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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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.
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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.
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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.
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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,
THE HOPF BIFURCATION AND ITS APPLICATIONS
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 HOPF BIFURCATION AND ITS APPLICATIONS
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].
THE HOPF BIFURCATION AND ITS APPLICATIONS
(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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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 HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
(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).
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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.
THE HOPF BIFURCATION AND ITS APPLICATIONS
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 <
Y
~
in the following sense:
since
k, we get a sequence of center manifolds
but their intersection may be empty.
See Remarks 2.6
~,
THE HOPF BIFURCATION AND ITS APPLICATIONS
20
regarding the differentiability of
M.
We will be particularly interested in the case in which bifurcation to stable closed orbits occurs.
=
~
before, assume that for
~O
~
(resp.
With
> ~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
~
~
<
0, Re
A(~)
A(~)
> O.
> 0, Re
Then there is a
ck - 2
<
0, for
~
=
Also assume
function
A(~)
and
0, Re
A(~)
d
R~ A(~)
~: (-8,8)
~
+
R
A(~)
such
0, and
I
~=O
>
O.
such that
* Simple means that the generalized eigenspace (see Section 2A) of the eigenvalue is one
dimens~onal.
THE HOPF BIFURCATION AND ITS APPLICATIONS
(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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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)
;\(~)
<
I}
has two isolated, and
A (~)
such
and the remaining part of
is inside the unit circle, at a nonzero distance
from it. (d)
dIA(~)11 d ~
>0 ~=O
•
Then (under two more "vague attractor" hypotheses which will be explained during the proof of the theorem), there is a
R2
24
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
<1>: 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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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)
<1>:
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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 <
00.
V
Then there exists a neighborhood submanifold
o and tangent to a)
of at
Y.
V
0,
of dimension
~.
If
and
d, passing through
x EM
and~(x)
E V,
(x) E M
(Local Attractivity): n = 0, 1, 2, ••• , from
Z
in
such that
(Local Invariance): then
b)
M
0
of
~n(x)
to
~n(x) E V
If
then, as M
n
~
00,
for all
the distance
goes to zero.
We begin by reformulating (in a slightly more general way) the theorem we want to prove.
We have a mapping
neighborhood of zero in a Banach space
Z
into
Z,
~
with
*This holds automatically if Z is finite dimensional or, more generally, if D~(O) is compact.
of a
THE HOPF BIFURCATION AND ITS APPLICATIONS
=
~(o)
O.
We assume that the spectrum of
D~(O)
29
splits into
a part on the unit circle and the remainder, which is contained in a circle of radius strictly less than one, about the origin.
The basic spectral theory discussed in Section 2A
guarantees the existence of a spectral projection
P
of
Z
belonging to the part of the spectrum on the unit circle with the following properties: i) and
P
(I-P)Z
PZ
D~(O).
The spectrum of the restriction of
D~(O)
to
lies on the unit circle, and
iii) to
so the subspaces
D~(O),
are mapped into themselves by
ii) PZ
commutes with
(I-P)Z
We let
The spectral radius of the restriction of is strictly less than one.
X denote
striction of D (0)
to
D~(O)
(I-P)Z,
D~(O)
Y.
Then
~(x,y)
to Z
X
=
Y
denote
and
X e Y
B
PZ,
A
denote the res-
denote the restriction of
and
(Ax+X*(x,y),
By + Y*(x,y»,
where A
is bounded linear operator on
X with spectral
radius strictly less than one. B
is a bounded operator on
Y with spectrum on
the unit circle.
(All we actually need is that the spectral radius of B- 1 is no larger than one.) k X* is a c + l mapping of a neighborhood of the origin in
X e Y
into
zero at the origin, i.e. DX(O,O) Y
Ck + l
is a in
=
X
0,
X with a second-order X(O,O)
=
0
and
and mapping of a neighborhood of the origin
e Y into Y with a second-order zero at the
30
THE HOPF BIFURCATION AND ITS APPLICATIONS
origin. ~
We want to find an invariant manifold for
Y
to
at the origin.
mapping into
u
X,
which is tangent
Such a manifold will be the graph of a
which maps a neighborhood of the origin in with
=
u(O)
0
and
Du(O)
Y
= o.
In the version of the theorem we stated in 2.1, we assumed that
Y
was finite-dimensional.
We can weaken this
assumption, but not eliminate it entirely.
(2.2)
Assumption.
¢ on
function
Y
1
which is
I Iyl
origin and zero for
k l c +
There exists a
on a neighborhood of the
I > 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 ~
c
X Ell Y, i.e. the graph of
in the sense that, i f
(xl'Yl)
II ylll
with b)
r u* = ((x,y)
The manifold
<
e:
The manifold
in the sense that, i f (~'Yn)
~n(x,y)
for all
n > 0,
Ilxll
then
<
e:
=
x
u*(y)
and
is invariant for
u*
~
and if
(u* (y) ,y)
Xl = u* (Yl) . is locally attracting for
r u*
are such that then
<
Ilyll
I
e:
Ilyll II xnll
<
<
e:
e:
,
and if IIYnl1
<
e:
~
THE HOPF BIFURCATION AND ITS APPLICATIONS
lim n+ oo
I IXn
- u*(Yn)
II
31
o.
=
Proceeding with the proof, it will be convenient to assume that than
1.
IIA
II
< 1
and that
IIB-lil
is not much greater
This is not necessarily true but we can always make
it true by replacing the norms on (See Lemma 2A.4).
X, Y
by equivalent norms.
We shall assume that we have made this
change of norm.
It is unfortunately a little awkward to exl plicitly set down exactly how close to one I IB- , I should be taken.
We therefore carry out the proof as if
1 IB-li
I
were an adjustable parameter; in the course of the argument,
I IB- l , I.
we shall find a finite number of conditions on
In
principle, one should collect all these conditions and impose them at the outset. The theorem guarantees the existence of a function
u*
defined on what is perhaps a very small neighborhood of zero. Rather than work with very small values of
x,y,
scale the system by introducing new variables calling the new variables again does not change make <
X*, Y*,
k + 1,
tiplying
A,B,
x
and
but, by taking
y). s
we shall
xis,
y/s
(and
This scaling
very small, we can
together with their derivatives of order
as small as we like on the unit ball. X*(x,y),
Y*(x,y)
by the function
Then by mul-
¢(y)
whose
existence is asserted in the assumption preceding the statement of the theorem, we can also assume that are zero when
I Iyl I
> 1.
sup
II x 11::1 y unrestricted
x*(x,y), Y*(x,y)
Thus, if we introduce sup jl,j2 jl+j2::k+l
THE HOPF BIFURCATION AND ITS APPLICATIONS
32
we can make
A as small as we like by choosing
E
The only use we make of our technical assumption on
very small. Y
is to
A
arrange things so that the supremum in the definition of may be taken over all
y
and not just over a bounded set.
¢,
Once we have done the scaling and cutting off by we can prove a global center manifold theorem.
That is, we
shall prove the following. (2.4)
Lemma.
Keep the notation and assumptions of the
A is SUfficiently small (and if
center manifold theorem. If liB-II
I
is close enough to one), there exists a function
defined and
k
u*,
Y,
times continuously differentiable on all of
with a second-order zero at the origin, such that a) invariant for b)
lim
n+ oo
I Ix n
ru*
The manifold
"xii
If
As with
n
u*(y),y E Y}
{(x,y) Ix
is
in the strict sense.
0/
- u * (y
=
)
II
=
I IB-li
<
0
I,
1,
and
y
is arbitrary then
(where (xn'Yn)
=
we shall treat
0/
n
(x,y)).
A as an adjustable
parameter and impose the necessary restrictions on its size as they appear.
It may be worth noting that
A depends on the
choice of norm; hence, one must first choose the norm to make I /B- l , I close to one, then do the scaling and cutting off to
A small.
make
To simplify the task of the reader who wants.
to check that all the required conditions on
I IB- l "
can be satisfied simultaneously, we shall note these conditions
*
with a
as with (2.3)* on p. 34.
The strategy of proof is very simple. manifold of
u);
M of the form we let
{x
= u(y)} (this stands for the graph
M denote the image of
some mild restrictions on
u,
We start with a
M under
0/.
With
we first show that the manifold
THE HOPF BIFURCATION AND ITS APPLICATIONS
~M
33
again has the form {x = u(y)} A
for a new function
u.
If we write ffu
for
u
we get a (non-
linear) mapping u 1+'9'"u
from functions to functions. and only if
u = ffu,
The manifold
M is invariant if
so we must find a fixed point of
do this by proving that
~
We
ff is a contraction on a suitable
A is small enough).
function space (assuming that
More explicitly, the proof will be divided into the following steps: I) II)
Derive heuristi~ally a "formula" for Show that the formula obtained in
Jv.
I) yields a
well-defined mapping of an appropriate function space
U
into
itself. III)t Prove that
ff is a contraction on
has a unique fixed point IV)
U
and hence
u*.
Prove that b) of Lemma (2.4) holds for
u*.
We begin by considering Step I). I)
i)
y.
=
This means that ii)
~(u(y),y),
we should proceed as follows
Solve the equation y
for
u(y),
To construct
Let
By + Y*(u(y) ,y) y
u(y)
is the
be the
Y-component of
of functions
~(u(y),y).
X-component of
Le.,
u(y) = Au(y) +X*(u(y),y). II)
(2.1)
(2.2)
We shall somewhat arbitrarily choose the space u
we want to consider to be
tone could use the implicit function theorem at this step. For this approach, see Irwin [1].
THE HOPF BIFURCATION AND ITS APPLICATIONS
34
. {u: Y .... X I Dk+l u
U
j
0,1, ••• ,k+l,
continuous;
all
y;
U{O)
for
=
=
Du{O)
o}.
We must carry out two steps: Prove that, for any given
i)
y
a unique solution
y E Y.
for each
that~,
Prove
ii)
u E U,
equation (1) has
And
defined by (2.2) is in
U.
To accomplish (i), we rewrite (2.1) as a fixed-point problem:
It suffices, therefore, to prove that the mapping _ Y 1+ B-1 Y -
is a contraction on
Y.
-) B-1 Y * (u (-) Y ,y
We do this by estimating its deriva-
tive: I IDy[B-1Y-B-ly*(U(Y),y)] I I ::11B-lll
by the definitions of
A
and
l + D2 Y*(U(y) ,y) II :: 2AIIB- ll U. If we require
l 2AIIB- ll
< 1,
Y
is a function of
y,
By the inverse function theorem,
of
y. Next we establish (ii).
and
< 1
for all
= 0,
D.%.(O)
First take
j = 0:
-%.(0)
IliVu(y)11
y.
Note
y
is a
Ck + l
function
By what we have just proved,
Thus to show ~ E U,
I I DjiVu(y) II
for each
*
depending also on the function
u.
5'u E c k + l •
(2.3)
y
equation (2.1) has a unique solution that
IID1Y*(u(y),y)Du(y)
y,
what we must check is that j = 0,1,2, ..• ,k+l
(2. 4) (2.5)
= O.
~ IIAII·llu(Y)11 + Ilx*(u(Y),Y)11
< IIAII
+ A,
so if we require IIAII + A < 1, then
I Wu{y)
II
< 1
for all
y.
(2.6)
*
THE HOPF BIFURCATION AND ITS APPLICATIONS
To estimate
D9U
we must first estimate
35
Dy(y).
By
differentiating (2.1), we get (2.7)
yU: Y
where
+
Y
is defined by
=
yU(y)
Y*(u(y) ,y).
By a computation we have already done,
I IDYu(y)!! Now
B + DY
u
~ 2A
B[I+B-1DY u )
(2.3)*), B + DY u
for all
and since
y. 2AI IB-11!
<
1
(by
is invertible and
The quantity on the right-hand side of this inequality will play an important role in our estimates, so we give it a name: (2.8)
1IB- l , I
Note that, by first making by making
A
y
small, we can make
very close to one and then as close to one as we like.
We have just shown that
II Dy (y) II
< y
for all
Differentiating the expression (2.2) for D5U(y)
=
y. YU(y)
(2.9)
yields
[A DU(y) + DXu(y») Dy(y); }
and
(2.10) (xu (y) = X*(u(y) ,y».
Thus 11D9U(y) II
<
(IIAII + 2A) .y,
(2.11)
36
THE HOPF BIFURCATION AND ITS APPLICATIONS
so if we require
(IIAII
+ 2A) Y
(2.12)
< 1,
we get 11D%(y)
II
< 1
for all
y.
We shall carry the estimates just one step further. Differentiating (2.7) yields
By a straightforward computation,
so
Now, by differentiating the formula (2.10) for
D~,
we get 2 D :§ilu(y)
so
If we require (2.13)
we have
IID2~(y)11 <1
for all
y.
At this point it should be plausible by imposing a sequence of stronger and stronger conditions on I I Djy(y) II
< 1
for all
y,
Y,A,
that we can arrange
j = 3,4, •.. ,k+1.
*
*
THE HOPF BIFURCATION AND ITS APPLICATIONS
37
The verification that this is in fact possible is left to the reader. ~u
To check (2.5), i.e. u = 0,
Du
yeo)
o
0)
IL%. (0)
o
D%
(assuming
we note that
since
0
is a solution of
AU(O) + X(u(O) ,0) = 0
~u(O)
0,
By + Y(u(y) ,y)
0
and
[A Du(O) + D1X(0,0)Du(0) + D2X(0,0)] ·Dy(O) [A· 0 + 0 + 0]· Dy (0) = O. This completes step II). Now we turn to III) III)
We show that
~
is a contraction and apply the
contraction mapping principle.
What we actually do is slightly
more complicated. i) norm.
Since
We show that U
~
is a contraction in the supremum
is not complete in the supremum norm, the con~
traction mapping principle does not imply that point in
U,
but it does imply that
the completion of ii)
U
has a fixed
~ has a fixed point in
with respect to the supremum norm.
We show that the completion of
U
with respect
to the supremum norm is contained in the set of functions from
to
y
X
th k
with Lipschitz-continuous
and with a second-order zero at the origin. point
u*
~
of
derivatives
Thus, the fixed
has the differentiability asserted in the
theorem. We proceed by proving i)
I I u l -u 2 I 10
=
Consider
i).
u l ' u 2 E U,
supl lu l (y)-u 2 (y) I I· y
Let
and let Yl(y), Y2(y)
the solution of y
u
1,2.
denote
38
THE HOPF BIFURCATION AND ITS APPLICATIONS
We shall estimate successively
11:i\-Y211 0' and
Subtracting the defining equations for
Yl'Y2'
II9"u 1 ~211 O. we get
so that (2.14)
Since
I I Du 1 1 10 2 1,
we can write
(2.15)
Inserting (2.15) in (2.14) and rearranging, yields 1 (l-n·IIB- 11) IIY1-Y211
I 1Y1 -Y 2 1I 0
i. e.
2. >.·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).
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
<
=
~(t)
is differentiable, that
II ¢ 1/ •II y 1//
for all
n, t
and that
for all
n, t l , t . 2
By this last inequality and the Arzela-
Ascoli Theorem, there exists a subsequence verges uniformly on every bounded interval. rily denote the limit of this subsequence by
hence, passing to the limit
j
->- 00,
we get
~'
n j (t)
which con-
We shall temporaX(t).
We have
THE HOPF BIFURCATION AND ITS APPLICATIONS
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 **
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
(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-
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
(2.7) Let
z
Theorem.
Center Manifold Theorem for Flows.
and let
Ft
o < t <
zero for Ft(X)
cO
be a
ck + l
is
C
0
semi flow defined in a neighborhood of Assume
T.
norm * away from
00
be a Banach space admitting a
jointly in
Ft(O) t
and
and that for
0,
=
x.
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 <
00.
Then there exists a neighborhood
V
of
submanifold
d
passing through
and tangent to
M CV Y
at
of dimension 0
0
in
Z
and
such that
(a)
If
x EM, t > 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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
-<
t
t
This may be achieved by joint continuity
B.
and use of a
borhood of
0, 0 < t
F
F
B as in the above proof.
has
G t
Since this does not affect
on a small neighborhood of zero, we get our result.
*In
linear semi group theory this corresponds to analyticity of the semigroup; it holds for the heat equation for instance. For the Navier Stokes eqations, see Sections 8,9.
t see Renz [1] for further details.
THE HOPF BIFURCATION AND ITS APPLICATIONS
Exercise.
(2.8)
for flows). d(x,y) dt
=
(nonuniqueness of the center manifold
X (x,y) = (-x,y 2 ).
Let
X(x,y)
49
Solve the equation
and draw the integral curves.
Show that the flow of Theorem 2.7 with
X
satisfies the conditions of
the y-axis.
y
center manifold for the flow.
Show that the
y-axis is a
Show that each integral curve
in the lower half plane goes toward the origin as that the curve becomes parallel to the y-axis as
t
t
~
and
->-
-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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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 <
denote its spectral radius defined by
IA E a(T)}.
r(t)
is also given by the spectral
radius formula: r(T)
limit I ITnl
I
lin
•
n+oo
(The proof is analogous to the formula for the radius of convergence of a power series and can be supplied without difficulty; cf. Rudin [1, p.355].) The following two lemmas are also not difficult and are
THE HOPF BIFURCATION AND ITS APPLICATIONS
51
proven in the references given: (2A.l)
Lemma.
tion and define (2A.2)
Lemma.
d(01,02) > 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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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<
r).
such that for
Then t > 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
<
Ilxll.2lxl.2
I T(x)
Hence
I l'7nl
r(T) = lim
I f we define
< "'.
r
II Tn+l(X) II n+l r
Then there is a norm
is given by
is a norm and that
I I Tnn " ) I I x I I .
(2A.5) on
r(t)
T.
rt
e rt > lim I I entAl I lin.
is
>
spectral radius
Set
n+'"
Ixl = sup n;::a
II entA(x) II e
rnt
t~a
and proceed as in Lemma 2A.4. There is an analogous lemma if
CJ r
< cr (A) ,
giving
e
rt
With this machinery we now turn to Theorems 1.3 and 1.4 of Section 1: (1.3) operator.
Let
Theorem.
Let
T: E
cr(T) C {zlRe z < a}
+
E
be a bounded linear
(resp.
cr(T) ~ {zlRe z > 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
vector field on a
=
X(PO)
then
O.
If
is asymptotically
stable. Proof. Po
=
O.
We can assume that P is a Banach space E
As above, renorm
I letA I I < e - d ,
where
A
E
and find
€ > 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
a. = €/2. Let
show that if remains in
B
be the open
X
E B,
o Band
....
r /2 ball about O. We shall 2 then the integral curve starting at X
0
exponentially as
t
.... +00.
o This will
prove the result. Let Suppose
x (t)
x(t)
be an integral curve of
remains in
B
for
0
X
< T.
starting at Then noting
that
*See
for instance, Hale [1], Hartman [1], Lang [1], etc.
x • o
THE HOPF BIFURCATION AND ITS APPLICATIONS
54
x
x(t)
= x
I: I:
+
o
+
o
x (x (s)) ds A(x(s)) + R(x(s)) ds
gives (the Duhamel formula; Exercise 2.9), x(t) = e tAx o + It 0 e (t-s)A R(x(s)) ds
I:
and so Ilx(t)11 Thus, letting
~e-tExo
f (t) = e
tE
Ct
Ijx(t)
~xO
f(t)
+
+
Ct
e-(t-s)Ellx(s)llds.
II,
I:
f(s) ds
and so
(This elementary inequality is usually called Gronwall's inequality cf. Hartman [1].) Thus
Hence
x(t) E B,
extended in x(t)
+
0
t
as
0 < t
< T
so
x(t)
may be indefinitely
and the above estimate holds. t
+
+00.
It shows
~
The instability part of Theorem 1.4 requires use of the invariant manifold theorems, splitting the spectrum into two parts in the left and right half planes.
The above analysis
shows that on the space corresponding to the spectrum in the right half plane, (2A.6)
PO
is repelling, so is unstable.
Remarks.
Theorem 1.3 is also true by the same
THE HOPF BIFURCATION AND ITS APPLICATIONS
55
proof if T is an unbounded operator, provided we know o (e tT ) etO(T) which usually holds for decent T, (e.g.:
if the spectrum is discrete) but nee,d not always be true (See Hille-Phillips [1]).
This remark is important for applications
to partial differential equations. require
which is unrealistic for partial
R(x) = 0(1 Ixl I)
differential equations. Assume F
in
t
However, the following holds:
O(DFt(O»
=
etO(DX(O»
and we have a
cO flow
cl
with derivative strongly continuous
(See Section 8A),
then the conclusion of 1.4 is true.
and each
t
Theorem 1.4's proof would
F
t
is
in the notation above,
This can be proved as follows: we have, by Taylor's theorem:
where
R(t,x)
as long as
is the remainder.
x
remains in a small neighborhood of -2Et < e and hence
because some
E > 0
R(t,x)
=
J:
Now we will have
and
x
in some neighborhood of
{DFt(sx),x - DFt(O) ·x} ds. x
and
00
+
0
Exercise.
(2A.7) F: E
+
E
a
exponentially as
cl
map with
inside the unit circle. U
of
0
Let
such that if
is a stable point of
E
t
+
for
remember
remains close to
0
~
be a Banach space and
F(O) = 0
and the spectrum of
DF(O)
Show that there is a neighborhood x EU, F(n) (x)
F.
-Et < e
Therefore, arguing as
in Theorem 1.4, we can conclude that Ft(x)
0 '"
this is
0;
+
0
as
n
+
00;
Le.
0
56
THE HOPF BIFURCATION AND ITS APPLICATIONS
SECTION 2B THE POINCAR~ MAP We begin by recalling the definition of the Poincare map.
In doing this, one has to prove that the mapping exists
and is differentiable.
cO
of
flows
Ft(X)
In fact, one can do this in the context such that for each
t, F
t
ck ,
is
as
was the case for the center manifold theorem for flows, but here with the additional assumption that t
t > 0.
as well for
Ft(X)
is smooth in
Again, this is the appropriate hypo-
thesis needed so that the results will be applicable to partial differential equations.
However, let us stick with the ordinary
differential equation case where vector field
X
F
t
is the flow of a
at first.
First of all we recall that a closed orbit flow
on a manifold
Yet)
X
L >
y
O.
ck
such that
The least such
M
of a
is a non-constant integral curve
y(t+L) L
y
Yet)
for all
is the period of
y.
t ER
and some
The image of
is clearly diffeomorphic to a circle. (2B.l)
Definition.
Let
y
be a closed orbit,
let
THE HOPF BIFURCATION AND ITS APPLICATIONS
m E Y,
say
m
Y(O)
=
and let
S
57
be a local transversal sec-
tion; i.e. a submanifold of codimension one transverse to
,
Y(Le.
Y (0)
is not tangent to
S).
'21 eM
Let
x IR
be the
domain (assumed open) on which the flow is defined. A Poincare map of (PM 1)
(PM 2)
there is a function
p(x) (PM 3)
if
=
o
->-
W l
where:
are open neighborhoods of k diffeomorphism; is a C
P
for all
P: W
S
WO'W l e and
is a mapping
Y
0: W
o
->-
IR
m E S,
such that
x E WO' (X,T-O(X)) E'21 and F(X,T-O(X)); and finally,
t E (O,T-O(X)), then
~
F(x,t)
W
o
(see Figure 2B.l). (2B.2)
Theorem (Existence and uniqueness of Poincare
maps). (i)
If
X
is a closed orbit of
is a
ck
vector field on
M,
and
Y
then there exists a Poincare map of
X,
Y. (in
->-
Wl
a local transversal section
S
at
m E
and
pI
are locally conjugate.
(ii)
S'
at
P: W o
If
m ' E Y),
then
P
is a Poincare map
That is, there are open neighborhoods m' E S I, W e Wo 2
and a
n
Wl '
k C
and
~
I:
0
pI
Y also
(in
,
of W of m E S, W 2 2 , H: W ->- W ' such that 2 2
diffeomorphism I W2 e Wo and the diagram
,
1 0- (W,)
; j
.W
W2----1.8....'--»,
commutes.
y)
of
n
0W,
THE HOPF BIFURCATION AND ITS APPLICATIONS
58
P(x)= F(x,t - 8{x»
Figure 2B.l Here is the idea of the proof of existence of further details see Abraham-Marsden [1). trarily.
By continuity,
borhood
U
o
assumption,
of
m.
of
at
x
U 2
t-differentiable at =
m
P P
tive of
FT-O(x) (x) E S.
This is
P(x),
will be as differentiable as at
m FT
arbi-
of
m.
t = 0
By
and is
and hence also in a neighborhood
Therefore, there is a unique number
such that tion
S
is
S
(for
is a homeomorphism of a neigh-
to another neighborhood
FT+t(X)
transverse to of
m
FT
Choose
P
F
o(x)
near zero
and by construcis.
The derivative
is seen to be just the projection of the derivaon
a diffeomorphism,
T S m
P
(this is done below). will be as well.
particular, this means
Ft
t > 0
FT
is
CJ
For partial differential equations, a semi-flow i.e. defined for
Hence if
F
is often just t (See Section 8A). In
is not generally a diffeomorphism.
THE HOPF BIFURCATION AND ITS APPLICATIONS
(For instance if
59
is the flow of the heat equation on L , 2 t is not surjective). For the Poincare mapping this means
F
F
t that we will have a for
t > 0),
morphism.
but
P P
(if
Ft(X)
is differentiable in
t, x
is just a map, not necessarily a diffeo-
This is one of the technical reasons why it is im-
portant to have the center manifold theorem for maps and not just diffeomorphisms. As we outlined in Section 1, there is a Hopf theorem for diffeomorphisms and this is to be eventually applied to the Poincare map circle for
P
P
to deduce the existence of an invariant
and hence an invariant torus for
Ft.
For partial differential equations, this seems like a dilemma since
P
need not be a diffeomorphism.
can be overcome by a trick:
However, this
first apply the center manifold
theorem to reduce everything to finite dimensions--this does not require diffeomorphisms; then, as proved in Section BA, (see BA.9) in finite dimensions
F
and hence P will autot matically become (local) diffeomorphisms. Thus the dilemma is only apparent. Let us now prove the fundamental results concerning the derivative of
P
of
Ft.
E,
and we can let
so we can relate the spectrum of
P
to that
It suffices to do the computation in a Banach space m = 0,
the origin of
We begin by calculating (2B.3)
Lemma.
Banach space.
Let
dP(O)
Let
in terms of
F : E + E be a t be on a closed orbit
0
E. Ft.
Cl
flow on a
y
with period
dF
f O.
Let
ated by
V
T
E = F
e
V
t ~ (0) It=o = V.
and let and
F
Let
V
be the subspace gener-
be a complementary subspace, so that 1 2 (Ft(X,y), Ft(X,y)). Let P: F + F Ft{x,y)
60
THE HOPF BIFURCATION AND ITS APPLICATIONS
be the Poincare map associated with
y
at the point
O.
Then
2
dP (0) = d1F 'I (0) . Proof.
Let
'I(X)
the above notation).
be the time at
Then by definition of P(x)
so
aF
dP(x) Letting
P(x)
1
(x)
in
P,
1
F 'I (x) (x, 0)
1
(X,O)d'I(X) + d1F'I(x) (X,O).
at
(x,O) = 0,
('I-O (x)
we get a F1
1 (O)d'I(O) + d1F (0) . 'I 2 aF ) (O,V) by construc(0), at'I (0)
at
dP (0) =
1 aF 'I However d t (0) = (a F a F1 'I tion,so a t (0) = o. Thus,
-IT-
~ dF
(2B.4)
Lemma.
Proof.
~ (0,0)
(0,0)
ds = dF
'I
So,
dF of 'I s ds
Is=O
dF t = dt
(0,0)
It='I
v
(O,O)ls=o
(d1F'I(0,0)'0 + d 2 F 'I (0,0)V)
(0,0) (O,V) d F (0,0)V 2 'I ( 2B• 5 )
0
d F (0,0)V = V. 2 'I
dF 'I + S
Is=O
1 d1F'I(0).
dP (0)
V.
Lemma.
0 CJ (dF'I (0 , 0) )
CJ (P (0» U {l}.
(This is
true of the point spectrum, too). Proof.
*
where Let
A
The matrix of
dF (0,0) 'I
is
indicates some unspecified matrix entry.
E~.
Then
A
E CJ(dF'I(O,O»
iff
dF'I(O,O) -
AI
is not
THE HOPF BIFURCATION AND ITS APPLICATIONS
I - l o r is not onto.
But
(dF T (0, 0) - AI) (~) = (dP ( (dP (0) 0: - Ao:, Clearly,
°)0:,
1 Ea(dFT(O,O».
(0:,6)
we can choose
6
Since
The former implies A 'I 1,
for any
0:, There-
1'1 A Ea(dP(O}}.
On the other hand, let
is not onto, then clearly neither is
dFT(O,O} - AI. 0.
A Ea(dFT(O,O».
so that the second component is onto.
A Eo(dP(O)}. dP(O) - AI
and
such that the expression
Assume the latter.
=
A '11
is zero or the map is not onto.
fore,
+ 6) + (- A0: , - A6 ) (2B. 1)
Assume
A E a(dP(O».
If,
* (0:)
* (0:) + 6 (1 - A».
Then either, there exists (2B.l)
61
Suppose there exists
Then choosing
6
=
*(o:) '-1'
0:
such that
we see that
dP(O}o: - Ao:
A E o:(dF (0,0».
