~

(il )

+ 2Q (y -

0

,

X.0 ).

(5.9)

( 5.10)


Since we may assume that

~O

187

f 0 (5.11)

~o

O.

where at least one of the coefficients is not

If we set

(5.12) it follows from (4.10),

(5.6) and (5.9) that

.

w

.!: (~),

thus (5.13)

If one forms the combination

(5.7) - P(4.11) - 0(5.10), in which

L'

cancels out, and sets

then, using (5.11) and (5.12), one obtains: S2~ + u

~(~) + 2P(2~(XO'Yl) + ~(Yo'Yo,yO))+~

(5.14)

with

If we now apply the bracket criterion of the previous section to (4.10) and (5.9), it follows from (5.13) that

If we apply it to (5.14), in which follows from (4.6)

* The

P

and

o

(with

~

has the period

TO' it

z = yO) that

and 0 in (5.11) are unrelated to the symbols as used in Section 2.

P

188


Hence, by (4.16),

Likewise, it follows that

a 62 = From this, either

-2p(T

6

Im(a') ) a •

2 +]12

is given by (1.4)

2

]12 f 0) or else

not zero since

6

=

(and then

O.

2 In either case

2 is completely determined (in the second case

p:a

6

is

p = 0).

To check that the first case really occurs we must undertake a somewhat longer consideration.

One may think of

the process as schematized in the following manner.

6

equation for equation (5.4» sis.

and

v

The

(namely the equation which follows

should be divided by the factor in parenthe-

It is then once again of the form

v + 6v

~t,E:(~'>

with L

-t,E: where on not

t

~O

+

E:~l

+ E:

2

~2

+ ••• ,

i > 0, depend is a constant operator, while L -0 with the period The coefficients of 1, E: are

altered by the division.

Introduction of the power

series leads to

* One

~O

~O(~),

v -1

~O(~l) + ~l (~O),

*

does not really have to assume 61 = O. criterion this is a consequence of (5.17) •

( 5.15)

From the bracket


and so forth.

The situation is the following.

there are two solutions

. z

~,

E

=

0

Furthermore

az

p~ +

v -0

with period

For

189

(5.16)

and (5.17)

for both bracket subscripts. ~l =

p~

with fixed periodic

+ ah + ~

and

It follows that p'~

(5.18 )

+ a'z

h.

For the third equation of

(5.15), the bracket criterion gives 13 P 2 S2 a

AlP + Bla

(5

·19)

A P + B a 2 2

with A.

1

[~l (~)

+ L (z) 1 . ,

Bi

[~l (~)

+ ~2 (~)] i '

while (5.17) implies that

-2 -

P', a'

1

(5 .20 )

drop out.

The situation

13 , P, a 2 To them belong two

now is that the equations (5.19) with the unknowns

13 • * 2 linearly independent pairs (p,a). Each of the two solution

have two distinct real solutions

systems leads now to a unique determination of the v.

-1

* In

Si

and

through the recursion formulas, if one suitably normalizes

the gen~ral case, that is if the special condition (5.17) is not fulfilled, the splitting into two cases occurs already at the second equation (5.15). The solution of the problem in this case is found in F. R. Houlton, Periodic Orbits. Compare Chapter 1, particularly pages 34 and 40.

190

v.


~ ~

To this end choose a constant vector

way that

~o

(5.16).

.

=

a

1

=

(t

0) for both pairs

0

in such a

(p,a)

in

One concludes then that v

v. that is, -1.

.

0

~ ~

1,

~

~

t

at

0) for

(t Z

C,

0,

~

(t

D

Then, for either of the two values of

o.

i >

Let

0) • 13

2

, the system of

equations

o

(5.21)

Cp + DO = 1

uniquely determines the unknowns p, a,

~o

are determined.

p

and

a.

Up to now

Using the definition of

~,

13

2

,

~

and (5.18), one obtains from the third equation of (5.15) ~2 = p'~ + a'~ + p"~ +

a"i + ••• ,

where the terms omitted are already known.

(5.22) From the fourth

equation of (5.15) one obtains the equations

by using (5.18), Since

~1

. a

=

(5. 20), 0

(t

=

(5. 22) and the bracket criterion. 0), we add to these equations the

equation CP' + DO'


191

6 , p', cr' 2 With the help of (5.21), the

Through the three equations, the three quantities are now uniquely determined. determinant is found to be

It is not equal to zero, since by hypothesis, 63' P ' , cr ,

From this

distinct solutions

(5.19) has two and

are

~l

determined. 6 , p", 4 are determined by equations with exactly the same left It is now easy to see that at the next step

cr"

hand side, and that by the further analogous steps everything is determined. We return now to the special problem which interests us, and assume that by suitable normalization two different formal power series pairs

(6,~)

exist which solve the equa-

tion (1 - T 8 2

2

+ •••

)v-

= -t,8L (v).

+ 6v -

On the other hand it was previously demonstrated that under the assumption

$

6

0, two actual solutions exist, of which

one is known, namely (5.8).

Under this assumption the second

(normalized) solution can thus be represented by the power series and the formula (1.4) for

6

does in fact hold. To 2 dispose of this completely we must still show that 6 = 0 ~2

cannot occur if schematic problem. and the second

6

2

~

O.

We show this also in terms of the

Since (5.19) has the solution ~

6

2

=

p

=

0, 0,

(5.23)

0

192

If


S

= 0,

were

then (5.4) would have a solution with the

properties given there.

X,

Setting in the power series for w ~o + -0

~O (~O)

v

~O (~l) + ~l (~o)

-1

w + -1

v -2 + ~2

w

gives

(5.24)

L (w ) + ~l (~l) + ~2 (~o)· -0 -2

We have w -0

p~

+ crz.

(5.25)

Since

is also of this form, according to the bracket

criterion

~o

analogously that

~l

=

~l =

O.

Similarly, as in (5.18), we find

p~ + cr~ + p'~ + cr'z.

It has been demonstrated above that tion

(y)

By (5.17), it follows

must be equal to zero.

of period

v

L (v) -t,E: -

TO' unique up to a factor.

has a soluThus we

certainly have v

-2

=

As above, using (5.20), application of the bracket rule to (5.24) gives the equations

(in which

p', cr'

once again fallout).

it thus follows that

p

=

A

v = O. Ac-2 If one subtracts from the second

cording to (5.25) w = crz. -0 equation of (5.4) the solution by

According to (5.23)

0, and from this

cry of

and divides

E:, then the whole process can be repeated, and we find


~i

successively that the it is demonstrated that

= B

0, and thus

v

=

O.

193

With this

cannot be equal to zero.

The verification of the formula (1.4) is thus complete under the assumption placed by

~

$

O.

~2

f O.

This assumption could be re-

The considerations would be changed only

in that in the calculation of the coefficients the case of splitting will occur later. The difficulties of these considerations could be avoided in the following manner.

One first calculates purely

formally as above the coefficient of the power series for and

v

and then shows the convergence directly by a suit-

able application of the method of majorants.

This would cor-

respond to our intention of facilitating the application to partial differential systems.

But one can also carry out

the discussion of the case of splitting and the proof of

· 1 y Wlt . h d etermlnants. . tttt (1 • 4) exc 1 USlve

tttt

B

See editorial comments in Section SA.

194


SECTION SA EDITORIAL COMMENTS BY L. N. HOWARD AND N. KOPELL

(t) 1. fied.

Hopf's argument can be considerably simpli-

After "blowing up" the equation (1.1) to (2.3), one

wishes to show that for each sufficiently small a

]..I (E) ,

T(E)

a period

E

there is

y

and initial conditions

°

(E)

(suitably normalized), so that (2.16) holds; the family of solutions to (1.1) asserted in the theorem is then

°

Ey(t,]..I(s),E,y).

yO ~

=

°(E)

~o.

Now (2.16) is satisfied i f

~(t,E)

]..I = E = 0,

Hence, the existence of the functions

]..I (E) , T(E),

follows from an implicit function theorem argument,

provided that the

(*. ~, :~1It has maximal rank.

n x n

matrix

= TO' " = 0,

(Here

lY.-

is an

dyO

presenting the derivative of

,

X

0, y

°

n x (n-2)

with respect to

matrix re(n-2)

initial directions; there are two restrictions on the initial


conditions from the normalization.)

195

We show helow how the

rank of this matrix may be computed more easily. ~

Let

~

and

be the right and left eigenvectors corL ; by rescal-

responding to a pure imaginary eigenvalue of

O

ing time, we may assume that this eigenvalue is

T

are eigenvectors for ~

Let

= 1.

-i.)

(£:

i.

and

We may also assume that

L'

We note that hypothesis (1.2) may be rephrased: ~(~)

(To see this, let of

L~

be the eigenvector

which corresponds to the eigenvalue

pure imaginary eigenvalue, normalized by ~(O)

at

~

de

~

~

near a

e = 1, so

with respect to

0, we get

=

L Now

a(~)~

Differentiating

= r.

a(~)

O

+ L'r

d~

de a' (O)~ + a (O)d~

(SA.l)

de

.

Qjl= 0 ~

plied by

and

.8:.L O

a (0).8:..

on the left, we get

Let

'1.

be defined by

x:

Hence, i f (SA.l) is multi-

a' (0).)

~·L'r

t

YEo

is replaced by

s = t/(l+T), where

T

is to be adjusted (for each

that the period in

s

is 2w.

For each

~

we construct the solution with initial

E,T

condition where

~.

and

Then (1.1) becomes

'1.(0), normalized by requiring z =

r .

z = O.

W = (.8:.

~.8:.)

y(O)

(Hence, the initial conditions are

parameterized by points in the _

E) so

n-2

dimensional space

..L

Note that by the simplicity of the imaginary

eigenvalues, W is transverse to

~ t!l

£:.)

This solution we

196


~(S,T,~,~,E).

denote by

Y(T,]l,~,E) = y(21T,T,~,~,E)

Let

~

T

v

O.

=

E

=

= Q,

we have

y(s)

X.(O,T,~,~,E).

= yO(s)

To show that there is a family of

tions with show that ~

0, ~

-

T

=

T(E),

~

3~3 (T,~,~)

=

~(E),

~

n

at

Re(£e is )

=

and

21T-periodic funcit suffices to

~(E),

has rank

At

~

=

=

T

E

=

0,

= O.

3y

Let

v (s) =

---r

~(s,O,O,O,O).

3T

Then

-

satisfies the

variational equation

with initial condition YT(O) = Q. The solution to this dyO 3V i 21T-(r-r) equation is Y..e = s ds which implies that 3T 2 --21TIm r. 3y __ (s,O,O,O,O), which satWe next calculate Y (s) -

3~

]1

isfies the variational equation

with initial condition y~

is the real part of

~

(0)

~,

Q.

=

where

Since ~

L'y

Re L ' £e is ,

o

satisfies (SA.2)

with initial condition

~(O)

O.

A particular solution to (SA.2) is Qe

is

, where

b

~

=

S(~·L,£)~~is +

is any complex vector which satisfies: (SA.3)

Now since

(i-L O) 2

is singular, but (SA.3) may be solved for

O.

The solutions

b

b

all have


£0 + kr

the form of

for which

b

for any 9,'Re b

197

There is a unique such value

k.

= I"

=

Re b

O.

(Take

-9,' (b + ~).) We use this value of b. The solution to -0 (SA.2) satisfying 21(0 ) = 0 then has real part

=

k

S!..Y as s

where

av ajl

= Land

01.

Hence

-Re b.

1. (0)

27f Re(E:.·L'!:.)!:. + .r(27f) - .r(0).

=

=

1.(27f) - 1. (0) follows since

=

-9,'Re b

1.1

satisfies

~(!'1.)

!'Yl

=

Similarly,

=

9,'Y 1

O.

(This

Since

= i!·1.'

9,'L Y 0-

0, !·1.(s) - O.

We note that

!.1. (0)

o. )

!.1. (s) -

av

8Z

Finally, we compute

in

~

due to the variation

iY(s)

satisfies

9,'oz

'I'oz

Now

OZ

O.

d ds(~)

=

Let

OZ

iY

in initial conditions.

aV

This implies that

=

OZ, and 2 7f L O E(~) = (e

LO(OY), ix.(0)

is in the subspace

be the variation

-

9,

W orthogonal to

Then

-

and

I)~.

T.

Since there are no other pure imaginary eigenvalues for (in particular, no integer multiples of 27fL (e

O

-

L

O

±i) , the matrix

av I)

Now Re rand

W.

is invertible on

Hence

Rn ,is the direct sum of

az

has rank

n-2.

W and the span of

(This follows from the simplicity of the 27fL O pure imaginary eigenvalues.) The range of (e - I) is W, so

ava

Re(!·L'!:.)E 2.

Im r.

(T til ,~)

has rank

are independent.

n

if and only if

This is true if

Im rand Re(!·L'!:.)

~

O.

The argument in this section does not require

analyticity; it merely sets up the hypotheses of an implicit function theorem.

Hence this argument provides a proof for

198 a

THE HOPF BIFURCATION AND ITS APPLICATIONS Cr

that

version of this theorem. !(~,~)

is

r

More specifically, suppose

times differentiable in

x

~.

and

is r times differentiable Then the right hand side of (2.3) r-l in E. The function in x and ~, but only C l r in E and at least Cr Y(T , ~ '!.' E) defined above is c in the other variables.

Hence, the implicit function theorem

says that the functions

T

tions to (1.1),

(tt)

(E),

(E), !.(E) r-l are all C ~

X(t,E) = Y(l+~(E)

namely

and The periodic solu,E), are

r

C .

The uniqueness proved in this argument is

weaker than that of Theorem 3.15 of these notes.

That is, it

is not proved in Hopf's paper that the periodic solutions which are found are the only ones in some neighborhood of the critical point.

For example, Hopf's argument does not rule out a

sequence of periodic functions the associated

xk(t)

such that

maxlxk(t)

I

+0,

T + 00. Such behavior k is ruled out by the center manifold theorem, which says that ~k +

0, and the periods

any point not on the center manifold must eventually leave a sufficiently small neighborhood (at least for a while) or tend to the center manifold as

t

+

00.

Thus the center manifold

contains all sufficiently small closed orbits; since the center manifold is three dimensional (including the parameter dimension), the uniqueness of the periodic solutions is a

conse-

quence of the uniqueness for the two-dimensional theorem.

(ttt)

Formulas equivalent to Hopf's but somewhat easier

to apply can be obtained in a simpler manner. point is to use the "e

is

"

The main

form of the solutions more ex-

plicitly and thereby avoid introducing the bracket criterion.

199


We again assume that time has been scaled so that the pure imaginary eigenvalues of the notation introduced in (t). rescale time by

=

t

L

±i, and we use

are

O

Following Hopf we further

=

(l+T(E»S, T(O)

0, and let

~

=

EX.

Then (1.1) becomes

where

and

Q

terms when

K

II

are respectively the quadratic and cubic O.

Let the 2n-periodic solution of (SA.4) be :L(S,E) 2 is £), and the E~l + E ~2 + ... , where, as before, ~O = Re(e ~i

are

~'~i (0)

2n-periodic with ~i

(Since the

= !'~i

:Ll 1

'2

T

We use the fact that

=

II

= A~l Re[e

0

~'~i(O)

2 E ll2 +

We find that

+ Q(:LO'~O)' and

2is

Q(£,£)].

~ + Re(~e2is)

Q(~O'~O) =

~l

O. )

series for ~i' the

0, so

III

l

i > 1.

for

is inserted in (SA.4) and like powers of

lected. and

=

are real, we may simply require

To get recursive equations for the

:L(s,d

(0)

Y.o

E T

are col2 E T

2

+ ...

should satisfy

1

-

'2 Q(£,£) +

A periodic solution to this equation is

where

~

~

and

are constant vectors satis-

fying -L

a

0-

=

!2 -Q(r,r"l - 1

2" (Since

-L

O and

do determine

~

2i - L and

o

~.)

9.(£,£).

are non-singular, these formulas Thus

Re(C reisj, where the complex number 1-

so that

~.

finds that

~l

(SA.S)

C

l

is to be chosen

(0) = 0; using equations (SA.S), one readily

+

ZOO

Now

THE HOP? BIFURCATION AND ITS APPLICATIONS

~Z

.

~Z

is a periodic solution of LO~Z + ~ZL'~O + ZQ(~'Yl) + ~(~o'Yo,yO) + 'ZLOYO

Hence

These equations have a periodic solution if and only if there is is no resonance, which requires that the coefficient of e in this last formula should be orthogonal to criteria in disguise).

~

(the bracket

Thus

Hence we get the formulas for

~Z

and

(SA.6)

where

~

and

£

are the solutions of

are unchanged if the eigenvalue is

iw

(SA.S).

[These formulas

instead of

cept that, instead of the second equation (SA.S), solution of

(Ziw - L O)£

= ~ Q(£,£).

given above should be divided by The formulas

i, ex-

£

is the

Also the value of

C l

w.]

(SA.6) are equivalent to Hopf's (4.16).

The determination of the left eigenvector

~

and the solution


of the linear equations (SA.S) for

~

and

£

201

takes the place

of finding the adjoint eigenfunctions and evaluating the integrals implied by the bracket symbols.

(tttt) 1.

The translators must admit that .they have

found this section somewhat less transparent than the rest of the paper.

In their article, Joseph and Sattinger [1]

point out an apparent circularity in a part of Hopf's argument; they also show there that it can be rectified rather easily. 2.

