Dynamical Systems: Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Bressanone (Bolzano), Italy, June 19–27, 1978 [1st ed.] 978-0-8176-3024-9;978-1-4899-3743-8

Table of contents :
Front Matter ....Pages 1-3
Bifurcations of Dynamical Systems (John Guckenheimer)....Pages 5-123
Horseshoes for continuous mappings of an interval (M. Misiurewicz)....Pages 125-135
Various Aspects of Integrable Hamiltonian Systems (J. Moser)....Pages 137-195
Hopf Bifurcation for Invariant Tori (A. Chenciner)....Pages 197-207
Lectures on Dynamical Systems (Sheldon E. Newhouse)....Pages 209-312

Citation preview

C. Marchioro ( E d.)

Dynamical Systems Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Bressanone (Bolzano), Italy, June 19-27, 1978

C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy [email protected]

ISBN 978-0-8176-3024-9 ISBN 978-1-4899-3743-8 (eBook) DOI 10.1007/978-1-4899-3743-8

©Springer-Verlag Berlin Heidelberg 1980

Originally published by Springer-Verlag Berlin Heidelberg New York in 1980. Reprint of the 1st ed. C.I.M.E., Ed. Liguori, Napoli & Birkhäuser 1980 With kind permission of C.I.M.E.

Printed on acid-free paper

Springer.com

CONTENTS

J. GUCKENHEIMER: Bifurcations of Dynamical Systems M. MISIUREWICZ.: Horseshoes for Continuous mappings of an Interval Various aspects of integrable J. MOSER Hamiltonian systems Hopf Bifurcation for Invariant Tori A. CHENCINER Lectures on Dynamical Systems S.E. NEWHOUSE

Pag. 5 Pag. 125

Pag. 137 Pag. 197 Pag. 209

CENTRO INTERNAZIONALE MATEMATICO ESTIYO (C.I.M.E.)

BIFURCATIONS OF DYNAMICAL SYSTEMS

JOHN GUCKENHEIMER

Bifurcations of Dynamical Systems John Guckenheimer University of California, Santa Cruz

§1

Introduction The subject of these lectures is the bifurcation theory of dynamical

systems.

They are not comprehensive, as we take up some facets of bifurcation

theory and largely ignore others.

In particular, we focus our attention on

finite dimensional systems of difference and differential equations and say almost nothing about infinite dimensional systems.

The reader interested in

the infinite dimensional theory and its applications should consult the recent survey of Marsden [66) and the conference proceedings edited by Rabinowitz [89) .

We also neglect much of the multidimensional bifurcation

theory of singular points of differential equations.

The systematic ex-

position of this theory is much more algebraic than the more geometric questions considered here, and Arnold [7,9) provides a good survey of work in this area.

We confine our interest to questions which involve the geometric orbit

structure of dynamical systems.

We do make an effort to consider applications

of the mathematical phenomena illustrated.

For general background about the

theory of dynamical systems consult [102).

Our style is informal and our

intent is pedagogic.

The current state of bifurcation theory is a mixture of

mathematical fact and conjecture. proved is small [11).

The demarcation between the proved and un-

Rather than attempting to sort out this confused state

of affairs for the reader, we hope to provide the geometric insight which will

8

allow him to explore further. The problems we deal with concern the asymptotic behavior of a dynamical system as time tends to finite

dimensional

(1)

Q)

C

There are three kinds of systems we examine on a manifold

M:

smooth, continuous time flows

~:

Mxm

~

M obtained from in-

tegrating a vector field or solving a system of ordinary differential equations on

M,

(2)

smooth, discrete time flows obtained by iterating a diffeomorphism

f: M ~ M, and (3)

non-invertible, discrete time semiflows obtained by iterating a

smooth map

f: M ~ M.

Similar problems arise in each of these situations, and we shall pass rather freely from one to the other as we select the most convenient setting for each problem we study.

In some applications, we shall find examples of all three,

A common procedure will be to pass from (1) to (2) when examining periodic phenomena and from (2) to (3) as a singular limit or projection. these steps, the dimension of the state space

At each of

M is reduced which makes our

analysis easier and our geometric intuition keener. Dynamical systems theory has placed a major emphasis upon the elucidation of the typical, or generic behavior of dynamical systems.

we shall

say that a phenomenon is robust if it persists under perturbations of the system.

Bifurcation theory goes one step beyond the study of robust pro-

perties to examine those which are "almost" robust. lated ways of doing this.

There are several re-

One way is to examine those properties of para-

metrized families of dynamical systems which persist under perturbation of the family.

Another outlook is to search for the typical way a robust pro-

perty disappears or changes into another robust property.

This corre-

sponds to looking at the boundaries of regions in the space of dynamical

9 systems which enjoy similar properties.

A third viewpoint is to search for

low codimension hypersurfaces in the space of systems with qualitatively similar properties.

We shall rely more upon the first approach t han the

others, but all of them will appear at times during the discussion. Attempts to provide a truly coherent approach to bifurcation theory have been singularly unsuccessful.

In contrast to the singularity theory for

smooth maps, viewing the problems as one of describing a stratification of a space of dynamical systems quickly . leads to technical considerations that draw primary attention from the geometric phenomena which need description. This is not to say that the theory is incoherent but that it is a labyrinth which can be better organized in terms of examples and techniques than in terms of a formal· mathemat ical structure.

Throughout its history, examples

suggested by applications have been a motivating force for bifurcation theory. Hoping to convey this spirit we shall try to avoid as much abstraction as possible.

The penalty is that we shall leave the generality of many r esults

unspecified.

The sophisticated reader desiring a precise accounting of the

state of the art will not find it here. Let us lay out our elementary bifurcations.

pl~n

of action.

We begin with a description of the

These are bifurcations which involve the mildest

lack of robustness in the local behavior of a dynamical system. trate both the continuous and discrete cases. tinuous flows can be described in the plane.

We illus-

All of the examples for conIn the plane or on the two

spter e , comp licated recurrent behavior does not occur. proceeds much more smoothly than in other situations.

Thus the theory Sotomayor [104] has

given a systematic account of the codimension one phenomena encountered on the sphere, and our exposition of the elementary bifurcations is largely drawn from his work. The two dimensional theory has been carried further to the analysis of higher codimension bifurcations of singular points.

The most complete

10 results are those of Dumortier [24].

Takens' concept of nonnal forms [111]

plays a central role, so we describe it in some detail and illustrate with an example.

The example we use is a codimension two bifurcation studied by

Takens

and Bogdanov [16].

[~14]

this bifurcation.

They independently computed the unfolding of

Holmes and RanQ [SO] have used this example in a problem

involving non-linear oscillations, and we shall exhibit it in the differential equations of a continuous, stirred tank chemical reactor. After describing these local results, much of our attention will be devoted to bifurcations which involve homoclinic behavior of dynamical systems. Topological constraints prevent homoclinic phenomena from occurring in two dimensional continuous flows, but it can be found with three dimensional vector fields, two dimensional diffeomorphisms, and one dimensional maps. Two examples, the forced van der Pol equation and the Lorenz attractor provide the setting for us to describe the relationship between these three contexts.

The interest in homoclinic behavior comes from its role in dynamical

systems which have complicated asymptotic behavior (called "chaos" in some of the applied literature) and sensitivity to initial conditions [91].

Math-

ematical models with these properties are appearing in a rapidly growing list of disciplines which now includes chemical kinetics, geodynamics, fluid mechanics,

electric~ !

circuit theory, ecology, and physiology .

The bifurcation behavior of one dimensional maps has been an area of substantial advances in the past few years.

We shall describe these results

in the language of kneading sequences introduced by Milnor and Thurston. The order properties of the line restrict the order in which various bifurcations can occur.

This is the only situation for which it is possible

to give explicit relations between large sets of bifurcations.

We shall

formulate topological results about rotation numbers for homeomorphisms of the circle in these terms.

11

Armed with the kneading theory., we reconsider the three dimensional vector fields introduced earlier.

The bifurcation theory of the Lorenz

attractor gives us an example of a vector field which has moduli for topological equivalence.

The Lorenz vector field is structurally unstable and

cannot be perturbed to a structurally stable vector field.

Nonetheless, the

topological equivalence classes of all of the nearby vector fields are c~aracterized o~

by two parameters.

In the case of the van der Pol equation,

analysis suggests sensitive phenomena which have not been observed

numerically. The next section is a brief introduction to bifurcations of population models.

We describe several population models which have complicated dy-

namics and bifurcation theory.