0
T
A
Consult Abraham-Marsden [1] Abraham-Robbin [1] or Hartman [1, Section IV. 6, IX.IO] for additional details on this and and the associated Floquet theory. One of the most basic uses of the Poincare map is in the proof of the following. (2B.6) with period
Theorem. T.
Let
Let
m E y
be a closed orbit for
y
and suppose
dFT(m)
inside the unit circle except for one point circle.
Then
y
Proof.
1
F
t has spectrum
on the unit
is an attracting (stable) closed orbit.
By Demma 2B.5, the condition on the spectrum
means that the spectrum of the derivative of the Poincare map P
at
m
is inside the unit circle.
an attracting fixed pojnt for struction of
P
that
y
P.
Hence from 2A.7
m
is
It fo110ws from the con-
is attracting for
Ft.
o
62
THE HOPF BIFURCATION AND ITS APPLICATIONS
(2B.7)
Exercise.
(a)
Give the details in the last step of the above
(b)
If, in 2B.6,
proof. P
has an attracting invariant
circle, give the details of the proof that this yields an attracting invariant 2 torus for the flow.
THE HOPF BIFURCATION AND ITS APPLICATIONS
63
SECTION 3 THE HOPF BIFURCATION THEOREM IN R
2
n AND IN R
The center manifold theorem is used to reduce bifurcation problems to finite dimensional ones as follows.
~~: Z
Consider a one parameter family of maps a Banach space
Z, where
E R or an interval in
~
1+ 'I'~(x)
taining
O.
Assume
for all
~.
Assume that for
(~,x)
~
< 0
is
ck + l
and
R
~
=
0
Z
on
con-
'I'~(O) = 0
the spectrum of
is strictly inside the unit circle, for
+
D'I'~ (0)
the spectrum
splits in two pieces as in the center manifold theorem and for
~
> 0
the spectrum has two pieces, one inside and one
outside the unit circle.
f-L
See Figure 3.1.
f-L=O Figure 3.1
64
THE HOPF BIFURCATION AND ITS APPLICATIONS
Consider'!': R x Z ..,. R x Z,
(fJ,x)
/+
(fJ,'I'fJ(x».
The
derivative at zero is D'l'(O,O) (v,h) =
v,
a'I' fJ -a-
[ =
(since
'l'fJ(O) = 0
(x}v
I
fJ
fJ= 0 x=O
(v,D 'l'fJ (O) 'h)
x
for all
consists of the spectrum of
fJ}.
Thus, the spectrum of
Dx'l'fJ(O)
plus the point one.
Therefore, we can apply the center manifold theorem to produce an invariant manifold in The
fJ
=
D'I'(O,O)
'I'
to
R x Z.
constant slices of this invariant manifold
produces a one parameter family of invariant manifolds for 'I'fJ.
These manifolds have the same dimensionality as the
eigenspace of the piece of the spectrum crossing the unit circle, and this is often finite dimensional. There is an entirely analogous reduction possible for flows using the center manifold theorem for flows. It should be cautioned that the center manifold, while containing all the local recurrence, is not globally invariant nor need it be attracting in the strict sense.
However, if
care is exercised, and if a pair of eigenvalues crosses the unit circle or the imaginary axis if we are thinking in terms of the vector field, then we are in the two dimensional case. (We remark--see Section SA that a semiflow of
k l C +
a finite dimensional space automatically has a
k C
maps on generating
vector field, so that after the reduction is made we may usually assume that the generator of the flow is smooth.) Hence we shall next examine, in detail, the finite dimensional case.
(Details of the above reduction process are
given in Section 4.)
THE HOPF BIFURCATION AND ITS APPLICATIONS
65
First, we consider the two dimensional case.
(The
n-dimensional case is treated in Theorem 3.15 below.) 2 The Hopf Theorem in R The following theorem is essentially due to Andronov (1930) and Hopf (1942) and was suggested in the work of
..
Poincare (1892)
Theorem.
(3.1)
field on
R
is also
k
c
+
2
such that Let
•
Let
271/IA(0)
1
A (fl)
and
for all
0
I
k-2* C
(C)
(X ,0)
fl
> 0
fl
Then
function
~
of
fl :
X
for
R
->-
(-8,8)
~
xl f
0
= o.
There is a neighborhood
0
=
is on a closed orbit of period
such that any closed orbit in
Furthermore, if
X
such that for
A (fl)
and radius growing like fl(O)
vector
and
fl
d (Re A(fl) ) > O. dfl fl=O
there is a
(B) R
=
(k > 4)
have two distinct, complex
fl
(xl,O,fl(X » l
and such that
3
fl
Also, let (A)
such that
X (0)
ck
be a
fl
dX (0,0)
conjugate eigenvalues Re A(fl) > O.
X
U
U
for all
(0,0,0)
in
is one of those above.
is a "vague attractor" for f1(x l ) > 0
of
xl f
0
X ' then O and the orbits
are attracting. The meaning of "vague attractor" will be spelled out as we go along and the detailed calculations involved in this condition are worked out in Section 4 (see also Section SA). In Section 3A it is shown that "vague attractor" can be
+The present version of Theorem 3.1 is due to Ruelle and Takens [1]. The n dimensional case is due to Hopf. Section SA for comparisons.
* If
X is analytic, then see Section 5).
f1
See
will also be analytic (Hopf [1];
66
THE HOPF BIFURCATION AND ITS APPLICATIONS
weakened to "attractor" in the usual Liapunov sense.
In any
case, this condition is usually not obvious in examples and will be discussed extensively below. Our proof follows Ruelle-Takens [1], with the details included.
At the end of the section we shall discuss what
happens if
d(Re_A(~»/d~
Proof.
= 0
(see Section 3A).
The essence of the proof is an application of
the implicit function theorem. We show that for small ~, k l there is a C £unction which takes the point (xl'O,~) (P(xl'~) ,O,~)
to the first intersection (xl'O,~)
and
=
under the flow of
P(xl'~)
P(xl'~)
of the orbit of
X with the xl-axis such that
have the same sign (Figure 3.2).
- xl·
V
Let
xl
V(xl'~)
is a displacement function.
We use the implicit function theorem to get
11- = 11-0
Figure 3.2 a curve
(xl,O,~(xl»
orbits of the flow of
of zeros of
V, i.e., a curve of closed
X.
(xl,O)
The map
~
(P(xl,~(xl»'O)
THE HOPF BIFURCATION AND ITS APPLICATIONS
67
is the Poincar~ map associated with the closed orbit through
(xl,~(xl».
We use standard results about Poincar~ maps to
find conditions under which the orbits are attracting.
The
uniqueness of the orbits is essentially the uniqueness of the implicitly defined function in the implicit function theorem. (Proving uniqueness of the closed orbits is more complicated in higher dimensions, see Section S and Section SA, page 198. Step 1: that
By a ~-dependent change of basis on
Re A (~) Im A (~) ) dX (0,0) = [ -Im A(~) Re A(~) ~
so that
Im
A(~)
> O.
A(~)
In the new coordinates, X
~
continuous derivatives up to order may not exist.
where
2 R , we may assume
k
will have
akx~/aj1k
except that ~,
Furthermore, for each
variant under the change of basis.
is chosen
the xl-axis is in-
(That is, the new xl-axis
is the same as the old one, and we are only changing the x -axis). 2
Let us now note a few simple lemmas:
function from
U C R
eigenvalues for all
ck
functions from Proof.
+
4 R •
~
E [a,b] C U.
Let the matrix have two distinct
(a,b) C R + ~.
By the quadratic formula, the eigenvalues are
all + a 22 ± l(a ll +a 22 )2 - 4a l2 a 21 -=-=---..;;::.::..---=-:::2-=e::----....::.::-=- 4a
a l2 21
values are (3.3)
Then the eigenvalues are
2 By assumption
is bounded away from zero on
ck
functions of
Lemma.
Let
22
)
(a,b), so the eigen-
on this interval.
T: ~2 + ~2
(all +a
0
be a linear transforma-
tion that is real on real vectors and has no real eigenvalues.
68
THE HOPF BIFURCATION AND ITS APPLICATIONS
Let
vI + iV
be an eigenvector with eigenvalue
2
(~)
is an eigenvector Proof. vector of
+
i(~)
with eigenvalue
show that there is a
z
=
There
which has the same eigenvalue.
Any complex multiple of
T
A.
A.
x + iy
[:) +
vI + iV
2
is an eigen-
Thus, it is sufficient to such that
i[:).
This is equivalent to
solving the pair of equations: I
o i.e.,
The columns of this matrix are independent over
=
R
since if
vI + iV 2 = (l+iclvl· Therefore, vI (l+ivl-l(vl+iv2) is a real eigenvector, which cannot be.
v2
cv l ' then
Thus, the equation can be solved. [] (3.4)
Proof. TV
I
TV
Let
T
be as in the previous lemma.
= Re T(vl+iv2l
I Re[A(vl+iv2l]
=
1m AV
Lemma.
because
Re AV I - 1m AV 2 •
TV 2
T =
is real. Im[A(vl+iv2l]
+ Re AV 2 • CJ
I
Using .the preceding lemmas, we see that if is an eigenvector of
g)
and
a (~l ) [ B(~)
dX~(O,Ol
with eigenvalue
[~) A(~l,
+
i(~~~l) then
are independent vectors such that the matrix
THE HOPF BIFURCATION AND ITS APPLICATIONS
of
dX~(O,O)
Re A (~) 1m A (~) ] • [ -1m A(~) Re A(~) a
[~)
with respect to
a (~») [ s (~)
and
69
is
We now show that the vector
function of
Let
~.
We solve the equation
= A (~)
all
[
(~)
l+ia(~»)
•
From this we obtain the equations
is(~)
=
Re A (~)
Therefore, Re A (~) - all a(~) =
(~)
1m A(~)
Because the change of coordinates is. linear for each because
a
and
coordinates
akx
S
X
are
~,
functions of
will have continuous k
th
~
and
in the new
partials except that
. I ax ax are In partlcu ar, -a-- and a xl x2 functions in the new coordinates. From now on, we will assume
a-k
may not exist.
~
that the coordinate change has been made, i.e., that
=
dX (0,0)
~
Step 2:
of
•
There is a unique
ck - l
X
Iji: R
such that map
Re A (~) 1m A (~) ) [ - 1m A (~ ) Re A (~)
Iji*X
Iji (r, e) =
~'
where
x~
vector field 2 -;. R2
(r cos e, r sin e) , and
on
R
2
is the polar coordinate Iji*
is the differential
Iji.
X
Let
* Note that
X e
component of
is the "angular velocity" of X
along a unit vector
direction which is what
X e
X, and not the
in the
ee often stands for.
a/ae
THE HOPF BIFURCATION AND ITS APPLICATIONS
70
Then
-r sin r cos x
\l
:) [~:)
.
implies
sin e] [X\ll] cos e x ' r
t- o.
r
\l2
-
Thus, if the vector field X can be extended to be \l 2 on all of R , it is clearly unique. X\lr cos e X\ll + for all
sin e X\l2' which is
(r, e) •
Consider
-sin e x\ll(r cos e, r sin e)
~~r~
+ co~ e X\l2(r cos e, r sin e),
t- o.
r
Then
-sin e lim r .... O
lim X\le(r,e) r .... O
+
because
cos e lim r .... O
O.
X (0,0) \l
x\ll(r cos e, r sin e) - X\ll (0,0) r x\l2(r cos e, r sin e) r
-
X\l2 (.0, 0)
Thus
(-sin e)dX\ll (0,0) (cos e, sin e)
+ (cos e)dX\l2 (0,0) (cos e, sin e) (-sin e) (Re A(\l)COS e + 1m A(\l)sin e)
+ (cos e) (- 1m A(\l) cos e + Re A(\l) sin e) -1m A(\l).
We therefore define
THE HOPF BIFURCATION AND ITS APPLICATIONS
71
(cos e xjll(r cos e, r sin e)
+ sin e xjl2(r cos e, r sin e»~r xjl(r,e)
+ co~ e xjl2(r cos e, r sin e»~e ' r
a
- Im A(jl)ae-
r
to
0
=
-sin e X (r cos e, r sin e) + r jll
+ c~s e xjl2(r cos e, r sin e),
Xjle(r,e) - Im A(jl),
To see that
Xjle(r,e)
r
is
1
- X l(r cos e, r sin e) r jl
ck - l
=
r
t
0
0
ck - l
, we show that the functions
!r
and
X (r cos e, r sin e) jl2
are
when extended as above. Lemma.
(3.5)
Let
2 A: R .... R
be
Then
1
-
A(x,y)
A(O,O) =
I0
Ai (0,0) =
A(tx,ty) Y dt. ay
aA(tx,ty) x + ax
0
1 A (x,y) l
I
1
aA(tx,ty) dt , ax
dA(O,O) dX
Proof.
Let
and
and
A (X,y) = 2
A (0,0) = 2
I0
aA(tx,ty) dt. 3y
aA (0,0) ay
The first statement is true by Taylor's theorem
and the second statement is easy to prove by induction. lemma,
!r
l ax , (rt cos e, rt sin e)
X . (r cos e, rt sin e) = cos e jlJ l
+ sin e
By the
Io
J ....JU.
dt
0 ax
ax . (rt cose, rt sin e)
....JU. ay
dt
for
j
1,2.
Since all k
th
72
THE HOPF BIFURCATION AND ITS APPLICATIONS
dX
partials of
X
, too.
Step 3:
D
The Poincare Map (see Section 2B for a discussion of
Poincare maps). respectively.
=
-
Let the flows of
X
and
X
be
~t
and
~t
It is elementary to see that
Consider the vector field we have
the functions dfl and so the integrals are
Ck - l
under the integral sign are
ck - l
--rk '
are continuous except
~flt(O,e)
=
X.
Since
X(O,e)
~021T/[A(O)
(O,e-Im A(fl)t,fl).
(O,O-IA(O) 121T/[A(O) 1,0)
=
I (0,0)
(Figure 3.3).
(O,-21T,0)
8
r
-
-
-
-
-
8=-27T
Trajectory of (0,0,0) under ~Ot Figure 3.3 Because
X
is periodic with period
21T, it is a
k C -l
vector
field on a thick cylinder and the orbit of the origin is closed.
We
can associate a Poincare map
P
with this orbit (Figure 3.4).
THE HOPF BIFURCATION AND ITS APPLICATIONS
Poincare map on cylinder
/
/
domain of P
/
<0,0,0)
8=-27T ~
range of P Poincare Map in R Figure 3.4
3
73
74
THE HOPF BIFURCATION AND ITS APPLICATIONS
That is, there is a neighborhood
~
and
E
such that the map
(-8,E)}
p(r,~)
where
U =
is the
r
{(r,O,~)
I r E (-8,8)
P(r,O,j.J) = (P(r,~) ,-211,~),
coordinate of the first intersection
of the orbit of (r,O,~) with the line e = -211, is defined. k l This map is c - • The map T(r,~) , which is the time t when k-l ~t(r,O,~) = P(r,O,~) is also C • Note that under 1JJ, the Therefore, the displacement map k is defined and c - l on the
r-axis becomes the xl-axis. (xl'O,~) 1+
(xl+V(xl'~) ,O,~)
neighborhood
e = {(xl'O,~)
(xl+V(xl'll)'O,~) (xl'O,~)
I
xl E (-8,8)
with the xl-axis such that the sign of
(3.6)
E (-8,8) }
is the first intersection of the orbit of
P(xl,j.J) = xl + V(xl,j.J)
sign of
~
and
Remark.
xl
and the
are the same.
Using uniform continuity and the fact
is e-periodic, i t is easy to see that there is a t 2 2 neighborhood N = { (r, 8 ,j.J) I r + ~ < o} such that no point that
of
N
of
¢t
is a fixed point of in
(0,0, ll) •
(3 • 7 )
Lemma.
dP
(xl ,ll) dX
l
I xl=O ll=~
= (a llt (x l ,x 2 ), b~t(xl,x2»' flow satisfies the following equation: Proof.
Let
~t(xl,x2)
T(Xl,ll)
f
o
The
Xlll(a~t(xl'O), b~t(xl,O»dt.
We differentiate this equation with respect to
xl
to get the
desired result; the differentiation proceeds along standard
THE HOPF BIFURCATION AND ITS APPLICATIONS
75
lines as follows V(xI+6xI'~) -
V(XI'~)
6x
I
I
6x
I
From this we see that
aT(xI'~)
+
a
Xl (a T ( ~
x2
~
xl'~
) (xl,O), b T ( ~
In case
Xl = 0, we can evaluate this expression.
a~t(O,O)
=
b~t(O,O)
=
0
and since
XI~(O,O)
=
xl'~
) (xl' 0) ) .
Since
0, the second
term on the right-hand side vanishes. Recall that T(O,~) = aX I 2TI/Im A(~). By the chain rule, ~ (a t(O,O), b t(O,O) = xl ~ ~ aX aa~t ax aXI~ ab aa-l (0,0) ax-- (0,0) + ax ly (0,0) + ~ ax yt (0,0). Since I I I aXl~
aa
(0,0)
(0,0) = Re
A(~)
and
is a fixed point of
and because ~~t'
we can evaluate the derivatives
76
THE HOPF BIFURCATION AND ITS APPLICATIONS
of the flow here.
=
dflt (0,0)
[-::
Re A(fl)cos Im A(fl)t
e
Re A(fl)sin Im A(fl)t
e
aa
Therefore,
~ aX l
(0 0) '
=
A (fl)
rm ,(.,
-Im A ()l)
Re A (fl)
expt[ Re
exp[tdX (0,0)] fl
rm't.,t]
t
Re A(fl)sin
t
Re A(fl)cos Im A(fl)t
e tJRe A(fl)cos Im A(fl)t
II
and
So we have
av aX
(O,fl)
l
2 'IT/lm A (fl) t Re '( ) 1\ )l (Re A(fl) cos ImA(fl)t-Im e 0
A(fl)sin Im A(fl)t)dt
J
e 2 'IT (Re A (fl) ) / Im A (fl ) _ 1.
Step 4.
Use of the
Implicit
0
Function Theorem to Find
Closed Orbits The most obvious way to try to find closed orbits of
V.
is to try to find zeros of
av
(0,0)
aX l
av ail
or
(0,0)
Since
t
V(O,O) = 0, if either
were not equal to
0, the conditions
for the implicit function theorem would be satisfies and we could have a curve of the form such that
2.Y.... aX
V
(0,0).
=
0
(xl()l) ,fl)
along the curve.
Instead of
V
or
(Xl,fl(X )) l
Unfortunately,
we use the function
l
av
aX l
(O,fl)
o
~~ (0,0)
0
THE HOPF BIFURCATION AND ITS APPLICATIONS
(tx l ,)1)x dt
v
because
l
o
k-2 C .
is
Recall that
Proof.
II ~~l
v
Lemma.
(3.8)
77
k-l C .
is
V(O,)1)
V(X ,)1) l V(x ,)1)
0,
l ---,;-,,-=-aX l
1
Io
av aX
t
(tx l ' )1)dt , x 1
l
The function
k 2 C -
easily seen to be Lemma.
(3.9)
0.
by induction.
3iJ
and
N 2
there are neighborhoods )1: N ->- N 2 l
function
Proof. since
o.
1 [av lim aX )1->-0 )1 l
~[e2n(Re
d ll
(0,)1) -
N l
such that
av aX
V(O,O)
Re A(0)
of
O.
Therefore,
and a unique
0
and such that
)1 (0) = 0
lim )1->-0 2 a v a)1aX
(0,0) )
-
t
(0,0)
o
l
l
A()1))/Im A()1)
>"
[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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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)<0.* 3x 3 1
*This
condition is computable; see Section 4.
THE HOPF BIFURCATION AND ITS APPLICATIONS
(3.13)
Lemma.
If
(0,0)
Xo' then the orbits through ~(xl) > 0
show that that
~
(xl,~(xl»
~(xl)
To show that
~"(o}
>
o.
V(xl,~(xl»
=
O.
and evaluated the result at 2
~"(O)
=
_a3~ aX l
are attracting and
=
xl
=
=
O.
~
0, we
0, this shows
Again we differentiate
0, we get
~
av (0,0) -~--~-- (0,0). / - oj.loX l
that the orbit through
=
~'(O)
xl
Having done this three times
xl
21T dReA(~)1 > 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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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.
There is a neighborhood
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
There is a neighborhood
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'
THE HOPF BIFURCATION AND ITS APPLICATIONS
? 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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
<
I}
and our assumptions plus continuity will imply
the same for map is the spectrum of
Since the spectrum of the Poincare d¢
T,~
(x)
restricted to a subspace
transverse to the closed orbit, this shows and the orbit is attracting.
~
a (dP (x» C {z
I Iz'
< I}
THE HOPF BIFURCATION AND ITS APPLICATIONS
(3.17) X (x ,X ) = 2 V l Theorem 3.1. (3.18) ordinates.
Exercise.
83
Show that the vector field
2 (x ,V(l-x )x -x ) l 2 l 2
Exercise.
satisfies the conditions of
All equations are given in polar co-
Match each set of equations to the appropriate
picture and state which hypotheses of the Hopf bifurcation (If you get stuck, come back to this
theorem are violated.
problem after reading Section 3A.)
1-
3.
.
2
r
-r (r+v)
e
1
r
r(r+ll) (r-ll)
e
1
5.
2.
4.
. e
r
1
r
vr(r+v)
e
1
r
222 -v r (r+l1) (r-l1)
e
1
y
222 r (fJ-r ) (211-r )
-----*'""------"':~y
x
(A)
2
(B)
84
THE HOPF BIFURCATION AND ITS APPLICATIONS
y
x
-----*----"7-"y
x
(0)
( E)
THE HOPF BIFURCATION AND ITS APPLICATIONS
85
SECTION 3A OTHER BIFURCATION THEOREMS Several authors have published generalizations of the Hopf Bifurcation.
In particular, Chafee [1] has eliminated
the condition that the eigenvalue axis with nonzero speed.
A(~)
cross the imaginary
In this case, bifurcation to periodic
orbits occurs, but it is not possible to predict from eigenvalue conditions exactly how many families of periodic orbits will bifurcate from the fixed point.
Chafee's result gives a
good description of the behavior of the flow of the vector field near the bifurcation point.
See also Bautin [1] and
Section 3C. Chafee's Theorem We consider an autonomous differential equation of the form (3A.l)
x
where ~
x
> 0
and
and
X
vary in real Euclidean space
is a small parameter (called P
is a real
n x n
matrix.
~
Rn
(n > 2),
in previous sections),
We assume the following
86
THE HOPF BIFURCATION AND ITS APPLICATIONS
hypotheses. There exist numbers
that
P
EO > 0
such
is continuous on the closed interval
[0, EOl
and
is continuous on the domain E
For each
(HZ)
that the origin
in
r
> 0
and
(HI)
O
X
Bn(r ) x [0, EOl • O [O,EOl
X(O, E) = 0
we have
so
is an equilibrium point of (3A.l).
x = 0
(H ) For each r in [O,rOl there exists a k(r) > 0 3 such that on the domain Bn(r) x [O,E ] the function X is O uniformly Lipschitzian in x with Lipschitz constant k(r); moreover,
k(r)
(H 4 )
+
as
0
For each
r E
+
in
O. [O,EOl
complex-conjugate pair of eigenvalues
the matrix
P(E)
a(E) ± ib(E)
has a
whose
real and imaginary parts satisfy the conditions a(O) = 0,
a(E) > 0
b (E)
The other eigenvalues
> 0
Al (E),
Az (E), •.. , An _ (E) Z
have their real parts negative for all (H )
For
S
E = 0
E
of
P (E)
in
[O,E ]' O the equilibrium point of (3A.l)
at
the origin is asymptotically stable in the sense of Liapunov (Lefschetz [11, p. 89, and Section 1 above). Hypotheses (HI) and (H ) are sufficient to guarantee 3 the usual properties of existence, uniqueness, and continuity in initial conditions for solutions of (3A.I).
In that which
follows the solution of (3A.l) assuming a given initial value at
t
o
will be denoted by
x(t,xO,E).
In connection
with this notation we should mention the well-known autonomous property of (3A.I): initial value
xo
the solution of (3A.I) assuming a given at a specified value of
t,
say
to'
is
THE HOPF BIFURCATION AND ITS APPLICATIONS
87
x(t - to' x O'£)' The hypothesis (H ) replaces the S "vague attractor" hypothesis considered earlier.
given by
(3A.l)
Theorem.
Let
(3A.l) satisfy the hypotheses £0
be as in (H ). l such that o < r
Then,
there exist numbers
r l ,r 2 ,
~
and such that the following assertions
r O'
~
0 < £1
£0'
2
~
r
l
are true. (i)
£ E (0'£1]
For each
there exist for Equation
(3A.l) two closed orbits
Yl (£) and Y2 (£) (not necessarily distinct) which lie inside a neighborhood of the form Bn(r(£», ~
r2
and
r (£)
0 < r (£)
Moreover, 2 Yl (£) and Y2 (£) lie on a local integral manifold M (£) 2 homeomorphic to an open disk in R and containing the origin x = O. Regarded as closed Jordan curves in M2 (£), Yl(E) and
Y2 (E) Y (E) 2
are concentric about the origin with, say,
->-
0
as
£
0 +.
where
->-
Y (E) inside l
when these curves are distinct. (ii)
lies inside
E E (0, El ] that part of M2 (E) which is filled by solutions of (3A.l) which
For each Yl(E)
approach the origin as equilibrium point
t
ar x
->-
0,
and which, except for the
_00
approach
Yl(E)
as
t
->-
+00.
No
other solutions of (3A.l) remain in (iii)
For each
outside
E E (O,E ] l but contained in
Y2 (E) of (3A.l) which remain in which approach
Bn(r ) for all t < O. l that part of M2 (E) lying
Bn (r ) is filled by solutions 2 M2 (E) n Bn(r ) for all t > 0 and l t ->- +00.
Y (E) as 2 (iv) For each E E (O,E ] l (3A.l) which approach the origin
there exist solutions of x
=
0
solutions fill a local integral manifold Mn - 2 (E)
homeomorphic to an open ball in
as (=
t
->-
+00
and these
invariant manifold)
Rn - 2
and containing
88
THE HOPF BIFURCATION AND ITS APPLICATIONS
the origin
x
=
O.
E E (O,E ]' x(t,XO,E) is a soluO n tion of (3A.l) for which X E B (r 2 ), then x(t,xO,E) reo mains in Bn(r ) for all t > O. Moreover, if x(t,XO,E)~O l ~ t + +00 (§gg (iv)) then as t + +00, x(t,XO,E) approaches (v)
If for a given
the closed invariant set n(E) consisting of those points in 2 M (E) which lie on Yl(E) or Y2 (E) or between them. The solutions which approach n(E) contain in their positivelimiting sets one or more closed orbits (which mayor may not coincide with
Yl(E)
or
Y2(E)).
See
Figure 3A.l.
___ stable manifold of x=O invariant set
Figure 3A.l
Chafee [2] has also proved a theorem parallel to 3A.l for the case in which the vector field at time the flow at time where
x
t
t - a
= x(t-a).
for some
a > 0;
i.e.,
t
depends on x = F(t,x t )
THE HOPF BIFURCATION AND ITS APPLICATIONS
89
The following example (see Chafee [1]) shows that one cannot predict the number of distinct families of closed orbits to which the flow bifurcates.
In cylindrical coordinates, let
2 n 2 n 2 n 2 n r (E-r) l (2E-r) 2 (3E-r) 3 .•• (mE-r ) m
dr dt
de
rf(r"ll
=
1
dt dz dt
J (JA.2l
- z
where
nl+···+n m
is odd.
This equation is thermore, it has
m
COO
in rectangular coordinates.
Fur-
distinct families of closed orbits which
bifurcate from the origin at
E
=
0
(i.e., at
r
2
=
jE,
Z
=
In rectangular coordinates, the derivative of the vector field at the origin is
where a ± i.
a =
En. . J (E J )
n· TI(j J). J
By varying
m
The eigenvalues are
and the
n. 's J
-
1
and
one can vary the number
of distinct closed orbits independently of the order to which a
vanishes at dr dt
O.
For example,
2 2 2 r(E-r) (2E-r ) (3E-r) dr dt
or
dr dt
=
2 3 r(E-r)
or
2 ? ? r(E-r) (2c-r-)-.
Chafee has shown us another example proving that differentiability with respect to the parameter is necessary to insure uniqueness of the closed orbits. let
In polar coordinates,
0).
90
THE HOPF BIFURCATION AND ITS APPLICATIONS
elr
at dB dt (3A.2)
1
Exercise:
Show that although
d RedA (E)
I E=O
2 > 0, bifurcation the two distinct periodic orbits (at r
=
El / 3 ,
2E l / 3 )
(3A.3)
occurs. Remarks.
the situation where considered.
X
In the paper of Jost and Zehnder [1], depends on more than one parameter is
See also Takens [1].
Some interesting recent results of Joseph give another proof of Chafee's result that one does not need but only that the fixed point at Section 3B).
~
=
0
V"
is stable.
I
~
(0)
0,
(See also
Joseph also is able to deal with the case in which
a finite amplitude periodic orbit arises. for details, and Joseph [2].
See Joseph-Nield [1]
For the case of more than one
parameter, Takens [1] also obtains finite amplitude bifurcations. Alexander and Yorke [1] prove, roughly speaking, that if a vector field
X
increases either (i) period of point.
y~
has a closed orbit
~
y~,
then as
~
.\
y~
remains a closed orbit;
becomes infinite or (iii)
y~
(ii)
shrinks to a fixed
This is done without regard to stability of
another proof, see Ize [1]. this to our attention.)
the
y~.