The relationship of

S, the Floquet exponent near

zero (of the periodic solution), to the coefficient

~2

can

be found with relatively little calculation, as follows. The argumented system F

x

~

(x)

-

(SA.7)

o has the origin as a critical point.

There are three eigen-

values of this critical point with zero real part; a zero eigenvalue with the

~-axis

as eigenvector, and the conju-

gate pair of imaginary eigenvalues

±i

(after suitably re-

scaling the time variable) with eigenvectors

£

r.

and

All other eigenvalues are off the imaginary axis, so this critical point has a 3 dimensional center manifold. center manifold must contain the

~-axis,

This

the periodic solu-

tions given by Hopf's Theorem, and any trajectories of

(SA.7)

which for all time remain close to the origin; it is tangent to the linear space generated by the and imaginary parts of

r.

Let us set

~-axis

x

and the real

202


O. for

E~

0

is regarded as a replacement

given by the function

~,

of HOpf's Theorem:

~(E)

~ = ~(E) = ~2E2 + ••• , where we are now assuming that ~2f

~,

O. and

(~,~)

Thus we may think of the real and imaginary parts of E, as parameters on the center manifold. For any 2 on this manifold ~2 = 0 (E ) since it is at least ~'x

quadratic in

=

E~

and

r· x =

E~.

equations of the center manifold as

E2~(~,~,E), where ratic in

~

and

~.~ ~.

= !.~ =

0

Thus we may write the

and

X = -2 is at least quad-

~(E),

~

~

For any trajectory on the center mani-

fold we then have, with the notations of (t) and (ttt), rt +

r

~ + E(~~~ + ~~~)

= i~r

- i~ £

+ E

LO~ + ~2E2L' (~£ + ~ £) + EQ(~£+~ £)

+ 2E

2 2 3 Q(~£ + ~ ~,~) + E C(~£ + ~ r) + O(E ).

By mUltiplying on the left with

i,

i~ + ~ E2~'L' (~r + ~ 2

-

-

(SA.8)

we obtain r)

2

-2

--

+ E~' [~ Qt.!>£) + 2S~Q(£,£) + ~ Q(£,£)] 2

+ E where

y

-

3

y(~,~)

is cubic in

(SA.9)

+ O(E ) ~

and

r.

We now introduce the function

As we will see below, I(~,~)

is approximately invariant

along trajectories lying on the center manifold.

For any


203

trajectory on the center manifold we have, using (SA.9) and its complex conjugate,

where

°

~

is quartic in

and

~.

The terms of order

in the above are £Re{~3~.Q(£,£) _ ~3T'Q(~,~) + _ 2 ) _2 (_) _ 2_ - } ~~ l'Q(~,~ + 2~~ l·Q ~,~ - 2~~ l'Q(~,~) = O.

~~2~.Q(~,~)

where

°1

order

£

2

and

is also quartic in

-

Thus

]l2£2Re[~1·LI (~r + ~ :£)1 + £201 (~,~) + 0(£3)

dI dt

£

dI dt

Thus

(SA.10) is of

•

(1 = ~1~12 is also an approximate invariant, but 0 dIO dI 2 is 0(£) whereas is only 0(£). If we consider dt dt a trajectory on the center manifold starting at t = 0 with ~

ce

= c, it

we see from

(SA.9) that it is given to

; thus, after a time of approximately

once again return to

1m

~

=

O.

by

2TI, it must

This arc is a circle to

0(£), but is more accurately described (to curve of cOnstant

0(£)

0(£2»

as a

I.)

We see from (SA.10) that the change in I around this way is given, to

2

0(£), by

in going once

204


where c

=

f2~ it -it 0 01 (e ,e )dt.

1

2~

02

However, we know that if

we get the periodic solution, for which

1

sequently

02

= -~2Re(~·L'£)

= -~2Re(a'

(0»

=

~I

0; con-

as noted in (t).

Thus, in general, ~I =

Since

c

=

1

2

2~£ ~2Re(a'(0)(c

2

4

3

(511..12)

-c) + 0(£).

gives the periodic solution, the

is also divisible by

(c-U.

Thus, for

c

[-2~2Re

a' (0) + 0 (c-l)].

0(£3)

near

1,

part (5A.12)

may be written 2

6I = 2TI£ (c-l) For small ~

0(1)

=

(5A.l3)

£, any trajectory on the center manifold with must keep going around an approximate circle.

However, it cannot be periodic unless it passes through ~

= 1.

~2Re

Hence, it is apparent from (5A.12) that, when

a' (0) > 0, all trajectories on the center manifold (at

a given

£, i.e.,

~)

which are inside the periodic solution

must spiral out towards it as ~2Re ~I

=

a' (0) < 0) • (~(2~)-c)c.

Since

I

t

->-

+00

(or as

is approximately

t

->-

if

2

~I ~ 1 ,

Thus (5A.13) implies that these trajectories

asymptotically. approach the periodic solution with exponential rate

S

= -2£2~2Re

a' (0) + 0(£3), and this must thus be

the numerically smallest non-zero Floquet exponent. 3.

Equation (5A.12) actually tells us more; it implies

that we may approximately describe" the trajectories on the center manifold as slowly expanding (or contracting if ~2Re

a' (0) < 0) circles whose radius

the formula

c

varies according to

20S


c where

t

2

ttl + tanh(E2P2Re a' (0) (t-t l »)

is the time at which

l

The function

4.

I

c

2

= 1/2.

is also of some use in relating

the above to the "vague attractor" hypothesis. P

=

0, the

n-dimensional system

x=

FO(x)

If we set

has a two-

dimensional center manifold for its critical point at

0,

tangent to the linear space spanned by the real and imaginary parts of

r.

As in a previous paragraph, we set

x = E(~_r + ~ _r) + x_ ' where ~,x = y·x = O. On this center - -2 - -2 2 manifold x = 0(E 2 ) and is at least quadratic in ~ and -2 "f; E is now an arbitrary scaling parameter. For any trajectory on this center manifold one obtains the same formula (SA.9) except for the omission of the

P L'

term - the other

2

terms written down all come from the F

If we then consider the function

p'

p-dependent pairs of I

for a trajectory

on this center manifold, we obtain (SA.IO) again, with the P2 term omitted, but the. same get (SA.ll) without the 02

=

-P 2 Re a' (0)

manifold at

P

01; integrating this around, we

P2

term but the same

02'

Since

we have for trajectories in the center 0, to order

2

E,

or 3 L'>(EC) = -21f P Re a' (0) (EC) • 2

Since

L'>(EC)

where

EC

V"'(O)

=

=

is xl

V(x ) (the Poincare map minus identity), l is the coordinate Re(~'~)' we see that

-21fp Re(a'(0»'6 2

=

-21fRe(a'(0»·3p"(0).

This relates

the calculations here to the stability calculations done in §4.

206


SECTION 6 THE HOPF BIFURCATION THEOREM FOR DIFFEOMORPHISMS Let

X

be a vector field and let

bit of the flow sociated with

¢t

y.

of

X.

(See §2B).

that is invariant under

P.

Let

P

be a closed or-

y

be a Poincare map as-

Suppose there is a circle Then it is clear that

is an invariant torus for the flow of

X

a

U ¢ (a) t

t

(see Figure 6.1).

r

-Figure 6.1 If we have a one parameter family of vector fields and closed


207

and Y , it is quite conceivable that for small Il 11 might be stable, but for large 11 it might become

orbits

X

11, Y ll

unstable and a stable invariant torus take its place. call that

is stable (unstable) if the eigenvalues of

the derivative of the Poincare map < 1

(>

Re-

1).

have absolute value 11 The Hopf Bifurcation Theorem for

(See §2B).

P

diffeomorphisms gives conditions under which we may expect bifurcation·to stable invariant tori after loss of stability The theorem we present is due to Ruelle-Takens [1];

of

we follow the exposition of Lanford [1] for the proof. In order to apply these ideas, one needs to know how to compute the spectrum of the Poincare map

P.

Fortunately

this can be done because, as we have remarked earlier, the spectrum of the time

map of the flow is that of

T

P U {I}.

(See §2B). Reduction to Two Dimensions We thus turn our attention to bifurcations for diffeomorphisms.

The first thing to do is to reduce to the two

dimensional case.

*

This is done by means of the center mani-

fold theorem exactly as we did in the previous case; i.e., assume we have'a one parameter family of diffeomorphisms ~Il:

Z

+

Z,

~Il

(0) = 0

and assume a single complex conjugate

pair of simple non-real eigenvalues crosses the unit circle as

11

increases past zero.

applied to

~:

three manifold

*As

(x,ll)

+

M; the

Then the center manifold theorem

(~Il(x)

11

,11)

slices

yields a locally invariant Mil

then give a family of

remarked before, for partial differential equations, P may become a diffeomorphism only after the reduction, and be only a smooth map before.

208


manifolds which we can identify by some fixed coordinate chart.

Then on

M we have induced a family of diffeomor]1 phisms containing all the recurrence. Thus we are reduced to the following case:

(modulo

"global" stability problems as in the last section). We have a one parameter family

~ : ~2 ~ ~2

of diffeo-

]1

morphisms satisfying:

A (]1)

I A (]1) I

a)

~]1

b)

d~]1 (0,0)

(0,0) = (0,0)

c)

d IA (]1) d]1

'I

I A (]1)

]1 < 0

such that for > I

A(]1)

has two non-real eigenvalues , < I

and for

and

]1 > 0

> O.

]1=0

We can reparametrize so that the eigenvalues of d ~ ]1 ( 0 , 0 )

are

( I +]1 ) e ±is (]1).

By rna k 1ng . a smoo th

dent ]1- d epen

change of coordinates, we can arrange that:

=

COS

8 (]1)

(1+]1) (

sin 8(]1)

-sin

8 (]1») cos 8 (]1)

The Canonical Form The next step is to make a further change of coordinates to bring cal form.

~]1

approximately into appropriate canoni-

To he able to do this, we need a technical assump-

tion: * e im8 (0) 'f 1 (6.1) make a smooth

*As

Lemma.

m

1,2,3,4,5.

(6.1)

Subject to Assumption (6.1), we can

]1-dependent change of coordinates bringing

~]1

D. Ruelle has pointed out, only m = 1,2,3,4 is needed for the bifurcation theorems as can be seen from the proof in §6A.


209

into the form: (x) = N (x) + O(/xI ]1

]1

S

)

where, in polar coordinates,

The proof of this proposition uses standard techniques and may be obtained, for example, from §23 of Siegel and Moser [1].

*

We give a straightforward and completely elemen-

tary proof in Section 6A.

As indicated above, we think of

as an approximate canonical form for

. ]1

Note two

special features of i)

The new

r

depends only on the old

ii)

The new

¢

is obtained from the old

r-dependent rotation.

r, not on by an

¢

We now add a final assumption: f

l

(0)

> O.

(6.2)

This assumption implies that for small positive N

]1

¢.

has an invariant circle of radius

rO' where

r

O

]1,

is ob-

tained by solving

Le., r

*The

2

O

]1

f

l

(]1)

•

canonical form is important in celestial mechaniOs for proving the existence and stability of closed orbits near a given one; i.e. for finding fixed points or invariant circles for the pqincare map. In the Hamiltonian case this map is symplectic (see Abraham-Marsden [1]) so Birkhoff's theorem applies, as are the results of Kolmogorov, Arnold and Moser if it is a "twist mapping".


210

We shall verify shortly that this circle is attracting for

N~~.

~~

not

surprising that

Since

differs only a little from ~~

N~~,

it is

has a nearby invariant circle.

The Main Result (6.2)

(Ruelle-Takens, Sacker, Naimark).

Theorem.

As-

sume (6.1) and (6.2) * . Then for all sufficiently small positive ~,

~~

has an attracting invariant circle. Before giving the proof, let us look at what happens if

(6.2) is replaced by the assumption ~,

small positive 2

~.

For

~

< 0,

N~~

N~

~

f l (0) < O.

has no invariant sets except

of Ruelle and Takens to ~

{O}

and

does have an invariant circle, but it

is repelling rather than attracting.

~

Then, for a

~

-1

By applying the result

, we prove that, in this case,

~

has a nearby invariant circle.

Thus, we again find the

usual situation that supercritical branches are stable and subcritical branches unstable.

~~: (y,¢)

->-

If

((l-2~)y - ~(3y2+y3) + ~20(1), ¢ +

e(~)

+

f 3 (~)

~~

2 (l+y)

+

~

2 0(1)).

1 ~

Finally, we scale

y

again by putting y

IiJz;

then

*

It would be interesting to explicitly compute fl(O) in terms of ~~ directly as we did in §4 for the bifurcat~on to invariant circles. However the labor involved in the earlier calculations, and the promise of a harder, if not impossible computation has left the present authors sufficiently exhausted to leave this one to the ambitious reader. The calculation of fl(O) in terms of ~~ rather than Xu is not so hard and has been done by Wan [preprint] and Iooss [6].


211

we rewrite this last formula as (z,<j»

->-

«l-2]1)z +]1

The functions

3/2

H]1(Z,<j», K]1 (z,<j»

-1 < z < 1,

o

< <j> < 27f

are smooth in

3/2

K]1 (z,<j»).

z, <j>,]1

on

< <j> < 27f,

for some sufficiently small

o

H]1 (z,<j», <j> + 8 (]1) +]1 1

]10; the region

-1 < z < 1,

corresponds to an annulus of width

the invariant circle for

N~]1

(which has radius

O(]1)

about

0(]1).

We

are going to produce an invariant circle inside this annulus. The qualitative behavior of off:

~]1

is now easy to read

can be written as

plus a small perturbation. rotation in the dire~tion.

~]1

<j>

The approximate

~

is simply a

direction and a contraction in the

z

Note, however, that the strength of the contrac-

tion goes to zero with

]1.

If this were not the case, we

could simply invoke known results about the persistence of attracting invariant circles under small perturbations.

As it

is, we need to make a slightly more detailed argument, exploiting the fact that the size of the perturbation goes to zero faster than the strength of the contraction. We are going to look for an invariant manifold of the form {z

u(<j»},

212


where i) ii) iii)

stant

u(¢)

is periodic in

lu(¢)

I

u(¢)

~ 1

for all

¢

with period

2n

¢

is Lipschitz continuous with Lipschitz con-

1

The space of all functions will be denoted by

u

satisfying i), ii) and iii)

U.

We shall give a proof based on the contraction mapping principle.

In outline, the argument goes as follows:

We

start with a manifold u (¢) },

M = {z with

u E U, and consider the new manifold

acting on small, <jllJM

M with

<jl lJ

.

We show that, for {z = {i(¢)}

again has the form

Thus, we construct a non-linear mapping

<jl M obtained by lJ lJ sufficiently for some

ff of

U

~ E U.

into it-

self by ffu

u.

We then prove, again for small positive contraction on

U

lJ, that

ff

is a

(with respect to the supremum norm) and

hence as a unique fixed point

u* •.

is the desired invariant circle.

The manifold

{z

u'" (¢)}

As a by-product of the

proof of contractivity, we prove this manifold is attracting in the following sense: Izl ~ 1, and let

Pick a starting point denote

n <jllJ (z, ¢) •

(z,¢)

with

Then

lim z - u*(¢ ) = O. n+ co n n It is not hard to see that the domain of attraction is much


Izl ~ 1.

larger than the annulus

213

In particular, it contains

everything inside the annulus except the fixed point at the center, but we shall not pursue this point. To carry out the argument outlined above, we must first construct the non-linear mapping

~

To find

9u(¢), we

should proceed as follows: A)

Show that there is a unique

component of

lJ

(u(¢),¢)

3/2

such that the

¢-

¢, i.e., such that

is

¢ - ¢ + 6 1 (lJ) + lJ

¢

KlJ(U(¢),¢) (21f).

*

(6.3)

and B)

Put 9u(¢)

equal to the

z-component of

(1-2lJ)u(¢) + lJ3/2 H (u(¢) ,¢). lJ

~u(¢)

(6.4)

In the estimates we are going to make, it will be convenient to introduce sup {I H O
(E

~

for each

0, then we may

near

make another perturbation to further reduce the dimension of E

o

f

Thus we see that by arbitrarily small perturbations of we can eventually ensure that

one real eigenvalue

~

consists of either or one pair of comk-l ~, with C depen-

~,

for all small

~.

Case 1. all small T

for all small

A~, A~

plex eigenvalues dence on

A~

Spec

~,

Spec with

antisyrnmetric.

consists of one real eigenvalue for

A = S + T, S symmetric, O Using the fact that the A are isometries Write

g

of Hilbert space, one checks that

Sand

T

.commute with

A • We choose an orthonormal basis for EO with respect to G which S is diagonal; then with respect to that basis, T ..

~J

Lemma 1Spec (AO+e:T) Proof.

If

T f 0, then for arbitrarily small

e: > 0,

has more than one point. Suppose

point for all small

Spec (AO+e:T)

e: > O.

consists of exactly one

Then for all small

e: > 0,

230


det(ZI-(AO+ET»

(z-A(E»n

=

and

where the coefficient of entries of

ET

z

are

n-l

n

o.

0:) one sees that

L

But the coefficient of

+

S .. S ..

i<j 11 JJ

of

E

L

i<j

unless

(T .. +ET. ,) lJ

2

lJ

By considering

z

(0)

n-2

Tr AO

=

(the diagonal nA(E).

for all small

But E > O.

is constant for small in this polynomial is

, which is not a constant function

o

O.