These models raise a number of practical

questions that we mention in passing.

The final section introduces a number

of other bifurcation phenomena which involve global bifurcations of dynamical systems more than the material of the preceding sections.

We touch upon

loops of saddle separatrices for plane vector fields, the wild hyperbolic sets of plane diffeomorphisms studied by Newhouse [74], and the moduli of saddle connections discovered by Palis [85]. Before embarking upon our exposition, it work for our discussions.

i~

necessary to set the frame-

This requires that we provide a minimal back-

ground from the theory of dynamical systems and that we outline the general strategy of bifurcation theory.

After establishing this context, we give a

list of basic examples in the next section. Let M.

~:

MX~ +

M be a continuous time, smooth flow on a smooth manifold

Recall that this means that

diffeomorphisms of fined by

X(x)

d

M.

= ~(·,t)

is a one parameter group of

Associated with each flow is its vector field de-

= dt(~(x,t)).

a bijection when

~t

The map

M is a compact manifold.

from flows to vector fields is This is a consequence of the

existence and uniqueness theorem for ordinary differential equations. curves

{

6, individual strata must be made up

from finite dimensional families of orbits described by parameters called moduli.

Nonetheless, there is an elegant theory here which allows one to

classify all functions with a "finite" degree of degeneracy, in principle. Transverse sections to orbits are called universal unfoldings.

Their inter-

section with the various strata show how the functions of the orbit can be approximated by less degenerate functions.

t6 One would like to imitate this singularity theory for dynamical systems, but there are considerable obstacles:

Let us indicate some of these.

An

initial attempt would consider the action of the diffeomorphism group on the space of flows (by conjugation ·of a flow as a group action of

~

or

lR .)

Unfortunately, this attempt founders immediately because there are no stable orbits.

The eigenvalues associated with singular points and periodic orbits

are invariants of the group action. same eigenvalues.

All flows in the same orbit have the

Moreover, in the continuous time case, the periods of

periodic orbits give continuously varying invariants of the action.

No flow

with either a singular point or periodic orbit can be stable for this group action. This difficulty prompts the intrOduction of weaker notions of equivalence between flows. morphism of

Two flows are topologically equivalent if there is a homeo-

M which maps orbits to orbits (preserving their orientation.)

This is the equivalence relation upon which the definition of structural stability is based.

A flow is structurally stable if it is an interior point 1

of its topological equivalence class (with respect to the C spac .3 ·of flows.)

topology on the

Now, there are examples of structurally stable flows, but

apart from very low dimensions they are not dense.

So we still cannot begin

systematically defining a global stratification of the space of flows with the first stratum consisting of structurally stable flows.

Moreover, the

equivalence relation we are studying is no longer given by a nice group action for which general theorems imply that the orbits are manifolds. There have been various attempts to pursue these matters further by looking at local bifurcations and by focussing upon restricted classes of flows.

As far as I am aware, none of these efforts have located a setting

in which there is a theory whose mathematical elegance remotely resembles that of the singularity theory for smooth mappings.

In Section §3, we

discuss the best effort, the classification of singular points of two

17 ~imensional

vector fields.

otherwise, efforts in this direction have en-

·countered the counterexamples which have proved the bane of attempts to find generic properties of dynamical systems. use in the theory.

Still, there is much of practical

Proceeding as if there were hope for a systematic theory

amid a hopeless sea of pathological examples has contributed significantly to our general.understanding of dynamical systems.

Throughout our journey

through this zoo, we shall place emphasis upon unfoldings.

When a mild form

of degeneracy has been located, we ask for information on the kinds of qualitative changes in dynamical behavior which occur for perturbations. These lectures are not self-contained.

They presume a modest ac-

quaintance with the theory of dynamical systems.

Nonetheless, I hope that

they will prove useful not only to mathematicians but also to worke r s in a number of disciplines which make substantial use of systems of differential and difference equations having complicated dynamical behavior.

18

§2

co-dimension one examples In this section we shall consider the simplest kinds of bifurcation of

singular points and periodic orbits for vector fields and maps.

We shall

freely introduce special coordinate systems, called normal forms, leaving for Section

the formulation of invariant expressions which are sufficient for

§3

the existence of the normal forms .

There are three kinds of bifurcations

called the saddle-node, flip, and Hopf bifurcation , which we consider .

They

correspond to the various ways in which a singular point or periodic orbit of a dynamical system can change its stability. Recall that a singular point if the Jacobian of a map circle.

f

(DX)x

x

of a vector field

X is Wypetbolic

has no pure imaginary eigenvalues.

is hyperbolic if

(Df)x

A fixed point

x

has no eigenvalues which lie on the unit

(Both of these derivatives are well defined linear endomorphisms of a

particular tangent space).

The bifurcations we look at in this section come

from the simplest kind of breakdown of the hyperbolicity requirements in a one parameter family.

To formulate the conditions properly, we need to

establish a certain amount of notation. Let meter

X : M ~ TM 1.1

1.1

e:

Let

lR .

be vector field which depends smoothly upon the para1.10

be a value of

a non-hyperbolic singular point of

x e: M be such that

and

X

is

The easiest ways for this to come

X 1.10

about are that a single eigenvalue of of eigenvalues of

(DX

)

1.10 X

(DX

1.1

) 0

X

are pure imaginary.

is

0

or that a single pair

The first of these situations

will be a saddle-node if it occurs in as transverse a way as possible, the second will be a Hopf bifurcation.

Rather than immediately stating the

necessary transversality conditions, let us first give normal forms for these bifurcations. Example 1: defined by

The saddle-node.

Consider the vector field

X : JR 2 1.1

~

JR 2

19

X (x,y) 11

The singular points of 2

(0, 0) = (11+X ,±y).

X

X (x,y) 11

For

11 > 0,

X

11

11

(x,y,l1)

(x,y) space is

has no singular points,

0,

X

11

and the other is a node .

ist for

y = 0.

X

X has two singular points at (±~,0). The Jacobian of 11(2x 0 ) , is given by which has real non-zero eigenvalues at ±2y 0 11


For

ll


0, this equation has a single limit cycle which is globally

The only initial condition whose trajectory does not lead to the As

limit cycle is the singular point at the origin.

becomes very large,

£

the character of the limit cycle becomes that of a relaxation oscillation. This means that the velocity along the limit cycle is very far from being uniform.

For the van der Pol equation with large

lustrated in Figure 4.1.

£

1

the limit cycle is il-

Approximately, the limit cycle trajectory alter-

nately follows sections of the curve

y2

= yl

3

/J - yl

and horizontal seg-

ments passing through the critical points of the cubic curve.

The velocity

along the horizontal segments is very large compared to the velocity along the cubic curve.

If one watches the flow, a point moving along the limit

cycle seems to jump from one section of the cubic curve to the other. Our principal interest here is the forced van der Pol equation, obtained from (1) by adding a periodic time dependent term:

45 2

••

•

Y- E(l-y) y + y

bw E COSW t.

(3)

Equation (3) can be interpreted as defining the vector field on

~2 x s 1

with the equations

(4)

6 .. w where

6

s1 •

is a variable along the unit circle

The dynamical behavior of this equation (4) is much more complicated than that of the unforced van der Pol equation (2) . history of study of this equation.

There is a substantial

It has served well as a model problem for

questions about nonlinear oscillations.

It is easy to build electrical de-

vices governed by the forced van der Pol equation, so that experimentation can be carried out readily.

Cartwright and Littlewood [20] made the first ob-

servations which indicated that basically new dynamical phenomena are present. Further analysis of this system by them [62,63] and of a similar system by Levinson [60] provided basic motivation for the development of Smale's ideas about dynami cal systems [102].

All of this work has yet to yield a complete

analysis of the dynamical behavior of the forced

V!n

der Pol equation.

It

remains a good model system to study in terms of dynamical system theory. Since the speed of the forced van der Pol vector field is constant in the 6

direction, all of the periodic orbits of this vector field have periods

which are multiples of

2n/w •

Solutions of period

2nn/w

wi th

n > 1

are

The fundamental observation of Cartwright and Littlewood

called subharmonic .

was that there are parameter values for which two different stable subharmonic solutions were present with different pe riods.

For the prope r parameters,

different initi al conditi ons can lead either to a s tabl e subharmoni c of period (2n-1)2n/w

or

(2n+l)2n/w •

This behav ior is robust, as it mus t be

46

since stable periodic orbits are robust.

Along with the two stable sub-

harmonic solutions, they deduced that there must be an infinite set of saddlelike periodic orbits.