For
(We thank L. Nirenberg for bringing
THE HOPF BIFURCATION AND ITS APPLICATIONS
91
SECTION 3B MORE GENERAL CONDITIONS FOR STABILITY Here we shall prove that the least
aIV aX nl
(0,0) f
0
n
for which
is odd, and that i f this coefficient is negative,
the periodic orbits obtained in Theorem 3.1 by use of the im-
plicit function theorem are attracting and occur for (we assume we have enough differentiability so that
~
V
> 0 n C ).
is
We also show that if the origin is attracting in the sense of Liapunov for the flow of
X '
O
then the periodic
orbits obtained in Theorem 3.15 (conclusions (A) and (B»
(3B.l)
k > 2.
Let the vector field
Then the function
j ~ 2(k-l)
there is a least
Lemma.
j
~(~l)
such that
is
X
c 2k
be
c 2 (k-l).
for
Assume that
~(j) (0) f O.
Then the
for which this is true is even.
Proof. ~ (j) (0) f
> O.
~
attracting and occur for
are
O.
that for all
Let
n
Assume xl
with
be the least integer
~ (n) (0) > 0 Ixll
< E:,
and choose
such that
j E:
~ (n) (x ) > O. 1
> 0
such
Then by the
THE HOPF BIFURCATION AND ITS APPLICATIONS
92
~(xl) = ~(n) (an )x a " ' a Il n-l
mean value theorem we have
o
where
(or xl < a j < 0) for all j. Suppose n is j < xl odd, then sign(x ) = sign(xlal···a _ l ). Therefore, for all n l < a
xl
with
xl < 0,
Ixll then
xl > 0,
< E,
if
~(xl)
< O.
n
For let
y~
such that ~(x~) = Il(Y~)
as
fore,
n
is even.
->-
then for each
00
Y~
and
~(n) (xl)
argument shows that if
and if
However, we know that this cannot
occur. < 0
~(xl) > 0
then
->-
< 0,
there is a
a
as
n
must be even.
n
->-
The same
00
There-
This also shows that bifurcation occurs
above or below criticality. [] (3B.2)
Lemma.
X
j ~ 2(k-l)
that there is a n
Let
c 2k
be
for
k > 2.
l.l{j) (0) f 0,
such that
axJ
2
for all
j
< n
Proof.
1 - 3 ~ (0 O)ll(n) (0). a xla II '
and
Upon differentiating the equation
j times and evaluating at (*)
and let
ajy (O,O)
In this case
be as in the preceding lemma.
Assume
xl
=
0,
is a sum of terms of the form
o because (0,0)
=
0.
V{O,Il) To find
we get
aj~
axJ
A~ll{~) CO) =
°
V(Xl,Il(X l » (0,0) + (*)
Ifor
for all
a n+l v --n+I
=°
(0,0),
ll.
~
< j
=
=
0.
and
Therefore, we must find
a xl the coefficient of
O.
~ en) (0)
(0,0) + (*l
This coefficient is easily seen to be 2
3
a ~
(0, 0) II ' (0).
a~
_3
a 2v d Xld II
°)
(°, ~ (n)
Therefore, (0) •
(0, 0)
(0,0) +
°
THE HOPF BIFURCATION AND ITS APPLICATIONS
Theorem.
(3B.3)
integer
j
<
is
~(xl)
the function
least such integer.
If
~(j) (0) I O.
Let
n
be the
a n+l v
- - - (0,0) < 0, ~ n+l o xl and sufficiently small. Furthermore, the
xl I 0
for all
2k Let X be c for k > 2. Then 2 c (k-l). Assume that there is an
such that
2(k-l)
93
periodic orbits obtained from the implicit function theorem are attracting. Proof. small
xl
~(xl)
We have already seen that
such that
xl I O.
for all
> 0
To show that the periodic orbits
are s.table, we must show that the function
aaxlVI
f (xl) =
o
has a local maximum at
(See
(xl'~ (xl»
Step 5 above).
a j+lv ~ a x 1J A
Q.
~ (n
(0,0) + (*) (0)
for
a 2v a xla ~ (O,O)~(n) even.
f(j) (0)
Q.
(0)
=
(*)
where < j.
0
for all
j
< n
because
is a sum of terms of the form n+l
fen) (0) = ~ (0,0) + a x n +l n+l 1
= ~ a n+~ a xl
(0,0)
<
O.
Recall that
Therefore, the mean value theorem shows that
has a local maximum at (3B.4)
Theorem.
f(j) (0)
xl
=
is
n
f(x ) l
O. CJ
Let the conditions of Theorem 3.15
be satisfied so that conclusions (A) and (B) hold.
Further-
more, let the origin be Liapunov attracting for the flow of X • Then the periodic orbits obtained from Theorem 3.15 are O attracting and occur for ~ > O. Proof. 85) holds.
Under these conditions, Chafee's Theorem (page
Therefore, the periodic orbits occur for
~
>
O.
94
THE HOPF BIFURCATION AND ITS APPLICATIONS
Since conclusion (B) of Theorem 3.15 holds, the orbits are unique, that is
Y = Y • Z l
Under these circumstances, Chafee's
theorem implies that the orbits are attracting.
~
95
THE HOPF BIFURCATION AND ITS APPLICATIONS
SECTION 3C HOPF'S BIFURCATION THEOREM AND THE CENTER THEOREM OF LIAPUNOV by Dieter S. Schmidt Introduction In recent years numerous papers have dealt with the bifurcation of periodic orbits from an equilibrium point.
The
starting point for most investigations is the Liapunov Center Theorem [1] or the Hopf Bifurcation Theorem [1].
Local results
concerning these theorems were published by Chafee [1], Henrard [1] Schmidt and Sweet [1] among many others, noted in previous sections, whereas Alexander and Yorke [1] discussed the global problem of the bifurcation of periodic orbits.
They showed in
their paper that Liapunov's Center Theorem can be derived as a consequence of Hopf's bifurcation theorem. J~
A. Yorke suggested that one should show also on the
local level that Liapunov's theorem can be obtained from the one of Hopf.
For this we provide an analytic proof of Hopf's
theorem based on the alternative method as outlined in Berger's article in Antman-Keller [1] which is general enough to include
THE HOPF BIFURCATION AND ITS APPLICATIONS
96
the center theorem as a corollary.
In addition our proof of
Hopf's theorem is simple enough to allow the discussion of some exceptional cases. The Hopf Bifurcation Theorem We consider the n-dimensional autonomous system of differential equations given by x
(3C.1) ~.
which depends on the real parameter possesses an analytic family that is
= o.
F(x(~) ,~)
x
=
x(~)
x
=
0,
that is
suppose that for a certain value of Fx(O,~)
is.
value
a(~)
If is
+
Fx(O,O)
is(~)
say
~
= O.
= 0,
the
Theorem (Hopf).
E
f o.
Under the above conditions
with
~
=
~(E)
~(O)
=a
and
T
T(O)
such that there arenonconstant periodic solutions of (3C.1) with period as
E +
tiS
is an integral multiple
a' (0)
there exist continuous functions depending on a parameter
We
is the continuation of the eigen-
then we assume that
(3C.1)
~,
F(O,~)
has two purely imaginary eigenvalues
and no other-eigenvalue of of
of equilibrium points;
Without loss of generality we assume
that this family is given by
matrix
We assume that (3C.1)
T(E)
=
=
T(E)
2nS
-1
X(t,E)
which collapse into the origin
O. (3C.2)
Remark.
Our assumptions for the Hopf theorem
are slightly less restrictive than they are usually stated as we do not require Furthermore
F(x,~)
the other eigenvalues to be non imaginary. does not have to be analytic for the
proof to hold but a certain degree of differentiability is
THE HOPF BIFURCATION AND ITS APPLICATIONS
97
required. Proof, form T
=
Y = S(~)t
form
Through a linear change of coordinates of the and by a change of the independent variable
S(~)x
we can bring equation (3C.I) into the following
.
(u(~)
+ i)YI
+
y,
~)
Y2
(u(~)
-
+
y,
~)
Y
B(~)y
YI
and y.
i)Y 2
+
"
~(YI' Y2 ,
y,
(3C.2)
~)
are the first two complex components of the vector YI = Y2. The remaining yare real and denoted by
Real solutions are only given if
n - 2 y.
components of the vector is a real
B(~)
n - 2
square matrix not necessarily in
normal form and the functions
~2'
and
quadratic in the components of the vector
are at least
y.
We introduce now the following polar coordinates YI
=
r e
is
r e -is
rTl
and arrive at the following system
.
r
u(~)r
S
I
.
Tl
+
B(~)Tl-
+
I
Re {e -is ¢l} (3C.3)
Im {e-is
r
u(~)Tl + .! r (¢
-
Re {e -is
Into this system we introduce the scale factor ~
=
8
by
r =
8p
8~1·
Because of
S
=
I + 0(8)
we can use
S
as a new indepen-
dent variable to overcome the autonomous character of the given system. ing form
The resulting differential equations have the follow-
THE HOPF BIFURCATION AND ITS APPLICATIONS
98
dp
e: R(8,p,I1,El
ere
(3C.4)
+
B (0)
and we are searching for
2TI - periodic solutions of this sysB(O)
By our assumptions on the eigenvalues of
tern.
that for
=
V(8,P,I1,e:)
e:
const, 11
= =
the only
0
2TI
-
periodic solution is
This solution persists for
O.
we find
=
P
Po
i f the
e: "I- 0
following bifurcation equation holds (see Berger [1]) 2TI
fo In this expression
P
and
11
solution of the given system e:0
=
R(8,p,I1,e:)d8
O.
represent the
2TI
periodic
(3C.l), but the terms of order
are already known and we can evaluate the term of the same
order in the bifurcation equation.
This leads to the equation
)11 u' (O)P O + O(e:) which can be solved uniquely to yield the implicit function theorem, since and
Po "I- 0
)11 = )1l(e:) = O(e:) u' (0)"1- 0
by
by assumption
because we are looking for nontrivial solutions.
Therefore the function found.
0
)1
The period of the solution in the original
nate system as found from the expression for T
2TI STOf
T(e:)
(1 + O(e:
2
».
d8 dt
x-coordiis
D
The Liapunov Center Theorem (3C.3)
Theorem (Liapunov). x
where
f
= Ax
Consider the system
+ f(x)
(3C.5 )
is a smooth function which vanishes along with its
THE HOPF BIFURCATION AND ITS APPLICATIONS
first partial derivatives at
x = O.
admits a first integral of the form where
S
=
ST
±is, A3 , .•• ,A n "',n
and
det
S t O.
S
t
O.
Then if
99
Assume that the system
=~
I(x)
Let
A
x T S x + .••
have eigenvalues
Aj/iS t integer for
j
3,
=
the above system has a one parameter family of periodic
solutions emanating from the origin starting with period
2w/S.
For the usual proof, see Kelley [1]. Proof.
As announced in the introduction we will show
that this theorem is a consequence of the Hopf bifurcation theorem.
To this end we consider the modified system x
=
Ax + f(x) +
~
(3C.6)
grad I(x)
and we will show that all conditions of Hopf's theorem are met and that the nonstationary periodic orbits can only occur for ~
=
O. The second part is easily done by evaluating
dI/dt
along solutions of (3C.6), which gives dI dt
=
grad I(x), Ax + f(x) + ~Igrad I(x)1 2 •
<
~
grad I(x) >
The second equality holds because for
Dc.s).
grad I(x(t»
Therefore
=
I dI ]l dt
0, which gives
I(x)
is an integral
is monotonically increasing unless x(t)
=
x(O)
that is a station-
ary point. In order to apply the theorem of the previous section we only have to verify the condition concerning the real part of the eigenvalue near
is.
Again through a linear change we
will bring the linear part of system (3C.S)
into a normal form.
We assume that this has been done already and the matrix
A
THE HOPF BIFURCATION AND ITS APPLICATIONS
100
has thus the following real form
-
where
A
is a real
=a
ATS + SA
with
a
a f
n-2
square matrix.
that the matrix
since
det
S
It follows from
in the integral has the form
S f O.
From the matrix
S 0
A + ).IS =
C
-S ).I a
0
0
A +
0
"S)
it follows at once that the eigenvalue near ~().I)
=
a).l
and therefore
a' (0)
=
a f
O.
is
has real part
CJ
An Exceptional Case for the Hopf Bifurcation Theorem Our proof of Hopf's theorem is easy enough to
al~ow
us
to discuss the case where the real part of the eigenvalue does not satisfy the condition derivative is nonzero
a' (0) f 0,
a"(O) f O.
but instead the second
The term with
).11
in the
bifurcation equation is zero and we will have to evaluate some higher order terms. We use the same normal form as given earlier in equation ~l
(3C.2) and for simplicity we assume that analytic functions in their variables.
and
~2
are
We assume that in a
preliminary nonlinear transformation mixed quadratic terms involving inated.
Yl
or
and a component of
have been elim-
This can be achieved with a method similar to the one
THE HOPF BIFURCATION AND ITS APPLICATIONS
101
used in the Birkhoff normalization of Hamiltonian systems, that is by a transformation of the form (cf. Section GA). Yl
->-
Y2
->-
Yl + Yl Ct Ty + Y2 ST-Y -T _ -T_ Y2 + Yl S Y + Y2 Ct Y
Y ->- y. The
n-2
dimensional complex valued vectors
Ct
and
Scan
be determined uniquely to eliminate the terms under question, because the matrix nor
2i
B(~)
in system (3C.2) does neither have
as eigenvalue for small
0
~.
In the function
we need to know the quadratic and
cubic terms made up of
Yl
and
2 2
~+
They are
Y2 •
3 2 2 3 CtY l + SY I Y2 + AY 1 Y2 + oY 2 + •.•
The terms not written down either only involve the or are of higher order. the parameter
~
y
variables
The coefficients depend of course on
and we write
a
= a(~) =
a O + al~ + O(~2)
similarly for the other coefficients. The differential equation of interest in the variables is the one for
dr d8
r
which reads
+ 1 + r- l u(~)r
This time we scale by s and obtain
dn d8
.
2 B(O)n + O(s )
~l
8, r, n
and
102
THE HOPF BIFURCATION AND ITS APPLICATIONS
with P P
2
Re{aOe
3
i6
+bOe
-i6
+cOe
-3i6
+ ..• }
{2i6 -2i6 -4i6 (Re aOe +SO+Yoe +oOe + ... }
Re {aOe
i6
+bOe
-i6
+cOe
-3i6
+ •..
}
{ i6 i6 Im aOe +bOe
+cOe -3i6 + ••• }) R 2 R3
p2
i
Re{alei6+ble-i6+cle-3i6+ .•• } u"(O)p.
The dots in the functions volving the
n
R ' Rl and R stand for terms inO 2 variables. Because n = O(g2) for 2n
periodic solutions those terms will be insignificant in evaluating the bifurcation equation, which has the same form as earlier and is given by
In evaluating this integral it is seen at once that there is no constant term.
Nevertheless care has to be exercised in
integrating
R because it will contribute to the O due to the form of the solution of p which is p =
g2
term
cOi (e-3i6_1)} + O(g 3 ) Po + g 2 Po2 Re{aOi(l-e i6 ) + bOi(e -i6 -1) + --3-
Due to our preliminary transformation the
n
variables appear
quadratic on
RO and therefore they will only contribute to higher order terms in g and The integration leads to the following bifurcation equation
The implicit function theorem allows us to state the following result:
If
u"(O) Re{SO+i aOb } < 0 O
there are two distinct
103
THE HOPF BIFURCATION AND ITS APPLICATIONS
solutions of the above bifurcation equation of the form ~l
= 0(8).
The solutions correspond to two families of periodic
orbits emanating from the equilibrium. u"(O) Re{So+i aOb } > 0 O
In case
there are no such solutions.
Finally,
if the discriminant is equal to zero higher order terms are needed to decide what is happening.
=
u(n-l)
=
0
u(n) (0) ~ 0
In the case
we scale by
r
=
u(O)
u' (0)
~
8n p
~l
8
and after identical computations we arrive at the bifurcation equation
D
Call
=
u(n) (0) Re{SO+i aOb } • O
If
n
is odd and
1 = ~l (8) small to the above bifurcation equation. I f n
there is always a solution of the form 8
then there are two such solutions if
(3C.4)
Theorem.
equations (3C.2) Assume n
=
u(O)
1,2,···
odd and
D
=
0
0
0(8)
~
D < 0
~
for
is even
and none if
D > 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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
~,
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
{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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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 HOPF BIFURCATION AND ITS APPLICATIONS
"-
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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, <
0, the
have absolute value
1, so all the eigenvalues of the Poincar~ map are
inside the unit circle and so the orbit is attracting (see Section 2B). Summarizing:
We have shown that the stability problem for
the closed orbits of the flow of
X~
the closed orbits of the flow of
x~, where
1
(X~{xl,x2,f(Xl'X2'~»'
is the same as that for
X~(Xl'X2)
2
X~(xl,x2,f(xl,x2'~»)'
=
Coordinates are
chosen so that
x ,x are coordinates in the eigenspace of 2 l and the third component is in a complementary
dXO(O,O,O) subspace
The set
F.
in a neighborhood of
{(xl,x2,f(xl,x2'~)'~)
(O,O,O)}
(xl,x2'~)
for
is the center manifold.
Outline of the Stability Calculation From the proof of Theorem 4.5, we know that the closed orbits of
X
~
will be attracting i f
V'"
(0) < 0
(or more
generally, see Section 3B, i f the first. nonzero derivative of V
at the origin is negative).
can be computed from those of two steps.
First we compute
The derivatives of X
o
at
V'" (0)
(0,0,0).
V
at
(O,O)
We do this in
from the derivatives of
" ' the vector field pushed to the center manifold, at X O
(0,0)
using the equation: T(X l )
V(x 1 ) =
fa
xl(at(xl,O),bt(Xl,O»dt (Note that in the
(4.2)
111
THE HOPF BIFURCATION AND ITS APPLICATIONS
x.)
two-dimensional case, X of
Xo
at
A
X (x ,x )
=
Then we compute the derivatives
(0,0) from those of X o at (0,0,0). Since 1 2 (X (x ,x ,f(x ,X ,1l», X (x ,x ,f(x ,X ,1l»), what l 2. 2 l 2 l 2 l II II
2 ll l we need to know is the derivatives of (0,0,0).
f
at the point
We can find these using the local invariance of the
center manifold under the flow of
X.
We use the equation
(see page 107):
(4.3)
A
Calculation of V"
I
(0) in Terms of X
We now calculate at
(0,0) using (4.2).
V'" (0)
from the derivatives of
Xo
We assume that coordinates have been
chosen so that
dXO(O,O)
ax l0 aX l ax 20 aX l
(0,0,0)
(0,0,0)
ax 0l aX 2 ax 20 aX 2
(0,0,0)
0
I A (0) I • (4.4)
(0,0,0)
-I A (0) I
0
This change of variables is not necessary, but it simplifies the computations considerably and, although our method for finding
V'" (0)
will work if the change of variable has not
been made, our formula will not be correct in that case. From (4.2) we see that
112
THE HOPF BIFURCATION AND ITS APPLICATIONS
d
"1
+ T' (xl) dX
[X
l
(aT (Xl) (xl,O) ,bT(X ) (xl,O»] l
l
+ Til (Xl)X (aT (Xl) (xl'0) ,bT(X ) (xl,O» l
d
"1
+ T' (xl) dX
[X
l
(aT (Xl) (xl,O) ,bT(X ) (xl,O»]. l
Using the chain rule, we get: T (xl)
J
°
2
~
[xl(at(Xl,o),bt(xl,O»]dt
dX l
Differentiating once more,
THE HOPF BIFURCATION AND ITS APPLICATIONS
db
t
[ dt
d
2 l
X
+ db 2
+
db
[db
-rr-t -rr-t
:~" [:>
db t ]
I
T (xl) + dX
T' (Xl)
+
I
:::it~
113
114
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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.
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
(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
THE HOPF BIFURCATION AND ITS APPLICATIONS
"']
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
Le.
(where all derivatives are taken at the origin).
()3 b
To compute
~ (0,0), we use the equation: ()x
l
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
+
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
~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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
dX~(a(~))
For
satisfy:
~ <~o' cr(dX~(a(~)) C {zlRe z
complex conjugate, simple eigenvalues ~
=
~o'
A(~)
zero speed and
A(~)
and
A(~O)
dX~(a(~))
and
has two
~.
At
cross the imaginary axis with non-
f O.
The rest of
cr(dX~(a(~))
remains
in the left half plane bounded away from the imaginary axis. I.
Under these circumstances
(A)
Bifurcation to periodic orbits takes place, as
described in Theorem 3.1. Choose coordinates so that
x
where
~O
are coordinates in the eigenspace to
and
is the coordinate in some complementary subspace.
Choose the coordinates so that*
I A (~O) I
0
dX
~O
(a
(~O)
-I A (~O) I
)
0
0
(B)
0
If the coefficient
0
3
(a(~O))
0
d_,X
V'" (0)
computed in (II)
~
~O
below is negative, the periodic orbits occur for are attracting. ~
< ~O'
If
V'"
(4.1)
and
(0) > 0, the orhits occur for
are repelling on the center manifold, and so are un-
stable in general. (C)
If
V'
I ,
(0)
0, the test yields no information
* See Examples 4B.2 and 4B.8.
Computer programs are available for this step. See for instance, "A Program to Compute the Real Schur Form of a Real Square Matrix" by B.N. Parlett and R. Feldman; ERL Memorandum M 526 (1975), univ. of Calif., Berkeley.
133
THE HOPF BIFURCATION AND ITS APPLICATIONS
and the procedures outlined in Section 4 must be used to pute
v(5) (0).
II.
com~
Good luck.
Write out the expression
v· • • (0)
d 2 xl
d 2 X2
d 2x l
d 2x 2
11 0 + --2- (a l (11 0 ) ,a 2 (11 0 » d xl
11 0 --2- (a l (11 0 ) ,a 2 (11 0 » dx 2
11 0 --2- (a l (11 0 ) ,a 2 (11 0 » d xl
11 0 --2dx 2
'a1 "0> ,a, "0> »
.
(4A.2)
134
THE HOPF BIFURCATION AND ITS APPLICATIONS
If your space is two dimensional, let
(A)
2
x • V' ,
~l = xl, ~2
Take off the hats; you are done with the computation of I
(0)
and the results may be read off from I. Otherwise, go to Step B. In expression (4A.2) fill in
(B)
for
i, j = 1,2. (4A.3) for
i,j,k = 1,2. (4A.4)
and
(C)
In this expression you now have, fill in
didjf
given by: dldlf(al(VO),a2(vO»
d l d 2f(a (va) ,a2 (v O» l d d f(a (v ),a (v » 2 2 l O 2 O
211. (v
2
o) 1
-!A(V O) Id3x~ (a(v o»
(4A.6)
o
2/ A (v )
O
I2+ (d3Xv3
(a (v ) ) ) O
o
2
THE HOPF BIFURCATION AND ITS APPLICATIONS
where
/';
(Note:
if (D)
v'··
135
(0)
x3
is linear, all derivatives of
f
are zero.)
If you have done it correctly your expression for
is now entirely in terms of known quantities and in
an explicit example, is a known real number and you may go to Step I to read off the results. Remarks.
S. Wan has recently obtained a proof of the
stability formula using complex notation, which is somewhat simpler.
It also yields information on the period (it is
closely related to the expression
So + iaOb O from Section 3C;
stability being the real part, the period being the imaginary part).
The formulas have been programmed and interesting
numerical work is being done by B. Hassard (SUNY at Buffalo) •
136
THE HOPF BIFURCATION AND ITS APPLICATIONS
SECTION 4B EXAMPLES We now consider a few examples to illustrate how the above procedure works.
The first few examples are all simple,
designed to illustrate basic points.
We finish in Example 4B.8
with a fairly intricate example from fluid mechanics (the Lorenz equations). (4B.l)
Example (see Hirsch-Smale [1], Chapter 10 and
Zeeman [2] for motivation). tion
2 d x dx + x - - + ( ddXt) 3 - a dt dt 2
equation.
Consider the differential equa-
0, a special case of Lienard's
Before applying the Hopf Bifurcation theorem we
make this into a first order differential equation on dx y = dt.
Let
a
and
Then we get the system dx dt = y,
Let
2 R •
Xa(x,y)
3
~
dt
(y,-y +ay-x). dXa(O,O)
-y
3
Now
+ ay - x. Xa(O,O)
o
for all
THE HOPF BIFURCATION AND ITS APPLICATIONS
2 a ± /a _4 2
The eigenvalues are
In this case
lal < 2.
ll-a 2
~+ i - 2 2
for
1m A(a) "I 0, where
Furthermore, for
a = 0, Re A(a) = a
d(Re A(a» da
Consider
Ia=O -- 2"1
137
such that a + ,!a2_ 4 A(a) = 2
a
-2 < a < 0, Re A(a) < a
and
2 > a > 0, Re A(a) > a
and
and for
Therefore, the Hopf Bifurcation theorem
•
applies and we conclude that there is a one parameter family of closed orbits of
X = (Xa'O)
in a neighborhood of
(0,0,0).
To find out if these orbits are stable and i f they occur for 3 XO(x,y) = (y,-y -x).
a > 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
THE HOPF BIFURCATION AND ITS APPLICATIONS
(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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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.)
THE HOPF BIFURCATION AND ITS APPLICATIONS
(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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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.
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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 HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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)
200
Figure 4B.l We also investigate the behavior of the system for fixed
b > 0
as
a
and McLaughlin [lJ.
+
00.
This has also been done by Martin
Our result agrees with theirs.
We
148
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
(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
.
•
qua d ratlc equatlons, eq: x
=x
2
t.
[Note that most
do not have global
t
solu tions. ] (4B.13)
Exercise. The following equations arise in the
oscillatory Zhabotinskii reaction (cf. Hastings-Murray [1]):
150
THE HOPF BIFURCATION AND ITS APPLICATIONS
x
=
2
s(y-xy+x-qx )
Y
51 (fz-y-xy)
z
w(x-y)
(compare the Lorenz equations!).
Let
f
be the bifurcation
parameter and let, eg: s = 7.7 x 10, q = 8.4 x 10 -6 , 1 w 1.61 x 10- • Show that a Hopf bifurcation occurs at f
f
c
where
Show that for the above vaJ.ues, the bifurcation is subcritical. S. Hastings informs us that the bifurcation picture looks like that in Figure 4B.3 (the existence of stable closed orhits for supercritical values is proven in Hastings-Murray [1]).
bifurcation parameter
, .... stable closed orbits
----
---
....... .......
, _surface of closed orbits
subcritical Hopf"_ _.\----r'/'
----~ ....~--closed
orbits switch stability here
Figure 4B.3
THE HOPF BIFURCATION AND ITS APPLICATIONS
151
SECTION 4C HOPF BIFURCATION AND THE METHOD OF AVERAGING by S. Chow and J. Mallet-Paret The method of averaging* provides an algorithm for preparing a bifurcation problem, that is, putting it into a normal form.
Once this is done, one may more readily determine
certain qualitative features of the bifurcation, by means of the implicit function theorem (or contraction mapping principle) and the center manifold theorem. Consider first the Hopf bifurcation problem f(z,cx)
z
about the equilibrium f
takes values in
Rn
z
= and
O.
(4C.I)
Thus assume f(O,cx) = O.
For simplicity, an
ordinary differential equation is considered although we could just as well consider a partial differential equation.
* The
In
method has been used by a number of authors, such as Halanay, Hale, Meyer, Diliberto, etc. cf. Kurzweil [1] and articles in Lefschetz [1].
152
THE HOPF BIFURCATION AND ITS APPLICATIONS
this case spaces X2 •
Xl
z
and and
z X 2
belong to (generally different) Banach and
f
is smooth from
Xl x (-aO,a O)
to
We may write (1) in the form z = A(a)z + g(z,a)
I
/g(z,a)
= o{lzI
2
(4C.2) ).
The standard hypotheses (see Sections 1,3) on the spectrum of A{a)
hold, namely that it posses a pair
A(a),A(a)
of complex
conjugate eigenvalues of the form A(a) = y(a) + iw(a) where y (0)
0,
v d~f y' (0) "I 0
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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.
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
=
;
<
a < ;
+ E
the solution bifurcates
0, is stable, and has the asymptotic form
z(t)
K(a - ;)1/2cOS(; t) + O(a - ;) 40 1/2 (31T-2) ~ 2.3210701.
K
(2)
Diffusion Equations.
Linear equations with non-
linear boundary conditions, such as
u
t
=
u
t
xx
~
o
0
<
x < 1
ag(u(O,t), u(l,t»
occur in various problems in biology and chemical reactions. (See, for example, Aronson [2].)
g(u,v)
Here we take
2 2 au + Bv + O(u +v ),
so the linearized equation around
u = 0
has the boundary
conditions
ux(O,t)
o
For appropriate parameter values
a[au(O,t) + Su(l,t)].
(a,S)
a pair of eigenvalues
of this problem crosses the imaginary axis with non-zero speed as
a
passes the critical value
a • O
The stability of
THE HOPF BIFURCATION AND ITS APPLICATIONS
161
the resulting Hopf bifurcation can be determined by averaging. The power of the averaging method is that it can handle a rather wide variety of bifurcation problems.
We mention
two here.
(3)
Almost Periodic Equations.