T

o

dim E .

=

on each side of (2.1)

Tr A = nA(O). Therefore A(E) = O This implies that det(zI-(AO+ET» E >

(2.1)

A , if T f 0 Proposition 2 G allows us to make small perturbations h of flu x J for

Since

which

T

commutes with

Spec DhO(O)

contains more than one point.

This allows

us to make another perturbation further reducing the dimension of have

T

o

E •

=

O.

f

Therefore for some perturbation of

we must

A = ±I. In fact, for some O A we must have ]1 = A]1 I for all small

In this situation

perturbation of

f

]1, A]1E lR, A = ±l. O Furthermore, for some such perturbation of

o

f , AG

is

"irreducible of

real type," i.e., the only elements of Hom EO O that commute with A are the multiples of I . To see this, G suppose

RE Hom EO, R

AO + ER

commutes with

of

°

Hence one can construct a perturbation

I.

flu x

commutes with AG and R f AI. Then 0' G A for all E and is not a multiple

J

such that

' ' t ur b a t lon re d uClng

DhO(O)

= AD

+ ER

h

of

and then another per-

dl"m EO.

Therefore for some arbitrarily small perturbAtion of f

we have


(1)

AO is irreducible of real type and

(2)

A j.l

G

Aj.l I

for all small

fE g

But the set (1 )

231

satisfying

j.l, with

1. 0

±l.

and (2) is open.

(1)

Therefore

and (2) hold generically. Case 2.

Spec

consists of one pair of complex

Aj.l

eigenvalues for all small

j.l.

For each

j.l

we have the

following commutative diagram: EO Aj.l

i

t

Fj.l

°

o

'IT

C

u

""'

IA" i

3
1 ~ -1 z ~ z + A PO(z) leaves the compact manifold S invariant O and is normally hyperbolic*to it. Suppose S is also invariIA I

Z

ant under the transformations for small folds

> 0

~

<en S~ C

ze

ia

(all real C!/,(S,<e n )

e~ E

there exist maps

e

is a diffeomorphism of

S

(2)

S

is invariant under

and

~

~

S

(3)

S

(4)

If

such that

~

a) .

Then

and mani-

such that

(1)

hyperbolic to

<en

1+

~

cI>~

S~.

is normally

.

-+ 0 A

cI>~

cI>~

onto

as

~

-+ O.

is a group of unitary transformations of commutes with

A

for all

~

and

AS

=

S,

*Normally

hyperbolic is defined as follows. Let S be a differentiable submanifold of a normed vector space E invariant under a diffeomorphism f: E -+ E. Let B -+ S be a subbundle of TE/S that is invariant under Df. Define P(DfIB) = lim sup sup 1 /Dfn(x) jBI Il/n. n-+ OO xES

f

is said to be normally hyper-

234


then

8

In fact, each +~]1

]1 then

]1 + 8]1 Ck(S,C n ).

is continuous from

is continuous from

A.

commutes with

]1

°

{]1:

R

k

C , k
O.

If we assume

IA]11

< 1

tain similar information for

~-l.

for ]1
1, and

for

o

is easy to see that each Let

=

IAol

and

z.

in

1

P (z) is A -invariant. ]1 G homogeneous polynomial of degree

Write

P(z)

=

(P (z),P (z», where l 2

2_ 2_ 2_ 2_ + CZ z + Dz z + Ez z 2 2 + Fz z Az lZl + Bz z l 2 l l 2 2 2 l 2 l

z

(z)

z

and P (z) 2 Since

=

2 2_ 2 2Qzlzl + Rz l z 2 z l + sZ2zl + Tz l z 2 + uz l z 2 z 2 + VZ 2 Z2 ·

p(azl,a

-1

z2)

=

(aPl(z),a

-1

P (z»

for all

a

2

lal = 1, we see that Since E

=

R.

P(z2,zl)

=

B

= C =0 =F =Q = S =T = U

(P 2 (z),P (z», it follows that l

A

with

o. v

and

Thus

One more calculation shows we can write

P(z)

in the form

2 2 2 2 P(z) = a(lzll +l z 2 1 ) (zl,z2) + b(lzll -l z 1 ) (zl,-z2); 2 a,b E 1, one can check that each A = {A g.. g E G} G

Ag

-1

x), where

and

Dkg

x E A. nkg-

l

=

°

is a linear isometry

of

E.

Thus

of

E.

The physical symmetries of the situation imply that Ag X]1

X]/g

for all

g E G.

assume that for small

~le

is a group of linear isometries

point for

]1 ,

is an attracting fixed ° can be proved; cf. Serrin [1,2)

X. (Actually this ]1 and Sections 2A and 9.) This is consistent with the experimental observation that Couette flow is stable for small rl •

2

Assume that at the first bifurcation point

a finite number of eigenvalues of

]10 >

°

, l only rl

(0) reach the imagin]10 ary axis, each of finite multiplicity. Then generically the stable manifold of the bifurcation gent at

(0,]10)

on which

A G

EO x R, where

to

OX

veE x R EO

must be tan-

is a subspace of

E

acts irreducibly.

Let us assume: (1)

EO -= R 2

(2)

If

g E SO (2) x 1, then

(3)

If

gEl x 0(2) , then

Ag A g

fixes acts on

EO

pointwise.

EO

exactly

like the corresponding element of the full orthogonal group 2 of 1R • Thus plane.

is essentially the full orthogonal group of the (0,]10)

is a "vague attractor" for

X]1 , then ac-

°

cording to Section 4, Example 1 (modified for vector fields), for each small

]1 >]10

near the origin of

we expect the following invariant sets

E x {]1}

(See Figure 7.1):


(1)

The origin, a zero of

(2)

A circle

is invariant under set is attracting in

S].l

X].l , non-attracting.

of zeroes of

.... 0 /\'G' S].l

243

as

X].l

in

V].l"

].l .... 0, and each

Each S

].l

S].l as a

E x {].l}.

,u-oxis

I

;I

I

,u=O

Figure 7.1 and Y = /\'gY , g E l x 0(2). Y 'Y E S].l 2 1 2 1 Y (x) = DgY (g-l x ). This means that Y and Y differ 2 2 1 1 Suppose

Then

only by a vertical rotation and/or flip of /\.SO(2) x 1

is the identity on

is fixed by the elements of

A.

Also, since

o

E , we see that each

/\.SO(2) x l ' i.e., each

YE S].l Y E S].l

exhibits horizontal rotational symmetry. This description of the

S].l

is consistent with the

experimentally observed phenomenon of Taylor cells.

When

].l

reaches some critical value one commonly observes that Couette flow breaks up into cells within which the fluid moves radically from inner cylinder to outer cylinder and back while it

244


continues its circular motion about the vertical axis (see Figure 7. 2) •

Figure 7.2 Taylor cells are a stable solution of the Navier-Stokes equations with horizontal rotational symmetry.

Since any Taylor

cell picture is no more likely to occur than its vertical translation by any distance (assuming the ends of the cylinder identified), we see why a whole circle of fixed points, interAI x 0 (2) , blossoms out from Couette flow. As recedes from (O,~) in our model; this corincreases, S

changed by

~

~

responds to the Taylor cells becoming "stronger," i.e., stronger radial movement. is still a zero of

X~'

Note that for

~

> ~O

Couette flow

but an unstable one.

(The above does not account for the fact that a Taylor cell vector field is invariant under a finite number of vertical translations, or, in our model, under a nontrivial subAl x SO(2)' To get this result one should assume 2 as follows: that for g E l x 0(2) , Ag acts on EO - lR View 0(2) as generated by numbers e, 0 < e < 271, where e group of

represents rotation through

e

degrees, and'by a flip

r.


We then represent

1 x 0(2)

by the homomorphism

(1,6)

~

in

0(2), the isometries of

n6,

(l,r)

representation is onto, we get the ever, each

S

~

47T

H = SO(2) x {O , n27T

n

variance under the new

, ... , AO

r.

2 (n-l)7T }. n

~2,

Because this

as above.

j.l

is invariant under

Yj.l E Sj.l

245

Now, how-

A , where H This additional in-

would be preserved in the follow-

G

ing. ) We now study the second bifurcation of

Xj.l'

tion is complicated by the circumstance that for zeroes of

X j.l

sional sets.

The situaj.l > j.l0

the

in which we are interested occur in one-dimenIn what follows we assume we have made a change

of coordinates so that Let

Yj.l E Sj.l'

eigenvalue

0

trum lies in Y~ =

AgYj.l

AgX(Y lI ) .

Re z < O.

for some

1

Notice that.if

DX(Y')·A j.l g Hence

Al x 0(2)

A(l,l)

Y~

E Sj.l

X(Y~)

also, then

=

XAg(Yj.l)

A 'DX(Y ), so g j.l

DX(Y') j.l

EO

and

DX(Y~).

Spec DX(Yj.l) = Spec

acts on

gonal group of the plane, Yj.l Al x 0(2)' namely

=

has an

and the rest of its spec-

g E l x 0(2), so

are conjugate. Assuming

of

j.l > j.l0' DXj.l(Yj.l)

of multiplicity

Therefore

~

DX(Yj.l)

For small

like the full ortho-

is fixed by just two elements and

A(l,r)' where

just the flip about the line determined by

Yj.l'

AO is (l,r) Since EO

is pointwise fixed by fixes

Yj.l

A • DX (Y }

g

j.l

is

A x l ' the subgroup of SO(2) Therefore DX(Y ).A ASO (2)x{1,r}' j.l g

for all

Suppose for is contained in multiplicity

1.

g E SO(2) x {l,rL j.l0 < j.l < j.ll

Re z < 0

and

YES j.l j.l , Spec DX j.l (Y j.l )

except for one eigenvalue

Suppose also that for

Yj.l 1

E Sj.l , 1

0

of

246


Spec

DX~

(y~

1

) n Re z = 0

consists of a finite number of iso-

1

lated eigenvalues (including y~

Fix 0(2)

1

S~

E

and let

r

Y

~l

, tangent to

Ty

~l

X

has a

Ell E

S~

l

Ell

~-axis,

1

is a finite-dimensional space and

~SO(2)x{1,d·

invariant under

local recurrence of U~S~

X

near

Since

y~

in a neighborhood of II: E x IR

Let

~

Then for each that

~SO(l)x{l,r}.

is fixed by

center manifold

of

now denote the unique element of

1

such that

where

0), each of finite multiplicity.

IIY / ~

->-

II Y II

~l

there is a unique

Y~.

is

~SO(2)x{1,r}.

y~

E

S~

such

Take this now as the de-

~l

finition of exactly

•

II Y II.

= IIY lI / "'1

~

contains all the points

, W

y~

contains all the

1 be projection onto the first factor.

E

near

1

W

is

Then the subgroup of

~G

Y~

that fixes

As in Section 1 we may identify a neighborhood of in E

l

W

with a neighborhood

Ell ~-axis

xlv

and

U

(Y

of

~l

,0,0)

with a vector field on

in

Ty

S ~l

~l

Y~

1

Ell

U, still called

X, in such a way that: (1)

y

corresponds to

~

(2) metries of (3)

~

X

,O,~).

1

S~

~l

~l

acts as a group of Hilbert iso-

SO(2)x{1,d

Ty

(y

Ell E •

1

on

U

cular, DX lI (y ,O,~) ... ~l

commutes with commutes with

~SO(2)x{l,r}·

In parti-

~SO(2)x{1,d·

Generically we have one of two situations: (1)

Spec

DX~ (y~

,O,~)

1

consists of the eigenvalue

0

247


with eigenspace

and a single real eigenvalue

A

)1

with (2)

Spec DX)1 (Y)1 ,0,)1) consists of the eigenvalue 0 1 with eigenspace Ty S and a single pair of complex conju)1 )11 gate eigenvalues A)1~ with Re A)1 = O.

X;

Let us assume (2) holds. Then for each )1 near 2 there is a unique subspace E of T U such that )1 (Y)1 ,0,)1) 1 2 DX(Y)1 ,0,)1) leaves invariant and DX(Y ,0,)1) IE has )11 )1 1 2 If we now regard E as a subonly the eigenvalues )1 set of

T

Y)1

S 1

the action of

)11

Ell E

1

Ell )1-axis, then

~SO(2)x{1,r}

because

is invariant under DX(Y)1 ,0,)1)

is.

1

2 Generically ~SO(2)x{1,r} acts irreducibly on E )11 2 2 1 _ _/ 2 _ .2 Assume E - R , so E = R"" and each E = R. Assume )11 )1 ~SO(2)x{1,r} acts on E~l like the rotations of the plane. 2 In other words, we assume ~ fixes E pointwise (l,r) )11 2 and ~SO(2) x 1 acts on E like the rotations of the plane. )11 2 2 Since ~ pointwise, it follows that E)1 fixes E (l,r) )11 1 2 is invariant under X)1' Because E is almost parallel to 1 )1 and is ~SO(2)X{1,r}-invariant, we can conclude that , 2 2 is in fact parallel to E)1l' hence E)1 is X)1-invariant.

Because of the last paragraph we see that we should consider the bifurcation of a vector field 2

E)1

x )1-axis, where 1

X' )1

XIE~

X'

defined on

(here we identify

2 E )1

with

2 x {)1 } ) • E We have that X' commutes with the rotations of )1 )11 E 2 - If and Spec OX I (0,)1) = {A,X"} where Re A < 0 for )1 )11 )1 < )11' Re A = 0 for )1 = )11' and we assume F.e A > 0 for

248


]J > ]J 1 •

If we assume

(O,]Jl)

we get a "Hopf bifurcation with symmetry" : of

X']J

that appear for

X' , ]Jl the closed orbits

is a vague attractor for

]J > ]Jl

are geometric circles (they

are invariant under rotations) and the motion on these circles is with constant velocity. Globally, the circle of fixed points cated into an attracting torus

T]J' for

has bifur]Jl ]J > ]Jl' where T]J S

is composed of closed orbits: one bifurcating off each fixed point of

S

The closed orbits are invariant under ]Jl ASO (2)x{1,r} and are interchanged by the elements of

AlxO (2)'

This bifurcation corresponds to the following experimental observation:

As

]J

increases, Taylor cells commonly become

"doubly periodic," i. e., a horizontal wave pattern is introduced, and this wave pattern rotates horizontally with constant velocity (see Figure 7.3).

Figure 7.3 This argument does not account for the horizontal periodicity commonly observed in the wave pattern.

*A

This

similar invariant set occurs for flow behind a cyclinder.


249

periodicity can be explained by assuming the representation of 2 _ 2 SO(2) x 1 in 0(2), the isometry group of E = R , is of

Vl

the form

{8,l)

~

n8.

A more serious problem is that our

argument predicts that a vertical flip symmetry should still be present after the second bifurcation-- the elements of are invariant under experimentally.

~(l,r).

This symmetry is not observed

The whole situation deserves further study.

The methods discussed here seem very fruitful towards this end.

TV

250


SECTION 8 BIFURCATION THEOREMS FOR PARTIAL DIFFERENTIAL EQUATIONS As we have seen in earlier sections, there are two methods generally available for proving bifurcation theorems. The first is the original method of Hopf, and the second is using invariant manifold techniques to reduce one to the finite (often two) dimensional case. For partial differential equations, such as the NavierStokes equations (see Section 1) the theorems as formulated by Hopf (see Section 5) or by Ruelle-Takens (see Sections 3,4) do not apply as stated.

The difficulty is precisely that the

vector fields generating the flows are usually not smooth functions on any reasonable Banach space. For partial differential equations, Hopf's method can be pushed through, provided the equations "parabolic"type.

~e

of a certain

This was done by Judovich [11], Iooss [3],

Joseph and Sattinger [1] and others.

In particular, the

methods do apply to the Navier-Stokes equations.

The result

251


is that if the spectral conditions of Hopf's theorem are fulfilled, then indeed a periodic solution will develop, and moreover, the stability analysis given earlier, applies.

The

crucial hypothesis needed in this method is analyticity of the solution in

t.

Here we wish to outline a different method for obtaining results of this type.

In fact, the earlier sections were

written in such a way as to make this method fairly clear: instead of utilizing smoothness of the generating vector field, or

t-analyticity of the solution, we make use of

F~.

smoothness of the flow

This seems to have technical

advantages when one considers the next bifurcation to invariant tori; analyticity in

t

is not enough to deal with the

Poincare map of a periodic solution (see Section 2B). It is useful to note that there are general results applicable to concrete evolutionary partial differential equations which enable the determination of the smoothness of their flows on convenient Banach spaces. in Dorroh-Marsden [1].

These results are found

We have reproduced some of the rele-

vant parts of this work along with useful background material in Section 8A for the reader's convenience. We shall begin by formulating the results in a general manner and then in Section 9 we will describe how this procedure can be effected for the Navier-Stokes equations.

In

the course of doing this we shall establish basic existence, uniqueness and smoothness results for the Navier-Stokes equations by using the method of Kato-Fujita [1] and results of Dorroh-Marsden [1]

(§8A).

It should be noted that bifurcation problems for partial

252


differential equations other than the Navier-Stokes equations are fairly common.

For instance in chemical reactions (see

Kopell-Howard [1,2,5]) and in population dynamics (see Section 10).