Levinson proved their existence for a slightly dif-

ferent vector field whose qualitative behavior is similar to that of the van der Pol vector field.

The construction of Smale's horseshoe was motivated by

Levinson's analysis. Here we shall see how this example leads to questions of bifurcation theory.

To put these questions into a framework amenable for analysis, we

want to descend a hierarchy which begins with three dimensional vector fields, end with one dimensional maps, and has two dimensional diffeomorphisms sitting in the middle.

The process of converting questions about periodic orbits of

an n-dimensional vector field to ones about an n-1 dimensional diffeomorphism has become routine in dynamical systems.

It involves taking cross-sections

to the flow and then looking at the return maps as described in Section §1. Generally, there are obstructions to a global construction of this sort, but Fix a plane de-

for the forced van der Pol equation it is easy to do this. fined by

a~

Observe that the time

constant (say 0).

2n/w

map of the flow

maps this plane i nto itself, and thus defines a diffeanorphism ~: The asymptotic behavior of flow, and we may study

JR2 -+

JR2

corresponds to the asymptotic behavior of the

~

instead of the flow.

~

The reduction of our problem from the study of a two dimensional diffeomorphism to a one dimensional map of the circle is not so clear cut. are substantial problems that we discuss later. about qualitative properties of

~.

We start with observations

Consider an annulus A of moderate size

surrounding the limit cycle of the van der Pol equation. properly, then

~

will map

A

annulus which is much thinner. parameter

€

increases.

There

If

A is chosen

into its interior and the image will be an The thickness of the image decreases as the

Figure 4.2 illustrates

A and its image under

In passing to a map of the circle we want to approximate

~

~.

by a map with

47 thickness; i.e., by a map

0

an image of

We assume that an approximating

image.

~

of rank 1 with a one dimensional can be f?und with the property

~

that each inverse image of a point is a smooth curves which connects the two boundary components of Now the map

~

A.

is homotopic to the identity map of

in which it is embedded.)

A once.

he core of

This implies that the image of

~on -l:

and

C

n +

C

and a corresponding

A onto one of its boundary components

of

C.

The map

is well defined since the inverse images of points for The map

are the same.

~

winds around

However, numerical computations show that the image has

~ to establish a coordinate system on A as s 1 x I

7TO

~

Let us use the curves which are the inverse images of points for

two "kinks".

projection

(through the flow

A

f

= no~on

not a homeomorphism due to the kinks of we will discuss at some length.

-1 ~

7T

is a map of the circle which is •

f

It is this circle map

which

We will want to see what the bifurcation

theory of circle maps tell us about the behavior of the solutions to the forced van der Pol equation. The second example which we consider in this section is the Lorenz attractor.

This is a global attractor for the flow of the following system of

differential equations: X = -lOx + lOy

crx -y - xz

z - -8z/3 where

a

is near 28.

(4.5)

+ xy. This system of differenti al equations was studie d by

Lorenz in the 1960's [64] as a model of a fluid dynamics problem.

During the

past few years, this system has been studied intensively by a number of people from the point of view of dynamica l systems theory [38,42 , 55,90,118] .

We

shall s ummarize the setting in which the equations (5) arise and some of the :r·esults about them.

In Se ction §6 , we consider the ir dynami cs i n more detail.

4X

Consider a fluid in a horizontal layer which is heated from below.

The

colder fluid at the top is denser than the warmer fluid at the bottom, so there is a buoyant force pushing the lighter fluid upward.

If the density

gradient :s sufficiently large, then this force actually causes the fluid to move.

Different flow patterns are possible, one consisting of cylindrical

rolls of rotating fluid.

If the heating is sufficiently rapid, then this

flow pattern will itself become unstable because the fluid will be rotating faster than heat can be dissipated.

The temperature distribution around the

roll will be complicated, and the direction as well as the velocity of the rolling motion will fluctuate.

The Lorenz equations model this situation by

expanding the velocity and temperature distribtuions in Fourier series and inserting them into the partial differential equations of motion.

This

yields an infinite set of ordinary differential equations for the coefficients of the series, but one truncates this set by assuming that all but three coefficients can be neglected.

There is little known about the relationship

between the solutions of this truncated set of equations and the soluti ons of the original set of partial differential equations which they are supposed to approximate.

This issue is of interest in the study of fluid turbulence.

For the parameter values chosen in equations (5), the solutions correspond to fluid

mo~ion

which is irregular .

directions for varying lengths of time.

The fluid rotates in alternate

The time intervals between successive

changes of direction vary with changes in the initial condition.

Within cer-

tain constraints, there will be initial conditions which lead to a random choice for the lengths of time between changes of direction. This behavior can be understood readily from a geometric description of the flow of the Lorenz equations.

There are three singular points of the

Lorenz system, a l l of which are in the plane

y

= x.

One is the origin, and

it is a saddle point with two dimensional stable manifold containing the z-ax i s and one dimensional unstable manifold tangent to the x-y plane.

The

4Q

other two singularities lie in the intersection of the planes

y

=x

and

Each is a saddle with one dimensional stable manifold and two di-

z = a-1.

mensional unstable manifold. away from the singular points.

The flow in these unstable manifolds spirals If one looks at the plane

y

= x,

then one

sees a vertical barrier directing points left and right away from the origin as they dgscend through the upper half plane, and one sees orbits spiraling away from the other singular points.

See Figure 4.3.

This picture is filled

in by extending the branches of the unstable manifold of the origin around the other singular points so that they descend near the singular point on the opposite side of the barrier.

See Figure 4.4.

Williams [118] has shown how we can understand this behavior in two dimensions.

To make a construction of this sort for a two dimensional flow,

one must allow a two sheeted surface (called a branched manifold) and throw away the singular points which are not at the origin. Figure 4.5.

This is illustrated in

The flow on this branched manifold reflects the fluid behavior

described above.

A trajectory spirals around one of the holes in the figure

until it crosses the vertical axis.

Then it spirals around the other hold

until it crosses the vertical axis again.

Any pair of trajectories which

start close to one another eventually separate and pursue their independent ways. The actual picture of the Lorenz attractor in three dimensions is considerably more complicated than the branched surface shown in Figure 4.5. Instead of being a surface of two sheets joined

along a single cut connecting

the two holes, the Lorenz attractor has an uncountable number of sheets which are joined

along their boundaries.

There are two ways to develop this picture.

First, one can use an inverse limit construction on the branched surface with its semiflow to eliminate the non-uniqueness of trajectories followed backwards.

This produces an accurate representation of the Lorenz attractor.

other approach is to study the Lorenz attractor via a cross-section in the

An-

so

horizontal plane of the singular points.

See Figure 4.6.

This allows us to

study a two dimensional map, whose properties are in turn determined by a one dimensional map. The cross section is illustrated in Figure 4.6.

The stable manifold of

the origin intersects the cross-section of the flow in a curve which do not return to the cross section at all. y

of points

y

The points to the right of

return to a figure which is greatly compressed in one direction, stretched

in the other direction, and pointed at the end which comes from points near The points which started to the left of

y •

y

return to a similar set.

Thus the return map would a homeomorphism of the c:ross-section into itself apart from the fact that it has a discontinuity along

y •

The attractor will

be the intersection of iterated images of the cross-section if the original size of the cross-section is

caref~lly

chosen. To do

We want to obtain a one-dimensional map from the cross-section.

so, we assume that there is an invariant foliation of the cross-section which contains the curve (x,y)

y.

In more prosaic language, we assume that coordinates

can be chosen for the cross-section so that the return map S (x,y)

has the form

(g(x ; , h(x,y))

continuity and positive slope. as

(x,y)

approaches y

perties of the return map study of the dynamics of

with

g

being a function with a single dis-

The function

ah ay

should tend to

0

uniformly

With such a representation, many of the pro-

e g.

and the attractor can be determined from a We shall return to a discussion of these

matters in Section §6 after taking up the properties of one dimensional maps in Section §5.

51

Figure 4.1 The van der Pol limit cycle for large

e

52

0

Figure 4.2

The annulus A and its image under the van der Pol return map

4>

53

Figure 4.3

The singular points of the Lorenz system viewed in the plane

y

=x

54

Figure 4.4

The unstable manifold of the saddle point in the Lorenz at tractor

55

....... ....

branch

' '\

\

curve

stable manifold of singular point

Singular point

Figure 4 . 5 The branched manifold for the Lorenz attractor

56

--=:

y

-Fiqure 4.6

A cross section to the Lorenz attractor

57

§5

Kneading Sequences and Bifurcations of one Dimensional Maps The examples

of the Lorenz attractor and the forced van der Pol equation

have led us to the study of one dimensional maps which are not homeomorphisms. A third application which provides impetus for the study of one dimensional maps lies in population dynamics, discussed in Section §7.

one dimensional

maps are subject to severe constraints which come from the order properties of the line.