Z(t)
where
A(a)
Consider
A(a)z + g(z,t,a)
is as before, g
(4C.13)
is almost periodic in
t
uni-
2
in compact sets and g = O( I z I ). Suppose 271 in addition that the periods for N = 1,2,3,4 are Nw bounded away from the fundamental periods for g. Then an formly for
(z,a)
averaging procedure similar to that described above yields a normal form in scaled coordinates given by (4C.ll).
Here
however the higher order terms (but not the constant almost periodic in
t.
section
t
=
Thus this manifold can be thought of (x,t) E
as a cylinder in the const.
(4)
pn
l
x R
space, where each
is a circle near
is almost periodic in as those in
K)" are
t
x
=
O.
The cylinder
with the same fundamental periods
g.
A special case of (3) occurs in studying the
bifurcation of an invariant torus from a periodic orbit of an autonomous equation.
In an appropriate local coordinate
system around the orbit, the autonomous equation takes the form (4C.13) where around the orbit and
t
represents the (periodic) coordinate z
the normal to the orbit.
dition on the fundamental periods of
g
The con-
reduces to the
162
THE HOPF BIFURCATION AND ITS APPLICATIONS
standard condition that the periodic orbit have no characteristic multipliers which are Nth roots of unity, for N K
1,2,3,4, ~
0
The invariant cylinder that is obtained if
is periodic in
t
and thus is actually a two
dimensional torus around the periodic orbit.
THE HOPF BIFURCATION AND ITS APPLICATIONS
163
SECTION 5 A TRANSLATION OF HOPF'S ORIGINAL PAPER
BY L. N. HOWARD AND N. KOPELL
"Abzweigung einer periodischen Lasung von einer stationaren Lasung eines Differentialsystems" Berichten der MathematischPhysischen Klasse der Sachsischen Akademie der Wissenschaften zu Leipzig. XCIV. Band Sitzung vom 19. Januar 1942. Bifurcation of a Periodic Solution from a Stationary Solution of a System of Differential Equations by Eberhard Hopf Dedicated to Paul Koebe on his 60th birthday 1.
Introduction Let
x.
~
= F.(Xl, ••• ~
,x ,ll) n
(i
1, ... ,n)
or, in vector notation, (1.1)
164
THE HOPF BIFURCATION AND ITS APPLICATIONS
be a real system of differential equations with real parameter
Il, where G
F
I Il I
and
is analytic in < c.
For
IIl I
and
x
x
in a domain
let (1.1 ) possess an analytic
< c
family of stationary solutions E:.(~(Il) ,Il)
for
Il
i. (Il )
x
lying in
G:
= O.
As is well known, the characteristic exponents of the stationary solution are the eigenvalues of the eigenvalue problem
where
L stands for the linear operator, depending only on -Il Il, which arises after neglect of the nonlinear terms in the series expansion of
F
about
x
= x.
The exponents are
either real or pairwise complex conjugate and depend on
Il.
Suppose one assumes simply that there is a stationary solution
x
-0
in
G
for the special value
none of the characteristic exponents is
Il
=
0
and that
0; then, as is well
known, it automatically follows that there is a unique stationary solution
~(Il)
x -x = -0 is analytic at
in a suitable neighborhood of
11l1,
for every sufficiently small
and
~(Il)
Il = O. On passing through
Il
=
0
let us now assume that none
of the characteristic exponents vanishes, but a conjugate pair crosses the imaginary axis.
This situation commonly occurs
in nonconservative mechanical systems, for example, in hydrodynamics.
The following theorem asserts, that with this
hypothesis, there is always a periodic solution of equation (1.1) in the neighborhood of the values
x
-
= -0 x
and
Jl =
O.
THE HOPF BIFURCATION AND ITS APPLICATIONS
Theorem.
~
For
0, let exactly two characteristic
=
exponents be pure imaginary. a(~), a(~)
165
Their continuous extensions
shall satisfy the conditions
a (0)
-
-I- 0,
a(O)
Re(a' (0»
-I- O.
(1.2)
Then, there exists a family of real periodic solutions ~
=
~(t,E),
~(t,O)
=
=
~(O),
E -I- O.
small E
~
0
~(E)
which has the properties
-I-
~(t,E)
but E(~)
and
~(t,E)
T(E)
L
and
I~I
such that for
and
are analytic at the point (t,O).
The same
and 2 1T /la(0)1.
T(O) For arbitrarily large
= 0
for all sufficiently
and correspondingly at each point
holds for the period
b
~(~(E»,
~(O)
there are two positive numbers
a
< b, there exist no periodic solu-
tions besides the stationary solution and the solutions of the semi-family
E > 0
which lie entirely in
whose period is smaller than I~-~(~) ~,
For sufficiently small exist only for
~
> 0
I<
a.
*
the periodic solutions generally ~
or only for
that they exist only for
Land
~
< 0;
it is also possible
o.
=
As is well known, the characteristic exponents of the periodic solution
~(t,E)
are the eigenvalues of the eigen-
value problem
v+ where
~(t)
ltv
L
--t , E
has the same period
(v)
(1.3)
-
T
T(E)
as the solution.
* The other half-family must represent the same solution curves.
166
L
THE HOPF BIFURCATION AND ITS APPLICATIONS
is the linear operator obtained by linearizing around the
periodic solution. period
T
and at
It depends periodically on E =
0
is analytic in
t
and
characteristic exponents are only determined depend continuously on for
F
with the E.
The
mod (2rri/T)
and
One of them, of course, is zero;
E.
does not depend explicitly on 0,
t
t, so
v
is a solution of the eigenvalue problem.
For
E ~
0
the ex-
ponents, mod(2rri/T )' go continuously into those of the sta-
O
tionary solution
~(O)
of (1.1) with
~
= O.
By assumption
then exactly two exponents approach the imaginary axis. of them is identically zero. real and analytic at
E =
The other
0, B(O)
=
O.
B
=
B(E)
One
must be
It follows directly
from the above theorem that the coefficients
~l
and
Bl
in the power series expansion
~l
satisfy
= Bl = O.
In addition to that it will be shown
below that the simple relationship (1. 4)
holds; I have not run across it before. In the general case
~
2
I 0, this relationship gives
information about the stability conditions. for
~
<
solution
0
If, for example,
all the characteristic exponents of the stationary x
!(~)
have a negative real part (stability,
THE HOPF BIFURCATION AND ITS APPLICATIONS
x
a small neighborhood of
collapses onto
then there are the following alternatives.
x
as
167
t
+
00),
Either the periodic
solutions branch off after the destabilization of the stationary solution
(~>
0); in this case all characteristic expon-
ents of the periodic solution have negative real part (stability; a thin tube around the periodic solutions collapses onto these as
t +
fore, that is for unstable.
00). ~
Alternatively, the family exists be-
< 0;
then the periodic solutions are
*
Since in nature only stable solutions can be observed for a sufficiently long time of observation, the bifurcation of a periodic solution from a stationary solution is observable only through the latter becoming unstable. tions are well known in hydromechanics.
Such observa-
For example, in the
flow around a solid body; the motion is stationary if the velocity of the oncoming stream is low enough; yet if the latter is sufficiently large it can become periodic (periodic vortex shedding).
Here we are talking about examples of non-
conservative systems (viscosity of the fluid).+
In conserva-
tive systems, of course, the hypothesis (1.2) is never fulfilled; i f
is a characteristic exponent, -A
always is as
well. In the literature, I have not come across the bifurcation problem considered on the basis of the hypothesis
* In
n = 2
dimensions, this is immediately clear.
+1 do not know of a hydrodynamical example of the second case. One could conclude the existence of the unstable solutions if, with the most careful experimenting, (very slow variation of the parameters) one always observes a sudden breaking off of the stationary motion at exactly the same point.
168
(1.2).
THE HOPF BIFURCATION AND ITS APPLICATIONS
However, I scarcely think that there is anything es-
sentially new in the above theorem. developed by
Poincar~
The methods have been
perhaps 50 years ago, * and belong today
to the classical conceptual structure of the theory of periodic solutions in the small.
Since, however, the theorem
is of interest in non-conservative mechanics it seems to me that a thorough presentation is not without value. to facilitate the extension
In order
to systems with infinitely many
degrees of freedom, for example the fundamental equations of motion of a viscous fluid, I have given preference to the more general methods of linear algebra rather than special techniques (e.g. choice of a special coordinate system). Of course, it can equally well happen that at a real characteristic exponent tion
~(~)
a(~)
~
0
of the stationary solu-
crosses the imaginary axis, i.e., a(O)
0,
a' (0)
I- 0
*Les methodes nouvelles de la mecanique celeste. The above periodic solutions represent the simplest limiting case of Poincare's periodic solutions of the second type ("genre"). Compare Vol. III, chapter 28, 30-31. Poincare, having applications to celestial mechanics in mind, has only thoroughly investigated these solutions (with the help of integral invariants) in the case of canonical systems of differential equations, where the situation is more difficult than above. Poincare uses the auxiliary parameter £ in Chap. 30 in the calculation of coefficients (the calculation in our §4 is essentially the same), but not in the proof of existence which thereby becomes simpler. In a short note in Vol. I, p. 156, Painleve is touched upon: Les petits mouvements periodiques des systemes, Comptes Rendus Paris XXIV (1897), p. 1222. The general theorem stated there refers to the case ~ = 0 in our system ( • ), but it cannot be generally correct. For the validity of this statement F must satisfy special conditions.
THE HOPF BIFURCATION AND ITS APPLICATIONS
while the others remain away from it.
169
In this case it is not
periodic but other stationary solutions which branch off.
*
We content ourselves with the statement of the theorems in ~
= ~*(E),
of stationary solutions, different from
;, with
this simpler case. ~ = ~*(E)
=
~(O)
0, x*(O)
There is an analytic family,
= x(O).
the solutions exist for acteristic exponent
~l
If ~
>
SeE)
I 0
(the general case) then
0 and for
< O.
~
For the char-
which goes through zero, the ana-
log of (1.4) holds:
If
x
is stable for
~ <
0
just the opposite holds for ~
x*.
0, than one will observe
<
ceptional case ~2
and unstable for
x*
~
with
< O.
E
>
(If one observes for
~
>
0.)
0
then
x
for
In the ex-
~l = 0, the situation is different.
I 0, then the new solutions exist only for
for
~
~
E -2~
2
negative) .
or only
> 0
There are then two solutions for fixed
positive, one with
If
~
,
(one
Here we have
a.' (0) •
From this one can obtain statements about stability analogous to those above.
In this case either both solutions
x*
are
stable or both are unstable.
2.
The Existence of the Periodic Solutions. Without restriction of generality one can assume that
the stationary solution lies at the origin, i.e.,
*An
example from hydrodynamics is the fluid motion between two concentric cylinders (G. I. Taylor).
170
THE HOPF BIFURCATION AND ITS APPLICATIONS
~(O,].I)
Let the development of
F
O.
in powers of the
x.
J.
be
L (x) + Q (x,x) + K (x,x,x) + ••. ,
-].1--
fl-
-].1---
where the vector functions K (x,y,z), .•• -].I - - -
are linear functions of each argument and also symmetric in these vectors. The substitution (2.2)
x
carries (1.1) into
The right hand side is analytic in point case
=
8 8
=a
].I
=
0, y
=
yO
(yO
8,].1,
arbitrary).
~
at the
We consider the
in (2.3), that is, the homogeneous linear dif-
ferential equation
-z
(2.4)
= L
(z). -].1-
For the question of existence, this has the deciding significance. The complex conjugate characteristic exponents
a (].I) ,
a(].I) , which were referred to in the hypothesis, are simple for all small
1].11.
In the associated solutions at e
at_ a
of (2.4), the complex vector
~
e
~'
up to a complex scalar factor;
a
(2.5)
is consequently determined is the conjugate vector.
THE HOPF BIFURCATION AND ITS APPLICATIONS
171
Furthermore, there are no solutions of the form (2.6) a(~)
tor ~
~ ~
0
~(~)
O.
~
is analytic at
One can choose a fixed real vecI~I, ~
so that for all small
.
~ ~
0
~ ~
for
O.
is then uniquely determined by the condition 1
(~
~
f 0).
(2.7)
By hypothesis, a(O) ~(~)
~
O.
(2.8)
O.
~
is analytic at
= -a(O)
The real solutions of (2.4), which are linear combinations of (2.5), have the form ce ata + ce ata with complex scalar
c.
(2.9)
They form a family depending on two
real parameters; one of these parameters is a proportionality factor, while the other represents an additive
const~nt
in
t
(the solutions form only a one parameter family of curves). Because
For
c
e
1,
~,
we have z·e
c a.e + -c a.e
z. e
c a a.e + c a a·e
}
at
t
O.
is
(2.9)
(2.10)
because of (2.7), this t
0:
z
satisfies the conditions: z·e
0,
~(z.e) dt - -
1.
(2.11)
172
THE HOPF BIFURCATION AND ITS APPLICATIONS
This is the unique solution of the form (2.9) satisfying these conditions; for from t
and from (2.9), ~
0:
z·e
z·e
o
(2.7) and (2.8) it follows that
c = 0; thus
o. By hypothesis, for
~
=
0, a, a
are the only ones
among the characteristic exponents which are pure imaginary. ~
Hence, for
0,
=
tions of (2.4).
(2.9) gives all the real and periodic solu-
Their period is 21f o = I a (0) I ~ = 0, (2.10)
(2.12)
T
In particular, for
is the only real and periodic
solution with the properties (2.11). For later use we also notice that, for
~
0,
(2.4)
can have no solutions of the form t E(t) + 9.(t) where
£
and
cally zero.
9.
have a common period and
12.
is not identi-
Otherwise (2.4) would break up into the two equa-
tions
and
12.
would be a nontrivial linear combination of the solu-
tions (2.5).
The Fourier expansion of
g(t)
would then lead
to a solutinn of the form (2.6). By differentiation of (2.4) with respect to ~
0
~
at
one obtains the non-homogeneous differential equation 0,
(2.13)
THE HOPF BIFURCATION AND ITS APPLICATIONS
~-derivative
for the
z'
t(a'e
The factor of
at
t
~
173
of (2.10):
(at + -a'eat~) + e a
I
+ e
at-a ' )
is a solution of (2.4).
(~
0) •
If one expresses
it linearly in terms of the solution (2.10) and
~,
it follows
from (2.8) that
z'
t(Re(a')~
+
1m (a I)
a
•
~)
+
( 2.14)
~(t)
with (2.15)
~(t) •
Now let
be the solution of (2.3), which satisfies the initial condition
y = y
°
for
t =
o~
According to well known theorems
it depends analytically on all its arguments at each point
°
(t,O,O,y).
It is periodic with the period
T
if and only if
the equation
°)
y(T,~',E,y
is satisfied.
If one denotes by
° ° z ° the fixed
- y
of the fixed solution (2.10) of (2.4),
~
(2.16) initial value
0, then (2.16) is
=
satisfied by the values E =
The problem is: ~
and
°
y.
for given
These are
n
E,
0,
° z°
Y
(2.17)
solve equation (2.16) n + 2
equations with
for
T,
unknowns.
In order to make the solution unique, we add the two equations
y where
~
o
0,
yO. ~
1
is the real vector introduced above and where
(2.18)
174
yO
THE HOPF BIFURCATION AND ITS APPLICATIONS
=y
for
t
=
O.
The introduction of these conditions im-
plies no restriction on the totality of solutions in the small, as will be demonstrated in the next section.
For the
~ = E = 0, yO = ~o, it follows from (2.11)
initial values
that these equations are satisfied by the solution (2.10). Now for all sufficiently small
IE/,
(2.16) and (2.18)
have exactly one solution T
=
T(E),
~
~
=
y
(E) ,
o (E)
o
Y
in a suitable neighborhood of the system of values
y
0,
~
if the following is the case:
o
o
z
(2· 20)
the system of linear equations
formed by taking the differential (at the place (2.17» respect to the variables for given
dE.
T,~,
dT
E,
with
is uniquely solvable
Equivalently, there are such functions (2.19)
if these linear equations for solution
Yo
= d~ = dyO =
O.
dE
=
0
have only the zero
This is the case, as will now
be shown. We have
y
= ~~
(y),
o).
(2. 21)
y(t,~,O,y
y
In particular y(t,~,O,~
is the solution to (2.10).
o
) =
(2. 22)
~(t,~)
The differential
dy(t,~,O,y
o)
is
the sum of the differentials with respect to the separate arguments when the others are all fixed.
If we introduce for
the differentials dt, as independent constants or vectors the notations:
THE HOPF BIFURCATION AND ITS APPLICATIONS
175
a,
p,
then, the differential referred to becomes
pz + a:t' y
Here and
~
y'
and
=
dl!d~
are taken at
T
0, Y..
o
z
is the solution of
with the initial value
y
+ u.
= ~(t,O).
u
If one sets
o
for
t ~,
:t'
=
o.
According to (2.22),
then
~(t)
is the solution
o.
(2.23 )
of
The linear vector equation arising from (2.16) is then (2.24)
0, ~(t)
where
denotes the solution (2.10) of (2.25)
~(t)
is any solution of this homogeneous linear differential
equation with constant (2.23). p =
a
~O'
and
~(t)
is the solution of
We show now that (2.24) is possible only if
o
and
o.
~(t)
Now for all
t
o. This is true because
~(t)
has period
(2. 26)
TO' so it
follows from (2.23) that the square bracket is a solution of (2.25).
+
z
is also a solution of (2.25)+, so the whole left
In the original, this number is (2.23).
o
176
THE HOPF BIFURCATION AND ITS APPLICATIONS
side of (2.26) is a solution of (2.25). fact that
=
~(O)
By (2.24) and the
0, the initial value of this solution is
zero, and thus it is identically zero.
Now from (2.13) and
(2.23) it follows that ~(t)
9:.
~'(t) + 9:. (t),
•
= ~O (~O
Thus, by (2.14) and (2.15), the square bracket in (2.26) has the value
TO[Re(al)~(!) + Im~al) itt)] + [9:.(t+T O) - 9:.(t)]. If one sets
+ a9:.
~
= wand
aT Re(a')z(t) + [p+aT
o
-
O
!. (t)
Im(a')]z(t) a -
,
it follmvs that ~(t) + ~(t+TO) - ~(t) = 0,
where (2.25).
~(t)
~(t)
is a solution and
a periodic solution of
This means that t
~(t)
with periodic
q.
However, as we stated before, such solu-
tions cannot exist unless Since
+ q(t)
~(t)
TO
~,!
z
O.
=
are linearly independent, it follows from
(2.27) and from the hypothesis (1.2) that Thus, by (2.24),
~(t)
Finally, since
has period dy
a
=
~
a
a
=
0
and
p
at
t
0,
=
O.
TO.
, and
. a
d~
~,
it follows from the equations (2.18) that 0, A periodic solution of
u = ~O(~)
at
t
O.
with these properties must
THE HOPF BIFURCATION AND ITS APPLICATIONS
177
vanish, as we have stated above.
With this the proof of t the existence of a periodic family is concluded. The solutions (2.19) are analytic at T
TO(l + TIE +
II
lllE + ll2 E
The periodic solutions
2
E
=
0
(2.28)
+
y(t,E)
of (2.3), and the periodic
family of solutions (2.29)
~(t,E)
of
(1.1)+, are analytic at every point
(t,O).
One ,obtains exactly the same periodic solutions if one begins with a multiple
mT
of the period instead of TO' O that is if one operates in a neighborhood of the system of
values T instead of (2.20).
3.
= mT O'
II
=
o
(2.30)
Completion of the Proof of the Theorem.
numbers
a
and
periodic solution
in
z
Nothing essential is altered in the proof.
For arbitrarily large
than
o
0, Y
b
L > TO
there are two positive
with the following property.
~(t) f 0
L, which belongs to a
Every
of (1.1)+, whose period is smaller II
with
I~I < a, belongs to the family
Illi < b
(2.29),
a suitable choice is made for the origin
and which lies
(2.28), E > 0 of
if
t.
If this were not the case, there would be a sequence t
+
See editorial comments in §SA below. In the original paper, this number is (1).
178
THE HOPF BIFURCATION AND ITS APPLICATIONS
of periodic solutions
~k(t) ~ O++having bounded periods
T < L, and of corresponding k (t) k = Maxlx t -k
K
and such that no pair
I
~-values,
~k + 0
0,
+
with
~k(t), ~k
(3.1)
belongs to the above family.
We let
is a solution of (2.3), with
e:, and
instead of
satisfies Maxl:L (t) t k
I
= l.
One considers first a sUbsequence for which the initial values converge, have
Xk (t) ....
~
~(t),
the maximum of
I~I
of the form (2.9) TO. where
+
,
o .... z o. where
= c
~
= ~O(z)
z
1, ~
e
=
=
~(O)
and
o
z.
Since ~
is not identically zero.
is
0, and it has the fundamental period
If one shifts the origin of ~.
It I < L, we
Then, uniformly for
0, one finds that
t
~(t)
in
z
~ ~
0
to the place there.
This
quantity can be taken to be positive; otherwise, since
one could achieve this by shifting from
t
o
Consequently, 0,
.0 z
~ >
O.
From this it follows that in the neighborhood of ++
In the original paper, the sequences indexed.
+In the original, this number is (2.8).
zO
and for
x k ' T , etc. are not k
THE HOPF BIFURCATION AND ITS APPLICATIONS
small
K
(2.3)
(K
and
I~I, all solutions of the differential equation
instead of
s) cut the hyperplane
In this intersection let ~k(t),
~k
k,
K
t
=
O.
k
+
0,
~ =
0
once.
Then, for the sequence
Zk O
~O.
+
Also e
K
~.
under consideration, with this choice of ori-
gin, we always have
and
179
O.
~k +
z·0
+
p > 0
If one now sets
then
y is a solution of (2.3)+, for the parameter values -k For it, we have sk > 0 and
v •
....k
e
-
0,
~.
~ =
1,
at
t
O.
(2.18)
The periods in the sequence of solutions must converge to a TO' mT • Furthermore sk + O. However, this O implies that from some point on in the sequence one enters
multiple of
the neighborhood mentioned above of (2.20) or (2.30) in which, for all sufficiently small
s, there is only one solution of
the system of equations under consideration.
The solutions
of our sequence must then belong to the above family and in fact with
s > 0, which conflicts with the assumption.
Thus
. .~s prove d . tt t h e assert~on
From the fact we have just proved it now follows that ~(s)
if
from
+ tt
o
1 0, then the first coefficient which is different in
~
=
~ls
+ V2s
2
+ ••.
is of even order; the same
In the original, this number is (3). See editorial comments in §SA below.
180
THE HOPF BIFURCATION AND ITS APPLICATIONS
holds for the expansion
T
=
TO
(1 +
T
e: +
T
e:
2
1 2 For the solutions of the family corresponding to the associated
and
~
among those for
+ ••• ) . e: < 0, and
T-values, must already be present
e: > 0:+
In particular we have
o. The periodic solutions exist, for sufficiently small I~I, only for
and ~
4.
~ > 0, or only for
I~I
~ < 0, or only for
= O.
Determination of the Coefficients. We shall need the following result, which gives a
criterion for the solvability of the inhomogeneous system of differential equations (4.1)
(~
where
~(t)
has a period
TO'
Let (4.2)
be the differential equation which is adjoint to the homogeneous one; L*
is the adjoint operator to
L
(transposed
matrix), defined by v
~* (~).
u
Then (4.1) has a periodic solution
w
+
with the period
TO'
i f and only if
++ And indeed with a shift of the
t-origin of approximately T /2. O
+In the following, the inner product of two complex vectors a, b is defined by Iaib i .
THE HOPF BIFURCATION AND ITS APPLICATIONS
TO
fo
Sl:~*dt =
181
(4.3 )
0
for all solutions of (4.2) which have the period
TO.
This result follows from the known criterion for the solvability of an ordinary system of linear equations. necessity follows directly from (4.1) and (4.2).
The
That the
condition is sufficient can be shown in the following way: The adjoint equation has the same characteristic exponents and therefore it also has two solutions of the form
e at a*,
a (0)
(4.4 )
-a(O) ,
from which all periodic solutions can be formed by linear combinations.
Furthermore, the development of
~(t)
in Fourier
series shows that it suffices to consider the case ~
= e
-at
and the analogous case with
b
a
instead of
-a.
In (4.1) let
us insert w
e
(aI +
~)£
-at
c.
(4.1) then becomes b.
(4.4) and (4.2) imply (aI + 1) *~* = 0 while (4.3) says
b·a* = O.
From this everything follows
with the help of the theorem referred to. Secondly, we shall need the following fact. solution
~ ~ 0
always a solution
z=
of z*
~(~)
having period
For any
TO' there is
of the adjoint equation, with the same
182
THE HOPF BIFURCATION AND ITS APPLICATIONS
period, such that T
O
Jo
~: ~*dt
.
Otherwise, the equation ~,
and
w - tz
+
w =
'I-
O.
~(~)
*
+
~
would have a solution
would be a solution of the homogeneous dif-
ferential equation, which contradicts the simplicity of the characteristic exponent Let
z*
-1
and
a.
be two linearly independent solu-
z* -2
tions of (4.2) with the period
[~]i
=
JTO~.z~ o
-~
TO'
Let
(i
dt
1,2) •
Then the criterion for solvability of (4.1) under the given conditions ·is
O. We also note that
zi, z*
(4.5 )
can be chosen in such a way that
2
1, where
~
[z]
-
2
=
[21]
- 1
is the solution (2.10) of (2.4) with
o ~
(4.6 )
0
(biorthogonalization). The problem of the determination of the coefficients for the power series representation of the periodic family can now be solved in a general way. dependent variable
s
If one
define~
the new in-
by
t
(4.7)
then according to (2.28) the period in the family of solutions
*Also,
the integrand is always constant.
+In the original, the statement reads "w+tz", which is incorrect.
THE HOPF BIFURCATION AND ITS APPLICATIONS
y
y(S,E)
=
s
(or
is constantly equal to
t) and
TO'
183
y, as a function of
E, is analytic at every point
(s,O).
One
has (4. 8)
where all the
Yi
respect to
will again be denoted by a dot.
s
have the period
TO'
The derivative with We write for
simplicity
Then, using (3.2), and inserting (4.7) and (4.8) in (2.3), one obtains the recursive equations
YO
~(yO)
(yo = ~)
(4.9) (4.10)
Yl = ~(Yl) + 9. (yo ,yo) -T
2Yo
+ •
Y2
~(Y2) + )l2~' (yO) + 29. (yo 'Yl)
(4.11)
+ !S.(Y.o'YO'Yo)
............. from which the
::Li ' )l 1.. , T.1. are to be determined. In addition to these, we have the conditions following from (2.18)
J!.k
for
k
=
stead of
1,2, ••• s.
. e = Y.k .
~
=
0,
at
s
=
(4.12)
0
In the equations, we again write
By (4.10) and (4.12), Yl
as a periodic function with period
TO'
t
in-
is uniquely determined From (4.11)
must first be eliminated with the help of (2.13).
L'
Since the
parenthesis in the first summand of (2.14) is a solution of ~
~(~),
(2.13) can be written
184
THE HOPF BIFURCATION AND ITS APPLICATIONS
Re
(0:
I )
~
+
I1n(o:')
z'
•
+ h
0:
(4.13)
Let v
-
"-2
(4.14)
)12h = v, --
which, according to (2.15), has the period
z
=
TO'
Since
YO' it follows that
(4.15)
Thus, by (4.6) (4.16) -[2Q(Y O'Yl) + K(YO 'YO'YO)]2' By hypothesis (1.2),)12
and
One then solves (4.15) for and (4.12), k
=
T
v
2
are determined from this.
and obtains
from (4.14)
2, in a unique way.
In an analogous fashion all the
higher coefficients
are obtained from the subsequent recursion formulas. )12 f
general
o.
If
tions exist only for holds for
5.
In
is positive then the periodic solu)1 > 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)
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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)
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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.
THE HOPF BIFURCATION AND ITS APPLICATIONS
~ ~
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'
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
~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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
~(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
THE HOPF BIFURCATION AND ITS APPLICATIONS
£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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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.
THE HOPF BIFURCATION AND ITS APPLICATIONS
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".
THE HOPF BIFURCATION AND ITS APPLICATIONS
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].
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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<¢<2rr lJ
A =
I V IK I
-1- -
lJ
-l
lJ
but remains bounded as
lJ
We now prove that (6.3) has a unique solution.
+
O.
To do
this, it is convenient to denote the right-hand side of (6.3) temporarily by
x(¢):
x(¢) = ¢ + 6 (lJ) + lJ 1 We want to show that, as *i.e., ¢
differs from
integral mUltiple of
¢
3/2
KlJ (u(¢) ,¢).
runs from
0
to
21f, x(¢)
¢ + 6 (lJ) + lJ3/2KlJ(U(¢),¢) 1 21f.
by an
runs
214
THE HOPF BIFURCATION AND ITS APPLICATIONS
exactly once over an interval of length u(¢), K~(Z,~)
icityof
=
x(O) + 2TI.
We therefore only have to show that ing.
Let
-
~l < ~2'
From the period-
~, it follows that
in
x(2TI)
2TI.
x
is strictly increas-
Then
Now
IK~(U(~),;P)-K~(~l)'~l)1 ~ 2AI~-~11
=
<
A[lu(~2)-u(~1)1 + I~ 2-~111
2A(¢2-~1)'
(The second inequality follows from the Lipschitz continuity of
u.)
Thus
1 - 2A~3/2 > 0, x
(6.5)
is strictly increasing and (6.3) has a unique solution.
thus
get
~
as a function of
above estimates that
~
~,
and it follows from our
is Lipschitz continuous: (6.6)
The definition (6.4) of
~
E U.
therefore makes sense, and
we next have to check
that~
to 6.7 is immediate.