Problems in other subjects are probably of a simi-

lar type, such as in electric circuit theory and elastodynamics (see Stern [1], Ziegler [1] and Knops and Wilkes [1]).

It

seems likely that the real power of bifurcation theorems and periodic solutions is only beginning to be realized in applications.

The General Set-Up and Assumptions. We shall be considering a system of evolution equations of the general form dx/dt where

X

~

= X~(x),

x~o)

given,

is a densely defined nonlinear operator on a suit-

able function space E, a Banach space, and depends on a parameter and

~.

For example, X

may be the Navier-Stokes operator

~

the Reynolds number (see Section 1).

~

assumed to define unique local solutions semiflow

Ft

which maps

x(O)

to

This system is

x(t)

x(t), for

and thereby a ~

fixed,

t > O.

The key thing we need to know about the flow our system is that, for each fixed ping on the Banach space general).

E

(F

F

F

t

of

00

is a C mapt is only locally defined in t,~,

t We note (see Section 8A at the end of this section)

the properties that one usually has for shall assume are valid:

F

t

and which we


(a)

F

(b)

Ft +s

(c)

Ft(X)

253

is defined on an open subset of

t

=

F

0 F (where defined ); s t is separately (hence jointly [§8A]) con-

t, x E R+ x E.

tinuous in

We shall make two standing assumptions on the flow. ~he

first of these is

fixed

(8.1)

Smoothness Assumption.

t, F

is a

t

COO

Assume that for each E

map of (an open set in)

to E .

This is what we mean by a smooth semigroup. we cannot have smoothness in

t

Of course

since, in general, the gen-

will only be densely defined and is not a of F t 11 E to E. However, as explained in Section 8A smooth map of

erator

X

it is not unreasonable to expect smoothness in t

> O.

11, t

if

(This is the nonlinear analogue of "analytic semi-

groups" and holds for "parabolic type" equations).

We shall

need this below. In Section 9 we shall outline how one can check this assumption for the Navier-Stokes equations by using criteria applicable to a wide variety of systems.

general (For sys-

terns such as nonlinear wave equations, this is well known through the work of Segal [1] and others.) The second condition is (8.2) x

Continuation Assumption.

lie in a bounded set in

E

for all

Let t

Ft(X), for fixed for which

Ft(X)

254

THE HOPP BIPURCATION AND ITS APPLICATIONS

is defined.

Then

Ft(X)

is defined for all

t > O.

This merely states that our existence theorem for

F

t

is strong enough to guarantee that the only wayan orbit can fail to be defined is if it tends to infinity in a finite time.

This assumption is valid for most situations and in

particular for the Navier-Stokes equations. Suppose we have a fixed point of

i.e. , F (0) = 0 for all t > O. Letting t for fixed t, denote the Frechet derivative of F t is clearly a linear semigroup on E. Its genDFt(O) 0 E E;

sume to be OFt

Gt

=

F , which we may ast

erator, which is formally

DX(O), is therefore a densely de~

fined closed linear operator which represents the linearized equations.

*

Our hypotheses below will be concerned with the

spectrum of the linear semigroup

G , which, under suitable t conditions (Hille-Phillips [1]) is the exponential of the spectrum of

DX(O).

(Compare Section 2A).

The third assumption is: (8.3) family

F llt

Hypotheses on the Spectrum.

of smooth nonlinear semigroups defined for

an interval about in

t,x,ll,

Assume we have a

for

(a)

o

(b)

for

0 E R. t

> O.

Suppose

F~(X)

in

is jointly smooth

Assume: F ll .

is a fixed point for II < 0,

II

t'

the spectrum of

is contained in

* Even

if a semigroup is not smooth, it may make sense to linearize the equations and the flow. For example the flow of the Euler equations is

from

S

H

to

vative extends to a bounded operator on Marsden [1].

Hs - l , but the deris-l H ; cf. Dorroh-


[1

{z E C: (c)

1z

I

G)1 = D F)1 (x) I ; x t x=o t (resp. )1 > 0) the spectrum of

U, where

0

and an for

s > 0

)1 E [-s,s]

for

)1.

They are locally attracting and hence stable.

)l

> 0, one for each

near them are defined for all hood

and

There is a one-parameter family of closed orbits for

F)1 t

such

U

)1 > 0

t > O.

varying continuously with Solutions

There is a neighbor-

of the origin such that any closed orbit in

U

is

256


one of the above orbits. Note especially that near the periodic orbit, solutions are defined for all

t > O.

This is an important criterion

for global existence of solutions (see also Sattinger [1,2]). Of course one can consider generalizations:

.for in-

stance, when the system depends on many parameters with multipIe eigenvalues crossing or to a system with symmetry as was previously described.

Also, the bifurcation of periodic or-

bits to invariant tori can be proved in the same way. Proof of Theorem (Outline).

From our work in Section 2 we

know that the center manifold theorem applies to flows. for the smooth flow

Ft(X'~)

=

(F~(X),~)

we can deduce

the existence of a locally invariant center manifold three-manifold tangent to the tions of

G~(O).

~

Thus,

C; a

axis and the two eigendirec-

(The invariant manifold is attracting and

contains all the local recurrence, but

F

t

still is only a

local flow on this center). Now there is a remarkable property of smooth semiflows (going back to Bochner-Montgomery [1]; cf. Chernoff-Marsden [2]) which is proved in Section SA: flow

this is that the semi-

is generated by a COO vector field on the finite t dimensional manifold C; i.e., the original X restricts to a 00

C

F

vector field (defined at all points) on

C.

This trick then immediately reduces us to the Hopf theorem in two dimensions and back to Section 3.

[]

th~

proof can then be referred


257

Bifurcation to Invariant Tori. This can be carried out exactly as in Section 6. ever, as explained in Section 2B, we need to know that is smooth in

t,~,x

for

t > O.

How-

F~(X)

Then the Poincare map for

the closed orbit will be well defined and smooth and after we reduce to finite dimensions via the center manifold theorem as in Section 6, it will be a diffeomorphism by the corollary on p. 265.

Therefore we can indeed use exactly the same bi-

furcation theorems as in Section 6 for bifurcation to tori. To check the hypothesis of smoothness, one uses results of Section SA and Section 9.

258


SECTION 8A NOTES ON NONLINEAR SEMI GROUPS In this section we shall assemble some tools which are useful in the proofs of bifurcation theorems.

We begin with

some general properties of flows and semiflows (= nonlinear groups and nonlinear semigroups) following Chernoff-Marsden [1,2].

These include various important continuity and smooth-

ness properties.

Next, we give a basic criterion for when a

semiflow consists of smooth mappings following Dorroh-Marsden (1).

Flows and Semiflows. (8A.l)

Definitions.

is a collection of maps

Let

Ft:D

->-

D D

be a set.

defined for all

such that: 1)

FO

2)

Ft +s

Identity

and Ft

0

F

s

for all

A flow on

t,s E R.

t E R

D

THE HOP? BIFURCATION AND ITS APPLICATIONS

Note that for fixed F_

0

t

F

= Id, and t A semiflow on

Warning:

i. e. ,

D

is a collection of maps

a semiflow need not consist of bijections.

(8A.2)

Definition.

DC N. q)

where

is one to one and onto, since

F : D + D t t > 0, also satisfying 1) and 2) for t,s > O.

defined for

and

t, F t

259

Let

A local flow on

is open in the

that for all

where defined.

D

be a topological space

is a map

F: q)CIR

D

+

D,

= {x E D I (t, x) E q)} t satisfies (1) and (2)

and i f

!?)

F : q) + D, then F t t t The flmv is maximal i f

(s,Ft(X»E q)='!> (s+t,x) E q).

l
t x

F

write and

F

tn

we can choose

(x ) = F t tx n

n

(x ) n

0

F t +t -t (x n )· x n

Ft - t

=

x

(8A.5) 1)

t, x

~

F

is assumed open

so that the various

and Since

x n -> x, and

G

~:

G

=

Ft(X).

[]

be a separately continuous

on a metric space ~

Suppose that D

N, then the above ar-

is jointly continuous. DeN

is dense and that

Ft

which extends hy continuity to a flow on t

(x). x

is continuous, we have that

G x N -> N

gument also shows that

such that

Ft(x)

t

t-> t

be a topological group which is also a

Let

group action of

flow on

-

tx + tn -> F

(Yn) -> F _ (F (x» t tx tx

Let

2)

There

Remarks.

Baire space.

~

Ft(X)

is continuous for each

the same is true for each CO.

t

N.

is continuous at

is continuous at

Since for fixed Ft

F

Since the domain of definition of

(tx'x) • in

N.

= G =

Since this is a local

Therefore, we may assume

theorem, we may work in a chart. that

261

Indeed, let

x

x E N

xED.

is a M Then

and the extended flow is

x E D and x E N. Then n for fixed t, Ft(X ) -> Ft(X), so that t ~ Ft(X) is the n pointwise limit of continuous functions. Therefore, for each n

-> x, where

x E N, there is a second category set

Sx C R

such that if

x

262


t E Sx' then

~

t

Ft(X)

is continuous.

the proof of Theorem 8A.3 shows that 3)

Sx

The argument used in

=~

x E N.

for all

Many of these results can be generalized to the

case in which

N

is not locally metrizable; e.g. a manifold

modelled on a topological vector space (e.g,:

a manifold

modelled on a Banach space with the weak topology; - a "weak c.f. Ball [1].

manifold").

4)

The same argument also works for semiflows, at t > O.

least for

is also true at

If t

direct argument). fail at

=

t

0

=

N

0

is locally compact, joint continuity

a more

(Dorroh [1], but one can give

In general, however, joint continuity may

so it has to be postulated.

Using these methods we can obtain an interesting resuIt on the

t-continuity of the derivatives of a differenti-

able flow. (8A.6)

Theorem.

Let

N

be a Banach manifold.

cO

be a

flow (or local flow, or semiflow) on N. Let k be of class T for k > 1. Then for each j < k,

Ft

Tjpt: Tj (N) x E T

j

is jointly continuous in (Only

(N).

Proof.

t > 0

are working in a chart.

Therefore, TFt(X,V)

variable clearly ~

We may also assume that we

By assumption, this is continuous in the space

x, so we need to show it is continuous in

t.

But

lim n(F (x + ~) - F (x». Thus n+ oo tnt is the pointwise limit of continuous functions

DxFt(X)'V

DxFt(X)'V

and

By induction and Theorem 8A.3 we are reduced k = 1.

DxFt(X)'V).

t E IR

for semiflows.)

immediately to the case

t

Let

=

so has a dense set of points of

t-continuity.

The rest of

263


the proof is as in Remark 2 of (8A.S).

[]

The Generalized Bochner-Montgomery Theorem. For simplicity we will give the next result for the case of flat manifolds.

But it holds for general manifolds

M, as one sees by working in local charts. (8A.7)

Theorem.

(Chernoff-Marsden). Let

Ft

be a

jointly continuous flow on a Banach space IE. Suppose that, k for each t, F t is a C mapping, k > 1- Assume also that, for each

x E IE,

is the operator norm. in

t

and

x.

-+

IIDFt(X)-III Then

0

Ft(X)

t

as

-+

0, where

" . II

ck

is jointly of class

of the flow is k-l an everywhere-defined vector field of class C on IE. Proof. DFt(X)

Moreover the generator

X

Under the stated hypotheses, we can show that

is jointly continuous as a mapping from

R x IE

S((IE, IE), the latter being all bounded linear maps of IE

equipped with the norm topology.

¢(t,x)

for

DFt(X).

into IE

to

In fact, if we write

The chain rule implies the relation

¢(s+t,x) = ¢(s,Ft(x»·¢(t,x). We have separate continuity of

¢

(8A.I)

by assumption, and then

we can apply Baire's argument as in Theorem (8A.3), together with the identity (8A.I) to deduce joint continuity. Now let support.

Define

¢(t)

be a

J¢: IE

-+

COO E

function on

R

with compact

by (8A.2)

By joint continuity, we can differentiate under the integral

264


sign in (8A.2), thus obtaining

DJ~(X) = Now if

~

foo _00

~(t)DFt(x)dt.

(8A.3)

I IDJ~(X)-II I

approximates the a-function then

small; in particular

is

is invertible.

DJ~(X)

J~

function theorem it follows that

By the inverse k is a local c diffeo-

morphism. Moreover,

COO~(sjFS+t(X)dS J:oo~(S-t)Fs(X)dS. The latter is differentiable local

O.

Ck

ID

t

diffeomorphism, Ftx

and

is jointly

But then the flow identity shows that

for all

Since

ck ~he

for

J~

is a

t

near

same is true

0

t. (8A. 8 )

1)

x.

Remarks.

The above result is a non-linear generalization of

the fact well known in linear theory that a norm-continuous linear semigroup has a bounded generator (and hence fined for all

t E R, not merely

is de-

t > 0).

Furthermore, the same argument as above applies to semiflows and to local flows.

This has the amusing consek quence that a semiflow which is C and the derivative is norm continuous in

t

at

t

=

0

has integral curves which

are locally uniformly extendable backwards in time (since the k-l generator is C ). This is most significant when combined with the next remark.

265


2) of

If

DFt(X)

E

to

hypothesis.

is finite dimensional, the norm convergence I

follows automatically from the smoothness

Indeed, Theorem (8A.6) implies that

in the strong operator topology, i.e., DFt(X)V

DFt(X) v

7

7

I

for each

v; but for a finite-dimensional space this is the same as norm convergence. Accordingly if

is a finite-dimensional manifold, a k flow on M which is jointly continuous and c in the space k variable is jointly c • The latter is a classical result of Montgomery.

M

There is a generalization, due to Bochner and

Montgomery [1] for actions of finite dimensional Lie groups. This generalization can also be obtained by the methods used to prove Theorem (8A.7)

(cf. Chernoff-Marsden [2]).

Let us summarize a consequence of remarks (8A.8) that is useful. (8A.9)

Corollary.

on a Banach manifold

N.

Let

F

t

be a local

Suppose that

a finite dimensional submanifold

F

MeN.

Ck

semiflow

leaves invariant

t

Then on

locally reversible, is jointly Ck in t k-l generated by a C vector field on M.

and

x

M, F

t and is

is

Another fact worth pointing out is a result of Dorroh [1].

Namely, under the conditions of Theorem (8A.7), F

actually locally conjugate to a flow with a (rather than

l Ck - ).

Ck

t generator

is

266


Lipschitz Flows. (8A.IO)

Definitions.

flow) on a metric space that

Ft

Let

F

be a flow (or a semit M, e.g. a Banach manifold. We say

is Lipschitz provided that for each

constant

M t

t

there is a

such that

The least such constant is called the Lipschitz norm, We say that for every

o and a number

of

x,y E %'

for all

to E R, there is a neighborhood %' E: > 0, such that

and

t E [to-E: , to+E:] .

If

to be any bounded set, we say that the flow I

C

%'

can be taken

F

is semi-

t (This term was introduced by Segal. )

Lipschitz.

ILip '

is locally Lipschi~z provided that,

F

t E M and

X

I IFtl

Note that

flows are locally Lipschitz. Let

F

t

be

a continuous Lipschitz flow, and let

Mt = I IFtl I Li . Then (just as in the linear case) we have an p estimate of the form M < Me(3l t l t where

M, (3

are constants.

multiplicative:

Ms + t of the flow identity.

~

Indeed, note that

(8A.4) M t

is sub-

Ms'M t ; this is an immediate consequence

Moreover, we know sup d(F x,Fty)/d(x,y). xfy t

Thus

M

t

is lower semicontinuous, being the supremum of a

family of continuous functions.

In particular, M t

is meas-


urable.

267

But then an argument of Hille-Phillips [1, Thm.

7.6.5] shows that (8A.4) holds for some constants

M, B.

Uniqueness of Integral Curves. It is a familiar fact that integral curves of Lipschitz vector fields are uniquely determined by their initial values, but that there are continuous vector fields for which this is not the case.

*

On the other hand, it is known that integral

curves for generators of linear semigroups are unique.

The

following result shows that such uniqueness is a consequence of the local Lipschitz nature of the flow (cf. van Kampen's Theorem; Hartman [1, p. 35]). (8A.ll) Banach manifold

Theorem.

M, with domain

locally Lipschitz flow (a) xED, t

~

(b) borhood

F

t FtX

Ft.

X

be a vector field on the D.

Assume that

X

has a

More precisely, assume that:

is a group of bijections on is differentiable in

For each of

Let

D, and, for each

M, with

X E M and to E IR there is a neigho in M and an s > 0, such that in local

charts, 2 3 *The famous example is X(x) = x / on the line. In a Frechet space E, a continuous linear vector field S: X 7 X may have infinitely many integral curves with given initial data; viz. S(xO,x l ' ••• ) = (x l ,x 2 ' .•. ) on E the space of real sequences under pointwise convergence, or may have no integral curves with given initial data; viz. S(f) = df/dx on E = COO functions on [0,1] which vanish to all orders at 0 and 1. The result (8A.ll) is generalized significantly in DorrohMarsden [1].

268

for


x,y E

, and

t E [t -£' t +£]. Here the constant C o o is supposed to be independent of x, y and t. (In other ~

words, the local Lipschitz constant is supposed to be locally bounded in

t.