In this section, we shall consider these constraints with

particular attention to how they affect the bifurcation behavior of the maps. A concrete family of maps which has motivated much of this work is the one dimensional family of quadratic maps Throughout this section terval

[c 0 ,c~]

This means that

f: I

f

\l

I

~

(x) =

\l

-

x

2

•

will be a continuous map of the in-

with a finite number of turning points f

c 1 < c 2 Am (y) = -am (y).

These two cases prove the proposition. The proposition gives us the means of relating the order of the line to order properties of the itineraries of individual points. itineraries of the ineraries

! (x)

ci,O

as

x

~

i

t,

~

varies in

the kneading sequences of

f.

In particular, the

determine completely the set of it-

I.

The itineraries of the

are called

There are consistency conditions which must be

satisfied by the kneading sequences coming from the proposition above.

We

shall see later how these limit the kneading sequences which actually occur. To fix ideas before treating the general case, let us consider maps which satisfy

I= [0,1], · f(O)

which must be a maximum. by

f

~

(x) =

4~x(l-x),

~

= f(l) = 0,

and there is a single turning point

c

A family of maps with these properties is given e (0,1).

We shall illustrate how the kneading sequence

of the critical point I.

c

.The critical value

of f(c)

f

determines the itineraries of all points in

is the largest point in the image of

means that the itinerary of every invariant coordinate of f(c).

f(x)

x

f.

This

must satisfy the condition that the

is no larger than the invariant coordinate of

Conversely, if a prospective itinerary has this property, then there

is indeed a point with this itinerary. be a sequence of addresses such that

(1)

if

i > 0

sequence f(c), (2)

and

Bi-l

C,

~

{Bi,Bi+l'Bi+2 , ... }

the invariant coordinate associated to the is smaller than the invariant coordinate of

and Bi-1 = C then the sequence

if

the critical point

is the itinerary of

f(c).

x.

is the itinerary of Proof:

1 the invariant coordinate associated to

Denote by

is open. It is nonempty unless o.

Let

X

e: L

B.

j

and let

thesis (2) implies that

for all

i, but then

J

i

~

z,

A. (x) 1

implying that

B is the itinerary of

be the smallest integer

with

is not the turning point

is not the turning point, then there is a

A. (x)

for

Io

1

i:l(z) •

~·

Thus

z

L

>

x

c

!:1 (x) < ~· .f or

with

is open i f

Hypo-

i < j.

If

A. (z) = A. (x) 1

1

A . (x)

is not

J

A. (x) is the turning point, then B. is not the J J turning point, and there are z > X such that Aj(z) = B. (z). Moreover, the J continuity of f implies that the itinerary of fj+l(z) tends to the the turning point.

itinerary of

If

'+1 fJ (x) = f(c)

in the sense that, given any

A. (fj+l(x)) (provided that one of the addresses are 1

has a maximum at

x,

so

(Z)

e:j+l

= -1

for

hypothesis, the invariant coordinate of

z

c).

k,

A. (fj+l (z)) 1

Note that

slightly larger than

x.

By

is smaller than

60 !!f(c)), 8j+l

hence there is a smallest 1,

k

>

0

with

Bj+k

this implies that there is a

Z

>

with !(z)

L

is open.

I

is connected, we conclude that there is an

Similarly, if

U

m

{xj!l!_ < !!x)} ,

X

then x

Since

Aj+k(x).

~




but

c as an address, but then end with the itinerary

Ai+k(x)

A. (f(c)) l.

and

A. (x) J

~

c

for

j

~

k,

There are examples which show either possibility occurs.

interval containing exist.

f,

We cannot determine without further information whether sequences

occur which do not contain of

the itineraries of

~characterizes

On

tends to a stable periodic orbit, then such

the other hand, if every interval containing

which contains not exist.

c

c

c

for some If some x

will

has forward image

again, then points with the property indicated above will

The quadratic map which has exactly one periodic orbit of each

power of 2 is an example of such a map (see Section §6). The proposition allows us to determine information about the periodic orbits of

f.

Each periodic itinerary is represented by at least one periodic

orbit, but information from itineraries and kneading sequencies cannot specify how many.

This

pr~pts

points

and

y

x

same itinerary.

the definition of monotone equivalence:

are monotone equivalent if all points in

two periodic

[x,y]

have the

Counting monotone equivalence classes of periodic orbits can

be accomplished by counting periodic itineraries.

Moreover, information about

periodic orbits gives information about bifurcations. -For a map

f

with a single critical point, the maximum possible number

of monotone equivalence classes is such that classes of fixed points.

~

has

2n

The minimum possible number is

monotone equivalence 1.

Consider a one

parameter family which begins with a map having only one periodic orbit, necessarily a fixed point.

If a map

f

\.1

in this family has many periodic

61 orbits, then there must have been bifurcations in the family which gave rise to each periodic orbit.

These bifurcations cannot occur in an arbitrary

Define an ordering of periOdic orbits by calling the larger periodic

order.

orbit the one with the largest.point: with

x

y

>

for every

y E

e2 •

If

if there is an A and

ineraries with the invariant coordinate of ordinate of

~,

periodic orbit itinerary

A.

B

are two periodic it-

A less than the invariant co-

then the preceding proposition implies that a map having a

e1

with itinerary

B

also has a periodic orbit

This means that a bifurcation which gave rise to

before the bifurcation which gave rise to the periodic orbit

e1 .

e2

e2

with occurred

The order-

ing of periodic orbits for a map in the family gives the same ordering for a sequence of bifurcations giving rise to these orbits.

Thus we obtain a

combinatorial algorithm for deciding the order of the bifurcations gi ving rise to a pair of periodic orbits:

it is the order of the invariant co-

ordinate of the largest points in these orbits.

Given two periodic it-

ineraries, comparing their invariant coordinates involves comparing sequences whose length is bounded by the product of the periods of the two itineraries. There are two types of bifurcations of periodic orbits which are generic for one parameter families of one dimensional maps:

saddle-nodes and flips .

Hopf bifurcations cannot occur on a one dimensional space.

A typical bi-

furcation structure one encounters is illustrated in Figure 5.1.

An

initial

saddle-node gives rise to a pair of periodic orbits , one stable and one unstable .

For the stable orbit to persist but lose its stability, it must pass

through a flip bifurcation. twice

the period.

This generates a new stable periodic orbit of

This new orbit must also go through a flip if it is to

persist but lose its stability.

The result is an infinite sequence of flip

bifurcations following the saddle-node.

At each flip a new periodic orbit

of twice the period of the preceding one is born.

For the quadratic family,

one can prove that each saddle-node is followed by a sequence of flips before

62

bifurcations involving other periodic orbits occur. These results tell us much about the order in which bifurcations occur far families which are qualitatively similar to the family of quadratic maps. They do not give quantitative information about how large the bifuraction set might be.

In particular, the outstanding question in this area is whether

the bifurcation sets of such one parameter families have measure zero (generically or for specific examples). results related to this question.

Jacobson has just announced the first We discuss it more fully at the end of

Section §6. There are three directions in which we would like to extend the theory we have just outlined for maps with a single critical point.

First, we would

like to allow more critical points .

Second, we would like to consider circle

maps as well as maps of an interval.

Third, we would like to consider maps

which have discontinuities. by examples.

All three of these considerations are motivated

Both the first and second are necessary for our analysis of the

forced van der Pol equation.

The third is needed to understand the Lorenz

attractor. We want to describe the periodic orbits of a function having several critical points.

There are two aspects to this question which we need to con-

sider.

~ssumes

The first

that a map is given together with the kneading se-

quences of its critical points and then asks for the itineraries of points· in the domain of the map.

The second asks for a list of which kneading se-

quences do occur for maps.

Once these two questions are answered, we want

to embed the information in a picture which describes the relationship between the bifurcations of periodic orbits with various itineraries. Lemma: I~

Consider a map

[c0 ,c~].

If

x

f: I

+I

which has turning points

has the itinerary

A and

x

inequality of invariant coordinates is satisfied:

£

c 1 , •••

,c~-l

and

I, then the following

63

Proof: as

The map is

~(Ii)

+1

f

is monotone on

or

-1 .