For (ii), note that
<
1 - 2~ + ~3/2 •
Condition i), corresponds
We
THE HOPF BIFURCATION AND ITS APPLICATIONS
Thus,
I~u(~)
I
1
<
~
for all
215
provided (6.7)
Finally,
IffL1(~1)-YU(~2)!
< (1-211) !u(¢1)-u(¢2)!
3 2 + 11 / ;1d!U(¢1)-u(¢2)! +
~
3 2 (1-211 + 211 / A)
u.
by the Lipschitz continuity of for
so
!¢1-~21]
1~1-¢21
Inserting estimate
(6.6)
1¢1-¢2 1 , we get
~u
is Lipschitz continuous with Lipschitz constant
1
provided (l-211+211 3 / 2 A) (1_211 3 / 2 A)-1 < 1.
(6.8)
Evidently (6.8) holds for all sufficiently small positive
11,
so (iii) holds. The next step is to prove that Thus, let
u l ,u 2
E
~,
U, choose
~
and let
is a contraction.
-~l' -~2
denote the
solutions of ~l + 6 1 (11)
-
~2 + 6
respectively.
+ 11
1 (11) + 11
3/2 3/2
--
K11(ul(~l)'~l)
-K11 (u 2 (~2) '~2)'
SUbtracting these equations, transposing, and
taking absolute values yields
216
THE HOPF BIFURCATION AND ITS APPLICATIONS
3/2 -1¢1-¢2 1 2..]1 /K]1 (U l (¢l) '¢l) - K]1!U 2 (¢2) '¢2) 1 2.]1
3/2
A[lu l (¢1)-U 2 (¢2) 1 +
(6.9)
-
1¢1-~2I).
Now
IU1(~1)-U2(~2)1
<
IU1(~1)-U2(~1)1
<
I lu l - U2 1 I
+
!U2(~1)-U2(¢2)1
1~1-~21.
+
Inserting this inequality into (6.9), collecting all the
!~l-~ 21
terms involving
on the left, and dividing yields (6.10)
Now we use the definition 1 ~ul (¢)-ffu 2 (¢) +]1 <
(1-2]1) IU l
3/2
(~1)-U2
(¢2) 1
-IH]1 (u l (¢l) '¢l) - H]1 (u 2 (¢2) '¢2)
(1-2]1)[llu l - u 2 " +]1
<
I ~
3/2
~u:
(6.4) of
+
1~1-~21]
A[llu l - u 2 11 + 21¢1-¢2
I lu l - u 2 1I{ (1-2]1)
I
1]
(1+]13/2 A (1_2]13/2 A)-1]
+ ]13/2 A [1 + 2]13/2 A (1_2]13/2 A)-1]}. Let
a
denote the expression in braces. 1 -
so we can make
a < 1
2]1 + 0(]1
by making
3/2
]1
Then
), small enough.
If this
is done, we have (6.11)
i.e., ~ is a contraction on
U
and hence has a unique fixed
THE HOPF BIFURCATION AND ITS APPLICATIONS
point
u*. To
prove that the invariant manifold
attracting, we pick a point
(zl'~l)
(by (6.7)), so ~l
217
(z,~)
u*(~)}
{z =
in the annulus
is again in the annulus.
is
Izi < 1,
Now let
denote the solution of
~l'
The definition of
on the other hand, needs
Subtracting these equations and then estimating and re-arranging as in the proof of -
(6.8), we get
3/2
I~l-~I~)l
A (l-2)l
3/2-1 A)
Iz-u*(~)I·
Now subtract the equations u*(~l) = YU*(~l) =
-
(1-2)l)u*(~1) +)l
(1-2)l)z +)l
3/2
and again imitate the proof that Y
with the same
a
as in (6.11).
Iz -~*(~ ) n n
I .::.
a
n
Iz-u(~)
3/2
-H)l (u*(~l) '~l)
H)l(Z,~)
is a contraction to get
By induction,
I ....
0
as
n"" "'.
In our proof, we used only the continuity of
H)l' K)l
218
THE HOPF BIFURCATION AND ITS APPLICATIONS
and their first derivatives, and we obtained a Lipschitz continuous
u*.
Closer examination of the argument shows that
we needed only Lipschitz continuity of more differentiability of more differentiability for Specifically, let u(~)
of class i)
ii)
Uk k
C
u*.
K~,
K~.
If we have
we would expect to obtain
This is indeed the case.
denote the set of periodic functions
satisfying
lu(j)(~)1 lu(k)
H~,
H~,
(~) I
,::,1, j
= O,l, ... ,k; all
~.
is Lipschitz continuous with Lipschitz
constant one. If
H~,
K~
" . h ave Llpschltz contlnuous
k th derlvatlves, .. a
straightforward generalization of the estimates we have given shows that for self.
~
sufficiently small,
It may be shown that
Uk
~
maps
Uk
into it-
is complete in the supremum
norm (as in the proof of the center manifold theorem) , so the fixed point of ~ must be in Uk' i.e. , u* has Lipschitz th d ' . continuous k erlvatlve. I f we make the weaker assumption th that H~, H~ have continuous k derivatives, slightly more complicated arguments show that u* also has a continuous th k derivative; we proceed with the proof by showing that the set of
u's, whose
k th derivatives have an appropriately
chosen modulus of continuity, is mapped into itself by
~.
THE HOPF BIFURCATION AND ITS APPLICATIONS
219
SECTION 6A THE CANONICAL FORM We shall give here, following Lanford [1], an elementary and straightforward derivation of the canonical form for the mapping
~~: ~2
Recall that we have already ar-
+ ffi2.
ranged things so that COS
8(~)
sin
8(~)
(lW) (
-sin
8
(~»)
cos 8(~)
(x)
+
Y
We want to organize the second, third, and fourth degree terms by making further coordinate changes. to identify
R
2
It will be convenient
with the complex plane by writing
z
x + iy.
Then (1 +~ ) e
From now on we shall leave
~
i8
(~)
.
out of our notation as much as
possible. The higher-order terms in the Taylor series for
~
may
THE HOPP BIFURCATION AND ITS APPLICATIONS
220
be written as polynomials in
z
and
z, i.e.,
Az + A (z) + A (z) + ••• , 2 3
where, for example, 2\'
L
2 J. 2 -J. a.z z.
j=O J
Let us begin in a pedestrian way with a new coordinate degree
z' = z + y(z) , where
2, i. e., has the same form as
relation between z
z
and
z'
Y A2 •
A2 •
We choose
is homogeneous of We can invert the
as
z' - y(z') + higher order terms.
Since for the moment we are only concerned with terms of degree
2
or lower, we calculate module terms of degree
3
and higher and replace equality signs by congruance signs (=). Thus we have z-z'-y(z')
(I-y) (z') •
In terms of the new coordinate we have
- ( I+y)
(I+y) [Az 1
- (I+y) [/..2'
Ay (z') + A (z 1 -y (z' ) ) ] 2 Ay (z') + A (z' )] 2
- AZ '
Ay (z') + A (Z') + y(AZ'-AY(ZI)+A (Z')) 2 2 Ay (z' ). - AZ' + A (ZI) + Y (AZ') 2 Now y(z' ) y(Az') - AY(Z') On the other hand,
I 12 ,2 _2 y (A 2 -A)Z' 2 + y (A -A)Z'Z'+y (A -A)Z' 2 1 O·
THE HOPF BIFURCATION AND ITS APPLICATIONS
A (Z') 2 so, if we put 2 -a 2 Y2 2 A -A we get <1>'
221
2 2 ,2 2-=02 a z + a z'zr + aOz , 2 1
,
2 -a l 2 /AI -A
Y l
(z')
AZ' +
2 O X-2-A -a
,
YO
,
3
o( I z' I ).
We must, of course, make sure that the denominators in our expressions for the there is no problem for
do not vanish. ~
1
Since ~-dependent
0, but we want our ~
coordinate change to be well-behaved as
+
O.
This will be
the case provided 2i8 (0)
oJ.
ere
i8 (0)
,
F
1
e
i8 (0)
e- 2i8 (0)
F e i8 (0)
i. e., provided
e
3i8(0)
1
1.
Thus, if these conditions hold, we can make a smooth dent coordinate change, bringing
A to zero. 2 we have made this change and drop the primes:
3
We assume that
Az + A (Z) + ••. 3
(z) (A
~-depen-
is not the original
A ) and see what we can do about Ar 3 This time, we take a new coordinate z' = z + Y (z), Y
homogeneous of degree degree
4
and <1>'
higher.
3, and we calculate modulo terms of Just as before, we have
(z') _ (I + y)(z' - y(z'»
= Az' Again, we write out
+ A (z ') + Y (A z ') - Ay (z' ) • 3
222
THE HOPF BIFURCATION AND ITS APPLICATIONS
Y (z')
Y (\z') -
AY (z ' )
3 Y a' 3
2_ _ 2 - 2 + Y2 z z' + Ylzz' + YOz'
Y (A 3 -A)Z,3 + Y (IA/--l)AZ,2 Z ' 3 2
+ Y (IAI 2X-A)Z'z,2 + Y (X3 -A)Z,3 O l A
3
(z')
= a
3
3
z'
a , a , a 3 O l
Y ' Y ' YO' we can cancel the 3 l
terms provided
e The
2_
2
By an appropriate choice of
333
3
+ a z' z'
3
2i8 (0)
f 1,
e
4i8 (0)
term presents a new problem.
f 1.
For
V f 0, we can, of
course, cancel it by putting
Y3
=
This expression, however, diverges as the value of
8(0).
+
0, independent of
For this reason, we shall not try to ad-
just this term and simply put ordinates
~
Y = O. 3
Then, in the new co-
(dropping the primes) (Z)
We next set out to cancel the 4th degree terms by a coordinate change
z' = Z + y(z), Y
homogeneous of degree 4.
A straightforward calculation of a by now familiar sort shows that such a coordinate change does not affect the terms of degree
< 3
and that all the terms of degree
celled provided e Thus we get
Si8 (0)
t-
1.
4
can be can-
THE
HOPP
'fez) =
BIFURCATION AND ITS APPLICATIONS
3
(;\ + a21z1
2
)z + o(lzl
3
223
).
This is still not quite the desired form.
To complete the
argument, we write ;\ + a
3/ z 1 2 2
=
(l+11)e
ie(]1)
f (]1) 2 2 [1 - -l- I z / + i f (]1) Izi ] 3 1+]1
(where 2 (l+]1-f (]1)/zl l
)e
f
l
,f
i[e(]1)+f 3 (]1)lz/
Thus ire (]1)+f
=
(l+]1-f
l
(]1)/z/2)e
are real)
3
(]1) Iz1 3
2
2
]
4 + O(lzl
).
5
] z + O(lzl
);
when we translate back into polar coordinates, we get exactly the desired canonical form.
224
THE HOPF BIFURCATION AND ITS APPLICATIONS
SECTION 7 BIFURCATIONS WITH SYMMETRY by Steve Schecter In this section we investigate what happens in the bifurcation theorems if a symmetry group is present.
This is
a non-generic condition, so special situations and degeneracies are encountered.
(See Exercises 1.16 and 4.3).
At the end of the section we shall briefly discuss how the ideas presented here apply to Couette flow (we thank D. Ruelle for a communication on this subject). Parts 0-4 of this section are based on Ruelle [3]. related reference is Kopell-Howard [3].
A
See also Sattinger [6].
In the first parts of the section we confine ourselves to diffeomorphisms.
There are entirely analogous results for
vector fields.
O.
Introduction. Let
E
be a real Banach space with
C~
norm,
~
>
3.
THE HOPF BIFURCATION AND ITS APPLICATIONS
x \+ Ilxll
(This means that the map G
is
cQ.
on
225
E - {O}.)
Let
be a Lie group and
A a (smooth) representation of G G as a group of linear isometries of E. We denote the elements
of
A G phisms f
0
by
g E G.
A , where g
f: E
E
+
that commute with
A G
AG, i.e., that satisfy
A 0 f for all g E G. g If we know f(x) and f commutes with
A g
=
y
is determined for all of
We wish to consider diffeomor-
A , then fey) G under the action
in the orbit of x 2 E = R with the Euclidean norm and
A • For example, if G is the group of all rotations of the plane (a representa-
=
tion of
G
that
takes circles to circles.
f
SO(2», then if
f
commutes with
A G
it follows
Notice: (0.1)
If
f
commute with
~G
then for all
mutes with
If
(0.3)
commutes with Because
=
0
a,b E R, af + bh
commutes with
E
that
also com-
¢: E
A G
.
and
f(O)
0, then
A G
¢: E
If
g E G, for all
0, ¢
f
commutes with
functions
hare diffeomorphisms of
AG.
(0.2) Df (0)
and
+
R
x E E, and
¢A (x) = ¢(x) and all g commutes with AG, then ¢f
satisfies f
AG. E +
has a ~
satisfying
¢ = 1
stant on each sphere centered at Given a
and a linear isomorphism
These
0, and ¢
¢
is con-
satisfy the
diffeomorphism
+
E
bump
f: E
+
E
A G Df(O), these bump functions
A: E
and is sufficiently close to
O.
cJI,
C
on a neighborhood of
outside a larger neighborhood of
conditions of (0.3) .
Q.
norm, we can construct
cQ.
that commutes with
226
THE HOPF BIFURCATION AND ITS APPLICATIONS
together with properties (0.1) to a new diffeomorphism
A
mutes with
and
G
h Ct-close to
Dh(O)
preserving maps
such that
Each
f
(b)
f
(c)
flJ(O) = 0
(d)
flJ
(e)
DfO(O)
is
lJ k c
is a
C
t
com-
satisfying:
diffeomorphism of
E.
in the second variable, lJ. for all
commutes with
=
h
denote the space of level-
f: E x (-1,1) + E x (-1,1)
(a)
Iz/
f
f
A.
1 < k < t, let j l
Given
values on
(0.3) allow us to perturb
lJ. for all
A G
lJ.
has a finite number of isolated eigen-
1, each of finite multiplicity.
(Since these
eigenvalues are isolated, the rest of the spectrum is bounded away from
Iz/
=
1.)
We wish to study how the qualitative picture of near the origin changes as
lJ
passes
0, for generic
flJ f E
~.
Y, to be ·de-
In fact, we will study an open dense subset of fined in Section 2. * 1.
Reduction to Finite Dimensions. Let
EO
be the finite dimensional subspace of
responding to the eigenvalues of
DfO(O)
Iz/
on
=
E
cor-
1.
* 5l is given the appropriate Whitney topology. set
B (f, <1»
containing
strictly positive ting
B(fi
(x,lJ);
a . II axJ.
= {h E
cO
f E
A basic open is given by specifying a
~
function
~I!
<1>:
E x
(-1,1)
/h(x,lJ)-f(x,lJ) II <
a J. f(x,lJ) h(x,lJ) - ---. axJ.
II
<
+
R
(x,lJ) for all
and setfor all (x, lJ)
i
and all
1 < i < t; and / I~ h(x,lJ) ajJ J. (x,lJ) and 1 < i < k}.
for
THE HOPF BIFURCATION AND ITS APPLICATIONS
(7.1)
Proposition.
A EO
(1)
EO
=
q
227
for all
g
E G.
can be given a Hilbert space structure so that remains a group of isometries. Proof.
(1)
Let
C
be a simple closed curve in
~
= C (where C denotes the complex conjugate of C), Spec Df 0 (0) n C = fJ and Spec DfO(O) n Int C = In E ~ ~, the complexification Spec Df 0 (0) n {z: I z I l} . of E, let P = ~ (zI - DfO(O) ® I)-ldz. P commutes with 1T1. C such that
C
J
AG ® I
because it is the limit of operators that do.
Real Spectral Splitting Theorem, P Q: E
so
A . But G be the closure of
1m Q
E.
is the complexification
of a real operator
-7
Clearly
Q
is invariant under
(2)
Let
r
The desired inner product on
EO
By the
is
A ,
commutes with
1m Q
G
= E o.
A~, a compact group. (x,y)
=
J
dy< x,yy>,
r
where
dy
is Haar measure on
product on
EO.
(7.1) fold
V
< , >
is any inner
[]
ck
f E ~ has a
Theorem.
near
rand
(0,0) E E x
local center mani-
(-1,1). tangent to
EO x
(-1,1),
satisfying: (1)
V
(V)l
n
(E x
{)l}»
is
cR-
and
A G
invariant. (2) (-1,1)
There is a level-preserving chart
satisfying
¢A
g
=
A ¢
All the local recurrence of Proof.
(1)
for all
g
f)l
near
g
°
¢: V
-7
EO x
E G. takes place in
V)l.
The point here is that the construction of
a center manifold in the center manifold theorem can be done in a
AG-invariant manner.
In order that
V
be
AG-invariant,
228
THE HOPF BIFURCATION AND ITS APPLICATIONS
it suffices that the first "trial center manifold" used in its construction be
AG-invariant.
(Consult Section 2 and the
center manifold theorem for flows.)
Take the first trial
center manifold to be the subspace of
n
Spec DfO(O)
{z:
Izl > I}, which is
E
corresponding to
AG-invariant by the ar-
gument of Proposition 1 (1) • (2)
=
(QIEO) x I, where
Q
is as defined in the
0
proof of Proposition 7.1 (1) • 2.
Generic Behavior. From now on, because of Theorem 7.1, we will think of as acting on a neighborhood of U x J, and we will let
A
(0,0)
stand for
II
in
E
°x
(-1,1),
Df (0) EO. ll
is now a group of isometries of the Hilbert space ~
We will now exhibit a tractable generic subset of The characteristic polynomial of factors of the form with
=
IAI
1.
Let
(x-I), (x+l), and A
of
(A
is a product of
2
(x _2 Re AX + AA),
be a complex eigenvalue of
similar argument applies if space of
A O
X
2
=
±l) and let
F
A
(a
be the null
k
- 2 Re AA + AXI) , where k is the exponent O O 2 (x - 2 Re AX + AX) in the characteristic polynomial of
A • Then F is AG-invariant because it is the null space O of an operator that commutes with A , and F~ is A G G invariant because F is and the A preserve the inner proq duct. Let Then
A
O
B: EO
+ EAOB
invariant.
+
EO
be orthogonal projection onto
commutes with
A + EAOBIF O and, for all E f 0, A + O
A and leaves F G has only the eigenvalues
EAOBIF~
F~.
F~
and A
and
has no eigenvalues on
X·,
THE HOPF BIFURCATION AND ITS APPLICATIONS
Iz I
229
Using (0.1) and (0.3) it is now easy to perturb
l-
flu x J
D(fIU xJ)O(O) = A + e:A B has only 0 O 0 the one pair of eigenvalues A, A on I z I 1. Since E
has a
so that the new
E
AG-invariant complement in
easy to extend this perturbation of tion of
f
such that the new
~
A and
Izi
on
=
flu x J
DfO(O)
Spec
A~
has only the eigenvalues
o
E.
This
If it is not the
consists of exactly one pair of com-
A~, A~
plex eigenvalues
to a perturba-
1, each with finite multiplicity.
perturbation reduces the dimension of case that now
EO @ ker Q', it is
(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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
(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<
EO
o
is the null space of
'ITj.l
C
.,.
[(Aj.l @ I) - Aj.l I]n/2
EO ~ C, a complex subspace of complex dimension x
~
1, and
'ITj.l
F j.l
o I
E
Here
E
..",.
is orthogonal projection.
t"
F
0
IIFj.l
j.l
in
~, i(x)
The inner product
induces a complex inner product on EO ~ C with O respect to which A ~ I is a group of unitary operators. on
EO
Fj.l O
is invariant under
G
A
G
® I.
~
AO el IIFj.l" G
~
I) Fj.l
AGO ~ I
because
is conjugate to
We now work with the
Aj.l
Aj.l
~
I
commutes with
and commutes with
~ el I/Fj.l
in analogy to
Case 1. We see that generically (1) AO ~ IIFj.l is irreducible and G
for all small j.l, where I denotes the c n / 2 and A is complex with F j.l j.l 11. 0 1 = l. In other words, generically EO may be regarded O as a complex inner product space and AG as an irreducible (2)
Aj.l
A I
j.l identity operator in
232
THE HOPF BIFURCATION AND ITS APPLICATIONS
group of unitary operators on ~,with
small 3.
A~
complex and
Results in the Case Eigenvalues on
A~
EO, while
OfO(O)
IAol
A~I
for all
1.
=
has One Pair of Complex
Izi = 1.
Ruelle's main result, which we state without proof, is Theorem 7.2 below, which helps one find invariant manifolds in Case 2 of Section 2. a neighborhood of
We now think of
(0,0)
in
flu x J
F x J, where
dimensional complex inner product space and val in
R
containing
(7.2) k C
in
with tion). k-3 ~
11,
is an inter-
flu x
is
J
C9,
for fixed
~
,
k > 3. Suppose also that OfO(O) = A I ~ 4 3 A I- 1 and A I- 1 (a generic assurnp0 0
but
1
J
is a finite-
o. Suppose
1 < k < 9"
I AOI
By a level-preserving change of coordinates that is 00
in
can put
where
Lemma.
F
as defined on
and f
P
C
for fixed
and commutes with
~
1\.0
G'
we
in the form
is a homogeneous polynomial of degree 2 in z Z, and Q~ is 0(lzI 3 ). In fact, 1 Q~ (z) I .::. 2 3 c (I z I) I z I and IOQ (z) I < c(lzl) Izl where c ( • ) is indeIl
11
pendent of If
~
and
=
o.
2 z E 0:: , z = (zl ,z2)' each
eous polynomial of degree is one of the form P (z)
-
lim c(u) u.... O
2
in
z
zi E 0::, then a "homogenand
1
in
z"
on
THE HOPF BIFURCATION AND ITS APPLICATIONS
By way of motivation one might note that for such for
A
E
~
with
family of
C!/,
parameter
~
=
A~Z
in
Z
IQ~ (z) I 2.
2. !/,
diffeomorphisms, 1
and
(z).
+
and
00, depending on a real
<
varying in an interval about
o.
P~(z)
is a continuous
complex function; 2
AP
P(AZ)
p(z)
Theorem.
(7.2)
~~(Z)
1, one has
IAI
233
1
+
Q~
~ ~ A~
(z), where
Suppose
P~
is a homogeneous polynomial of degree
in
Z
c(lzl)/z/3
is independent of
with coefficients continuous in
and and
o =1
1DQ~
2. c ( I z / ) I z I
(z) I
lim c(u) u-+O for ~ > O.
o.
=
2
where
c (. )
We also assume
Suppose the vector field and IA I > 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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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 <
to
< ]1 < ]10}
~,
to
Theorem 7.2 gives information on invariant manifolds for ]1 > O.
If we assume
IA]11
< 1
tain similar information for
~-l.
for ]1 <
°
]1 < 0, then we can obby applying Theorem 2 to
It is easy to check that
-]1
A- 3 X--Ip
-]1
-]1 -]1
(z ) + Q' (z). -]1
Therefore one should look for invariant manifolds of the vector field
4.
Examples. Let
l.
O A
A 1 be the only eigenvalue of Df O (0) 0 l. Let h = flv, V the center manifold for
and let
on
Iz I =
f
osition 7.l.
According to Theorem 7.1 we may think of n defined on a neighborhood of (0,0) in R x R, where: Each
(2)
h]1 (0) = 0
(3)
Dh]1 (0) = A]1I, AO = 1, A]1E
bolic to
S
commutes with
]1
for all
h
as
G
]1.
if there is a splitting of
TS ~ N_, such that
of Prop-
AO.
(1)
h
n R
EO
be the full orthogonal group of
G
R
for all
]1.
TEls, TEls
P(DfIN_) < min{l,p(DfITS)}
and
P(Df-IIN+) < min{l,p(Df-IITS)}. This means that iteration Df contracts every vector of N_ more than Df contracts any vector of TS, and under iteration Df expands every vector of N+ more than it expands any vector of TS.
THE HOPF BIFURCATION AND ITS APPLICATIONS
O A
Because that
contains all the reflections, it is easy to see
G
h~(x)
°
about
235
is a multiple of
to spheres about
p (Ixl)x, where for each ~
x.
0. ~,
u
h~
Also,
h~(x) = A~X
Therefore ~
p (u) ~
takes spheres
+ ~-l
is a real-valued
function of u with P (0) = 0. Write p~ ( Ix ~ 2 o(lxI ). The non-wandering set of h~ near
I)
°
I I
= a~ x
C
2
+
consists of
exactly those spheres that are taken into themselves; each such sphere consists entirely of fixed points. If f is at 3 least c in ~ and a < (i.e., (0,0) is a vague atO tractor), we can apply exactly the analysis of to see that
°
~
for small
>
°
there is a one-parameter family of such
spheres, one for each small
°
ing to
2. numbers
as
~ +
~
>
0, with the spheres converg-
0.
*
We think of
a
with
lal = I
0(2)
as generated hy the complex
and a reflection
r.
Suppose we F = a::
are in the situation of Case 2, Section 2, with the irreducible representation of
0(2)
a::
on
2
2
and
given by
We think of fjV as acting on a neighborhood of (0,0) in 2 a:: x R and denote this map by h. According to Lemma 7.2, after a change of coordinates we have a new map
where
* David
P~(z)
is homogeneous of degree
2
Fried explained this example to us. of Section 4.9 of Ruelle's paper.
in
h'
z
with
and
I
in
It is a reworking
236
THE HOPF BIFURCATION AND ITS APPLICATIONS
z, and
Q (z) is ]1 we assume IA]11 > 1
'I
uniformly in for
]1.
< 1
p(Z)
1.
P
l
A~-invariant
be a
z
in
2
1, and
]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:. Therefore
According to Theorem 2 we should now study the differential equation dz -1 2 2 dt = z ± AO {aO(lzll +l z 2 1 )(zl,z2) 2
(4.1)
2
+ bo(lzll -lz 2 ' ) (zl,-z2)}. We will look for invariant manifolds
S
of this differential
THE HOPF BIFURCATION AND ITS APPLICATIONS
237
and
equation which are also invariant under the action of under the maps
Z 1+
z
polynomial in z ..... az, 1a 1
=
q(z) = q(az) that
S
az,
and
=
lal
1.
is a
Now suppose
invariant under
O A
and the maps
G
o
(Le., q(z) = q(A z) for all
1
for all
with
Suppose further
1) •
lal =
and
g E G
g
is exactly the union of the orbits of (any) one of
Ag
its points under the actions of
and the group of maps d
z ~ az,
lal
z E S.
With this motivation, we will search for the manifolds
S
=
1.
Then it follows that
by studying these polynomials For
lal
1, q(zl,z2)
=
Hence for fixed
z2' q(zl,z2)
larly, for fixed q(zl,z2)
-1
z2)
=
for
2
q(a zl,z2)·
depends only on
zl' q(zl,z2)
0
depends only on
Izll2. Iz212.
SimiSince
q(z2,zl)' q is symmetric in zl and z2. Now let 2 2 I z and let d(zl,z2) = 21z1z21. Then 1 1 + I z2 1
s(zl,z2) sand
=
q(z). q(a z 1 ,a
=
dt q(z)
d
2
are a basis over
~
for the polynomials in which·
we are interested. Let
-1
-1
Re A a and let 6 = Re A boo Let O o O 2 2 1 2 ~2 1 1 1 2 z,z E,,with z = (zl,z2)' z (zl,z2)' where the i 12 12 12 Zj E
at
a
=
d~ dz -z] [z, dt] + [dt'
+ bo(lzlI2_lz212) (zl,-z2)}]
+
-1
[z ± A
O
2
{a (l z 1 1 +l z 1 2 O
2 )
(zl,z2)
2 + b 0 ( I z 1 1 -I z 212) (z l' - z 2) } , z] 2s ± (2 a s
2
2 2 + 26 (s -d )).
and
238
THE HOPF BIFURCATION AND ITS APPLICATIONS
z z
4Iz1/2Iz2/2 = 4z 1 1 z 2 2 , one calcu-
Also, since lates that
d
2(d ±
dt d 1 ds 2 dt
=
sd).
s ± (as
1 d "2 dt d
2
Therefore: 2 2 + S(S -d »
(4.2)
d ± asd.
We make the generic assumption that
a, S, and
nonzero.
are non-negative func-
Recalling that
d
and
s
a + S
are all
tions, we see that we have found three possibilities for invariant manifolds: (1)
d
0'0 s
(2)
d
0, s = :;:(a+S)
(3)
d
s = -+ a
0
-1
-1
Now we have the following commutative diagram
I
(C2
("d)
1R 2
Flow of (4.1)
-=--
I
«::2
(',d)
....
Flow of (4.2)
R
2
Because the vector field (4.2) is Lipschitz, its zeros sent fixed points of its flow.
~2
Hence their inverse images in
are invariant under the flow of (4.1).
Therefore we have
found the following invariant manifolds of (4.1): (1)
sell
(2)
S(2)
=
repre-
{o}
{z E «::2: d = 0, s = :; (a+S) -I} 2 2 :;'(a+S)-l} u {z E C :zl = 0, I z 2 1
THE HOPF BIFURCATION AND ITS APPLICATIONS
{Z E «;
(3)
S (3)
2
=
z2
0,
Izli
2
two circles or 50. 2 {z E iC : s = d -
1
}
+ 2a
=
-
+ (a+l3)
- l} +
a
-1
239
}
2
{z:
Iz2
Izll
2 1
50.
torus or
The vector field (4.1) is normally hyperbolic to each of these manifolds: (1)
The derivative of (4.1) at
0
(2)
The derivative of (4.2) on
s(2), with respect to
the variables (3)
s, d
is
0)
(-2
o
is
I.