This is the case for a globally Lipschitz

flow, for example.) Conclusion:

if

c' (t) = X (c (t) ), then Proof. we assume

is a curve in

= Ft

c (t)

~,

Given

to' let

and a neighborhood ~

£ > 0

of

X

X

=

o as in

c(to).

o should be small enough so that

It-tol.:. £.

if

Define T

such that

(c (0) ) .

a Banach space.

hypothesis (b)i in addition, £ c(t)E~

D

We can work in a local chart (see (8A.13», so

M

Then choose

crt)

h(t)

F

=

small, Ilh(tH) - h(t)

II

t

_tc(t).

o

11 F t

Then, for

t

near

to' and

_t_TC(t+T) - F t _tc(t) I I 0

o

11 F t _t_TC(t+T) - F t _t_TFTC(t) I I o 0

.:.cllc(t+T) - FTC(t)ll. Moreover,

~

[C(t+T) - FTc(t)]

~ [e(t) - FTc(t)] h

+

X(c(t»

is differentiable, and

is constant, whence

c(t)

From this the relation (8A.12) applies to

Cl

[c(t+T) - c(t)] +

- X(c(t» hI (t)

= O.

= 0

as

T

+

O.

It follows that

Thus h(t)

=

F _ c(t ) for t near to. O t to [] c(t) = FtC(O) follows easily.

Corollary. flows

=~

Ft.

The conclusion of Theorem (8A.ll)


Proof.

We shall verify condition (b) of the hypothesis.

In a local chart, our results (see (8A.6» show that

269

DFt(X)·y

on joint continuity

is continuous jointly in

t, x, and

Hence, by the Banach-Steinhaus Theorem, for a given to

there is a convex neighborhood

I IDFt

so that

(x) I I :5.. C

x,yE%'

and

of

X o

and an

It-tol :5..

and

xE%'

if

value theorem then shows that if

~

11Ft (x) - Ft (y)

Xo

E.

y. and

E >

0

The mean

II :5.. C II x-y II

0

It-tol < E.

The above results generalize classical theorems of Kneser and Van Kampen. Note.

They easily generalize to semi-flows.

An explicit example of a continuous vector

field with a jointly continuous flow

Ft

for which the con-

clusion of Theorem (8A.ll) fails is the following well known example.

Define

¢(y)

=

¢' (y)

On

t

R

=

let

X

be defined by

ly13/2 sgn y.

lyll/2.

is a flow for

sgn t.

c(t)

c(O)

=

O.

is differentiable, with

=0

X.

Ft(X)

In particular

=

Ft(O)

3 2 It1 /

is another integral curve with

See Hartman [1) for more examples.

(8A.13) 1)

¢

It is easy to check that

¢(t+¢-l(x» But

Then

Remarks.

In case one wishes to work globally on

M and not

in charts, one should use the proper sort of metric as follows: Definition. a Banach space

E.

Let

N

be a Banach manifold modelled on

Let

d

be a metric on

is compatible with the structure of

N, if

N. d

We say gives the

d

270


topology of

(%', cjJ)

a chart all

and if given any

N

x,YE~,

2)

X

o

E N, there is, about

x o'

a(x ) , S (x ) such that for O O (x) - cjJ (y) II -< Sd(x,y) .

and constants

d(x,y) < all

cjJ

This method is not the one usually employed to

prove uniqueness of integral curves, for example, in

n IR..

One usually assumes that the vector field is locally Lipschitz and then uses integration to prove this result. call how this goes. field on

R

n

Let

X

Let us re-

be a locally Lipschitz vector

(or any Banach space).

Let

d(t)

and

c(t)

be two integral curves of X such that d(O) = c(O). Then t t Id(t)-c(t) I If X(d(s)) - X(c(s))dsl .:. K Id(s)-c(s) Ids.

fo

o

One then uses the fact (called Gronwall's inequality) that if t Kt Ka(s)ds, then a(t) ~ a(O)e a is such that a(t) ~

f

Therefore, we have

d(t)

o

=

c(t).

For purposes of partial dif-

ferential equations, however, it is important to have the resuIt as stated in (8A.II) because one is often able to find Lipschitz bounds for the constructed flows, but rarely on the given generator. 3)

Another method sometimes used to prove uniqueness

of integral curves for a vector field is called the energy method. tion x

=

for

H: D x D y

+

R+

X

with domain

Suppose there is a smooth func-

such that

H(x,y)

=

r

if and only if

and such that for any two integral curves X, dH(C(~~,d(t)) .:. K(t,d(O),c(O))H(c(t),d(t))

is locally bounded in clude that

X

t.

DeN

c

and

d

where

Then as in Remark 2 we can con-

has unique integral curves.

This method is

directly applicable to classical solutions of the Euler and Navier-Stokes equation,

~or

example.

K

271


Measurable flows. Under rather general conditions, continuity of a flow in the time variable can be deduced from measurability. example, we have the following result. (8A.14) Let M.

Ft

Theorem.

Let

M

For

See also Ball [2].

be a separable metric space.

be a flow (or local semiflow) of continuous maps on x E M, the map

Assume that, for each

~

t

Ft(x)

is Borel

measurable; that is, the inverse image of any open set is a

R.

Borel subset of

Then

F

is jointly continuous

t

tively, jointly continuous fOr Proof.

Because

M

t

(respec-

> 0).

is separable, the Borel function

is continuous when restricted to the complement of some first-category set

C C R

to

to' note that

and a sequence

tn

+

(cf. Bourbaki [1].)

U [C-(to-t n )]

~

D; that is, t +

n Fs(X)

=

D

n=l

is of the first category, hence there exists an s

Given

- to + s when

~

n

C

for all

n.

s

E R with

Accordingly,

Now apply the continuous

+

to deduce that is separately continuous, and the conclusion follows from Theorem 1.

[]

Theorems of this sort are well known for linear semigroups (see Yosida [1]

for instance).

Some Results on Time Dependent Linear Evolution Equations. In order to study smoothness criteria we shall need to make use of some results about linear evolution equations. These results are taken from Kato [1,4,5].

We begin by de-

272


(This exposition is adapted from

fining an evolution system. Dorroh-Marsden [1].) (8A.15) T > O.

Definition.

A subset

{U(t,s) 0

(bounded operators on i)

U(t,t)

ii)

Let

=

be a Banach space and

t

S; or

Examples of this type of semigroup


277

are common. (8A.24)

(Trotter, Feller)

Remark.

For a given semi-

with generator A E G(X,M,S) the space X can t be renormed so that I IFtl I < est Indeed the new norm is group

F

Illxlll

suplle-StF (x) II. t t>O

=

One should note however that it is not always possible to renorm

X

so that two semigroups simultaneously become

quasi-contractive. (8A.25) norm on in

X

Theorem.

For each

t, let

J

I

I It

be a new

equivalent to the original one and vary smoothly

t; i.e.,:

satisfying Ilxll t

< e

clt-sl

,

x E X,

t < T.

0 < s,

IIxll For each

s A(t)

t, let

be the generator of a quasi-contracin the norm II' II t' Then 2cT r,l = e 0 < t < T, with

tive, semigroup with constant {A(t)}

is stable on

X

S

1"ith

respect to any of the norms

-

IJ

lit'

The proof is actually a simple verification; see Kato [3, prop. 3. 4) • There is another useful criterion for the hypotheses of (8A.2l) to hold as follows: (8A.26)

Theorem.

Let

i) and iii) of (8A.2l) hold

and replace ii) by ii") Y

onto

X

There is a family such that

{Sit)}

of isomorphisms of

27S


S(t)A(t)S(t)-l B(t) E B(X) Assume

S:

where

= A(t)

+ B(t),

B:

[O,T) + B(X) is strongly continuous. l [O,T) + B(Y,X) is strongly C .

Then the conclusions of (SA.21) hold «ii")

;>

(ii»

and moreover, the forward differential equation holds, and the

evolution system is

Y-regular.

Two important approximation theorems follow (see Kato [5]) •

(SA.27)

Theorem.

ses of (SA.21), n

Let

0,1,2, ...

=

{A (t)} n

where there are uniform sta-

bility constants in i) and ii).

Assume

IIA-O(t)-A-(tlll +0 n Y,X uniformly in

t.

uniformlv in

t,s E [O,T), and

as

n +

Then

M,S, Ilsllooy,x'

=

00

strongly in

B(X),

Ilu (t,s) - U (t,s) Ily,x + 0 n o

Bn(t) + BO(t)

strongly in

{A (t)} satisfy the hypo then where the primitive constants

11~-llloo,y,X, IIBlloo,x,x' n.

uniformly in

S (t) + SO(t) n

Let

0,1,2, ..•

chosen independent of

that

Un(t,s) + UO(t,s)

Theorem.

ses of (SA.26), n

n +

n+ oo

as

00

(SA.28)

as

satisfy the hypothe-

in

B(Y)

in B(Y,X)

Ilslloo,y,X

can be

I IAO(t) - A (t) I I + 0 n y,X t, as in (SA. 26) , and in addition Assume

B(X) , Sn(t) + SO(t) uniformly in

uniformly in

t.

t,s E [O,T).

in Then,

B(Y ,X),

279


A Criterion for Smoothness We now give a result which tells us when a semiflow consists of smooth mappings.

The result is powerful when

used in conjunction with the above linear results.

The pres-

ent theorem is due to Dorroh-Marsden [11. to which we refer for additional results.

Before proceeding, the reader should

attempt Exercise 2.9 to get a feel for the situation. We use the following notation. spaces with Dey

Y

X

and

Yare Banach

densely and continuously embedded in

is open and

X.

F

is a continuous local semiflow on D. t We let G: D + X be such that F is a semiflow for G. t For p,q E D, and the line segment {p+r (q-p) 10 ::. r < l} C D set 1

Z(q,p)

=J o

DG(p+r(q-p))dr

the averaged derivative of

G

Assumptions.

is

close to

a)

G: D

b)

for fixed

+

erator is an extension of

fED

and

g E D

sufficiently

in

X

whose

{Z(Fsg, Fsf):

X-infinitesimal gen-

°

0 => Log Isnl < -~TO < O.

the remainder of the spectrum of

I A (TO) c

other than

Thus, 1

is


contained in a disk of radius strictly less than

311

1.

Because

of the continuity of the discrete spectrum the same is true of

GE(T,O)

GE(T,O)

for

or

E

E~(O).

GE(T,o)

The other proper value of

is found by studying the degenerate

operator

where

E(E)

2;if

is

r

[~l-GE]-ld~

of sufficiently small radius about One uses the expansion G(E) = 0(1)

GE

=

I

r

with 1

and

being a circle GE

= GE(T(E) ,0).

Ac (TO) + EG(l) + EG(E)

where

and methods of the theory of perturbations to

G(El

obtain

- (1) G

(in the basis

The bijection

0" ->-

E

-1

makes a correspondence between the proper values of -(1) and G(E). The proper values of G are 0 and -4T Y !a(l) /2 O Or 1 - 8rrs(1) (A-A

-8 rr s(1)sgn yO.

This gives us

1

(0"

-1

)

G(E)

and

r C

)

+ O(E 2 )

as proper values of

GE(T,o).

We now state Theorem 4. fied and

If the hypotheses of Theorem 2 are satis-

YO> 0, the bifurcation takes place for r

A E

hood

a

0

~+(AC) • ~+(A )

E [O,T]

c

There exist such that i f

j1

> 0

and a right hand neighbor-

A E ~+ (A ), and i f one can find

c such that the initial condition (E =

IA-A ), c

U

o

satisfies

312


then there exists

0t E [O,T]

such that

°

IIU(t)- ~(t+Ot,E) /1

.... exponentially when t .... 00, U(t) 9 being the solution of (9A.l) satisfying U(O) = 00' This case is not so simple since there is the proper value, 1.

The theorem results from the following lemma:

Lemma 9.

Given

V

E V(o), that is to say, satisfying

o

/ IPovol /9

2

':::']J 1 E , then the equa-

tion, 3V = A (t+o)V + M(V V) 3t E ' admits a unique solution in IIV(t) 11

that

CO(O,OO; 9)

.:. ]J2E2e-cr/2 t,

9

V(O)

,

n

= Va'

Cl(O,oo;H)

V t > O.

We must identify some of the terminology. is the projection onto a vector collinear to Po = 1 - Eo' nt[···];E,O}.

such

First, Eo

3-~ 3~(O,E), and

"

~(PoVO'E'O) == Eo ~O{nT[WO(PoVO'E'O)'E'O],

We have

The notation on the right hand side is associa-

ted with the following problem: v

A

vet) = GE(t,o)W o + 9 (V,V;E,O) + 9 (V,V;E,O) t t with A

9

t

(u,V; E,O)

I°

t G (t-T,T+O)P",

~t(U,V;E,O) = _I t

E

M[U(c),V(T)]dT,

u+T

oo

W is such that o in the Banach space

where V

provided with the norm

G (t-T,T+O)E o M[U(T),V(T)]dT, E +T EoW O = 0, and where one searches for St 9 = {V: t .... e Vet) E CO(0,00;9)}, S , with Ivl = sup IleStv(t) 1/ S 9 tE(O,oo)

S = cr/2. These estimates can be shown:


/Gs(t,o)Wol s

.::.

MIIIWollg'

~

1

v

-1

l~t(U,V;S,O) 113 ~ M ya- lulSlvl 2 l~t(U,V;s,o) 113 ~ M2 ya where

Y

/Iw o II

llO

independent of

~ llos2, there exists a unique

the above problem. Now, V(O)

s'

luiS/viS

is a bound for the bilinear form

that there exists a

313

V

M.

It results

such that for in ~

The solution is denoted

satisfying

V(t)

=

n (WO,S,O).

v t ~O[nT(WO's,o), nT(WO's,o);s,o]

nO(WO's,o) = W +

o

which gives after decomposition: v

E0 ~ a [ nT (W o ' s , 0) , n T (Wa's' 0) ; s ,0] , v

PoV o

=

Wo + PO~O[nT(WO,s,o),nT(WO's,o);s,o].

a aw

After having remarked that it is easy to find / IPovOI

I~

[nT(WO's,o)] W = a Gs(t,o), O o independent of s such that if

III

llls2, then the second of the above equations is

solvable with respect to and

Wo

=

WO(PoVO's,o)

W o

satisfies

tion is completely explained. too difficult.

by the implicit function theorem, 2

IIWOllg ~ llOS.

The proof from here on is not

The uniqueness comes from the uniqueness of

the solutions to (9A.l) on a bounded interval for ently small.

The nota-

s

suffici-

The lemma will be demonstrated as soon as it is

shown that the solution of our above problem is the same as the solution of the equation in the lemma. immediately utilizing in evaluating

a at

%t GS(t-T,T+O)

~

~t(V,V;s,o)

is seen that the manifold V(o)

=

But this follows

As(T+O)Gs(t-T,T+O) v

and

~t ~t(V,V;s,o).

(It

is nothing more than the set

314 of

THE HOPF BIFURCATION AND ITS APPLICATIONS Va

such that

W can be found with a satisfying the equations for P8Va and

and

The techniques of Iooss are very similar to those of Judovich [1-12]

(see also Bruslinskaya [2,3]).

These methods

are somewhat different from those of Hopf which were generalized to the context of nonlinear partial differential equations by Joseph and Sattinger [1]. Either of these methods is, nevertheless, basically functional-analytic in spirit.

The approach used in these

notes attempts to be more geometrical; each step is guided by some geometrical intuition such as invariant manifolds, Poincare maps etc.

The approach of Iooss, on the other hand,

has the advantage of presenting results in more "concrete" form, as, for example,

%'(t,E)

in a Taylor series in

This is also true of Hopf's method.

E.

However, stability cal-

culations (see Sections 4A, SA) are no easier using this method. Finally, it should be remarked that Iooss [6] presents analogous results to this paper for the case of the invariant torus (see Section 6).


315

SECTION 9B ON A PAPER OF KIRCHGASSNER AND KIELH6FFER BY O. RUIZ The purpose of this section. is to present the general idea that Kirchgassner and Kielhoffer [1] follow in resolving some problems of stability and bifurcation. The Taylor Model. This model consists of two coaxial cylinders of infinite length with radii

r'

constant angular velocities

1

r'

and w l

rotating with

2

and

W •

2

Due to the vis-

cosity an incompressible fluid rotates in the gap between the riwI(r;-ri) cylinders. If A is the Reynolds number

v

(v

the viscosity), we have for small values of

tion independent of

A, called the Couette flow.

A a soluAs

A in-

creases, several types of fluid motions are observed, the simplest of which is independent of

¢

and periodic in

when we consider cylindrical coordinates.

z,

If we restrict our

considerations to these kinds of flows and require that the

316

THE HOPF BIFURCATION AND ITS

solution

v

A~PLICATIONS

be invariant under the groups of translations

°

generated by z + z + 2n/0 and ¢ + ¢ + 2n, 0 > and l we consider the "basic" flow V (Vr,V¢,Vz)'P in cylindriT

cal coordinates, N-S

(we assume

V,P

are given) we may write the

equation in the form (a)

DtU - 6u + AL(V)u + A6q = -AN(u),

(b)

V'u

°,

ul r=r

P

P + q,

l

,r

0, u(Tx,t) = u(x,t),

2

(9B.l)

where

V + u,

v

d

(ar'

L(V)u

N(U)

d

=

0, 32")'

(gradient)

LO(V)U + (V·V)u + (u·V)V,

(u·V)u + Q(u).