[ci-l'ci],

increasing or decreasing

The lemma follows from the monotonicity of the

invariant coordinates. This simple lemma gives the compatibility conditions which must be satisfied by the itineraries of orbits of obtained from the itinerary of the shift map i.

x

The itinerary of

by dropping the first term.

of symbolic dynamics:

a

f.

a(~)

=

if

(~)

Bi

f(x)

is

This is just

= Ai+l·

for all

To determine whether a given sequence occurs for a particular point, we

need to apply the lemma to the points in the orbit of the address of

fj(x)

must be satisfied.

is

x.

In particular, if

i, then the inequality

These inequalities can be evaluated from a knowledge of

the kneading sequences of the critical points and the itinerary of

x.

The

first of our results is a partial converse. Theorem:

Let

Assume that no Let

1

= a J..

»

'+1

< cli.)S(aJ 1

c 1 , ...

,c~_ 1 .

-

1

1

x

~

I

such that

with

i(j)

chosen so that

is the itinerary of ~

is the itinerary of

ci.

Then

x.

The proof proceeds in the same way as the case for a single

critical point.

= {xle 6(~)}

that there is an

are open.

X with

Consider the set

u.

= {xle

Therefore,

and

i I.

C g(Il'.)

l.~+l (A )ij

j.

for

is the number of

81

k g (I.)

times that

covers the interval

l.

I. • J

The number of unstable fixed points of Since

g

k

g

k

k

is given by the trace of k

is contracting at a fixed endpoint, each time

there is at least one fixed point in the interior of

g (I.) l.

A .

covers

Our assumption that

I .. l.

has only two stable periodic orbits implies that there will be exactly one

g

points of

g

k

Table 1 gives the number of unstable fixed

Ii.

fixed point in the interior of k < 15.

for

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0

0

3

16

5

3

28

64

39

45

176

307

260

392

1028

# unstable periodic orbits, period k 0

0

1

4

1

0

4

6

4

4

16

24

20

26

68

k # unstable fixed points gk

Table 1

r

The (one-sided) subshift of finite type

with transition matrix

is a topological space (again denoted E ) together with a map cr • ·Space consists of all sequences for all with The map

cr : E + r

0

i-

t

£.

i f i£

= jt

cr (~)

i

ji

with

set.

it+l.

The map cr

indices on a sequence, omitting the first term. g

d(~,il

This metric is comp lete on

1 if ii t ji

is defined by

equivalent to the map

The

i

A distance function can be defined by

o

A

r .

It shifts the is topologically

restricted to the unstable part of its nonwandering

This means that there is a homeomorphism

part of the nonwandering set of

g

such that

h

from

ho

= gh.

E to the unstable Put somewhat dif-

ferently, there is a unique nonattractive orbit in the nonwandering set of

g

82

for each sequence

i

with

g(I. ) , R. > 0. l.

R.

-

Thus, the subshift

A allows us to characterize the nonwandering set of

with transition matrix

A9

Note that

the nap · g.

C I . l. .H l

has only positive entries .

This implies that

o

E.

has a dense orbit in

Between the regions in the parameter space of the forced van der Pol equation where there are exactly two stable periodic orbits and the regions where there is only one stable periodic orbit, many bifurcations must occur. Provided the return map is well approximated by a rank 1 map these bifurcations will involve new stable periodic orbits which have not been observed numerically.

They should be so sensitive to changes in parameter values and

initial conditions that demonstrating their existence numerically might be In addition to the sensitivity of these stable orbits, one has

difficult.

to cope with the "stiffness" of the equation.

The wide range of velocities

encountered along solutions makes numerical computation quite difficult. It seems more likely · that one could actually locate these solutions by using the asymptotic methods of Grasman et al. [36]. forced van der Pol equatiqn for large hard to extract [ 62].

e

Information about the

based upon rigorous analysis is

Much more has been proved with less effort by

Levinson [60] and Levi [59] for a modified version of the van der Pol equation .

Again, the kneading theory is suggesting what one might find, but

there is much to be done before a complete picture of the bifurcation structure of the van der Pol equation emerges.

One aspect of the theory

about which there is little information of any kind is the way in which the region of moderate the large

e

e

in parameter space joins the small

behavior we have described here.

e

behavior and

This involves a third

mechanism (the others being the one described by Ruelle-Takens and the preturbulence of the Lorenz attractor) by which bifurcations lead to homoclinic behavior for flows.

The Hopf bifurcation for maps involves all of the same considerations as those described for the small An

bifurcations of the van der Pol equation.

£

s 1 , appears around a fixed point of the

invariant curve, homeomorphic to

map as the bifurcation occurs.

The behavior of the map on the invariant In a typical family, this rotation

curve depends on its rotation number.

number will not be constant, and a set of parameter values of positive When

measure will yield invariant curves with irrational rotation numbers. the rotation number is rational, there will usually be nearby periodic orbits with the same stability index of the fixed point before the bifurcation. An

exception to this last statement occurs in the "resonant" cases

where the rotation number if rational with a small denominator.

The most

striking of these cases is the one in which a Hopf bifurcation occurs with eigenvalues which are complex cube roots of

1.

To explain what happens, let us reconsider the normal form problem. 0, we want to find a

with a fixed point at

Gicen a map

JRn

local change of coordinates ation such a

which linearizes

(or its inverse) should satisfy is

~

at the origin.

f

the linearization of problem was to expand

f

and

¢

L

=

fo~

The equ-

where

L

is

The approach of Poincare to this

in Taylor series, thereby obtaining a re-

cursive system of equations for the coefficients of eigenvalues of

~oL

f.

¢.

If same of the

are roots of unity (or 0 ), then there is a obstruction

to finding a solution to this resursive system. The Hopf bifurcation problem takes place in we want to consider is rotation by is

L(x,y)

(-lx 2

+

.f3

1 21• --13? - 21'.

27T )•

JR 2 , and the linear map

L

The expression of the linear map

Writing the quadratic terms of

2 2 2 2 2 2 2 2 (a 1 (x +y) + a 2 (x -y) + a 3 (2xy), b 1 (x +y) + b 2 (x -y) + b 3 (2xy))

~

as

and those

84 of

as

f

leads to the following system of eguations for the

a.'s ~

and

b.'s. ~

- ../3;2

3/2

(

-13/2

./3;2

(

..£3;2)

f3/2 -..fi/2

--13/2

The last two pairs of these equations have rank 1, so the coefficients and

S3

a2

cannot both be eliminated and similarly with

Thus the

a3•

can be taken as

f

normal form of

and

a2

'fN on

y2

with

y2

-i

with

f

f;i

and

relative order of the points

fi(f;i)

ni

E

are non empty.

with the properties that y3

E

and that

f;l

is the

ni

We want to show that the

yl.

determine the quantity

log~/logA

in a way which is independent of all the choices which were made. It helps us to assume that the diffeomorphism points

p

and

q.

f

is linear at the saddle

This is not a severe restriction since there are co-

ordinate systems for which this is true, generically. assume that the cross-sections

and

Moreover, we may

lie in these neighborhoods.

With these assumptions, we can easily describe how the sequences tend to r. Measuring f;i/f;i+l

Near

Wu(p)

(and n.) by arc length fran

f;i

+

p, the coordinate along l

Similarly

~

geometric rate

~

-1

ni+l/ni

+

and the sequence

A

r

.

ni

along

y2,

this implies that

cl

Bi+l/Bi =

>

B.

i, j, m, n with the properties that

Pick integers

Bj+n B. >

Then

B.J+n >am+1+ . 1"

J

am+i+l a.

a

m+l

at the

with

and

Then we assert that the

determines the quantity

and

relative order of the

r

+

at the geometric rate A

r

Consider now two geometric sequences of numbers the property that

and ni ~.

is multiplied by

The sequence +

f;i

ai

am+i+l a.

loga/logB • and

>

Now

a

l

This gives the estimate

am-l >Bn > am+l

l

Taking logarithms (and remembering that logB/loga < estimate for

m+l

~ •

By picking

logB/loga

in the two sequences.

n

m-1

loga < 0), we obtain

m-1

-- < n

sufficiently large, we obtain as good an

as we desire fran the relative order of the points

This argument can be applied to the sequences that their relative order on

{~i}

determines the quantity

and

{ni}

to show

logiJ/log)..

The

relative order of the points will be preserved under topological conjugacies, so that quantity class of

f.

log1J/log>.

Melo [69]

is an invariant of the topological equivalence

has investigated this situation further.