213/a+13'
The derivative of ( . ) on
with respect to
-2 (a+213/a) the variable
s, d
, which has eigen-
is -2
( values
-2
and
In this case one can see that the vector field (4.1) contain no other compact invariant manifolds by examining the flow of (4.2) in the sd-plane. s
and
Because of the definition of
Now for definiteness suppose
Iz 1
is contained in Assume suppose
0 < d < s.
d, one needs only consider the region
11..]11 < 1 a < 0
Then for each
for and
]1 > 0
< 1
f]1 : E ...- E
and
except for the eigenvalues
]1 < 0 a + 13 < 0
and
11..]11 > 1
for
Spec Df]1 (0) A
]1, 1..]1"
]1 > O.
Also
(a "weak attractor" condition) .
we have the following invariant mani-
folds: Sill {a}, non-attracting for ]1 > o. ]1 S(2) two circles, invariant under f and the con]1 o nected component of the identity in A , interchanged by reG flections. They are attracting if S < 0, non-attracting if
240
S
THE HOPF BIFURCATION AND ITS APPLICATIONS
O.
>
S
SJ3), a torus, attracting if
S < O.
S(3) II
{(zl,z2) E
can be analyzed further.
~2:
z2
=
az l }
ITa)
The subspace
is pointwise fixed by
A~
IT 0
ro
~
O
A E
A~
-1
az , z2 ~ a zl)' Since the map 2 of Theorem 7.2 commutes with A0 we have that e (S (3) (J G' II
(this is the operator
ell
> 0, non-attracting if
C
that
ITa'
Since
h(S~3) (J
h
ITa)
zl
flv
=
s~3)
=
~
A~,
also commutes with (J ITa'
union of circles of the form
s~3)
Thus
s(3) II h, interchanged by the elements of
(J
it follows
is a disjoint
IT , each invariant under a
o
A. G
It seems that in this example "two Hopf bifurcations take place at once," resulting in invariant sets, for each II > 0, of the form
5.
(point U circle) x
Flow Between Two Cylinders.
Couette flow:
(point U circle).
Let us recall the set-up for
Suppose we have a viscous incompressible fluid
between two concentric cylinders.
Let
R l
be the radius of
the inner cylinder and
R the radius of the outer cylinder. 2 Suppose we forcibly rotate the two cylinders. Let n be the l
angular velocity of the inner cylinder and velocity of the outer cylinder. (i. e.,
Assume
n
n l
2
> 0
the angular and
we rotate both cylinders counterclockwise).
values of
nl
and
n
2
> 0
.For small
n
one observes a steady horizontal 2 laminar flow, called Couette flow. In fact, for arbitrary
n l , n 2 Couette flow is an explicit solution of the Navier Stokes equations, which, in cylindrical coordinates (r,¢,z), (see §l) 2 2 2 2 (n -n )R l R2 1 n2R2-nlRl l 2 r + r R2_R 2 R2 _R 2 2 1 2 1
is given as follows: v
e
THE HOPF BIFURCATION AND ITS APPLICATIONS
241
To get this solution one must ignore special phenomena that occur at the ends of the cylinders.
We will do this by identi-
fying the ends of the cylinders, so that the space
in
A
which our fluid sits is an annulus crossed with the circle. r Let E be the space of C vector fields y on A satisfying
div Y
= rl
We will think of
and
0
yl
dA
=
as small fixed and
l
will be our bifurcation parameter
~.
~
the vector fields
Y
on
A
topology.
rI
as varying: 2 For each ~ let
be the corresponding Couette vector field on each
cr
0, with the
A.
z~
Then for
satisfying
div Y = 0 Y form a sp.ace identify each
does not slip at
dA
which we may think of as
E
~
E~
with
Z
+ E.
~
in the obvious way.
E
We will
We will think
of the Navier-Stokes equations as defining, for each X~
vector field
on
the vector field
E.
X: E x
of
E x
m
,
a
We make the false assumption that
m~
E
X(Y,~)
given by
isfies the conditions of Ruelle's paper. (O,~)
~
*
= X~(Y)
sat-
Note that the point
corresponds to the Couette vector field
z~.
Because Couette flow is a steady solution of the Navier-Stokes equations, we have Let
G
=
X~
(0)
=
0
for all
~.
80(2) x 0(2), the rotations of the annulus
crossed with the full orthogonal group of the circle. the natural symmetry group of the situation. Couette vector fields
*To
z~
are
G-invariant.
This is
Notice that the For each
gE G
make this really work, one can use the methods of sections 8,9.
242
THE HOPF BIFURCATION AND ITS APPLICATIONS
define
Ag : E .... E
Because each for
g
by
(AgY) (x)
DgY (g
=
is an isometry of
A
k > 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):
THE HOPF BIFURCATION AND ITS APPLICATIONS
(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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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.
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
]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.
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
(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-
THE HOPF BIFURCATION AND ITS APPLICATIONS
[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
<
for
)1 = 0
the origin has two isolated simple eigenvalues
I Ie ()1)
Xl'iJT with
255
j = 1
the spectrum is in
D
I Ie ()1) I
(resp.
Ie ()1)
G)1
1
at
and
> 1) and the rest of
and remains bounded away from the unit
circle. (d)
(d/dt) jle()1) 11)1=0 > 0
(the eigenvalues move
steadily across the unit circle). Under these conditions, bifurcation to periodic orbits takes place. (8.4) holds, where
For their stability we make: Stability Assumption. V"
I
(0)
The condition
V"
'(0) < 0
is calculated according to the proced-
ures of Section 4 (see Section 4A). This calculation may be done directly on the vector field
X, since the computations are finite dimensional; un-
boundedness of the generator
X
causes no problems.
Bifurcation to Periodic Orbits. Let us recap the result: (8.5)
Theorem.
a fixed neighborhood
F~(X)
that x E V.
Under the above hypotheses, there is U
of
0
in
is defined for all
E
t > 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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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<
IR x D, such
topology induced on
(O,X) E q)
xE D,
so we can define
N
N
(t,x) E
q),
Similarly one defines a local
semiflow and a maximal semiflow. Now let with domain for all
D
N
is a map
xED.
xED C N.) (a,b) C IR
be a Banach manifold.
(TxN
X: D
+
T(N)
D
such that
is the tangent space to
An integral curve for
+
A vector field
such that
c
X
x(x) E Tx(N) N
at
is a curve
c:
is differentiable as a map from ,.
(a,b)
to
Nand
local flow) for D
c' (t) X
1+
A flow (resp. semiflow,
xED, the map
t
~
Ft(x)
is an integral
X. F
If
(t,x)
X(c(t».
is a flow (resp. semiflow, local flow) on
such that for all
curve of
=
t
Ft(x) F
If
t
is a flow on cO
is a
Ft : N
+
M, then
is a
cO
Ft If
Ft
is a
N
X
and
F Ft
F: R is a
flmv on
x
N
cO
+
N
via
flow on
extends to a con-
and the extension is a flow for
cO
such that
map, we say
is a flow for
tinuous map
N
cO
flow on
X. N
and for all
t
fixed
N.
260
THE HOPF BIFURCATION AND ITS APPLICATIONS
Ft : N
N
->-
is of class k C •
of class
k (resp. T ), then
Ck
(resp. Tk )
Ft : N
Here
->-
T 1 (N)
jth tangent map
F
is a flow
t
N is j T (N)
T
k
means
TjF : ->exists and is t k means that in each continuous for j < k; F : N ->- N is C t chart, the map x ->- djF j < k is continuous in the norm x (djF is the J.th total derivative of F at x. topology. x It is a j-multilinear map on the model space for N.) The that the
T
k
ck
case differs from the
case in that norm continuity
is replaced by strong continuity. Warning.
ck
A
flow is not assumed to be
variable, only in the the
ck
in the k A flow will be c
x-variable.
tin
t-variable only if it is generated by a smooth everywhere
defined vector field; see however the Bochner-Montgomery theorem stated below.
Separate and Joint Continuity. (8A. 3)
Theorem.
t
~
F
Ft(X)
Let
N
be
be a flow (or local flow) on N, t be separately continuous in x and t (i.e.,
a Banach manifold. and let
(Chernoff-Marsden [2]).
Let
F
is continuous for fixed
tinuous for fixed
t), then
F
x
and
x
~
Ft(X)
is conis
t
jointly continuous. For the proof, we shall use (8A. 4)
Lemma.
(Bourbaki [1] Chapter 9, page 18;
Choquet, [1] Vol. 1, page 127). Let
F, G
the following.
be metric spaces.
tely continuous, then for all
Let
Let
E
¢: E
be a Baire space. x
F
->-
G
be separa-
f E F, there is a dense set
THE HOPF BIFURCATION AND ITS APPLICATIONS
Sf
~
E
whose complement is first category such that if
e E Sf' then
is continuous at
¢
(e,f).
Proof of the Theorem (8A. 3).
N
We let
is a Banach space and E
JR, F
=
is a pense set of
IR
is a local flow on t xE U C N, t E (-E, E). F
Let
t x E (-E,E)
such that
N
t x close to compositions are defined. Let t n -> 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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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 HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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-
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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)
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
(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
<
T}
<
I
for
maps
U(t,r)
=
10
{U(t,x)
[O,T) x [O,T)
for
~
s
t
<
10
<
s < T}
=
B(X,X)
=
<
r
<
T}
s
<
in
<
t
X
<
T.
is said
f E X, the function
{U(t,s)}
of operators in
A(s)f
0
continuously into
X-infinitesimal generator of {A(s)
B(X)
0 < t < T, and
to be strongly continuous if for each U(·,·)f
of
X) is called an evolution system in X if
U(t,s)U(s,r)
An evolution system
s
<
X
X
X.
The
is the collection defined by
lim ~-l[U(s+~,s)f-f] ~!O
with
D(A(s))
consisting of all
f
for which this limit
exists, where the limit is taken in (8A.16)
Remarks.
a)
If
tinuous evolution system in
{U(t,s)}
sand b)
t Let
I IU(t,s)
10
T - a}
<
s
<
is the
{U(t,s)
10
s
<
X
T}, and let
t
<
<
with
0<
is bounded
a
T}
be a strongly continu-
X-infinitesimal generator <
T.
Then
{A(s+a)
10
~ s <
X-infinitesimal generator of the strongly
continuous evolution system (8A.17)
I Ix,x
in closed and bounded intervals.
ous evolution system in {A(s)
is a strongly con-
X, then it follows from the uni-
form boundedness principle that for
X.
proposition.
{U(t+a, s+a) Let
10
{U(t,s)
a-strongly continuous evolution system in
~
10 X
s ~ t <
s
<
with
T - a}.
<
t
<
T}
be
X-infinite-
THE HOPF BIFURCATION AND ITS APPLICATIONS
simal generator
o
{A(s)
s < T, and
<
10 2
A(·)f
s
<
maps
T}.
f E D(A(s»
If
[O,T)
273
for all
continuously into
X,
then (8/8s) [U(t,s)f] for 0
~
s
~
t < T, t
Proof.
If
0
>
o.
~
s
(8A.5)
-U(t,s)A(s)f
t
<
T, then
<
U(t,s+£)f - U(t,s)f
U(t,s+£) [f-U(s+£,s)f],
and therefore,
=
(8+/8s)U(t,s)f Thus for each
t
E (O,T) ,the function
tinuous right derivative on U(t,.)f
-U(t,s)A(s)f.
[O,t).
U(t,·)f
Thus the function
is continuously differentiable on
Yosida [1], p. 239), and (8A.5) holds for Since the derivative of at
U(t,·)f
[O,t) 0
(see
s
<
<
t
<
T.
has a limit from the left
t, it follows that
-v (t,s)A(s) f
(8- /8s) [U (t, s) f] for
0
<
=
s
t
<
T.
But, because of the domain of
this is what (8A.5) means when (8A.18)
Corollary.
Let
s
=
mal generator
{A(s)
< s < T, suppose
[O,T) into
into X.
{vet,s)
10 X
<
s
<
with
t
<
T}
be a
X-infinitesi-
10
<
f
is continuously differentiable from
X, and that
Then
V(t,·)f,
D
t.
strongly continuous evolution system in
o
has a con-
s
<
T}.
A(·)f(·)
Let
f(s) E D(A(s»
maps
[O,T)
for
continuously
274
THE HOPF BIFURCATION AND ITS APPLICATIONS
(a/as) [U(t,s)f(s)]
U(t,s)f' (s) - U(t,s)A(s)f (s)
0 < s < t < T, t > O.
for
Proof. continuity of IIU(t,s)
This follows from the Proposition, the strong {U(t,s)}, and the local boundedness of
Ilx,x'
0
We call (SA.5) the backward differential equation.
In
order that the forward differential equation (a/at) [U(t,s)f] hold, it is necessary that
=
(SA.6)
A(t)U(t,s)f
U(t,s)f E D(A(t», and this is a
more restrictive condition which may not be satisfied when the hypothesis of the above Proposition is satisfied. Now suppose
Y
is another Banach space with
and continuously embedded in (SA.19) in
X
maps
Definition.
is said to be Y
Y; i.e., if
evolution system in
Y
densely
x. An evolution system
{U(t,s)}
Y-regular if each transformation
continuously into
tinuous in
Y
Y, and
{U(t,s)}
{U(t,s)}
U(t,s)
is strongly con-
is a strongly continuous
as well as in
x.
Kato in [4] and [5] gives a variety of conditions on families
{A(s)}
of operators in
for these families to be the
X
which are sufficient
X-infinitesimal generators of
strongly continuous evolution systems in conditions are also sufficient for the be
x.
Some of these
evolution systam to
Y-regular and for the forward differential equation to
hold.
He also gives several convergence theorems for evolu-
THE HOPF BIFURCATION AND ITS APPLICATIONS
275
tion systems and upper bounds for the operator norms in terms of certain parameters of the infinitesimal generator. Since these results bear directly on our results later, jMe quickly summarize the fundamental points here for reference. Kato's papers should be consulted for details and related remarks. (8A.20) group
Definitions.
generators in
respect to
X.
Let
Y
A, or simply
A E G(X), the set of semi-
is said to be admissible with
A-admissible, if
Co
invariant and forms a semigroup of class A subset stants
M
and
G(X) S
A > Sand
let
M,
k
(AI-A.) IT J j=l
Al, ... ,A Theorem.
A(t) E G(X)
for
{A(t) 10
Y.
II
< M(A-S)-
k
elements of the suhset.
k
(Existence Theorem).
0 < t
< t
-1
< T}
< T,
Let
T > 0,
and assume that
is stable, say with constant
S; ii)
G(Y)
Y
is
A(t)-admissible for each
is the part of
A(t)
stable, say with constants Y C D(A(t»
iii) from of
in
Y
is said to be stable if there are con-
(8A.21)
i)
leaves
(called constants of stability) such that
II for
{etA}
[O,T) A(t)
to
into Y,
within
Y, then
t, and if {A*(t)}
A*(t)E is
M*,S*; and
for each
B(Y,X), where
t, and A-(t)
(called the part of
A-(·)
is continuous
is the restriction
A(t)
within
Y
to
Then there is a unique strongly continuous evolution
x).
276
THE HOPF BIFURCATION AND ITS APPLICATIONS
system
{U(t,s)}
tending
in
X
with
X-infinitesimal generator ex-
{A-(T)}; i.e., with infinitesimal generator
such that
B(t)
, [U(t,s)' (BA.22)
~
A-(t)
'x,x
for each
<~
S(t-s)
Remarks. a)
A
Furthermore,
o
for
If
the stability condition for
t.
{B(t)}
s < t
<
A(t)
T.
<
are independent of
t,
is the condition for the
Hille-Yosida theorem (Yosida [1]). b)
Actually in Theorem (BA.21) Kato shows that the
X-infinitesimal generator is precisely
{A(t)}.
We can add on any bounded operator to a family
{A(t)}
of generators and still get generators: (BA.23)
Remark.
Let
{A(t)
10 2
hypothesis of Theorem (BA.21), let
o
~
t < T, and let
for each
f E X.
B(·)f
map
t < T}
satisfy the
B(t) E B(X,X)
[O,T)
for
continuously into
X
Then there is a unique strongly continuous
evolution system in
X
with
X-infinitesimal generator ex-
{A(t) + B(t)}.
tending
In examples, it may be difficult to verify the stability condition (i).
To this end we have a useful criterion
given in the following proposition. Let
G(X,M,S)
stants
M,S:
denote the generators
I I (A_A)-kl I 2
to the semigroup condition if
M
=
1
A
on
M/(A_S)k, A > S
I IFtl' 2
Me
St
).
X
with con-
(corresponding In particular,
we have the generator of a quasi-contractive semi-
group, the condition being on the flow
First some notation:
I IFtl I 2
eSt.
I I (A_A)-l, 1 2
1/(A-S), A > S; or
Examples of this type of semigroup
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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-
°
<
s < T} g
(here
denotes a time of existence for Fsg and F sf). c) I Iug (t, s ) - u f (t, s) I IY,X + 0 as Ilg-flly+O
Tg
(see
(8A. 27)). (8A.29)
F : Y t U
l C .
f, there is a strongly continuous evolution system
{ug(t,s) 10 ::. s < t < T} g
f
X
along the segment.
X
+
Theorem.
Under these assumptions a), b), c),
is Frechet differentiable at
f
with
DF (f) t
(t,O). (8A.30)
Remarks.
The proof also shows that if
280
THE HOPF BIFURCATION AND ITS APPLICATIONS
"Ug(t,O)
I 'X,X
be
X + X 2.
is uniformly bounded, as
Lipschitz for
g
g
near enough to
A translation argument shows
DF
varies f t-s
Ft
in
will
Y. f
(F (f» s
= U (t,s).
Further Assumptions. d)
Ug(t,s)
is
Y-regular
and replace c) by c)' as
g
Ug('t,s)
converges strongly in
converges to
f
Y
to
f
U (t,s)
along straight line intervals (see
Theorem (8A.28». (8A.3l)
Theorem.
Under a), b), c)', d), F : Y t is Gateaux differentiable at f and DFt(f) (8A.32)
I lug(t,O) I Iy,y
Remarks.
= Uf 1.
+
Y
(t,O).
The proof also shows that if
is locally bounded, then
F : Y t
+
Y
is loc-
ally Lipschitz. 2.
If (8A.26) is used, we see that, in fact, DFt(f)
is locally bounded in
B(Y,Y)
the use of (8A.3l) to get that entiable, etc.
for F
fEY.
We can iterate
is twice Gateaux differ-
t
This will imply that
F
is in fact Coo. t (Gateaux differentiability and norm continuity of the derival tive implies c from f(x) - f(y) = Jl Df(x+t(x-y» (X-y)dti
°
also Gateau differentiability with locally bounded derivatives implies Lipschitz continuity). 3.
The derivative
bounded operator
DFt(f)
in (8A.29) extends to a
X + X.
Proof of (8A.29).
Let
°<
T'
< T • f
For
Ilg-f Il y
THE HOPF BIFURCATION AND ITS APPLICATIONS
sufficiently small, we define
w
on
[O,T']
F g - F f. Differentiating, and using s s G(q) - G(p), we have
by
281
w(s)
Z(q,p) (q-p)
=
w' (s) for
0 < s < T'.
If
0 < s < t
~
T', then by the Corollary
on p. 273. g (8/8s)u (t,s)w(s)
0,
so that
for
0 < t < T'.
Thus we have the estimates
and I/F t g - Ftfll Proof of (8A.31).
< Ilug(t,O)11
X -
x,X
Jlg-fll '
x
O.
As in the proof of (8A.29), we have
so that
and I/F t g - F fll y < Ilug(t,O) II Ilg-fll y ' t y,y This establishes the claims about Lipschitz continuity. we let
g
=
If
f + Ah, then we get
from which the differentiability claim follows directly.
[]
282
THE HOPF BIFURCATION AND ITS APPLICATIONS
Using these arguments, one can also establish ferentiability of ternal parameter.
in
Y
dif-
and differentiability in an ex-
t
We consider two such results from Dorroh-
Marsden [1]. (8A.33)
Corollary.
Under the hypothesis of (8A.29) f
or (8A.31) suppose that the evolution system
{U (t,s)}
with
f U (t,s)X C Y
{DG(F )} satisfies s for 0 < s < t < T · Then for fE D, G(F f) E Y for f t f (. ,0) g is Y-continuous on (O,T ) for u < t < 0 Tf · I f f each g E X, then F(.)f is continuouslv Y-differentiable on X-infinitesimal generator
(O,T ) • f
Proof.
Under these hypotheses one can establish a
chain rUle, so that by differentiating s
at
s
=
Ft+sf
=
Ft(Fs(f))
in
0, we get DFt(f)'G(f) U
which
f
(t, 0) • G (f) E
proves the first part.
Y
The second part follows be-
cause
( (
o
f
U (T,O)·G(f)dT.
s
The condition that t > 0
Ftf
be
Y-differentiable for
is a nonlinear analogue of what one has for linear
analytic semigroups (see Yosida [1]). For the dependence on a parameter, we assume z F t
depend on a parameter
z EVe Z
where
V
G(f,z),
is open in a
THE HOPF BIFURCATION AND ITS APPLICATIONS
Banach space.
283
F~(f)
Assume at the outset that
is conz tinuous in all variables, and for each z, F is as above. t To determine differentiability of FZ(f) in (z ,f) t we can use a simple suspension trick. Namely, consider the semiflow
H t
on
The generator is
D x V
defined by
K: D x V
+
X x Z,
K(f,z) = (G(f,z),O). If (8A.29) or (8A.30) applies to clude differentiability * of
in
H t
(f ,z) •
One of the key ingredients in (8A.29) concerning the linearized equations.
then we can con-
is hypotheses
Here
DK (f , z) (g, w ) = (D1 G (f , z) • g + D2G (f , z ) • w , 0 ) so we would be required to solve, according to (8A.29) or (8A.31), the system dw = 0 dt
i.e.
w
constant
dg (t) dt (similarly for systems involving the averaged generators
Z).
This is a linear system in g with an inhomogeneous term z D2 G(F (f),z)·w. The solution can be written down in terms of t the evolution system for in the usual way.
*A
via Duhamel's formula
For systems of this type there are theorems
more direct analysis, obtaining refined results, is given by Dorroh-Marsden [11.
284
THE HOPF BIFURCATION AND ITS APPLICATIONS
available to guarantee that we have an evolution system and to study its properties. Kato [4].
Note, for example, Theorem 7.2 of
In this way, one can check the hypotheses of
(8A.29) or (8A.31) for
We would conclude, respectively, from
differentiability of
D x V
to
X x V
(under the stronger conditions of (8A.31», from
D
x
and V
to
x V.
y
(For holomorphic semigroups, smooth dependence on a parameter can be analyzed directly, as in Kato [3], p. 487). (8A.34)
Remark and Application.
In Aronson-Thames
[1] the following system is studied: 2 u xx - q u = u t '
2
p
°
ux(O,t)
(O,t) = x 0 -pq(f v) (O,t),
vx(l,t)
pq{l- (fou) (l,t)}
ux(l,t)
Here
2 v xx - q v = v t
and
2 u /(l+u).
V
q
in
}
(0,1)
t >
are positive parameters and
x R+
° f(u) =
This system is related to enzyme diffusion in
biological systems.
They show that the eigenvalue conditions
'of the Hopf theorem are met.
In Dorroh-Marsden [1] it is
shown, using methods described above, that the semiflow this system is smooth.
of
It follows at once that the Hopf
theorem is valid and hence proves the existence of stable periodic solutions for this system for supercritical values of the parameters. These equations are usually called the Glass-Kauffman equations; cf. Glass-Kauffman [1].
Recent work of the dis-
crete analogue has also been done by Hsu.
THE HOPF BIFURCATION AND ITS APPLICATIONS
285
SECTION 9 BIFURCATIONS IN FLUID DYNAMICS AND THE PROBLEM OF TURBULENCE This section shows how the results of Sections 8 and 8A can be used to establish the bifurcation theorem Navier-Stokes equations.
for the
Alternatively, although conceptually
harder (in our opinion) methods are described in Sections 9A, 9B and Sattinger [5] and Joseph and Sattinger [1]. The new proof of the bifurcation theorems using the center manifold theory as given in Section 8 allow one to deduce the
resul~s
very simply for the Navier-Stokes equations
with a minimum of technical difficulties.
This includes all
types of bifurcations, including the bifurcation to invariant tori as in Section 6 or directly as in Jost and Zehnder [1]. All we need to do is verify that the semiflow of the NavierStokes equations is smooth (in the sense of Section 8A)
i
the
rest is then automatic since the center manifold theorem immediately reduces us to the finite dimensional case (see
286
THE HOPF BIFURCATION AND ITS APPLICATION
Section 8 for details).
We note that already in Ruelle-
Takens [1] there is a simple proof of the now classical results of Velte [1] on stationary bifurcations in the flow between rotating cylinders from Couette flow to Taylor cells. The first part of this section therefore is devoted to proving that the semiflow of the Navier-Stokes equations is We use the technique of Dorroh-Marsden [1]
smooth.
(see Sec-
tion 8A) to do this. This guarantees then, that the same results as given in the finite dimensional case in earlier sections, including the stability calculations hold without change. The second part of the section briefly describes the Ruelle-Takens picture of turbulence.
This picture is still
conjectural, but seems to be gaining increased acceptance as time goes on, at least for describing certain types of turbulence.
The relationship with the global regularity (or
"all time") problem in fluid mechanics is briefly discussed.
Statement of the Smoothness Theorem. Before writing down the smoothness theorem, let us recall the equations we are dealing with.
For homogeneous in-
compressible viscous fluids, the classical Navier-Stokes equations are, as in Section 1, 3v at (NS)
1diV v
Here
+ (v·V)v -
=
v~v
=
-grad p + b t ,
b
t
external force
v = 0
0
on
3M
(or prescribed on M, possibly depending on parameter ~)
a
M is a compact Riemannian manifold with smooth boundary
THE HOPF BIFURCATION AND ITS APPLICATIONS
3M, usually an open set in
287
3 R .
v
The Euler equations are obtained by supposing and changing the boundary condition to
I
3v + (v.\i)v 3t
(E)
div v
-grad p + b
0
vi 13M:
t
0
1
vii 3M
The pressure
p(t,x)
is to be determined from the in-
compressibility condition in these equations. The Euler equations are a singular limit of the NavierStokes equations.
Taking the limit
v + 0
is the subject of much recent work.
is very subtle and
The sudden disappearance
of the highest order term and the associated sudden change in boundary conditions is the source of the difficulties and is the reason why boundary layer theory and turbulence theory are so difficult.
This point will be remarked on later.
Note that Euler equations are reversible
in the sense
that if we can solve them for all sets of initial data and for
t
~
0, then we can also solve them for
hecause if t
<
t < O.
is a solution, then so is
This is
w
t
-v_ t '
o. For
s > 0
an integer and
I < P < 00, let
wS'P
de-
note the Sobolev space of functions (or vector functions) on M
~.,hose
derivatives up to order
of describing
Ws,p
are in
s
is to complete the
the norm
II f II s,p
L O.::.a.::.s
IIDafll
L
p
COO
L ; another way p functions
f
in
288
THE HOPF BIFURCATION AND ITS APPLICATIONS
where
Daf
is the
a
th
total derivative of
f.
Details on
Sobolev spaces can be found in Friedman [1]. We point out that in the non-compact case one must deal seriously with the asymptotic conditions and many of the results we discuss are not known in that case (see Cantor [1], and McCracken [2] however). The following result is a special case of a general result proved in Morrey [1].
For a direct proof in this case,
see Bourguignon and Brezis [1]. (9.1) above. Then
Lemma.
Let X
be a
X
Hodge Decomposition):
Let
ws,p
M, s > 0, p > 1.
vector field on
M be as
has a unique decomposition as follows: X = Y + grad p
div Y
where and
=
0
yl
and
laM.
Y E ws,p
It is also true that
p E ws+l,p. Let
wS'P = {vector fields
div X
o and
P: ws,p
+
Ws,p
xl laM}. via
X on
Mix E wS,P,
By the Hodge theorem, there is a map
X ~ Y.
The problem of solving the Euler
equations now becomes that of finding
v
t
E ws+l,p
such that
dV t dt + P«vt·V)v t ) = 0
(E)
(plus initial data). functions is
Ws,p
If so
s > ~ the p_roduct of two ws,p p' (v·V)v is in ws,p if v E ws+l,p.
This kind of equation is thus an evolution equation on as in Section 8A. Let
wS'P
THE HOPF BIFURCATION AND ITS APPLICATIONS
ws,p
{vector fields v
°
~ls,P
s
If
written
J
p
, div v
=
on
°
and
is of class
Mlv v
289
=
on
°
=
this actually makes. sense ° (see Ladyzhenskaya [1]).
3M}.
and the space is
The Navier-Stokes equations can be written: find such that dV
(NS)
t
dt
-
vP~v
+ P(vt·V)v
t
again an evolution equation in
W~'p.
t
=
°
In the terminology of
Section 8A, the Banach space X here is wO,p J p and 2 y = W ,p. The bifurcation parameter is often 11 = l/v, the
°
°
Reynolds number. The case
p
~
2
is quite difficult and won't be dealt
with here, although it is very important.
If
p = 2
one
generally writes -s 2
llil '
(9.2)
Theorem.
,
etc.