The Benard model consists of a viscous fluid in the strip between 2 horizontal planes which moves under the influence of viscosity and the buoyance force, where the latter is caused by heating the lower plane. the upper plane is is

If the temperature of

T , and the temperature of the lower plane l

TO

(TO> T ), the gravity force generates a pressure disl is balanced by tribution which for small values of T - T 1

°

317


the viscous stress resulting in a linear temperature distribution.

is above a critical point of

If however

value, a convection motion is observed. k

a, h, g, v, P,

Let

denote the coefficient of volume expansion, the thickness

of the layer, the gravity, the kinematic viscosity, the density and the coefficient of thermodynamic conductivity respectively.

We use Cartesian coordinates where the

8

points opposite to the force of gravity, perature and

p

the pressure.

By the

x 3 -axis

denotes the tem-

N-S

equations we have

the following initial-value problem for an arbitrary reference flow

V, T, P.

If

W = (u,8), v = V + u, P = P + q, 8 = T + 8,

we have (a)

DtW -

-

(b)

V'w

0,

(c)

where

~w

+ AL(V)w + Vq = -N(W),

wl x

3

=0,1

(9B.2)

0,

wi t=O = wO

A = ag(To-Tl)h3/v2

(Reynolds numJ::er or Grashoff num-

ber)

v ~ik L

0 ik

L(V)w N(w)

(a/ax , a/ax , a/ax , 0) , l 3 2 ? 2 2 a a (-2 + L 2 + -2) (oik +,L Pr °i4) , ax aX aX 2 l 3 -0

Pr

k/v

° _,L °i40k3' i3 k4 Pr

0 L W + (V·V)w+ (u' V)V, (u· V)w. Since experimental evidence shows that the convection

318


takes place in a regular pattern of closed cells having the form of rolls, we are going to consider the class of solutions such that q{Tx,t)

w (x,t) ,

w(Tx,t)

= q{x,t)

T E T , and l

where

T

l

is the group generated by the translations

x

where

....

1

T

2

u{Tx,t)

Tu{x,t)

q(Tx,t)

q (x,t)

e(Tx,t)

e (x,t)

is the group of rotations generated by

Ta

Note.

(c?s " S1.n a 0

-sin a cos a 0

0

It is possible to show that all differential

operators in the differential equation preserve invariance under

Tl

and

T • 2

An interesting fact is that a necessary

condition for the existence of nontrivial solutions is a = 21f/n, n E {1,2,3,4,6}, and that there are only sible combinations of terns

7

pos-

n,a,S, which give different cell pat-

(no cell structure, rolls,

rec~angles,

hexagons, squares,

triangles) • Functional-Analytic Approach. The analogy between (9B.l) and (9B.2) in the Taylor's and


319

Benard's models suggests to the authors an abstract formulation of the bifurcation

and stability problem.

In this part

I am going to sketch the idea that the authors follow to convert the differential equations (9B.l) and (9B.2) in a suitable evolution equation in some Hilbert space. We may consider ary

an open subset of

R3

with bound2 which is supposed to be a two dimensional C -mani-

dD

D

denotes a group of translations and ~ its fundal mental region of periodicity which we may suppose is bounded. fold.

T

Assume

D

T~.

U 'lET 1

Now we consider the following sets

{wlw: ct(D)

+

(ct

closure)

Rn , infinitely often differentiable

ct(D), w(Tx) = w(x), T E T }, l CT,oo(D), supp weD}, in

{wlw E

0, w

Defining (v,w) 'm

where

L

I Y I:.m

(DYv,DYW),

f

Ivl m

(DYv(x)·DYW(x))dx

w

and

Y

is a multiindex of length

ing Hilbert spaces

aT l,a

H

For the Taylor problem we have

3; one obtains the follow-

320


3, D

n

and for the Benard problem n

=

4,

=

D

n

2 R x (0,1),

R

the real numbers

[0,21T/a) x [0,21T/13) x

(0,1)

We may consider that the differential equations of the T

form (9B.l) or (9B.2) are written with operators in

L2 .

From H. Weyl's lemma it is possible to consider L~ T E& where G contains the set of V such that J q T aT T q E Hl . We may use the orthogonal projection P: L ->- J 2 aT

T G ,

D w - ~w + AL(V)w + t with the additional conditions of boundary and

for removing the differential equation AVq = -AN(w)

periodicity, to a differential equation in

If we con-

T

sider q E H , pVq == 0 and since we look for solutions on l aT aT J , we have Pu = u, and we may write the new equation in J as dw + P~u + APL(V)

with initial condition

(9B.3)

-APN(w)

at

o

wt=O = w .

The authors write (9B.3) in the form dw + A(A)w + h(A)R(w) dt where

-

A(A) =

P~

+ APL(V), R(W) = PN(w)

for Taylor's model and

heAl

o

w ,

0,

1

and where

heAl

A

for Benard's model.

Also l they show using a result of Kato-Fujita, that it is possible to define fractional powers of

-

A(A)

by


-

A(A)

-S

=

1 foooexP(-A(A)t)t S-l f(Ff dt,

321

13 > 0,

and that this operator is invertible. The ahove fact is useful in resolving some bifurcation problems in the stationary case.

If

A

= P~,

M(V)

=

PL(V)

the stationary equation form (9B.3) is (Sp)

Aw + AM(V)w + h(A)R(w)

where of

V

° V E Domain

is any known stationary solution with

A. If we consider the substitution K(V)

= v

-1/4 3/4 A M(V)A-

we may write (Sp) in the form v + AK(V) + h(A)T(v)

=

0.

If one uses a theorem of Krasnoselskii, it is possible to show the following theorem. Theorem 4.1. an eigenvalue of i)

Let be

K

A. E R, A. f J

°

be

(-A. )-1

and

J

of odd multiplicity, then

in every neighborhood of

there exists

J

°f

(A,W),

W E D(A)

(A. ,0)

J

in

such that

solves the

W

stationary equation (SP). ii)

if

(_A.)-l J

exists a unique curve a f

°

which solves

is a simple eigenvalue, then there (A(a),w(a»

(SP), moreover

Now if we assume that

weal f

such that (A(O),W(O»

V E C(IT)

and

dD

=

°

for

(A.,O). J

is a

00

C

manifold which is satisfied for the Taylor and B~nard Problem,

322


and we consider solutions of (Sp) on the space

Hm+ 2 m > 3/2, we may write the (SP) equation in the form

=

Amw + \Mw + h(\)R(w)

where

is an operator whose domain is

that in

R(w) = PN(w), N(w)

Hl,a'

°

(SP) ,

HI, a C PRm , Am (w) = AW, wED (A) •

n

n H m+2 If besides we consider

=

D(Am)

is an arbitrary polynomial opera-

tor including differentiation operators up to the order and

MA~l

K m

we may obtain the following theorem.

Theorem 4.2. let that

Let

~

cllwlm+2

for every eigenvalue plicity, tions

V E C(D), dD

be a

M such that there exists constants IMWlm+l

w

condition

2,

and

(_\.)-1 J

C l

/MA:lwl m+ l of

Coo-manifold;

~

and

C2

c2lwlm.

are in

Then

K, \. f 0, of odd multiJ

m

is a bifurcation point of (SP) '.

(\ . ,0) J

such

The solu-

and fulfill the boundary

cO)")

wl aD = 0.

We may note that this theorem shows that we may obtain a strong regularity for the branching solutions.

Besides,

we note that in theorems 4.1 and 4.2 the existence of nontrivial solutions of (SP) is reduced to the investigation of the spectrum of

K

or

Km•

Now, we are going to apply these theorems to the Taylor's and Benard models. Taylor Model. where

o v

For this model we take

is the Couette flow.

the solution is given by

v

°

K

=

A:l/4M(vO)A-3/4

In cylindrical coordinates

°

(O,v~,O)

where

°

v~ = ar + b/r


a

323

b

a ~ 0, v~ > 0, Synge shows that the Couette flow is

When

locally stable. For

a < 0, v~(r) > 0, Velte and Judovich proved the

following theorem for the Theorem 4.3. (r ,r ), T 2 l l

K

Let be

operator. a < 0,

°

v~(r)

>

°

r E

for

the group of translations generated by

2TI/a; ~ + ~ + 2TI, a > 0.

Then for all

z

+

z +

a > 0, except at

most a countable number of positive numbers, there exists a countably many sets of real simple eigenvalues K.

Every point

(Ai,O) E R x D(A)

(-A.) -1

of

1.

is a bifurcation point of

the stationary problem where exactly one nontrivial solution branch

(A(~),w(~»

emanates.

These solutions are Taylor

vortices. Strong experimental evidence suggest that all solutions branching off

(Ai,O)

where

Ai f Al

are unstable;

however, no proof is known. Benard model. a =

some

In this model if

(~2+S2)1/2, ~,. S

like on page3l8, it is known that for

Al (a), A E [O,A l ], w

=

°

is the only solution of the

stationary problem. For this model the bifurcation picture is determined by the spectrum of

K = A-l/4M(VO)A-3/4, where

in Cartesian coordinates by

vO

is given

324


In order to obtain simple eigenvalues, Judovich introduces even solutions

u(-x) =

(-u (x) ,-u (x) ,u (x», 8(-x) = 8(x), 3 2 l g(-x) = -q(x), and he shows in his articles, On the origin of

convection (Judovich [6]) and Free convection and bifurcation (1967) the following theorem. Theorem 4.6. prximately all

a

i) and

characteristic values

The Benard problem possesses for ap-

B

countably many simple positive

A.• 1

Furthermore (A.1 ,0) .

E R

x D(A)

is

a bifurcation point. ii)

If

n, a,

B

are chosen according to the note on

pg. 318, the branches emanating from

(Ai,O)

are doubly

periodic, rolls, hexagons, rectangles, and triangles. iii)

If

A l

denotes the smallest characteristic value,

then the nontrivial solution branches to the right of and permits the parametrization F: R

+

D(A)

W(A) = ±(A-Al)1/2 F (A)

A~

where

is holomorphic in

(A-A )1/2. l It is interesting to observe that since the character-

istic values are determined only by ferentials

a,

B

we may consider dif-

0,

with the same value of

0,

and to note that

we have solutions of every possible cell structure emanating from each bifurcation point. Stability. About this topic I am going to give a short description of the principal results. It is known that the basic solution loses stability as the assumption that

Ac

in the past section.

Under

simple, the nontrivial


solution branch emanating from A > Al

and is unstable for

(AI,O)

A < AI'

325

gains stability for This result can be de-

rived using Leray-Schauder degree or by analytic perturbation methods. V = Vo + w+

Precisely if we take

stationary solution of (SP) V

=

A(A)

A + AM(V),

W

strict solutions in

=

0

where

is called stable if for

is stable in sense with respect to

D(A S ), 3/4 < S < 1.

(Strict solution in

the sense of Kato-Fujita of the article), and unstable if it

is not stable in

the nontrivial solution branch lal < I

(A(O),V(O»

=

is a

(Alv O)

ST.

V

is called

If we consider that

(A(a),V(a»

R x

in

can be written in the form

a(A) r E N

F(A ) f 0 I (valid conditions for Benard's and Taylor's models (Theorem

where

F: R +D(A)

is analytic in

(A-AI)I/r

and

4.6 and Corollary 4.4), we have the following Theorem or Lemma 5.6. Lemma 5.6.

~

nonvanishing

Assume

A

C

=

Al

in the spectrum of

be a simple characteristic value of value of

Re

~

>

a > 0

G(A(AI,V O» ' K(V O)' 0

for all

Let

a simple eigen-

A

is

restricted to a suitable neighborhood

Al i)

V

o is stable for

A < Al

and unstable for

A > A I ii)

Al

A(AI,v O)'

Then, if of

and

veAl

is stable for

A > Al

and unstable for

326


For the B~nard's model, the assumptions of Lemma 5.6 n, a, S; A 1

are satisfied for fixed K(V O)

istic value of

is a simple character-

by Theorem 4.6 and

A

C

= A

1

follows

from Lemma 4.50f the article. For the Taylor's model only the simplicity of K(V )

a characteristic value of of

~

= 0

in

O

0(AA ,V ) 1 O

is known.

A 1

as

The simplicity

is an open problem.

However, we may give the following theorem. Theorem 5.7.

i)

For the Benard's problem, every solu-

tion with a given cell pattern (fixed

n, a, S)

exists in

some right neighborhood of in

D(A S ), S E (3/4,1).

A and is asymptotically stable 1 The basic solution V is asymptoti-

o

cally stable for ii)

A < A and unstable for A > A • 1 1 For the Taylor's problem, let the assumptions of

Lemma 5.6 on the spectrum of every period exists for

(0

fixed) V(A)

A(A;V )

O

be valid.

Then for

is asymptotically stable if it

A > A , and is unstable if it exists for 1

A < A • 1

Finally, we remark that these results can also be obtained using the invariant manifold approach. example, Exercise 4.3).

(See, for

That this is possible was noted a1-

ready by Rue11e-Takens [1] in their elegant and simple proof of Velte's theorem.

We also note that Prodi's basic

results relating the spectral and stability properties of the Navier-Stokes equations are contained in the smoothness properties of the flow from Section 9 and the results of Section 2A.

327


SECTION 10 BIFURCATION PHENOMENA IN POPULATION MODELS BY G. OSTER AND J. GUCKENHEIMER 1.

Introduction:

The Role of Bifurcations in Population

Models. Biological systems tend to be considerably more complex than those studied in physics or chemistry.

In analyz-

ing models, one is frequently presented with two alternatives: either resorting to brute force computer simulation or to reducing the model further via such drastic approximations as to render it biologically uninteresting. is

attractive.

Neither alternative

Indeed, the former alternative is hardly

viable for most situations in ecology since sufficient data is rarely available to quantitatively validate a model.

This

contrasts starkly with the physical sciences where small differences can often discriminate between competing theories. The situation is such that many ecologists seriously question

328


whether mathematics can play any useful role in biology. Some claim that there has not yet been a single fundamental advance in biology attributable to mathematical theory.

*

Where complex systems are concerned, they assert that the appropriate language is English, not mathematical.

A typical

attitude among biologists is that models are useful only insofar as they explain the unknown or suggest new experiments. Such models are hard to come by. In the face of such cynicism, perhaps mathematicians who would dabble in biology should set themselves more modest goals.

Rather than presenting the biological community with

an exhaustive analysis of an interesting model, it might be better to produce a "softer" analysis of a meaningful model. From this viewpoint, the role of mathematics is not to generate proofs, but to act as a guide to one's intuition in perceiving what nature is up to.

This is no excuse for avoiding

hard analysis where it can be done, but as models mimic nature more closely it becomes

harder to prove theorems.

In this spirit, we shall discuss several instances where some concepts of bifurcation theory have proved useful in ecological

modelling.

We shall discuss (briefly) bifur-

cation phenomena in three kinds of population models:

(i) dis-

crete generation populations modelled by difference equations, (ii) continuously breeding populations modelled by ordinary differential equations, and (iii) populations with age structure which require partial or functional differential equations.

* Perhaps excluding the Hardy-Weinberg law--which is trivial mathematically.


329

These represent three successive stages of increasing biological realism as well as mathematical intractibility.

Thus

we shall proceed from less realistic models based upon solid mathematical foundations to more realistic models based upon mathematical intuition.

In all cases, however, the useful-

ness of bifurcation theory transcends our ability to cite theorems.

By furnishing a qualitative modelling mechanism,

it provides a conceptual framework within which we can view a number of important ecological processes.

2.

Populations with Discrete Generations. (2.1)

once a year.

Consider an insect population which breeds A plot of the total number of individuals as a

function of time might look like Figure lO.la:

--L _ _----L

'--"'-'-.L......L_-..:..~!.....L.L.L:lo...----,,

.J-

---4._ t

Figure lO.la If we are only interested in either the mean, total or average number each year, we might consider an approximate difference equation model as shown in Figure lQ.ln:

330


N N orN tot' ov max

o

2

o

4

3

t

Figure 10.lb i.e., an equation of the form: (2.1)

Models of this kind are commonly employed in entomology (Hassell

&

May,

general, F(·)

[1]; Varley, Gradwell

&

Hassell, [1]).

In

will have the shape shown in Figure 10.2:

N

Figure 10.2 The reason for this is that as the

population density in-

creases crowding effects, such as competition for food, tend to increase deathrates and decrease birthrates.

331


Typical functional forms that have been employed in modelling insect populations are: r(l-Nt/K) Nt + l

Nte

Nt + l

Nt

(2.2a)

A (l+Nt)b

(2.2b)

Nt a 0, and perhaps the initial condition.

=

This, in turn, depends on the particular parameter values, such as

r

in (2.2a).

Let us consider (2.2a) as the proto-

type for our discussion. fixed point

N = K is

The eigenvalue of

=

A(r)

F'

tion rate increases past 2, A(r) circle and

N

F(·)

(N) = l-r.

at the

As the reproduc-

moves across the unit

ceases to be an attractor.

F

However, if

i

has a critical point--as we have supposed in the models (2.2)

F

--then the composition of

with itself will have at least

3 critical points and we can look at the period-2 fixed points: (2.3) F2

and the eigenvalues of

at these points: (2.4)

The stability of the pair of period-2 points, which have split off from the original fixed point as

r

crosses

2, is

determined by the eigenvalues,

A (r). Initially stable, 2 these period-2 points bifurcate when jA (r) I ~ 1. In this 2 case, the nature of the bifurcation depends on whether it occurs at

+1

or

-1.