He has

considered the question of whether there are additional moduli of the topological equivalence of two dimensional diffeomorphisms which are MorseSmale [82] except for the presence of one orbit of non-degenerate tangencies between the stable and unstable manifolds of a pair of saddles

p

and

q.

The answer depends upon whether there are other stable and unstable manifolds of saddles which intersect those of

p

and

q.

This gives an example (the

first?) in which the behavior of a diffeomorphism far away from the degeneracy which causes it to be unstable has a large impact on the topological equivalence class. Example:

The Newhouse Phenomenon [74]

The final example we consider in this section is the diffeomorphism counterpart of the loop for plane vector fields. diffeomorphisms

f: lR2 +

lR2

our concern will be with

having a saddle point whose stable and un-

stable manifolds have a point of tangency and with perturbations of these. Saddle points are often embedded in a larger set of homoclinic orbits [74]. Examples with this property have been extensively studied by Newhouse [72]. He has used them to provide counter examples to a number of different genericity conjectures for dynamical systems:

notably,

the finiteness of

the number of attracting periodic orbits of a system [72] and the manifold structure of stable and unstable sets [74]. The geometric phenomenon underlying these examples involves

prcp ~~ tie s

of Cantor sets of the line, so we must take a brief detour to discuss the

llO

F

interval

is a closed subset of a closed

C

A Cantor set

thickness of Cantor sets.

The complement

which has no interior and no isolated points.

of a Cantor set is a countable collection of disjoint open intervals, called the gaps of

C.

We are primarily interested in conditions which guarantee While all of the

that a pair of Cantor sets have a non-empty intersection.

Cantor sets we deal with have zero Lebesque measure, they nonetheless can have a thickness (defined below) which prevents two of them from being disjoint.

intersection is J

U

FJ. : F -

The sets

F .•

be an ordering of the open intervals which comprise

{Ui}i>l

Let

C.

to be the smallest closed interval containing the Cantor set

F

Take

C.

u.

form a defining sequence of closed sets whose

~

i~j

The thickness of

Denote the length of an interval

I. J

0

u

T ({F.}) J

J

u

I .l

and that

J

to be the

supremum of

I

from one of the intervals

uj+l

u.

c is defined in terms of the sequence

infT. j~l J

by

T.

J

:

I. J

(min~(I . 0 ), J

and the thickness

F .• J

of ~(I.

1

J

J

T (C)

J

Suppose that

))/~(U.).

of

is a finite

by deleting the

F.

J

J

taken over all defining sequences

T ({F.})

F.

Each

~(I).

is obtained from

union of closed intervals and interval

F-C.

I. J

Define

c to be the F.• J

Let us examine this rather complicated definition for a particular example.

Suppose

C

is constructed as a decreasing sequence of closed sets

c.

with

C.

the middle interval of length

J

J

by deleting from each component

obtained from

sequence we will have Therefore, all of the

a·~(I).

thickness of the Cantor set.

of

This means that for any defining

HI. 0 ) : ~(I.l) : cl;a). J J T. 's are the same number J

I

~(I.)

J l-ex 2a

and

~(U

.)

J

:

ex· ~ (U.)

and this is the

The fundamental result of Newhouse about the

thickness of Cantor sets in the following:

J

.

Ill

Proposition:

are two Cantor sets of the line with

c2

and

c1

Suppose

the following properties: are contained in disjoint closed intervals

(1)

and

(2)

is not contained in the closure of a gap of

c2,

and

c1,

is not contained in the closure of a gap of

The proof of the proposition proceeds by picking defining sequences

then proves inductively that

and

I

l.

n

G.

l.

'f

For

J

l.

l.

The argument is the following.

will not be removed.

I

least one endpoint of

in I-G

If we delete gap

is still in H,

n

(I-G)

is in

H

requires that

~ (Jl) ~(H)

to assumption.

J.

or

J

G

of

H

from

1

~(H)

At

See Figure 8.3.

Since

is not contained

I

of

J-H

is in

T ({G.})


n ) n::O

Let

Lemma 2.

For any

>t

is

+I> n-k)) ~ t • f • a

E

E

we

h(f,A)

Card (Enla) ~ Card An

, we have

log Card (Enl ) ~ h(!,A)

Denote:

a

ol..

0

:0::

oc n • log Card

I

(E0 a)

log ( ~ Card (Ani b)) MA'E

=

132

n

= 1,2, •••

• It is easy to see that n

Card (Anla) ~ 2] exp ( o£.k k=O

+ .6n-k I

is the

( k

)

smallest number for which the image of a given element Anla

is contained in an element of

under

For any

bE

A' E

we have

lim

sup

< h(f ,A)

, and hence

*

the definition of lim sup n~oo

E

~

I

= h(f,A)

Proof. such that p

. fn(e) .:::> b}

Fix a set

such that

implies that then

a € E

< u < h(f,A)

for

n ~ p

n~oo

*

a0 E E

such that

and a real number

n

*

• Suppose that there exists log Card {En\a) ) u

log Card (En la) ~ u

Lemma 3 • Therefore we have:

•

Assume that

Card (En+lla) ( 3 Card (En\a) lim sup

• From

, and thus, by Lemma 2,

• Then there exists an

log 3


0. Then

it follows· from 0 0

that x 3

= y3

the quadric

= 0 and the problem is reduced to one in R2 x R2 , where we have

x 1 y 2 - x 2y 1

= A.

This problem is still rotation invariant under

50(2) and we may use polar coordinates r,¢ and conjugate variables pr' p¢.

They can be definea by the canonical transformation r cos ¢

=W

yl r sin ¢ = W y2

= wr , p¢ = w¢ ¢-~sin¢ r

with W

pr

¢ + and x1 y 2 - x 2 y 1 = p ¢ = A.

gives

P¢ cos ¢ r

Thus p ¢ is the integral and H independent of ¢.

The reduced Hamiltonian is H

= -21

2) ( p 2 +A+ V(r) r 2 r

and the ¢-dependence is obtained separately from • ClH A ¢=-=()p¢ r2 This type of reduction was known to Jacobi who "reduced" the 3 body problem in R3 by using the invariance of this system under the Galilei contains the rotation group 50(3) as a subgroup.

group which

Eliminating the integrals

of the center of mass and the angular momentum one can reduce this system of 9 degrees

of freedom to one of 4 degrees of freedom, i.e. reduce the phase-

space from 18 dimensions to 8 dimensions.

Using, in addition, the conserva-

150

tion of energy

one has a vector field on a seven dimensional manifold.

(b) The moment map.

We describe the generalization of this reduction in

We consider a manifold M with a one-form 6 for which w = d6

abstract form.

is nondegenerate, so that (M,w) is a symplectic manifold. refers to

(M, 6)

as an "exact symplectic" manifold.

An

One sometimes

example is the cotan-

gent bundle M = T*N of a manifold N with the natural 1-form. If g

G

E G

is a Lie group we speak of a symplectic group action (

(1xi 2 IYI 2 -

=

o}

1)

and finally, since{~,IYI 2 } = 0, {~,} = 0 on M we have FO

Thus on F 0

= FM.

0 X=

y =

defines the geodesics on the sphere. (e) We observe that the manifold R2n+ 2 by the integral F 1 =

is an integral

(6) is

obtained by "reducing the phase space

~ (I y 12-1)" (in the sense of Section

3 ) • Indeed F 1

which belongs to the one dimensional symplectic group action (x,y) + (x + ty, y) , Thus

with the group G being R. (11)

The group G0 leaving

1jJ

F1

=

= { x,y

-1 (0)

=0

1jJ

F 1 is the moment map

E R2n+2 ,

invariant is evidently the whole group G

= R.

To form the quotient manifold 1ji- 1 (0)/G 0 we single out the point x on the line Thus

x + ty for which< fx,Y> = 0.

1jJ

-1

(O)/G 0

~M

and we arrive at the result:

If we reduce (4) by the group action (11) we obtain up to parametrization the geodesic fZow on f(x) = const. the unit tangent bundZe of ff

=

Moreover, F = 0 gives rise to the fZow on

o }.

Conversely we can view (4) as an ex-::s ,, :::e ::: Hamiltonian system of the geodesic

170 flow -- in the sense that the latter is obtained from (4) by a reduction via a symplectic group action. Clearly , there are many such extensions.

The use of such an extension is to

be compared with the use of ho110geneous coordinates to describe points on the sphere instead of spherical coordinates .