The Navier-Stokes equations in dimen-
sion 2 or 3 define a smooth local semiflow on
H~ C HO
= J.
This semiflow satisfies conditions 8.1, 8.2, and the smoothness in 8.3 of Section 8, so the Hopf theorems apply. (The rest of the hypotheses in 8.3 and 8.4 depend on the particular problem at hand and must be verified by calculation. ) In other words, the technical difficulties related to the fact that we have partial rather than ordinary differential equations are automatically taken care of. (9.3)
Remarks.
1.
If the boundaries are moving and
290
THE HOPF BIFURCATION AND ITS APPLICATIONS
speed is part of the bifurcation parameter sult holds by a similar proof.
~,
the same re-
This occurs in, for instance
the Taylor problem. 2. authors.
The above theorem is implicit in the works of many For example, D. Henry has informed us that
obtained it in the context of Kato-Fujita. proved by many authors in dimension
2
he has
It has been
(e.g. Prodi).
The
first explicit demonstration we have seen is that of Iooss [3,5] •
p I 2
3.
For the case
see McCracken [2].
4.
This smoothness for the Navier-Stokes semiflow is
probably false for the Euler equations on Ebin-Marsden [1]). pative term.
HS
(see Kato [6],
Thus it depends crucially on the dissi-
However, miraculously, the flow of the Euler
equations is smooth if one uses Lagrangian coordinates (EbinMarsden [1]). We could use Lagrangian coordinates to prove our results for the Navier-Stokes equations as well, but it is simpler to use the present method. Before proving smoothness we need a local existence theorem.
Since this is readily available in the literature
(Ladyzhenskaya [1]), we shall just sketch the method from a different point of view.
(Cf. Soboleoskii [1].)
Local Existence Theory. The basic method one can use derives from the use of integral equations as in the Picard method for ordinary differential equations.
For partial differential equations this
has been exploited by Segal [1], Kato-Jujita [1], Iooss [3,6],
THE HOPF BIFURCATION AND ITS APPLICATIONS
Sattinger [2], etc.
(See Carroll [1]
291
for general background.)
The following result is a formulation of Weissler [11. (See Section 9A for a discussion of Iooss' setup.)
EO' El , E2
First some notation:
will be three Banach
I I· I 10' 1/· I 11' I I· I 12 , with
spaces with norms
with the inclusions dense and continuous. spaces may be equal.)
Let
etA
be a
E2 C E
l
CEO'
(Some of the
cO
linear semigroup
which restricts to a contraction semigroup on E: · 2 Assume, for t > 0, e tA : E ->- E is a bounded linear map l 2 and let its norm be denoted II (t). Our first assumption is: on
EO
T
AI)
For
T > 0
assume
Jo
ll(T)dT <
co
vPL'l, and we can
For the Navier-Stokes equations, A choose either (i) or norm
(E)
EO =
J
2
-2
H = HI and IE 2 O li- l / 2 = completion of
, lEI
EO = lEI =
11(-vpL'l)
-1/4
uil L
, 2
E2
=
-1 H
=
domain of
J
with the
(-vPL'l) 1/2.
The case (ii) is that of Kato-Fujita [1]. The fact that
AI) holds is due to the fact that
is a negative self adjoint operator on (Ladyzhenska~a
with domain
[1]); generates an analytic semigroup, so the
/ (See Yosida [1].) 2 is ~ C t. However we have the Sobolev inequality {derived most easily norm of
e tA
J
from
-2 H
O
to
J
from the general Sobolev-Nirenberg-Gagliardo inequality from Nirenberg [1]:
where
292
THE HOPF BIFURCATION AND ITS APPLICATIONS
1 P
(if
i
n
+ a(l _ ~) + (l-a) 1 r
m - j - n
q'
n
is an integer
r
> 1, a = 1
is not allowed)},
that
~(t)
so we can choose
<
C/It, so
~(t)
in case (ii) one finds
=
Al)
C/t 3 / 4
holds.
Similarly
(Kato-Fujita [1]).
As for the nonlinear terms, assume A2)
J : E + E is a semi-Lipschitz map 2 l t Lipschitz on bounded sets), locally uniformly in J t (¢)
continuous in
(t,¢).
We can suppose
(i.e., t
Jt(O)
with =
0
for
simplici ty. Consider the "formal" differential equation
~ dt
= A¢
+ J
t
(¢)
(9.1)
in integral equation form (see e.g., Segal [1]) (9.2) where
(Adding an inhomogeneous term causes no real
t > to.
difficulties. ) (9.4)
Theorem.
Under assumptions Al) and A2), the
equation (9.2) defines a unique local semiflow (i.e., evolution system)
~
W(t,t ): E + E O 2 2 tinuously in t. Proof.
E
2
(in the sense of Section 8A) with
locally uniformly Lipschitz, varying con-
The proof proceeds by the usual contraction
THE HOPF BIFURCATION AND ITS APPLICATIONS
293
mapping argument as for ordinary differential equations (see Pick
o
constant of
Jt
from
and pick
such that
Lang [1]) :
T (
r-
t
< a
0
< a,
let
to
lEI
1E 2
0 !l (T)dT) (
0
Now choose
¢ EB
cO
of
maps
a
[O,T]
M
-a O
< 1 -
sup K (T» TE[tO,T] a
and let
aO of
Ka (t) be the Lipschitz on B the a ball about a
(9.3)
be the complete metric space
to
with
and metric d(¢,'JI) Define 5': M
sup tE[tO,T]
->- M
2
by
(t-tO)A ¢ +
5'<1> (t) = e
j1(t)-'JI(t)j1
t
Jto e
(t-T)A JT(¢(T»dT
From the definitions and (9.3) we have two key estimates: first
r t
(remember
JT(O) = 0
!l(t-T)K (T)·adT a
(9.4 )
o
=
a
~ maps
here), which shows
M
to
M
and, in the same way .a
d(~¢, 5''JI) 2 (1 - ~)d(¢,'JI)
(9.5 )
a
which shows
~
is a contraction.
The result now follows easily. (9.5)
Exercises.
1.
Lipschitz constant given by W(t,t )· O
[]
Show that a/aO.
W(t,t O)
Verify that
has
E2
W(t,s)W(s,t O)
294
THE HOPF BIFURCATION AND ITS APPLICATIONS
2. and
If
¢t
is a maximal solution of (9.2) on
T < 00, show lim sup I I¢ / /2 t .... T t
=
[O,T)
00; i.e., verify the continua-
tion assumption B.2. 3.
Use the Sobolev inequalities to verify that
P«u·V)u) above.
satisfies the hypotheses in case (i)
For case (ii), see Kato-Fujita [1]. Next we want to see that we actually have a solution
of the differential equation. A3)
Make
Assume that the domain of
(9.6)
Theorem.
If
A
as an operator in
AI), A2) and A3) hold, then any
solution of (9.2) solves (9.1) as an evolution system in with domain W(t,t ) O X(¢)
D
= 1E 2 ;
lEo
(in the terminology of Section BA,
is the (time dependent) local flow of the operator
= A(¢)
(9.2) in
+ J
E
2
Proof.
(¢), mapping t are unique. Let
¢ E E
E
2
to
lEO).
Solutions of
¢(t) = W(t,t )¢ E 1E be 2 O the solution of (9.2), so taking to = a for simplicity, 2
and
It is easy to verify that
~{¢(t+h)-¢(t)} = ~{ehA¢(t)_¢(t)} - Jt(¢(t»
I
t+h
+! h
writing
t
{e(t+h-T)AJ
(¢(T»-Jt(¢(t»}dT T
(9.6 )
THE HOPF BIFURCATION AND ITS APPLICATIONS
e(t+h-T)AJT(¢(T»
295
- Jt(¢(t»
e(t+h-T)A[JT(¢(T»-Jt(¢(t»] + et+h-TJt(¢(t»-Jt~(t» one sees that the last term of (9.6) and hence in A(¢(t»
A.
a
h
as
~
a
in
12:
~2'
the
1
The first term of (9.6) tend to
~o.
- Jt(¢(t»
domain of
~
~O
in
as
h
~
a
¢(t) E
since
[]
Thus we can conclude that the Navier-Stokes equations define a local semiflow on
and that this semi flow ex(via the integral equation).
tends to a local semiflow on
Smoothness. (9.7) A4)
Theorem.
Let
~2 ~ ~l
Jt :
continuously on
Al), A2) and A3) hold and assume 00
is
C
with derivatives depending
t.
Then the semiflow defined by equations (9.1) on is a
COO
semiflow on
W(t,t )
E ; i.e., each 2
O
--
derivatives varying continuously in
t
~2
00
is
C
with
in the strong topology
(see Section 8). Proof. take
X
= EO
by hypothesis.
We verify the hypotheses of (8A.31). and
Y
Since
=
E , with 2 Z(¢1'¢2)
D
= Y.
Lipschitzness of 2 clear. Hence
W(t,t )
W(t,t ): E
O
O
2
~ ~2
Certainly a) holds
is the same type of operator
as considered above, 9.4 shows b) holds. E
Here we
c ' ) holds by the
proven in (9.4) and d) is
is Gateaux differentiable.
296
THE HOPF BIFURCATION AND ITS APPLICATIONS
The procedure can be iterated.
The same argument ap-
plies to the semiflow
on
iE
2
. 2 Hence
-+ iE
W is
cl
(see (8A.32», and by induction is
D
COO.
In the context of equation (9.1) the full power of the machinery in Section 8A; in particular the delicate results on time dependent evolution equations are not needed.
One can
in fact directly prove (9.7) by the same method as (8A.3l). However it seems desirable to derive these types of results from a unified point of view. (9.8) AS)
Problem. A
Show that for
generates an analytic semigroup. t >
of every power of t
for
ness in
t > O. v
Assume, in addition that
if
a
and
A
and that
(Hint. A
¢ EEl' ¢(t) ¢(t)
Use (8A.33».
is replaced by
vA
lies in the domain
is a
COO
function of
Also establish smooth(see remarks follow-
ing (8A. 33) ) • W(t,t ) O in the context of the Navier-Stokes
A more careful analysis actually shows that are
COO
maps on
HI
equation (i.e., without assuming A3».
See Weissler [1].
Thus all the requisite smoothness is established for the Navier-Stokes equations, so the proof of (9.2) and hence the bifurcation theorems for those equations is established.
THE HOPF BIFURCATION AND ITS APPLICATIONS
297
The Problem of Turbulence. We have already seen how bifurcations can lead from stable fixed points to stable periodic orbits and then to stable 2-tori. tori.
Similarly we can go on to higher dimensional
Ruelle and Takens [1] have argued that in this or
other situations, complicated ("strange") attractors can be expected and that this lies at the roots of the explanation of turbulence. The particular case where tori of increasing
dimen~
sion form, the model is a technical improvement over the idea of E. Hopf [4] wherein turbulence results from a l6ss of stability through successive branching.
It seems however
that strange attractors may form in other cases too, such as in the Lorenz equations (see Section 4B)
[Strictly speaking,
it has only a "strange" invariant set].
This is perfectly
consistent with the general Ruelle-Takens picture, as
are
the closely related "snap through" ideas of Joseph and Sattinger [1]. In the branching process, stable solutions become unstable, as the Reynolds number is increased. lence is supposed
~o
Hence turbu-
be a necessary consequence of the equa-
tions and in fact of the "generic case" and just represents a complicated solution.
For example in Couette flow as one in-
creases the angular velocity
~l
of the inner cylinder one
finds a shift from laminar flow to Taylor cells or related patterns at some bifurcation value of lence sets in.
~l.
Eventually turbu-
In this scheme, as has been realized for a
long time, one first looks for a stability theorem and for when stability fails (Hopf [2], Chandresekar [1], Lin [1] etc.).
298
THE HOPF BIFURCATION AND ITS APPLICATIONS
For example, if one stayed closed enough to laminar flow, one would expect the flow to remain approximately laminar.
Serrin
[2] has a theorem of this sort which we present as an illustration: (9.9)
Stability Theorem.
domain and suppose the flow
Let
D C R
3
be a bounded ClD
is prescribed on
(this
corresponds to having a moving boundary, as in Couette flow). Let
V
v
= maxI
Iv (x) xED t t>O
viscosity.
I I,
d
= diameter
of
Then if the Reynolds number
is universally
L
2
D
R
and
=
v
equal the
(Vd/v)
~
5.71,
stable among solutions of the Navier-
Stokes equations. Universally
L2
stable means that if
is any
other solution to the equations and with the same boundary 2 conditions, then the L norm (or energy) of v~ - v~ goes to zero as
t
+
O.
The proof is really very simple and we recommend reading Serrin [1,2] for the argument.
In fact one has local
stability in stronger topologies using Theorem 1.4 of Section 2A and the ideas of Section 8. Chandresekar [1], Serrin [2], and Velte [3] have analyzed criteria of this sort in some detail for Couette flow. As a special case, we recover something that we expect. then
on
Namely i f vV + 0 t
as
t
+
00
in
ClM L
2
is any solution for
v
+
0
norm, since the zero solution
is universally stable. A traditional definition (as in Hopf [2], Landau-
THE HOPF BIFURCATION AND ITS APPLICATIONS
299
Lifschitz [1]) says that turbulence develops when the vector field
can be described as
where
vt(wl' ••. ,w n )
f(twl,···,tw n )
is a quasi-periodic function, i.e., f
is periodic
in each coordinate, but the periods are not rationally related. For example, if the orbits of the
v
t
on the tori given by
the Hopf theorem can be described by spirals with irrationally related angles, then
vt
would such a flow.
Considering the above example a bit further, it should be clear there are many orbits that the
v
could follow t which are qualitatively like the quasi-periodic ones hut which fail themselves to be quasi-periodic.
In fact a small
neighborhood of a quasi-periodic function may fail to contain many other such functions.
One might
desire the functions
describing turbulence to contain mos.t functions and not only a sparse subset. ological
s~ace
More precisely, say a subset S
U
of a top-
is generic if it is a Baire set (i.e., the
countable intersection of open dense subsets).
It seems
reasonable to expect that the functions describing turbulence should be generic, since turbulence is a common phenomena. and the equations of flow are never exact.
Thus we would want a
theory of turbulence that would not be destroyed by adding on small perturbations to the equations of motion. The above sort of reasoning lead Ruelle-Takens [1] to point out that since quasi-periodic functions are not generic, it is unlikely they "really" describe turbulence.
In its
place, they propose the use of "strange attractors."
These
exhibit much of the qualitative behavior one would expect from "turbulent" solutions to the Navier-Stokes equations and they are stable under perturbations of the equations; i.e., are
300
THE HOPF BIFURCATION AND ITS APPLICATIONS
"structurally stable". For an example of a strange attractor, see Smale [1]. Usually strange attractors look like Cantor sets
x
mani-
folds, at least locally. Ruelle-Takens [1] have shown that if we define a strange attractor
A
to be an attractor which is neither a
closed orbit or a point, and disregarding non-generic possibilities such as a figure 8 then there are strange attractors on
T4
in the sense that a whole open neighborhood of
vector fields has a strange attractor as well. If the attracting set of the flow, in the space of vector fields which is generated by Navier-Stokes equations is strange, then a solution attracted to this set will clearly behave in a complicated, turhulent manner.
While the whole
set is stable, individual points in it are not.
Thus (see
Figure 9.1) an attracted orbit is constantly near unstable (nearly periodic) solutions and gets shifted about the attractor in an aimless manner.
Thus we have the following
reasonable definition of turbulence as proposed by RuelleTakens: nonstationary solution of the
Navier- Stokes equations
Z
strange attractor in the space of all velocity fields
Figure 9.1
THE HOPF BIFURCATION AND ITS APPLICATIONS
301
" •.• the motion of a fluid system is turbulent when this motion is described by an integral curve of a vector field
x~
which tends to a set
A, and
A
is neither empty
nor a fixed point nor a closed orbit." One way that turbulent motion can occur on one of the tori
Tk
that occurs in the Hopf bifurcation.
This takes
place after a finite number of successive bifurcations have occurred.
However as
S. Smale and C. Simon pointed out to
us, there may be an infinite number of other qualitative changes which occur during this onset of turbulence (such as stable and unstable manifolds intersecting in various ways etc.).
However, it seems that turbulence can occur in other
ways too.
For example, in Example 4B.9
(the Lorenz equa-
tions), the Hopf bifurcation is subcritical and the strange attractor may suddenly appear as
~
crosses the critical
value without an oscillation developing first.
See Section 12
for a description of the attractor which appears.
See also
McLaughlin and Martin [1,2], Guckenheimer and Yorke [1]
and
Lanford [2]. In summary then, this view of turbulence may be phrased as follows.
Our solutions for small
~
number in many fluid problems) are stahle and as
(=
Reynolds
~
increa-
ses, these solutions become unstable at certain critical values of
~
and the solution falls to a more complicated
stable solution; eventually, after a certain (finite) number of such bifurcations, the solution falls to a strange attractor (in the space of all time dependent solutions to the problem).
Such a solution, which is wandering close to a strange
attractor, is called turbulent.
302
THE HOPF BIFURCATION AND ITS APPLICATIONS
The fall to a strange attractor may occur after a Hopf bifurcation to an oscillatory solution and then to invariant tori, or may appear by some other mechanism, such as in the Lorenz equations as explained above ("snap through turbulence"). Leray [3] has argued that the Navier-Stokes equations might break down and the solutions fail to be smooth when turbulence ensues.
This idea was amplified when Hopf [3]
in 1950 proved global existence (in time) of weak solutions to the equations, but not uniqueness.
It was speculated that
turbulence occurs when strong changes to weak and uniqueness is lost.
However it is still unknown whether or not this
really can happen (cf. Ladyzhenskaya [1,2].) The Ruelle-Takens and Leray pictures are in conflict. Indeed, if strange attractors are the explanation, their attractiveness implies that solutions
r~main
smooth for all
t.
Indeed, we know from our work on the Hopf bifucation that near the stable closed orbit solutions are defined and remain smooth and unique for all also Sattinger [2]).
t
>
0
(see Section 8 and
This is already in the range of inter-
esting Reynolds numbers where global smoothness isnot implied by the classical estimates. It is known that in two dimensions the solutions of the Euler and Navier-Stokes equations are global in remain smooth.
t
and
In three dimensions it is unknown and is
called the "global regularity" or "all time" problem. Recent numberical evidence (see
~am
et. al.
[1])
suggests that the answer is negative for the Euler equations. Theoretical investigations, including analysis of the
THE HOPF BIFURCATION AND ITS APPLICATIONS
303
spectra have been inconclusive for the Navier-Stokes equations (see Marsden-Ebin-Fischer [1] and articles by Frisch and others in Temam et.al.
[1]).
We wish to make two points in the way of conjectures: 1.
In the Ruelle-Takens picture, global regularity
for all initial data is not an a priori necessity; the basins of the attractors will determine which solutions are regular and will guarantee regularity for turbulent solutions
(which
is what most people now believe is the case). 2.
Global regularity, if true in general, will pro-
bably never be proved by making estimates on the equations. One needs to examine in much more depth the attracting setsin the infinite dimensional dynamical system of tpe NavierStokes equations and to obtain the a priori estimates this way. Two Major Open Problems: (i)
identify a ptrange attractor in a specific flow
of the Navier-Stokes equation (e.g, pipe flow, flow behind a cylinder, etc.). (ii) tractor,
link up the ergodic theory on the strange at-
(Bowen-Ruelle [1]) with the statistical theory of
turbulence (the usual reference is Batchellor [1]; however, the theory is far from understood; some of Hopf's ideas [5] have been recently developed in work of Chorin, Foias and others) .
304
THE HOPF BIFURCATION AND ITS APPLICATIONS
SECTION 9A ON A PAPER OF G. IOOSS BY G. CHILDS This paper [3) proves the existence of the Hopf bifurcation to a periodic solution from a stationary solution in certain problems of fluid dynamics. lar to those already described.
The results are simi-
For instance, in the sub-
critical case, the periodic solution is shown to be unstable in the sense of Lyapunov when the real bifurcation parameter (Reynold's number) is less than the critical value where the bifurcation takes place; it is shown to be (exponentially) stable if
this value is greater than the critical value in
the supercritical case. Iooss, in contrast to the main body of these notes, makes use of a linear space approach for almost all of what he does.
Specifically, his periodic solution is a continuous
function to elements of a Sobolev space on a fundamental domain
~
in
~3.
extensively used.
However, the implicit function theorem is The three main theorems of the paper will
THE HOPF BIFURCATION AND ITS APPLICATIONS
305
be stated and the proofs will be briefly outlined to illustrate this method. First, we formulate a statement of the problem. I
be a closed interval of the real line.
neighborhood in LA
iC
of this interval.
Let
V(I)
For each
is holomorphic of type (A) in
{LA}
[3), p. 375). Finally, LA
Also, each
~
is
LA.
Assume
H.
The
(see Kato
V(I)
m-sectorial with vertex-y A.
has a compact resolvent in
common domain of the that
LA
be a
A E V (I) ,
is a closed, linear operator on a Hilbert space
family
Let
K
H.
Let
~
be the
is a Hilbert space such
eKe H with continuous inj ections and such that y t
VU E K,
II I
(t)ull~ .::. ke A (l+t- a ) IlulIK,
0
.2
a < 1
is the holomorphic semi-group generated by M be a continuous bilinear form:
~x
9
->-
K.
where Let
-LA.
Now we can
state the problem:
au at
+ LAU - M(U,U)
0
U E CO(O,OO;~) n Cl(O,OO;H)
(9A.l)
U(O) = U E ~, U(O) = U(T) = U(2T) = ••• for some T > 0
o
IOOS5
shows that the equation of perturbation (from a sta-
tionary solution) for some Navier-Stokes configurations is of the above form (see also Iooss [5 ) for more details).
In
order to find a solution of (9A.l), it is necessary to make some additional hypotheses.
Let
sO(A)
=
sup
{Re
~}.
Then:
sEer (-LA)
and R, a left hand neighborhood V (A ) c c such that s (A ) = 0, a right hand neighborhood v+ (I.e) o c A E V- (A ) - {A }~ s (A) < 0, A E: v+ (A ) - {A }=$- s (A) > O. c c 0 c c 0 (H.l)
3 A E
306
THE HOPF BIFURCATION AND ITS APPLICATIONS
(H.2)
The operator
values are simple. sl
spectrum
cr(L A)
admits as proper values pure c
and f ' Also, these proper O o A E V(A) there exist two analytic
~O = in
imaginary numbers
functions
LA
For
c
E cr(-L A) l separates into
and
f
such that
sl (A C ) = sO'
The
{-sl} U {-f } U cr(L ). This A l separation gives a decomposition of H into invariant sub-
spaces: 'v'UEH,
U=X+Y,
X=EAU,
Y=PAU;
(LA+sl)U (A) = 0, (L~+fl)Wl (A) = 0, l (U (A),W(O»H = 1. The eigenvectors l
EA = E(-s ) + E(-fl ), 1 (Ul(A),W l (A»H = 1, Ul(A),
UlTAf
are a basis for
)nL(n)u,. U1 (') LA U + ~L ('_' A A A n=l c c
EAH.
For
A E V(A
LAU
)'
U (0) + (A-A)U (1) + •.• ; c
WI (A) = W(O) + (A-Ac)w(l) + •.. ; sl (A) In particular, we have
C
So +
(A-A
)
S (1)
C
sell = _(L(l)u(O) ,W(O»H' LA
+••••
+ c
sOU ( 0 ) = 0, L~
+ f W(0) = 0, (U ( 0 ) W(0) ) = 1. c O ' H make the hypothesis: (H.3) Re sell > O.
Re(L(l)u(O) ,W(O»H"I-
o.
Now, we can
By (H.l) this implies
These hypotheses are just the standard Ones for
the existence of the Hopf bifurcation. the theorem we also need to know that
For the statement of Y
o
= YO
r
+ iyO.
1.
_ (M(O) [U (0) ,L~lM (0) (U (0) ,U (0»] + c M(O) [U(O) ,(LA +2in l)-lM(U(0) ,u(O»], W(O»H o c
where
M(O) (U,V) = M(U,V) + M(V,U). We now state Iooss' Theorem 2.
If the hypotheses (H.l), (H.2), (H.3), and
THE HOPF BIFURCATION AND ITS APPLICATIONS
Yo r
'!-
°
trivial YO
307
are satisfied, there is a bifurcation to a nonT-periodic solution of (9A.I) starting from
Ac '
If
> 0, the bifurcation takes place for r
if
Y < 0, it takes place for Or
%' E c
O
(_00 ,00; 8!)
A E V+(A )' whereas c A E V-(A ). The solution c
corresponding to a translation in alytic with respect to
£'=
Finally, %'(t)
t.
/i~, c
A - A and one can write C £2 %,(2) (t) + ••• where %,(i) (t) is is the phase of the An
X(t)
%'(t,£) = £
( i
(t) +
Here
oscillations. Denote
+ inCA)
iE(-~I) -
Then the equations for the
is an-
~(l)
T-periodic.
outline of the proof will be given. i;(A)
a
the period analytic with
respect to
Arg a
Arg
is unique with the exception of
X
iE(-~I)'
and
Y
parts of
%' coming
from (9A.I) are
~t)
= Bt(X+Y.xX+Y;A),
dt -
nNAX
= i;X + EAM(X+Y,X+Y)
X(O) = X(T),
where
= F(X,Y,;A),
(9A.2)
and
Bt(U,V;A) = ft I (t-T)PAM[U(T) ,V(T)]dT. _00 A
By substitut-
ing
X(t) = A(t)U (A) + A(t)U (A) in the right hand side of I I the equation for Y we obtain a right hand side which is analytic with respect to (AC'O,O) respect to
~
in Y
x
(A,X,Y)
in a neighborhood of
{cO(_00,+00;8!)}2.
is zero at
(Ac'O,O).
And the
derivative with
Then, denoting
X(O) (t)
308
THE HOPF BIFURCATION AND ITS APPLICATIONS
A(t)U(O) + A(t)U(O), and using the implicit function theorem, one has
(A-A lin (i,j) (X(O) , .•• ,xeD»~ c t
l:
yet) = nt(X;A)
i,j':'2 n
where
(i, j )
(. , ... ,. )
t ous of degree
with
x(O)
We must now solve
j.
X(T), X E CO(_oo,oo;~).
=
assumed for
2 1T t
lx, !( T
0
-
NA
e T
X (t)
X + X(t) - X (t) + X(t)
- .2.:II!N T A X(t)dt e
Decomposing the equation for can solve for X(t)
q
The following form is
X:
(
l
is a continuous functional, homogene-
t
X(t)
=
aU (A) +aU (X) l l
X(t)
according to (9A.3), one
using the implicit function theorem
(X,A,T) t
where
q(i,j,k) (X;t)
pect to
X.
(9A.3)
X(t)
E [O,T],
is homogeneous of degree
is now replaced with
other equation (the one for
X).
k
qt(X,A,T)
with resin the
Splitting the result into
real and imaginary parts one obtains
n =
with
f(O,A,T)
=
21T T
+ g
g(O,A,T)
(I a I 2 ,A ,T) =
O.
=
0
The development in Taylor
series about the point
has first term
It is in this way that a non-zero value of solution for
and
T
%'(t,E).
allows a
by the implicit function theorem.
This completes the determination of fore
Yo r
X(t), yet)
and there-
THE HOPF BIFURCATION AND ITS APPLICATIONS
309
Now that we have the periodic solution it is desired to exhibit its stability properties. tion
U (t)
and set
satisfies
U(t) =
(lU'
%'(t+ o , E) + u ' (t).
Then
U' (t)
AE(t+o)U' + M(U' ,U')
at U' (0)
u'
We consider a nearby solu-
%'( 0, E) E E!
U0 =
o
E C (0,00; E!)
n c'
(O,oo;H)
where A (t+o) E
-LA + M
(0)
[%'(t+O,E),·], A
Therefore, in order to study stability one examines the properties of the solution of the linearized equation:
< 00
v(O) The solution of this equation is: t
vet) = I
(t)V A
o
+
f
IA(t-T)M(O)[%'(t+O,E),V(T)]dT.
0
We denote this solution as
The stability properties will come from the properties of the spectrum of
GE(T,o), which plays the role of the poincare map.
We can now state Iooss' Theorem 3.
The hypotheses of Theorem 2 being satis-
fied and the operator above, the spectrum of
being defined by the equation is independent of
0 E
~.
It is constituted on the one hand, by two real simple proper
310
THE HOPF BIFURCATION AND ITS APPLICATIONS
1, which are
values in a neighborhood of
1 -
8~~(1)
(A-A ) + O(A-A )'
and
On the other hand, the remainder
C
C
1
of the spectrum is formed from a denumerable infinity of proper values of finite mUltiplicities, the only point of accumulation being
0, there remaining in the interior of a disk
of radius
independent of
S < 1
E E ~(O) .
The following is then a direct result of Lemma 5 of Judovich [10]. Corollary.
The hypotheses of Theorem 2 being satis-
fied, if further
YO
<
0, the bifurcation takes place for
r
A E 0/- (A) c
and the secondary solution is unstable in the
sense of Lyapunov. Now the proof of Theorem 3 is outlined. is compact in~.
GE (T, <5) I,et that
0
E spectrum of oV
such that
G (T,8)V. E <5
2
To establish
Hence, its spectrum is discrete.
G (T, <5). E Then, i f
~\T
=
= G (nT-<5 ,<5)V
E
n E N
with
Hence, spectrum
such
(G (T,<5)) C E
Similarly, the reverse containment holds.