At

A < -1

each period 2 point bifur-

cates into a pair of stable points with period 4. cess continues as

r

increases:

bifurcations from

giving rise to pairs of attracting points of period bifurcations from

Ak(r)

=

+1

This proAk(r) 2k

either create or 'destroy

= -1

while

333


(c.f., Figure 10.3.)

periodic points.

bifurcation from A Ir)=+1 ./ k )f \

(

\ \

E

bifurcation from

\

,Jj

\';/ r

Alr)=-I k

I--

~ -

r

Figure 10.3 The orbit generated by

F(·)

becomes successively more com-

plicated with each bifurcation--the initially stable fixed point splitting to orbits of successively higher periods. This can continue indefinitely, with bifurcation points occuring closer and closer together.

As

r

is increased past

2, exciting successively stable higher periodic orbits, there can occur a limit point, re' beyond which completely aperiodic points appear.

That is, orbits are genetated -which do not

tend asymptotically to a periodic orbit.

Sufficient condi-

tions for such aperiodic orbits to exist has been given by Li and Yorke [1].

If

F(')

folds some interval onto it-

self as shown in Figure 10.4a, then there exist aperiodic points (i.e., initial conditions which do not lie in the domain of attraction of any stable fixed point).

334


1--: d

d

o .-J

~~ir-.-------------,-,--,----,c--..I...----==-age

Evolution of the host population toward a stable age distribution in the absence of the parasite. The initial waves were induced by the periodic addition of adult females.

c

....oo :J

a. o a.

10°1--------1.------.1...------===- Age Stroboscopic shot at one generation-time intervals (-51 days) of a "pulse" of hosts which evolves to a stable periodic solution. Figure 10.12.

Simulation of Host-Parasite System

352


leads us to conclude that an intuitive explanation for the population waves is a bifurcation phenomenon.

Moreover, it is

this mechanism which gives us a satisfying explanation for how the two populations are able to coexist in a homogeneous environment--the bifurcation phenomenon creates a "phase niche" within which the host can escape total annihilation by the parasite. (4.5)

Two other phenomena involving the age structure

deserve comment.

First, an examination of Figure lO.7b shows

that if a periodic signal (in this case a periodic food supply) is applied to the population system, the forcing frequency interacts with the natural resonant frequency to produce a "beat" frequency with a wavelength longer than either component (Oster and Auslander,

[1]).

This suggests a pos-

sible explanation for certain population periodicities observed in nature which do not appear to track any apparent environmental cycle.

Secondly, there appear

to be component

frequencies higher than that of the major resonance.

This

suggests that secondary bifurcations from the basic cycle may playa role in the dynamics.

If the age system is discretized

along the characteristics, the resulting

set of difference

equations corresponds to the Leslie model well known to demographers.

Beddington and Free [1] simulated such a discrete

age class model and found that, as certain parameters are varied, transitions to chaotic behavior occurred reminiscent of the aperiodic orbits discussed in Section 2 for single difference equations.

Thus it appears that the bifurcation

phenomenon can supply a satisfying mathematical mechanism for


explaining not only cyclic regularities in population dynamics, but perhaps some of the irregularities as well.

353

354


SECTION 11 A MATHEMATICAL MODEL OF TWO CELLS VIA TURING'S EQUATION BY S. SMALE (11.1)

Here we describe a mathematical model in the

field of cellular biology.

It is a model for two similar

cells which interact via diffusion past a membrane.

Each

cell by itself is inert or dead in the sense that the concentrations of its enzymes achieve a constant equilibrium. interaction however, the cellular system pulses

In

(or expressed

perhaps over dramatically, becomes alive:) in the sense that the concentrations of the enzymes in each cell will oscillate indefinitely.

Of course we are using an extremely sim-

plified picture of actual cells. The model is an example of Turing's equations of cellular biology [1] which are described in the next section.

I

would like to thank H. Hartman for bringing to my attention Reprinted with permission of the publisher, American Mathematical Society, from Lectures in Applied Mathematics. Copyright © 1974, Volume 6, pp. 15-26.


355

the importance of these equations and for showing me Turing's paper. The general idea of our model is to first give abstractly an example of a dynamical system for the chemical kinetics of four chemicals (or enzymes).

This dynamics repre-

sents the reaction of these chemicals with each other and has the property that every solution tends to one unique stationary point or equilibrium in the space of concentrations as time goes to

This is the sense in which the cell is dead,

where the cell consists of these four chemicals.

After a

period of transition, the chemical system stays at equilibrium.

We emphasize that our reaction process is an abstract

mathematical one and that we have not tried to find four chemicals with this kind of chemical kinetics. The next step is to give four positive diffusion constants for the membrane which could describe the diffusion of the four chemicals past the membrane.

The cellular system

consisting of the two cells separated by the membrane will be described by differential equations according to Turing. With our choice of the chemical kinetics and diffusion constants this new dynamical system will have a nontrivial periodic solution and essentially every solution will tend to this periodic solution.

Thus no matter what the initial con-

ditions, the interacting system will tend toward an oscillation (with fixed period).

After an interval of transition,

it will oscillate. Both the equilibrium of the isolated cell and the oscillating solution of the interacting system described above are stable (or are attractors) and even stable in a global

356

way.


But more than this, the equations themselves are stable

so that any equations near ours have the same properties. dynamical systems are "structurally stable."

Our

This gives them

at least a physical possibility of occurring. In Turing's original paper some examples of Turing's equations are given with oscillation.

However, these are

linear and it is impossible to have an oscillation in any structurally stable linear dynamical system.

Linear analysis

can be used primarily to understand the neighborhood of an equilibrium solution.

Development of linear Turing theory has

been carried very far in the very pretty paper of Othmer and Scriven [1]. Our example has reasonable boundary conditions, as one or more of the concentrations goes to

0

or to

00.

Also, a

complete phase portrait of the differential equation in eight dimensions for the cellular system is obtained. This example and Turing's equations as well go beyond biology.

The model here shows how the linear coupling of two

different kinds of processes, each process in itself stationary, can produce an oscillation.

This is the coupling of

transport processes (in this case diffusion across a membrane) and transformation processes (in this case chemical reactions).

In ecology, Turing's equations have another inter-

pretation; see, e.g., Levin [1].

Also S. Boorman's Harvard

Thesis has a related interpretation and analysis. We finally remark that our results could equally well be interpreted as putting a single cell into an environment which could start it pulsating.


(11.2) tions [1].

357

We give a brief description of Turing's equa-

These are sometimes called Rashevsky-Turing equa-

tions because of earlier work of Rashevsky on this subject. One starts from a cell-complex, in either the biologicalor mathematical sense of the word, e.g., as given in Figure 11. 1.

Figure 11.1 From the mathematical point of view this system is a cell-complex structure on a two- or three-dimensional manifold N

(e.g., an open set of

or

cells and they are numbered

Suppose there are

1, ... ,N.

It is supposed that the cells contain enzymes (or chemicals, or "morphogens" in the terminology of Turing) which react witp each other. chemicals.

Suppose there are

x

i

of these

Then the state space for each cell is the space 1 m (x , •.• , x ),

where

m

x

i

> 0, each

denotes the concentration of the

i},

i th chemical.

The state space for the system under discussion is the Cartesian product

P x

x P

(N times) or

(p)

N

•

state for this cellular system is a point, x E (p)N,

Thus a

358

x =


(xl' ••• ,x n )

with each

xi

E P

giving all the concentra-

i th cell, i = 1, ••• ,N.

tions for the

The dynamics for the

typical cell by itself is given by an ordinary differential equation on and

P; this can be described by a map

dx/dt = R(x).

This

R

R: P

+

Rm

describes how the chemicals react

with each other in that cell; the subject of chemical kinetics deals with the nature of of

dx/dt = R(x)

on

P

Most typically, the dynamics

is described by the existence of a

x E P

single equilibrium

x;

R.

such that every solution tends to

at least this will be the case if conservation laws have

been taken into account one way or another (as in the situation in Turing [lJ or Othmer and Scriven [lJ). A natural boundary condition on this equation is that if

x E P, x

ponent

k

R

(x)

=

(xl, .•• ,xm), with of

R(x)

xk

= 0,

then the

k th com-

is positive.

So far we have discussed each cell in some kind of hypothetical isolation.

The cells are separated from each

other by a membrane which allows for diffusion from one cell to adjoining cells.

In the simplest case of diffusion, if a

certain chemical has a bigger concentration in the

r th cell

than an adjoining well, then the concentration of that chemical decreases in the difference.

r th cell, at a rate proportional to the

This gives some motivation to Turing's equations

which add this diffusion term to give an interaction between the cells.

L

(T)

k

1, .•• ,N.

i E set of cells adjoining kth cell Let us explain (T)

in detail.

The first term above,


k th cell.

gives the chemical kinetics in the

R(X k )

359

The

2nd term above describes the diffusion processes between cells.

m xi - xk E R

Thus

represents the difference of the

concentrations of all the chemicals between the k th cells. to

Rm

Here

or an

~ik

m x m

is a linear transformation from matrix.

In the most

case, and the case we develop here, gonal matrix.

i th and

~ik

Rm

natural simple

is a positive dia-

Also the chemical kinetics for each cell is

considered the same.

(T)

is a 1st order system of ordinary

differential equations on the state space

(p)N

9f the bio-

logical system, and will tell how a state moves in time. We specialize to a case of 2 cells adjoined along a membrane which is the example pursued in the rest of the paper.

2nd cell

I st cell

Figure 11.2 R(Zl) + ~(z2-z1)' R(z2) + ~ (zl-z2)· This is an equation on Here

~

will be of the form

P x P

with

(zl,z2) E P x P.

360


with each

~i

O.

>

This is the most simple case.

also write (T 2 ) as given by a vector field

X

We may

on

P x P

where

(11.3)

Here we state our results.

Main Theorem.

a}.

zi > ~l'

...

Let

There exists a smooth

'~4 >

0

1 2 3 4 (z ,z ,z ,z ),

4

P={zER,z

4

00

(C)~

R: P

with the following properties (1),

R , and

->-

(3)

(2),

below: (1)

The differential equation

dz/dt = R(z)

on

P

is globallv asymptotically stable and is structurally stable. In other words there is a unique equilibrium

zE

P

-z

of the differential equation and every solution tends to as

t

->-

That

00.

equation

R

is structurally stable means that the has the same structural properties

dz/dt = RO(Z)

is a c l perturbation of R. (See O [1] for details on these kinds of stability and background on

as

dz/dt = R(z)

if

R

ordering differential equations.) (2)

On

P x P

with

z = (zl,z2) E P x P

the diff-

erential system, R(zl) + ~ (z2- z l)'

(T)

R(z2) +

~

~(zl-z2)J

is a "global oscillator" and is structurally stable. More precisely, a global oscillator on dynamical system which solution

y

P x P

is a

has a nontrivial attracting periodic

and except for a closed set

L of measvre

0,

..


y, as

everv solution tends to

t

(z,z)

solution tends to the unique equilibrium

Rk(ZO)

I

is

every

of (T).

The boundary conditions are reasonable in the

following way. for some

I

In fact, here

+

a six-dimensional smoothly imbedded cell and on

(3)

361

k

Let

satisfy

between one and four.

of

R(ZO)

z

k

o

k th component

Then the

is positive.

The above theorem gives mathematical precision to the statements made in Section 1 via the interpretation of (T) as in Section 2. The example is related to the phenomena of Hopf bifurcation; see Section 3. and

VIe

Our analysis is more global however,

have a complete description of the phase portrait. (11.4)

In this section we show how to construct the

differential equations of the previous section.

Q: P

Towards obtaining the vector field main theorem of Section 3 we will find a on

00

C

of the

vector field

and

with these properties:

(1) the equation (2)

II z II

has the origin, 0, as a global attractor for

Q

dz/dt = Q(x) There is a

~ K, then

(3)

On

on

K > 0

such that if

(z) = -z. 4 R x R4 the vector field Q

z

E R4 ,

362


is a global oscillator. Once such a Choose some I Izi

I

~ K.

E P

z Let

has been found we finish as follows:

Q

so that

z-z

R(z) = 2Q(z-z).

E P

for all

Then

R

z

with

will have the pro-

perties of Section 3. This changes the question from

P

to

where

'ole

can use the linear structure systematically (even though our equations are not linear). To find a

as needed above we first ignore property

Q

(2), or behaviour of (3).

at

Q

and concentrate on (1) and 4 4 S: R + R satisfying (1) and

In fact we shall find

(3) with

S

replacing

Q

and after that

S

is modified to

satisfy (2). Tmolard constructing this 11 = {(zl,z2)

4 E R4 x R

I

S, observe that the set

zl =z2} has the property that

the vector field (*) is tangent to and on

11

11.

is invariant under the flow

the flow is contracting to the origin.

Now suppose is odd.

That is, 11

S

satisfies

S(-z) = -S(z)

or that

Then on

the vector field (*) is invariant and has the form z

+

S(z) - )l(z)

S

THE HOPF BIFURCATION AND IT5 APPLICATION5

(up to a factor of

363

2).

From these considerations we are 4 motivated to seek an odd map 5: R4 + R which satisfies the following: (51)

5

has

0

has a global attractor and

is a global oscillator on

5 -

~

R4 . 4

(52)

~~

(53)

Boundary conditions can be made good.

is an attractor for (*) on

R

x

4 R .

The heart of the matter lies in (1); we consider that next.

First consider the matrix a

0

Ya

0

0

a

0

ya

-ya

0

-2a

0

-ya

0

-2a

iT 0

where

a < -1

2 x 2

y < 3/2, in linear coordinates

4

on

R.

~

Note that ~2'~3'~4'

~
0, 8 - (P+) < 0, and 8+(p_ ) > o.

properties

first intersections of with

371

x = P±.

WU(p)

with

R

occur at the points

Finally, it is assumed that the images of

are contained in the intervals

[±1/4, ±3/4].

g±

Figure 12.1

illustrates these essential features of the flow

X.

Figure 12.1 We remark that the conditions imposed on the eigenvalues of lim x ... 0

df±/dx

x

=

lim dg±(X,y)/ay o and x... 0 The reason for this behavior is given by

at

p

00

imply that

solving a linear system of differential equations near a saddle point. like a power of

The return maps x

arbitrarily close to

8±

acquire singularities

because the trajectories of

R±

come

p.

In the theorems which we now state, we assume that the

372


vector field

X

three manifold field

X.

is extended to a vector field on a compact M.

We continue to denote the extended vector

Note that the only properties used in defining

X

which do not remain after perturbation are the existence of the functions

f±

and

g±.

These functions are introduced

to simplify the discussion and are not essential properties of

X. (12.1)

Theorem. r C

in the space of


vector fields on

of second category in

M

%' such that if

%' of

X

(r ~ 1) and a set·~ y E

~,

then

Y

has

a singular point which is not isolated in its nonwandering set. (12.2)

hood

Theorem.

The vector field X has a neighborr ~ in the space of C vector fields on M (r > 1)

with the property that if ·~C %' is an open set in the space r of C vector fields, then there are vector fields in ~ whose nonwandering sets are not homeomorphic to each other. Theorem (12.2) states that of the set of topologically

X

~-stable

is not in the closure vector fields.

We attack the proofs of both of these theorems by giving a description of the nonwandering set of

X.

This des-

cription is given largely in terms of "symbolic dynamics" (Smale [4]). Consider the return map subsets of

e(R)

e

of

R.

We pick out four

which will be used in analyzing the sym-

bolic dynamics of the nonwandering set of

X.

Denote


8 (R+)

n

{p

8 (R+)

n

{O < x < f

8 (R_)

n

{f - (p+)

R = 8 (R_) 4

n

{O < x < p-} •

Figure 12.2 shows these sets.

The image of

R R

l 2

R 3

+

373

< x < O}

extends horizontally across

+

(p ) } -

< x < O}

R and R • 4 3 horizontally across R • Similarly, 8(R } l 3 extends across and

R under 8 l 8(R) extends 2

extends across

q

Figure 12.2 {a }'" of the integers 1, 2, k k=O 3, and 4 such that, for each k, (a , a + ) is one of the k k l pairs (3,1) , (4,1) , (1,2) , (4,3) , (1,4) , or (2,4) . The set Now consider sequences

of such sequences forms the underlying space shift of finite type" with transitio"n matrix

L

of a "sub-

374


0

1

0

1

0

0

0

1

1

0

0

0

1

0

1

0

(

\

)

Corresponding to each finite sequence

{a ' · · .,a }· constructed n O from "admissible" pairs listed above, the intersection n

k n e

(R

contains a component which extends horizontally

)

k=O

ak

across

R

a

For example, if

O

extend across

1, then the images of

a

O If a

need extend across

l

R 2 = 2, then only the image a

Hence

2

= 4,

extends

e(R ) 4

2

R , and e (R 4 ) extends across R . As n ina l 2 creases, the vertical height of these strips decreases expoacross

nentially. crossing

n

If

R a

contains an arc

k=O There are an

horizontally.

O

uncountable num4

n ek ( U R.) contains k=O i=l ~ an uncountable number of arcs extending across each R .•

ber of sequences in

L, hence

S =

~

We want to investigate whether nonwandering set of

e.

If each arc contained in

image under some iterate of R , then i

S

e

S

has an

which extends across each

e.

will be contained in the nonwandering set of

In these circumstances, we prove that the nonwandering set of

X.