Of course in homogeneous coordinates

one has to take account of the identification of x and Ax for A> 0. Similarly, in the Hamil toni an one has to eliminate the redundancy of the group action . (f) For the following we have to generalize the above considerations somewhat. It is, of course, not necessary that f (x) is a convex function. All that matters is that for some point~= x- t*(x,y)y {

on the line x + ty

cj>(x,y) = f(~) 0

= < fx(~) ,y>

Then cj>(x,y) = c is the equation for the tangents of f(x)

= c.

Since we are not interested in the parameter dependence of the Hamiltonian system we could take any function H(x,y,c) defining the tangent to f(x)

as a Hamiltonian, as long as

by

=0

H(x,y,c)

a ac

=c

H(x,y,c) does not vanish at these points.

Indeed since cj>(x,y) is obtained by solving H(x,y,cj>(x,y))

0

we have 0

H + H cj> y c y

1

and therefore

y

=0

- H

X

which is, up to parametrization, the above system. (g) We consider two surfaces f(x) and let cj>(x,y), W(x,y) line flows. PROPOSITION.

= 0,

g(x)

=0

with nonvanishing gradient

respectively be the Hamiltonians of the corresponding

We inquire when the corresponding flows commute. If for any line tangent to both f

< f X W g X = 1

For z < ·a 0 these are ellips:-id, but for a 0 < ·z < they are hyperboloids;

~

or

in fact one has n+l different types

172 of such quadrics corresponding to the intervals (-eo,a0 ), (a0 ,a1 l ... (an-l'an). Through every (2).

.

po~nt

x E Rn+l

n

r-f x

. th

w~

0

\1

~ 0 pass n+l such confocal quadratics

Indeed for a given .such x the rational function 1-Qz (x) has n+l distinct

roots, one in each of the above intervals and one can write 1 - Q (x) = z

{

(3)

n

z-

UV

V=O

z -

a\1

r-f -

,

uo < ao < ul < ••• < un < an

The equation z

= u\1

defines the desired n+l hyperquadrics intersecting at x;

it is well known that they intersect orthogonally. stronger orthogonality property

Actually they possess a

which is less well known -- and which will

be of interest to us.

= l, If L is a straight line which touches two confocal quadric Q zl at the points x(l), x( 2 ) then the normal of the quadrics at these

THEOREM.

points of tangency Proof:

(4)

are pe:tpendicular to each other, provided z 1

= z2 •

Let the points of L be given by x + ty where t varies over R, and let x + t.y. The condition J < V'Q (x(j)) ,y> = 0 ,

of tangency of L at x(j} is given by

z.

(j

l, 2}

J We want to show that (5}

vanishes.

< V'Q

(x ( 2 }} > (x (l}} , V'Q z2 zl For this pu:tpose we use the identity

Therefore, if Qz(x,y}

is the symmetric bilinear form belonging to Qz(x} we

173 have for (5)

This ·remarkable property implies that a line -- which generally touches n

confocal quadrics -- has n mutually perpendicular normals associated with it. These normals are clearly also perpendicular to the line, so that we have an orthogonal (n+l) frame associated with each such line. (c) To determine the differential equations of the geodesics of

th~

ellip-

soid we generalize this question in two ways: (i) We replace the ellipsoid by any of the confocal quadrics Qz (x) = m and (ii) determine the extended "line flow" of the previous section. For this purpose let u . (x) = z be any of these quadrics and consider all J

lines x + ty which are tangential to it. Q

z

m the condition for x + ty to

(x)

that for the

~

point of contact

=x

Since for the given value z we have

represent a tangent to Qz(x)

m is

+ t*y 0

or, if Qz(y)

~

0 we get from the second relation

t* Inserting this into the first

=-

Qz(x,y)/Qz(y) •

we see that x + ty is tangential to Qz(x)

m

if and only if

~z (x,y) = mQ z (y) - (Qz (x) Q (y) - Qz2 Cx,yl) z vanishes, always under the assumption Qz(y) j 0. Thus we can take to

~

z

~z(x,y)

as the Hamiltonian for the desired flow restricted

= 0.

As a consequence of

L~e

theorem and the proposition of the previous section

we have 0

if

0

I

0

1

174 i.e. the flows commute. For large values of m we have

2

Yv

n

I - + o(-ml3 av - z

!.~ m z

V=O

and it follows that the zeroes of

are distinct if m is large enough. For

~z

with distinct zeroes one can show now

~z

that the above .relation

{~.~}=0

zl z2 holds without further restriction, i.e. the ¢z ,¢z are in involution (see 1 2 Exercises 1, 2). This holds for large m, but the above expression being quadratic in m, it holds identically for all m, in particular for m

= 1.

I

If we express ¢z (for m = 1) in partial functions Fv(x,y)

(6)

av - z

V=O

one computes the F.J as residues to be (7)

F'V

Being functions

of~

zo

, •••

= Yv2

,~

+

~

(x v

£

~jllv

\T~

- x y )2 ~

v

av- a~

for n+l distinct z 0 ,z 1 , ••• ,zn we also have

zn

{ F'V,F~} = 0

(8)

Thus the fact that the Fv are in involution is a reflection of the geometrical proposition of Section 5 •

Of course, this fact can

also

be

verified by direct calculation (Exercise 3). The Hamiltonian of the line flow for the ellipsoid Q0 (x)

1 can be

expressed in terms of the Fv as ~

THEOREM.

The above

~0 -flow

0

=

is integrable and possesses the rational integrals

Fv given by (7) which are in involution.

COROLLARY.

The geodesic flow on the ellipsoid is integrable and has the

integrals Fv:

The energy takes the form

!. 2

COROLLARY.

n

l:rl 2 =!.2 \!=0 I ' Fv

The real components of the algebraic manifold M: Fv

(V=O,l, ••• n) is a torus, if dF'V are linearly independent of M.

cv

175 Exercise 1.

Define the polynomials

mn 1

A(z) =

zl

,P

(~-

V=O

Show that { p

N

z)

}

z2·

using the identity

Exercise 2.

Let P(z,x,y)

= zn

n-1 + p 1 (x,y)z + ••• + pn(x,y)

be a polynomial

with rational coefficients. If P(z,x,y) has distinct zeroes somewhere then

=0

and P

z2

=0

implies that {P

n

Hint: Factorize P = a point

(x,y)

n

(z -

V=O and for z 1

P

,P

} = 0

z2 in some neighborhood and show that at

= uv(x,y), {p

with c 'f 0 .

zl uv(x,y))

z1

,P

z2

one has

}

= c ·{uv ,u} =0 ~

Thus show that the zeroes u

are in involution, hence also

, p for all z 1 , z 2 ,x,y. zl z2 Note that the assertion of Exercise 2 does not hold if double roots occur If p

then for P

zl

=P

z2

Q 2

z

z

0 we always have {P

zl

,P

z2

}

= 4Q

Q

{Q

zl z2

zl

,Q

z2

}

=0

I

Exercise 3.

Show the functions (7) are in involution by direct calculation.

Exercise 4.

S)low that the n+2 functions (xvY~ - x~yv> 2

and

~-a~

are in involution. Therefore they must be functionally related; indeed n

I

V=O

Exercise 5.

Gv

= o.

Any line tangent to two confocal cones,

Q z.

J

(x)

=0

,

176 or tangential to such a cone and the sphere lxl

=1

has perpendicular normals at the points of contact. (This fac t leads to the expressions Gv of Exercise Exercise 6.

4.)

The tangents of any geodesics on an ellipsoid are tangent to the

same set of confocal quadrics, i.e. independently of the point on the given geodesic (see [5]). With minor changes one can show that the motion on an· ellipsoid under the influence of a potential lxl 2 is also integrable.

This was shown already by

Jacobi [1].

[1]

C. Jacobi,

Vorlesungen tiber Dynamik, Gesammel te

Werke, Supplement band,

Berlin, 1884. [2]

H. SchUth,

Stabilitat von periodischen Geodatischen auf n-dimensionalen

Ellipsoiden, Dissertation, [3)

A. Thimm, 1976 .

Bonn, 1972.

Integrabilitat bien

geodatischen Fluss, Diplomarbeit, Bonn,

(In this paper (in Theorem 4.1) it is shown that the geodesic

flow on the

ellipsoid admits "global" integrals in involution . This fact

is evident fr,_,m our rep res entation of the integrals.) [4]

K. Weierstrass,

Math.

[5)

D. Hilbert and Cohn 7.

An

Werke I, pp. 257-266.

Vossen, Auschauliche

Geometrie, Dover, 1955, p . 197.