1 E spectrum of
a%' GE(T,O) ~(O,E)
V "I- 0
Then there exists
nT, OW = GE(T,O)W.
spectrum (GE(T,O)).
The operator
a%' ~(O,E).
It can be shown that
1
G (T,<5), it suffices to show E
Note that
I A (TO) = lim GE(T,O). c E+O
2)
is a semi-simple (multiplicity
(TO). with all proper values sn of Ac correspond a finite number of proper values of
proper value of
I
If one removes from the spectrum of ing such that
LA
,
the proper values
c
si
are remain-
Re si > ~ > 0 => Log Isnl < -~TO < O.
the remainder of the spectrum of
I A (TO) c
other than
Thus, 1
is
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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:
THE HOPF BIFURCATION AND ITS APPLICATIONS
/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).
THE HOPF BIFURCATION AND ITS APPLICATIONS
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 HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
-
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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.
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
Nt + l
(2.2c)
l+e
Nt + l
= {
N < 1 t
ANt'
(2.2d)
AN 1 - b Nt > 1 t '
Each of these models has the origin as a fixed point and have the "l-hump" characteristic of Figure 10.2, Le., single critical point less than the positive fixed point.
Beyond
this, however, they are largely empirical, generated ad hoc by regression of one generation on the next.
*
By and large,
however, such simple-minded models have been surprisingly effective in reproducing the generation-to-generation variations in population levels. [ 1] 1 Varley, et. al., op.
*Note
that, for
r«
(Auslander, Oster, Huffaker, cit.)
1, equation (2.2a) is (2.2a*)
which is just the forward difference equation corresponding to the familiar logistic equation for population growth:
~~ =
rN(l -
~).
Thus, r in
(2.2) can be interpreted as the
net generation-to-generation reproductive rate. Although Equation (2.2a) can exhibit bifurcations, (2.2a*) cannot. (i.e., for large r, (2.2a) has a critical point, so it cannot be a finite version of (2.2a*».
332
THE HOPF BIFURCATION AND ITS APPLICATIONS
(2.2)
Whether or not the population, as modelled by
any of equations (2.2), settles down to a steady generationto-generation level depends on the stability of the fixed point, F(N)
N, N > 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
THE HOPF BIFURCATION AND ITS APPLICATIONS
(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
THE HOPF BIFURCATION AND ITS APPLICATIONS
1--: d
d
c
a
----~---............)
1 _-_-_-_--
Figure lO.4a Alternatively, if the population exhibits a "3-point cycle" wherein the population rises 2 years in succession and then crashes past the original level, then non-periodic motion will ensue,
(c.f. Figure lO.4b):
a
b
c
d
Figure lO.4b The consequences of this phenomenon for ecological modelling are profound.
We can have confidence in a model only if it
is subject to experimental validation.
If a series of yearly
censes are collected of some population, and they appear
THE HOPF BIFURCATION AND ITS APPLICATIONS
335
chaotic, exhibiting no perceivable regularities, then we can conclude one of three things:
(a) the system is truly sto-
chastic--dominated by random influences;
(b) experimental
error is of such magnitude that all regularities are obscured;
(c) a very simple deterministic mechanism is operat-
ing, but is obscured by the phenomenon described above.
As
an extreme case, the orbit generated by the simple map shown in Figure 10.5 is indistinguishable from a sequence of Bernoulli trials!
o
1/2 '---v---J---'- _ T - - - J It( 0 11
11
Figure 10.5 For systems of 2 interacting populations (e.g., predatorprey, parasite-host, etc.) the situation is even more delicate and little is known about the transition to aperiodic motion.
However, May [2] has simulated some 2-population
difference equation models.
He found that the population
trajectories exhibited chaotic behavior for quite reasonable parameter ranges. 3.
Populations with Overlapping Generations. (3.1)
If a population breeds continuously, so that the
336
THE HOPF BIFURCATION AND ITS APPLICATIONS
generations overlap, then the appropriate model is an ordinary differential equation of the form
dN dt
Nf(N)
N [ (per caPita) birth rate
per caPita) ( death rate
J
(3.1)
The number of such models in the literature is legion, and we shall comment only briefly on certain aspects pertaining to their bifurcation behavior. (3.2)
A recurrent theme in ecology is the phenomenon
of population oscillations.
The earliest prototype was the
predator-prey equations of Volterra and Lotka (see, for example, May,
[1]):
(3.2)
The solutions to (3.2) are indeed periodic (Hirsch and Smale, [1]), but are neutrally stable, the amplitude of the oscillations depending on the initial conditions. Recently, May [1,2] has shown that virtually all of the models for predator-prey systems possess either a stable equilibrium or a stable limit cycle (in the first quadrant). H~s
demonstration hinges on showing that most models fall
within the purview of a theorem by Kolmogorov [1], which is essentially an application of the Poincare-Bendixson Theorem to systems of the form (1).
[Essentially, any population
model such that 1) there is a single unstable singularity in the first quadrant and 2) the axes are invariant (e.g.,
THE HOPF BIFURCATION AND ITS APPLICATIONS
N=
337
~!(~)) will have a limit cycle since large radius orbits
must be directed inward due to the finite population limitation that must be imposed on any realistic model.] A typical predator-prey system which exhibits limit cycle behavior is:
(Rosenzweig,
[1]; May,
[1]):
-cN - kN (1-e 2 -fN l S(l-e )]
l
) (3.3)
The interpretation is that the prey, N , in the absence of l the predator, grows logistically and the predator, in the absence of prey, dies out exponentially.
The second term in
the first equation models a predator population whose capacity to capture prey gradually satiates. The equilibrium point of equation (3.3) can be computed explicitly: N 1
b -l/f
S)
In(l +
N = rN (1 2 l
-
N -l)/k[l K
(1+
b clf
!)
].
Then, computing the Jacobian at the equilibrium it is easy to check that the signs of the determinant and trace depend on the magnitude of the parameters
{r,K,k,c,b,S,f}
= n.
Thus, there exists a family of curves, parametrized by some combination of members of the RHP (c.f. Figure 10.6).
~,
carrying the eigenvalues into Since large radius orbits move
inward, the limit cycles are indeed generated by the Hopf mechanism.
We also note that, since the eigenvalue trajec-
tories are controlled by more than one parameter, the limit
338
THE HOPF BIFURCATION AND ITS APPLICATIONS
cycle can appear at finite, rather than zero amplitude.
Tr
sta ble
Figure 10.6 (3.3)
Predator-prey type equations have been used by
Bell [1] to model "populations" of antibody and antigen in the immune response.
Using Friedrichs'
[1] version of the
bifurcation theorem, Pimbly [1] demonstrated that Bell's equations exhibit periodic behavior which can be interpreted biologically in terms of the mechanism controlling infection. (3.4)
For systems of 3 or more species the possibil-
ityof higher order bifurcations raises the same operational problems as we encountered for difference equation models. Successive bifurcations beyond the first occur when the eigenvalues of the Poincare map passes outside the unit circle (Hirsch and Smale,
[1]), thus higher periods,
(and
aperiodic behavior) of this difference equation will produce quite chaotic-looking population records.
This phenomenon
is quite well known in Hamiltonian systems (Arnold and Avez, [1]).
Since it is generally much more difficult to obtain a
reliable experimental record for population systems, the
THE HOPF BIFURCATION AND ITS APPLICATIONS
339
existence of such "strange attractors" would imply that the model may well not be experimentally verifiable. Thus we find that bifurcation in model equations are a mixed blessing, explaining some phenomena and obscuring others.
4.
Age-Structured Populations. (4.1)
By using ordinary differential
and difference
equation models we have taken a naive view of population dynamics by assuming that the state of the population is specified by total population number alone.
A moments reflec-
tion shows that, in order to predict the growth of a population, account must be taken of internal variables such as age and size distributions.
Clearly a thousand individuals past
breeding age, or all of one sex do not constitute a viable population.
In this section we shall illustrate some of the
consequences of including the population age structure as a state variable. (4.2)
The equation of motion for an age-structured
population is easy to write down. age density function, i.e., N(t) tion.
Let
=
rt(a,t) = population
foon(a,t)da
a
=
total popula-
Then a conservation equation can be written for Cln + div J
at
n
loss by deaths.
, through age-time n 1, we can write
Since the flux of individuals, I vn, where
v
=
da dt
Cln Cln Clt + Cla
n: (4.1)
is just
(4.2)
340
where
THE HOPF BIFURCATION AND ITS APPLICATIONS
~(a,t,')
is the age-specific deathrate.
The special
feature of the equation is the boundary condition giving the birthrate, n(O,t) = where
b(a,t,·)
foo
o
b(a,t,')n(a,t)da
(4.3)
is the age-specific birthrate.
As we have
indicated, the birth and death rates are functions of other variables as well.
For example, population density frequently
affects mortality and fecundity, so that ~(a,t,N)
(4.4a)
b
b(a,t,N)
(4.4b)
N
f
(4.5)
where
OOo
is the total population.
n(a,t,N)da.
With appropriate smoothness and
boundedness assumptions, Gurtin and MacCamy [1] proved existence and uniqueness for the system (4.2) -
(4.5).
We note
that in engineering terms the age equations constitute a "distributed parameter positive feedback system."
This
easily implies that, as birthrates increase and/or deathrates decrease, the'system will pass from a stable to an unstable regime.
In the next subsection we examine the bifurcation
behavior of a single population feeding off a single resource.
Then we model a host-parasite system by coupling
two age systems together.
In both cases the existence of
bifurcations must be inferred from
qualitative and numerical
arguments, since direct verification is unavailable.
Never-
theless, we shall gain significant insights into some interesting ecological phenomena via our models.
THE HOPF BIFURCATION AND ITS APPLICATIONS
(4.3)
341
In one of the best known experiments in eco-
logy, the Australian entomologist A. J. Nicholson maintained a population of sheep blowflies on a diet of chopped liver and sugar for several years.
In Figure 10.7 we have repro-
duced a portion of his data.
The biological explanation for
the violent oscillations is straightforward:
Nicholson de-
liberately kept the food supply to the adult flies below the level required to sustain a population the size of one of the peaks.
At moderate population levels competition prevents
any individual from obtaining enough protein.
Protein starva-
tion, in turn, reduces the fecundity of each adult fly so that the next generation is much smaller.
For this smaller
generation the food supply is adequate and the fecundity rebounds to its maximum level. A model for this situation must include some accounting for the nutritional state of the adult flies since this governs the rate of egg laying. a variable,
~,
which measures the nutritional state (e.g.,
mass, "health") equation in
Accordingly, we shall define
(Oster and Auslander,
(t,a,~)
dn
at +
[1]).
A conservation
coordinates takes the form
dn aa
d ) + ~(gn = -1Jn
(4.5)
where g (t,a,Cf) is the growthrate of f(t).
~,
(4.6)
which depends on the food supply,
The birthrate is then
n(O,t,~)
=
ff da
d~'
nb(t,a,C,S).
The equation for the food abundance is
(4.7)
w
"'" '" 8
7 '"'3
::r:
6 ,-
t'J
5 ,-
::r: o
4
'ti ":l
3 ,-
tD
~
~ 2 ~I ~ a 0
:I:
H
":l
c:::
c
~ ~
100
H
~
I-
Z !Jl
s;
8
o
7
H
~ 6 .-
~
I..L.
'"'3
(f)
;:;
5
0 4 ,Cl: l1J C!l
~
'ti
I:"' H
3
()
~ o
2 ,-
H
zi Q
Z
(f)
a
100
400 DURATION
Figure 10.7.
Top:
500
600
700
EXPERIMENT (DAYS)
periodically forced system
Bottom:
constant food supply
343
THE HOPF BIFURCATION AND ITS APPLICATIONS
~~ where and
u(t) C(·)
= u(t)
(4.8)
- C(t,a,n)
is the rate food is supplied to the population is the consumption rate by the adult flies.
sonable empirical forms for the functions shown in Figure 10.8.
b(·),
~(.),
ReaCare
A careful numerical simulation of this
:--------- a
(a)
fecundity
(b)
mortality
C(f)
f (food available) (c) Figure 10.8.
consumption Constitutive Relations
model shows reasonable agreement with experiment, Figure 10.9 (Oster and Auslander, [1]).
However, we would like to see
how the model generates these oscillations; naturally, the mechanism of bifurcations suggests itself.
In order to exa-
mine this mechanism let us consider the simpler populationresource system
344
THE HOPF BIFURCATION AND ITS APPLICATIONS
- - simulation -- --- Nicholson data 10,0 oOr--------------------------......., I I'
....
<Jl ::;)
"o 5,000 ..... o o
,I
,I
:',
, I
,, \
, ,
I
I
Z
time, days
Figure 10.9 Comparison of simulated results with experimental data.
an an = -"n ar+aa: .. n(O,t) dR crt
J
(4.9a)
b(a,R)n da
F(R,n) •
(4.9b) (4.9c)
Regardless of the form of the functions the equations, linearized about an equilibrium state, will take the form (Oster and Takahashi, [1]) ax + ax 'dE" aa
(4.10a) (X+Y
x(O,t) dR dt
=
gR(t) + b
Ja
x(t,a)da
-AR(t) - By(t) + Cu(t)
(4.10b) (4.10c)
345
THE HOPF BIFURCATION AND ITS APPLICATIONS
where yet) =
g, A, B, C,
"" Jo x
da.
~
D
and
are linearization constants and
One way to obtain the response of system
(4.10) to various food supply schedules, u(t), is to compute the "transfer function"
(Takahashi, Rabins, Auslander, [1]).
That is, equation (4.l0a) can be written as L
is a linear operator.
ax dt
= Lx
where
Note that the initial conditions
for the linearized system are zero, thus taking the Laplace Transform with respect to time is equivalent to the eigenvalue equation
Lx(a,s) = sx(a,s), where
sEC.
Therefore,
we can compute the spectrum of the system (4.10) as follows. Taking the Laplace Transform of system (4.10) we obtain e-(s+i:i")aG(S)R(S)
X(a,s)
__--:C=-
R(s)
(4.11)
U(s)
(4.12)
S+A+ ~(s) s+~
where
X(a,s), R(s)
and
U(s)
are the transformed variables
and G(s)
gil _ ~{e-(s+i:i")a _ e-(s+~) (a+ y )} s+i1
Thus, the response, or "output", X(s,a)
(4.13)
can be expressed in
terms of the input, U(s), as X(a,s) where
g(s)
g(s)U(s)
e-(s+iJ)aG(s)
(4.14)
is the "transfer function".
The
response of the total population to food supply can be written yes)
1
s+i1
G(s)R(s)
(4.15)
346
THE HOPF BIFURCATION AND ITS APPLIC.1I.TIONS
where
Y(s) = fooX(a,S)da.
The characteristic equation for
a
the system is given by the denominator of equation (4.1): Bg
1 +
(s+A) {s+il - be
-(s+~)a
o.
(4.16)
-(s+V)y (l-e
}
The roots of (4.16) yield the system eigenvalues, as can be verified directly by substitution into (4.10). Bg
The product
can be interpreted as measuring (effect of population
on food level) x (effect of food level on birthrate).
Thus,
by varying the "gain", Bg, each of the infinite number of eigenvalues traces out a path on the complex plane.
Clearly
the system is asymptotically stable for
u(t)
population will eventually starve.
u(t) > 0, the dyna-
For
mics are controlled by, the parameter some idea of the effect of varying values by
(Bg).
(Bg)
a.
We can get
on the system eigen-
examining the special case of
i. e., all births occur at age
0, for the
b(a) = b*O(a-a),
The characteristic equa-
tion then becomes: Bg
1 +
O.
(4.17)
(s+A) (s+'il) (l-b*e- (s+V) a) At
Bg
=
0,
(4.17) has roots as
= -A,
s
s
= -~,
and those
satisfying e
Setting
b*.
a+ iw, the roots of
(4.18)
(4.18) are seen to be at
TI
2n , n = 1,2, ••• , where s = p is the real root a of (4.18). In Figure 10.10 we sketch the "root locus"
a
=
s =
(s+i1") a
p, w
(Takahashi, Rabins and Auslander, as
(Bg)
is varied from
a
to
[1]) for equation (4.17) 00
(Oster and Takahashi, [1]).
THE HOPF BIFURCATION AND ITS APPLICATIONS
347
-A
Figure 10.10 The branches start at Bg
+
Bg = 0
00, approach the asymptotes
(denoted by
w
= ±
xi and, as
2~n, n
=
0,1, ...
What is apparent from Figure 10.9 is that there is some range of parameter values for which the linearized model passes from stability to instability. parameter
(Bg)
That is, as the interaction
is varied in the appropriate range the lead-
ing pair of eigenvalues cross the imaginary axis.
At this
point the linearized system begins to exhibit small amplitude oscillations, which grow as the parameter is further increased.
(Of course, sooner or later other root pairs
cross to the RHP; these are associated with secondary frequencies, and will not concern us here.) Simulation studies of the model system (4.5 - 4.8) indicate that the oscillations do not grow from zero amplitude, but bifurcate to finite amplitude oscillations.
This
suggests that the bifurcation is controlled by 2 parameters rather than 1.
(c.f. Takens [1]) as shown in Figure 10.11.
348
THE HOPF BIFURCATION AND ITS APPLICATIONS
--
circular affractor 7/1
circular source
point
attractor
Figure 10.11 (4.4)
We can
use the age model to answer a puzzling
question in the ecological literature.
Over a period of
several years Professor C. B. Huffaker maintained an experimental ecosystem containing a parasitic wasp which lays its eggs in the larvae of a certain moth.
He noticed that, very
quickly after initiation, the populations settled into stable oscillations.
These oscillations were characterized by age
structures which were practically discrete generations.
Con-
ventional predator-prey models do not suffice to explain these oscillations since phase plane trajectories cross--the explanation lies in the age structure dynamics.
We can couple
two conservation equations like (4.2) by an age specific interaction that models the searching behavior of the parasite.
The resulting model looks like:
ap
at
+
.£.E.
aa
(4.19)
THE HOPF BIFURCATION AND ITS APPLICATIONS
349
(4.20)
p (0 ,t)
(4.21) ah + Y h bh(t,a',Hl(t-T»h(a',t)da' a
h(O,t) =
f
where
(4.22)
h
6+ 0
f6
HO(t)
Hl(t)
H(t)
=
h(a' ,t)da' = no. host larvae
(4.23 )
=
(4.24)
a +y
fahh
h h(a' ,t)da'
rhea' ,t)da'
o
fa p +Yp
no. host adults
(4.25)
total no. hosts
pta' ,t)da'
=
no. parasite adults.
(4.26)
ap
The form of the interaction between the populations can be derived by assuming a random search by each parasite for host larvae and employing a mean "area of discovery,"
A, for each.
If the hosts are distributed randomly (Poisson) in a plane, the inter-arrival times are distributed exponentially. the interaction takes the form:
Thus
(Auslander, Oster, Huffaker,
op. cit.) [no. hosts parasitized] (a) = bh (a) (l-e
-A(S)P l
)
(4.27)
This is added to the natural mortality (assumed constant) to obtain the total host mortality.
As indicated in equation
(4.22) the host birthrate includes a delayed effect that depends on the nutritional history of the host.
This is be-
cause fecundity is a function of adult size, which depends
350
THE HOPF BIFURCATION AND ITS APPLICATIONS
on the available food. The above model was simulated numerically using Huffaker's data (Auslander, Oster and Huffaker, op. cit.), and some of the results are shown in Figure 10.12.
First of
all, as the strength of the interaction is increased (e.g. by increasing the area of discovery, A) the
system undergoes a
transition from a state wherein all age classes are represented in both populations to one wherein only a few age class~s
are represented.
That is, the age profiles of both
species condense into "travelling waves," which propagate through the age structure in such a fashion that in a "stroboscopic photograph," the generations appear virtually discrete.
The phase relationship of the population waves in
each population determine the extent to which the populations can coexist.
If the parameters are adjusted so that all age
classes are represented, then the populations do not coexist: the parasite eliminates the host and then dies out itself. Following the same procedure outlined for the single population model, we can linearize equations
(4.19) -
(4.26),
Laplace transform and examine the roots of the characteristic equation as the coupling parameter is increased.
Clearly,
at zero coupling the parasite system is stable about the zero solution while the host population approaches a stable age distribution.
Simulation indicates that a stationary age
profile also exists with all age classes represented in both populations (continuous generations).
Furthermore, at suf-
ficiently high coupling strength the system is stable at zero. Thus, a root locus study, which reveals a leading root pair crossing the imaginary axis as the coupling is increased,
THE HOPF BIFURCATION AND ITS APPLICATIONS
351
c
o
....o :J
a. o a. 0>
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
explaining not only cyclic regularities in population dynamics, but perhaps some of the irregularities as well.
353
354
THE HOPF BIFURCATION AND ITS APPLICATIONS
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.
THE HOPF BIFURCATION AND ITS APPLICATIONS
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.
THE HOPF BIFURCATION AND ITS APPLICATIONS
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.
THE HOPF BIFURCATION AND ITS APPLICATIONS
(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 =
THE HOPF BIFURCATION AND ITS APPLICATIONS
(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,
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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,
..
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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'
~ <
and
1 4 y = ( y , •.• ,y)
has real positive eigenvalues say
This can be checked easily since
matrix.
iT
~l'
resembles a
Also there is a linear change of coordinates
which changes the matrix
into
~
~.
The coordinates of chemical concentrations are those in which ~
has the diagonal form.
centrations.
However the
Thus the y
do not represent con-
coordinates are much easier to
work with. We give now and a cubic map
53
5
as the sum of a linear map in terms of the
4 51: R
y-coordinates.
+
4 R
364
THE HOPF BIFURCATION AND ITS APPLICATIONS
Thus let
with
S = Sl + S3
Sl
l+a
1
ya
0
-1
a
0
ya
-ya
0
2a
0
-ya
0
2a
0
1 J
S3 (y) = (_ (yl)3 ,0,0,0). One notes now that duct
<Sy,y> <
the origin of
° R
is odd and since the inner pro-
y f 0
if
4
S
(an easy check) it follows that
is a global attractor for
The next step is to check that
S -
S. takes the form
~
o 4a
o Thus the
1
(
o
0) 4a
2
(y ,y )
2-dimensional sUbspace is a con4 tracting invariant subspace in R for S - ~, since a < 0; on this subspace, the equations for dy'/dt=y
2
-((y')
3
1
-y),
S -
~
take the form
2
dy/dt
1
-yo
This is Van der Pol's equation (see Hirsch-Smale [1]), which we know is a global oscillator. Thus
S -
~
is a global oscillator on
4
R.
The next step is to show that (S2) is true. be proved along the following lines. 4 The vector field X on R x R4
given by
This can
THE HOPF BIFURCATION AND ITS APPLICATIONS
can be written in the form Y (z) E ~~. 2
That
X
Y (z) + Y (z) l 2
where
365
Yl(z) E 6,
Then
points toward
Proof of Lemma.
~~
follows from the lemma.
Write
We already know
But
and that is
<
0
since, for any real numbers
a
and
b,
(a 3 +b 3 ) (a+b) > O. Finally one "straightens out" the flow 'of some large ball.
One uses a smooth function
o
in a neighborhood of
~
¢
~
1, ¢
large enough
~
0
S
¢: R+
0, and
¢(r)
outside R+,
+
~
1
for
r; then
Q(z) ~ (l-¢(I/lzll))s(z) - ¢(llz) I)z. It can be shown that
Q
satisfies (1),
suitable constants in the definition of (11.5)
(2),
(3) with
¢.
We end this note with some discussion of our
results. Various forms of Turing's equations, or reaction-
366
THE HOPF BIFURCATION AND ITS APPLICATIONS
diffusion equations have appeared in one form or another in many works and fields.
However, any sort of systematic under-
standing or analysis seems far away.
Before one can expect
any general understanding, many examples will have to be thought through, both on the mathematical side and on the experimental side.
This is one reason why I have worked out
this model. Moreover, the work here poses a sharp problem, namely to "axiomatize" the properties necessary to bring about oscillation via diffusion. ties does the pair
In the 2-cell case, just what proper(R,~)
need to possess (where
R
is
"dead") to make the Turing interacting system oscillate?
In
the many-cell case, how does the topology contribute? We have not hesitated to make simplifying assumptions here, because we were not making an analysis, but producing an example.
Because of the structural stability properties
of this example, one can use it to obtain more complicated examples, e.g., with as many cells (more than one) as one wants, as many chemicals (more than three) as one wants and complicated diffusion matrices.
But it is more difficult to
reduce the number of chemicals to two or even three.
Also
it is a problem to construct a model with three cells and two or three chemicals. There is a paradoxical aspect to the example.
One has
two dead (mathematically dead) cells interacting by a diffusion process which has a tendency in itself to equalize the concentrations.
Yet in interaction, a state continues to
pulse indefinitely.
THE HOPF BIFURCATION AND ITS APPLICATIONS
367
Several chemists have pointed out to me that interpreting the reaction
R
to be an "open system" makes the
model more acceptable. There is quite a history of numerical work on related systems which I will not try to cover here. Finally, there is a partial differential equation analogue to the version of Turing's equations studied here. This can be found in Turing's paper [1].
In this P.D.E. con-
text the recent work of L. Howard and N. Kopell on the Zhabotinsky oscillation bears strong analogies to the present work.
368
THE HOPF BIFURCATION AND ITS APPLICATIONS
SECTION 12 A STRANGE, STRANGE ATTRACTOR BY JOHN GUCKENHEIMER Examples have been given by Abraham-Smale [1], Shub [1], and Newhouse [2] of diffeomorphisms on a compact
mani~
fold which are not in the closure of diffeomorphisms satisfying Smale's Axiom A or in the closure of the set of stable diffeomorphisms (Smale [1]).
Q-
The suspension construc-
tion (Smale [1]) allows one to give analogous examples for vector fields on compact manifolds. This note gives another example of a vector field on a compact manifold which does not lie in the closure of Q-stable or Axiom A vector fields.
The interest of this
example is that the violation of Axiom A' occurs differently than in the examples previously given.
This example has ad-
ditional instability properties not verified for the previous Research partially supported by the National Science Foundation.
THE HOPF BIFURCATION AND ITS APPLICATIONS
examples.
A vector field
369
X
is said to be topologically Q l stable if nearby vector fields (in the c topology on the
space of vector fields) have nonwandering sets homeomorphic to the nonwandering set of cally
stable.
Q
X.
Our example is not topologi-
Moreover, it provides another negative ans-
wer to the following question about dynamical systems:
is it
generically true that the singularities of a vector field are isolated in its nonwandering set?
Previous examples of
Newhouse have nonisolated singularities in non-attractive parts of the nonwandering set. The example is based upon numerical studies of a systern of differential equations introduced by Lorenz [IJ.
The
system studied by Lorenz seems to have the dynamical behavior of our example, but we do not attempt to make the estimates I would like to acknow-
necessary to prove this statement.
ledge the assistance of Alan Perelson in doing the numerical work which underlies this note and conversations with R. Bowen, C. Pugh, S. Smale, and J. Yorke.
Finally, we mention
the explicit equations of Lorenz which display such marvelous dynamics (see Example 4B.8, p. 141): -lOx + lOy, We define a
Y COO
-xz + 28x - y, vector field
X
z
xy - 8/3 z.
in a bounded region
Inside the region there will be a compact invariant set
A
which is an attractor in the sense that
A
has a
fundamental system of neighborhoods, each of which is forward invariant under the flow of mensional. ordinates
X.
The set
To describe the construction of (x,y,z)
A
is two di-
X, we use co-
370
THE HOPF BIFURCATION AND ITS APPLICATIONS
7he vector field The first, p
X
is to have three singular points.
(0,0,0), is a saddle with a two dimensional s stable manifold W (p). The rectangle {(x,y,z) Ix = 0, s -1 '::y < 1, -< z < l} is to be contained in W (p). The =
°
stable eigenvectors of
()
of large absolute value and absolute value. segment from
p
at
X
()
are
with an eigenvalue of small
az
The unstable manifold
(~l,O,O)
to
with an eigenvalue
8Y
(1,0,0)
of intermediate absolute value.
WU(p)
contains the
and has an eigenvalue
Other conditions on
WU(p)
are imposed below. The other two singular points of ±1/2, 1).
=
q±
(±l,
These are saddle points with one dimensional stable
manifolds
The segments from
(±l, 1, 1) values of
X
at
q±
()
and
(±l, -1, 1)
to
The negative eigen-
are contained in
ing eigenvalues of by
are
X
have large absolute values. q±
The remain-
are complex with eigenspaces spanned
The real parts of these eigenvalues are
~.
small. Consider the square -1
~
8
is not defined when
Y
~
1, z
=
l}
R
=
{(x,y,z)
I
-1 ~ x ~ 1,
and its Poincare return map X
is
±l
or
°
8.
The map
since these points
lie in the stable manifold of one of the singular points. orbits in x
=
°
fined.
R
for
X = ±l
never return. Let
R+
R
be the set
be
8
never leave
R
R n {(x,y,z)
R n { (x,y ,z)
-1 < x <
restricted to
f±, g± and a number
while those for
At all other points of
be the set
R±. CJ,
> 1
~le
oL
The
R,G
is de-
x °Define <
<
l} and 8±
to
assume that there are functions
with the properties that
THE HOPF BIFURCATION AND ITS APPLICATIONS
The numbers
lim f± (x), denoted P± ' are assumed to have the x...0 The P+ < 0, p > 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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
There is a neighborhood
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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-
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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
THE HOPF BIFURCATION AND ITS APPLICATIONS
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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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