0

functions

f±

Denote by

f

determined by

yc

Ri

acting on the intervals the discontinuous map f±

(with, say, f(O)

(P+'p_).

Since

is not isolated in

Whether or not every arc in

has an image extending across the set

interval

is contained in the

S

depends only on the

(P+'O)

f:

and

(p+,p_) +

0.)

S

(O,p_).

(p+,p_)

Consider a sub-

df±/dx> a > 1, the sum of the


lengths of the components of fore, some image of

k > 0

and an

k

has more than one component.

y

y

x E

ca

is at least

only point of discontinuity for a

375

with

f

is

ThereThe

x = O. so there is

o.

fk (x) =

The map 8 has a periodic point of period 2 in R l 2 because 8 (R ) crosses R horizontally. Therefore, f l l has a point

r

of period

2.

Any neighborhood of

an image which eventually covers there is an open set (p+,p_). if

has

Now assume that

U C (P+,P_), none of whose images cover

Then no image of and

(P+'P_).

r

U

contains

p.

It follows that

are two open sets, none of whose images

cover

(p+,p_), then

U lJ U also has this property (be2 l is in none of its images.) Thus there is a largest

cause

r

open set

U C (p+,p_)

images cover

with the property that none of its

(p+,p_).

It follows that

f

-1

(U)

=

u

f (U).

We observed above that any interval contains a point which is eventually mapped to Thus

U

by the iterates of

0

contains a neighborhood of

hoods of

P± •

This implies that

U

0

is a dense subset of

f-l(U) C U

property

U

containing

o.

U.

(~_,O).

contains

Since

f

Notice that the

Let

(~_,~+) (~_,O)

U.)

U

must

be the component contains

f_(O) = P_, the images of

are endpoints of components of (~+,a)

Since these points

(p+,p_).

Some image of (Since

of

o.

implies that the components of

map onto the components of of

and, hence, neighbor-

contains a neighborhood

of each point which eventually maps to are dense, U

f.

0, 0

The first time an image

0, that power of

f

is continuous on

is orientation preserving, it follows that

376


is mapped by this power of a periodic point of for some power of

f. f

f

to

S.

We conclude that

8

images of the vertical lines

of x

finite set of vertical lines.

is

have images

p±

which are periodic points of

For the return map

s

Therefore

f.

R, this implies that the =

p±

each remain within a

Because

vertical direction, the intersections of

8

contracts in the R

with

WU(p)

have

8-trajectories which tend asymptotically to periodic orbits of of

8.

These periodic

8

trajectories lie on periodic orbits

Y , Y for the flow X. Because 8 is uniformly hyperbolic 2 l (apart from its discontinuity), these periodic orbits are hyperbolic with two dimensional stable and unstable manifolds. Applying the Kupka-Smale Theorem (Smale [1]), we note that it is a generic property of vector fields that the stable manifold of a hyperbolic periodic trajectory intersect the unstable manifold of a singular point transversally. not the case here.

This is

Thus we conclude that in the open set of

vector fields which we have described, those vector fields for which an" arc of

S

eventually extends across each

a set of second category.

R.

1.

form

I do not know whether there is an

open set of vector fields with this property. Proof of Theorem (12.1): is chosen so that every arc in

S

e

Let us assume now that

X

has the property that some image of

eventually extends across each

Ri •

If

w E Sand U is a rectangular neighborhood of w in R, then 8 k (U) extends across each R for k sUfficiently i k large. Also 8- (U) extends vertically across R for k sufficiently large because

8

contracts the vertical direc-


377

It follows that e-k(u) n ek(u) f ~ for k very 2k large. Thus e (U) n u f ~ and w is nonwandering. We

tion.

conclude that Since

x.

S

S

is contained in the nonwandering set of

intersects

WS(p), p

e.

is in the nonwandering set of

This proves Theorem (12.1).

[]

The nonwandering sets of the vector fields satisfying Theorem (12.1) have a two dimensional attractor

A

which con-

tains the origin.

R

contains

S.

A

with

We want to go further in describing the structure of can be done most completely when

~his

point with of

The intersection of

f

WU(p) C WS(p).

which map

p+ and

p

is a homoclinic

This happens when there are powers p_

to

O.

For purposes of definiteness, we shall describe the case that

f

2

(p±) = O.

and

p

to

O.

Now

A

in

Afterwards we indicate the mod-

ifications which are necessary when higher powers of p+

A.

RnA =

s.

If

fmap

f2 (p±) = 0, then

n e (R ) C "3 e(R ) C Rll) R · U R4 , e (R ) C R , e (R ) C R , and l 4 2 3 l 2 4 Consequently, i f {ak}oo is a sequence with a. E {1,"2,3,4}, J. k=O

n ek(R ) f ~ if and only if {a } E L. I f k k=O ak {a k } E L, then there is a segment extending across Ra

then

lies in

S

picture for to points of

and hence in A.

A.

which O This presents the following

There is a Cantor set of arcs, corresponding

L, each of which extends across some of the

Ri's.

These are joined at their ends by

12.3.

Note that points of

A_Wu(p)

which are homeomorphic to a 2-disk

WU(p).

See Figure

have neighborhoods x

Cantor set.

R

378


R

R

Figure 12.3 If higher powers of

f

map

p+

and

p

to

0, then

we construct another subshift of finite type as follows. the image of

8(R)

along vertical lines passing through each

point in the orbit. 8(R) the

matrix T

Let

8(p+)

and

8(p_).

~

T

Define

by

{ 1

ij

0

if

8 (R.) II R. i- ~ 1.

if

8 (R ) j

J

n

R.

1.

=

1'.

be the (one-sided) subshift of finite type with tran-

sition matrix

T.

Corresponding to each sequence in

there will be exactly one arc crossing attractor

A.

we remark that if

8

8k(~ )

which lies in the A

n

R

=

l'

if

k

is to be cut along components of

also contain points of

~,

{a } E~. Finally, k does not preserve vertical segments in

n

k=O

R

Ri

The closure of these segments will be

as before, because

R, then

This will divide Rl, ... ,R • n

into a number of components, say n x n

Cut

WU(p).

WS(p) n R

which


Proof of Theorem (12.2): two steps.

379

We prove Theorem (12.2) in

In the first step, we consider two flows, X

and

X, of the general sort considered in this paper such that, for the flow

X, WU(p) C WS(p), and for the flow

WS(p) = {p}.

We prove that

X

and

sets which are not homeomorphic.

X

n

X, WU(p)

have nonwandering

The second step demonstrates

that vector fields of each of these two classes are dense in r some open set in the space of C vector fields. We have described above the attractor vector field

X

for which

is path connected and

WU(p) C WS(p).

A_Wu(p)

A(X)

of a

In this case, A

is locally homeomorphic to

the product of a 2-disk and a Cantor set.

Furthermore, WU(p)

is homeomorphic to the wedge product of two circles, a "figure eight." Now consider the attractor for which

WU(p)

set containing then

A(X)

points

n

WS(p) = {p}

p.

If

w E A(X)

WU(p) C C

X

Cantor set.

=

P

x

w

which are the WU(p) - {p}.

A(X), then

of

An R

C

Since

to the left P+

=

x.

is homeomorphic WU(p) ~ WS(p) C-{p}

w-limit sets of the two trajectories in A single point which is the

traje?tory must be a singular point. X

C

is homeo-

WU(p) C C, there must be two points of

and

A{X),

It is easily seen that

or to the right of the line

is homeomorphic to

points of

Consider the set

such that no neighborhood of x

X

is a two dimensional

is to be homeomorphic to

to the wedge product of two spheres. for

A

of a vector field

since there are no points of

of the line A(X)

and

must be path connected.

morphic to a 2-disk

If

A(X)

A(X)

in

A(X)

other than

w-limit set of a

There are no singular p, so we conclude that

C

380


is not homeomorphic to the wedge product of two spheres. Hence

A(X)

and

A(X)

are not homeomorphic.

This concludes

the first step of the proof. We now prove that the sets of vector fields

X, X

of

the sort considered above are each dense in some open set. The Kupka-Smale Theorem implies that vector fields like in that

WS(p)

n WU(p)

= {p}

X

form a set of second category.

PEA

Since the set of vector fields with

A two

and

dimensional is a second category subset of an open set, there is a dense set of vector fields of the form of

X

in some

open set of vector fields. The only thing remaining to prove is that there is a dense subset of an open set of vector fields for which Wu(p) C WS(p).

Consider the effect on

tion

parallel to the

Y

of

of decreasing

X p

and increasing

of a perturba-

x-axis which has the effect p+.

Support of

y-x

WU(p)

/

--l

See Figure 12.4.

~

71

C)

.r------7l/ /~7 /

Figure 12.4


We examine successive intersections of for the vector fields

Y

and

are orientation preserving.

X.

381

WU(p)

The functions

with

f+ and

R f

Consequently, as long as the

corresponding, successive intersections for the two vector fields lie on the same side of the line

x

=

0

in

R, the

effect of the perturbation is push the intersections following

along

P

tions following map

e

WU(p) P+

to the left and to push the intersecto the right.

expands in the

x

Furthermore, since the

direction, the distance between

the corresponding, successive points of intersection grows exponentially.

The distance cannot grow indefinitely, so

after sometime, the corresponding points of intersection lieon opposite sides of the line bation intermediate between intersection of

WU(p)

with

(in both directions along WS(p)

x Y R

= and

O.

Thus, for some perturX, there are points of

which lie on the line

WU(p).)

This means that

for the intermediate perturbation.

x = 0

WU(p) C

We conclude that

there is a dense set of vector fields in some open set of the space of vector fields for which

WU(p) C WS(p)

to finish

the proof of Theorem (12.2). As is traditional in dynamical systems, we end with a question.

The vector fields described here are very pathologi-

cal from the point of view of topological dynamics.

Yet they

seem to preserve as much hyperbolicity as they possibly could without satisfying Axiom A.

There is now a well developed

"statistical mechanics" for attractors satisfying Axiom A (Bowen-Ruelle [1]).

How much of this statistical theory can

be extended to apply to the vector fields described here?

382


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405

INDEX

A-admissible, 275

in semigroups, 255-257

Almost Periodic Equations, 161-162

and symmetry, 225

stationary, 169 to a torus, 9 Aperiodic Motion, 333-335, 351

in 'I'1right' s equation, 159-160 Biological Systems, 11

Attractor, 4-6 Lyapunov, 4, 91, 93 strange, 24, 297, 338-339 vague, 66-67, 78-79, 205 Averaging, 155-159 for almost periodic equations, 161-162 and bifurcation to a torus, 161-162 in diffusion equations, 160-161 in Wright's equation, 159-160 Benard Problem, 314, 316-317 bifurcation in, 323-324 stability in, 324-326

age structured, 339-340 and bifurcation, 160-161, 327-329 with discrete generations, 329-335 and oscillations, 336-338, 341-347, 354-356, 360-361, 366-367 and travelling waves, 349-350 Center Manifold Theorem, 19-20, 28, 30-43 counterexample to analyticity, 44-46 for diffeomorphisms, 207-208 for flows, 46-48 nonuniqueness of, 44-46 for semigroups, 256 with symmetry, 227

Bifurcation, 7-11 in almost periodic equations, 161-162

Center Theorem of Lyapunov, 98-99

to aperiodic motion, 333-335

Chemical Systems, 11, 160-161

and averaging, 155-159 in the Benard problem, 323-324 in biological and chemical systems, 11, 160-161, 327-329 in the Couette flow, 322

Zhabotinskii reaction, 149-150 Couette Flow, 16-17, 315 bifurcation in, 322 stability of, 324-326 and symmetry, 240-249

Hopf, 8-11, 20-21, 23 in the Navier-Stokes equation, 14-15

Diffeomorphisms center manifold theorem for, 207-208

406

INDEX

and Hopf bifurcation, 207-210 Energy Method, 270

in Lienard's equation, 141-148 in the Lorenz equations, 141-148 multiparameter, 90

Euler's Equations, 13, 287 Evolution System, 272-278 infinitesimal generator of, 272 and stability, 275 Y-regular, 274 Flow, 258-260

in the Navier-Stokes equations, 306-312 and Poincare map, 66-67 and van der Pol's equation, 139 2 in R , 65-81 n in R , 81-82, 166-167, 201-204 for semigroups, 255

continuity and smoothness of, 260-267, 271, 278-284

stability formula for, 126, 130, 132-135

and energy method, 270

stability of, 77-80, 91-94

time dependent, 271-276

uniqueness of, 80-81

uniqueness of integral curves of, 267-269

for vector fields, 20-21

Generator, Infinitesimal

in Wright's equation, 159 Instability, 2, 5-6

A-admissible, 275 for evolution systems, 272

Karmen Vortex Sheet, 15

and stability, 275-278 Lienard's Equations, 136-137 Generic, 299 Hodge Decomposition, 288

Lorenz Equations, 141-148, 369 and turbulence, 148

Hopf Bifurcation, 8-11, 165 almost periodic, 161-162 and averaging, 155-159

Lyapunov, 6, 66, 91, 93 center theorem, 94-95

in biological systems, 361, 160-161

Lyapunov-Schmidt, 26

in chemical reactions, 149-150, 160-161

Navier-Stokes Equations, 12-18, 286

for delay equations, 88 for diffeomorphisms, 23, 207-210 • generalized, 85-90 global, 90

and Benard problem, 316-317 and Couette flow, 16-17, 316 global regularity of, 302-303 .

407

INDEX

Hopf bifurcation in, 306-312 and Karmen vortex sheet, 15 local existence for, 290-295 and Poiseuille flow, 18 semiflow of, 289, 29'5-296 stability of, 298 and Taylor cells, 17 turbulence in, 15, 24 Oscillations in Biological Systems, 336-338, 341-347, 354-356, 360-361, 366-367 Poincare Map, 21, 56-60 and Hopf bifurcation, 66-67 and stability, 61

invariant torus, 256-257 quasi contractive, 277 smooth, 253 and stability, 276-278 Spectral Radius, 50 Spectrum and stability, 52-55 Stabili ty, 3-6 asymptotic, 4 and averqging, 155-159 in the Benard problem, in the Couette flow, 324-326 for evolution systems, 275-278 formula for Hopf bifurcation, 126, 127, 132-135

Poiseuille Flow, 18

for Hopf bifurcation, 77-80, 91-94, 166-167, 201-204

van der Pol's Equation, 139

of invariant torus, 210

Prqndt1 Number, 142

in the Navier-Stokes equations, 298

and linear equations, 6-7

Rayleigh Number, 142 Reynold's Number, 12

omega (Q), 369-372 and the Poincare map, 61 for semigroups, 275-278

Semiflow, 259 of Navier-Stokes Equations, 289, 295-296 and smoothness, 279-284 semi groups center manifold theorem for, 256 generator of, 274 and Hopf bifurcation, 255

shift of, 9 and spectrum, 52-55 Strange Attractor, 24, 297, 338-339 Symmetry, 225 and center manifold, 227 and Couette flow, 240-249 and Taylor cellsj 243-249

408

INDEX

Taylor Cells, 17, 314, 315 bifurcation to, 322 stability of, 324-326 and symmetry, 243-249 Torus, Invariant, 206-207 and averaging, 162 in sernigroups, 256-257 and turbulence, 297, 299 Turbulence, 15, 24 and Couette flow, 297 and invariant tori, 297, 299 in Lorenz equations, 148 and Navier-Stokes equations, 297, 299-303 Turing's Equations, 354-361 Vague Attractor, 65-66, 81-82, 205 Wright's Equations,

159-160

Y-regular, 274 Zhabotinskii Reaction, 149-150, 367

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Vol. 2 l. Sirovich Techniques of Asymptotic Analysis ISBN 0-387-90022-5

Vol. 14 T, Yoshizawa Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions ISBN 0-387-90112-4

Vol. 3 1. Hale Functional Differential Equations ISBN 0-387-90023-3 Vol. 4 1. K. Percus Combinational Methods ISBN 0-387-90027-6 Vol. 5 R. von Mises and K. O. Friedrichs Fluid Dynamics ISBN 0-387-90028-4 Vol. 6 W. Freiberger and U. Grenander . AShort Course in Computational Probability and Statistics ISBN 0-387-90029-2 Vol. 7 A. C. Pipkin Lectures on Viscoelasticity Theory ISBN 0-387-90030-6 Vol. 8 G. E. O. Giacaglia Perturbation Methods in Non-linear Systems ISBN 0-387-90054-3 Vol. 9 K. O. Friedrichs Spectral Theory of Operators in Hilbert Space ISBN 0-387-90076-4 Vol. 10 A. H. Stroud Numerical Quadrature and Solution of Ordinary Differential Equations ISBN 0-387-90100-0 Vol. 11 W. A. Wolovich linear Multivariable Systems ISBN 0-387-90101-9 Vol. 12 l. D. Berkovitz Optimal Control Theory ISBN 0-387-90106-X

Vol. 15 M. Braun Differential Equations and Their Applications ISBN 0-387-90114-0 Vol. 16 S. Lefschetz Applications of Algebraic Topology ISBN 0-387~90137-X Vol. 17 l. Collatz and W. Wetterling Optimization Problems ISBN 0-387-90143-4 Vol. 18 U. Grenander Pattern Synthesis Lectures in Pattern Theory Vol. 1 ISBN 0-387-90174-4

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