Integrable System on the Sphere

(a) A point moving on the sphere Sn: lxl = 1

under the influence of a

quadratic potential U(x) system.

For n = 2

this was shown by

c.

Neumann in 1859 [1]

using Jacobi's

approach of separating variables in the Hamilton Jacobi equations.

First we

proceed differently and show that this system is obtained by reduction of

177 another

integrable system in R2 (n+l).

Then we will apply separation of

variables. The equations of motion are (1)

where A is determined in such a way that the particle says on the sphere. This leads to A=< ax,x> -

(2)

IAI 2

Inserting (2) into (1) we obtain the desired nonlinear system of differential equations which we want to establish as an integrable

one.

(b) We compare the above system with the Hamiltonian system y \)

(3)

=

Cv=O,l, ••• n)

with (4)

Note that this system has the integral lxl 2 since it is invariant under the ;yrnplectic action (x,y)

+

(x,y+2xs)

generated by lxl 2 •

sensible to reduce the above system by this integral, The isotropy group

Therefore it is which we fix at lxl

is G =Rand given by the action (x,y)

+

(x,y+2xs).

1.

We

characterize the quotient manifold {cx,yl

I

lxl =

1}/G

by picking the point of minimal distance in the line y + 2xs, obtaining as reduced manifold (5)

M = { (x,y)

To determine the reduced flow

I Ixl

= 1, < x,y>} •

we determine

in such a way that

The first condition is, of course, satisfied as lxl 2 is an integral of H, hence hence of H0 , but the second yields jJ

and there fore

1 ax,x> r
lxl 2

Dropping the term< x,y>

2

-

~ (lx! 2 jyj 2- 2)

+

(lx1 2-l) + ~ (!x[ 2 jyj 2 - 2) on M we obtain

whose gradient vanishes

and the reduced differential equations (on M)

xv

= Hy0

(I y 12

= y\)

y v lxl 2

+ < ax,x>) x\1 - a\lx'J +· 2~

avxv + ( - jyj 2)xv which is precisely the system (1), (2). The system (1), (2) is obtained from (3), (4) by reduc-

Thus we have shown: tion to the manifold M.

(c) The system (3), (4) is integrable, . and possesses the rational integrals (x

v

V"']..l

(6)

-

x y >2 }l v

a- a \)

ll

These are the functions of the previous section with x,y exchanged -- and therefore are in involution. To prove this assertion it suffi ces to show that H of (4) is a function of FO'. •. ,Fn.

But it is readily verified that 1

n

L

=2 V=O Thus this system and the geodesic flow H

related -- although not equivalent. tion" (x,y)

-+-

(y,-x)

and the

ellipsoid are closely

If one makes the "hodograph transforma-

then the integrals of one go into those of the other.

One is governed by the Hamiltonian (7)

the other by (8)

Although this problem was treated already in 1859 by C. Neumann arrive at the algebraic integrals. Devaney (3]

a few years ago.

he did not

They were found by K. Uhlenbeck (2] and

179 We show how this problem was solved by

(d) Hamilton JacoDi equations.

C. Neumann following the pattern of Jacobi, who used separation of variables ·This technique requires an appropriate

of the Hamilton Jacobi equation .

In this case these variables are "elliptical spherical

choice of variables. coordinates"

which are defined as follows:

(elliptische Kugel koordinaten)

n

For given a 0 < a 1 < ••• < ·an

u. = u . (x) as the solutions of the equation J J n x2 n (z-u,)/A(z) __v_ = (9) J j=l V=O z - av where the u. interlace the a as follows:

n

l:

r-f xv

and x = (x0 ,x1 , ... ,xn) ,

~ 0 define the

0

n

n0

A(z)

(z - av)

J

ao < ul < ·al < ••• < un < a n

(9')

According to

(9)

we have for z = u. (x) J

n '

~

{

2 ~ z-

~

=

Ln x2 V=O V

0 1

Thus the u. (x) can be viewed as "coordinates on the sphere". J

z = u . defines J

the intersection of the sphere with one of a family of confocal cones. From (9) one can express the

x~ rationally in terms of the uj -- e.g. by

computing the residue (10)

XV= A'

Similarly

like the

of coordinates.

U(z) =

where

(a)

These formulae express the x

nj=l n

U(~)

2

(z - u . ) J

on sn in terms of u 1 , u 2 , ••• , un.

elliptical coordinates the u. form an orthogonal system J

Indeed, one computes n

l: V=O

(dx )2

v

n

du~ \ J j;;l gj

(11)

1 U' (u . ) - - ___.J_

4

A(u.) J

To prove this we compute the coefficient of 2 dxv =

The coeffi cient of

duj

d~

L

duj du .

j uj - av xv is found to be

d~ in ds 2 • Since by

(10)

180

1 4

ifj gj

t-

k 1

4

n

I V=O and for j = k 2

n

I V=O

equal to 2

XV

(u . J

-

av)

i d

XV

-:rdziz-'\1

2

I

v

z=u.

1 d

4

= -

U(z) dz A(z)

I

_

U' _ (u.) 1 _ __ L

z=u.

-

4 A(u.)

J

J

J

which establishes (11) -- and the orthogonal character of these coordinates. To describe the differential equation

f

0

we use the variational principle

(T - V) dt

=0

where, by (11), T V=

1

2 1

2

n

n

1

1~1 2

2

• 2 g.u.

I

J J

j=l

2 avxv

I V=O

1

2

n

1 n

I "' - -I 2 1 V=O

u.J

The second equality follows by comparing the coefficients of z- 2 in (9). If we introduce the canonically conjugate variables v. by aT v. =--=g.u. J

~~e

J J

a~.

J

Hamilton function becomes 1

J

n

I

H=T+V=2 j=l where we dropped the unessential constant (12)

uj) Thus the equations of

l!Ption become

u. =

- H

H

v. J

H ( u, as> au

const.

vj and the Hamil ton Jacobi equations J

More precisely, we ask for a solution S

u. J

S(u,n) depending on

n = of the equation H(u, as> = nl au

(13)

Then the canonical mapping (u,v) (14)

....,..

defined by

as vj = au . ,

.

.

J

as

~j =an.

J

takes the Hamiltonian into H = n1 and the differential equation (15)

~j = ojl ,

into

181

This is the standard use of the Hamilton-Jacobi equations. (e) Separation of variables.

The success of this approach depends on the

possibility of solving the above equation (13) which takes the form n

j=l where gj is given

in

(as ) 2

-1 (·gj

I

(16)

au .

- uj)

J

= 2n1

(11).

To solve this equation by separation we employ an identity frequently used by Jacobii n-2 + n2z + ••• + nn is a polynomial then P(u . ) n u~ ---=:J_ = (17) u. ~= nl and U' (u.) j J J=l J J To prove this note that the left-hand side of the first relation is the sum of I f P (z)

n1 z

n-1

,I

the residue

of 1

27fi

J P(z) U(z)

.

I

I

dz

which makes the relation evident. The second can be proven similarly. We rewrite (16)

using the expression (11) for gJ. and B.=- 4 A(u.)S

In

(

B.

J

)

J

2

uj

~-uj=l u;(uj) j

Using the identities (17)

this can be written in the form

I urT--J

j=l

j uj

(B . J

u~.J

2 P(u.>) = 0 • J

Thus we can solve the equation by setting each individual term

s 2u. - ujn- 2 P(u.) J

4 A(u.)

J

Setting

the equation is separated into

(~)2 au . J

which is solved by

s

1 =-

Introducing the polynomial R(z) become

0 .

J

Q(u.) - .....=..__L

4A(u.) J

n

I

2 j=l

= -A(z)Q(z)

the differential equations (15)

182

1

(18)

2

n

Clearly the nl,n2, ••• ,nn to relate them to the

du. for p = 0,1, ••• ,n-1. --1. = 0 p,n-1 IR::) lz=u . dt

I j=1

J

are integrals in involution and it is interesting

rational integrals derived before.

For this purpose

we write

2J&= A(z)

(19)

as partial fractions. n1 , ••• ,nn

It is clear that the coefficients are functions of the

and hence integrals in involution.

We claim

that the Mv

F\1 are

the previous integrals. We sketch the proof:

Since both Q(z)/A(z) and n

I v=O

(20) are integrals of the motion orbit.

z - a \1

it suffices to identify them for some point of an

For this we take the zeroes, say z = q. of Q(z). J

has u = 0 from (18) and hence x

0

n

and therefore the last expression becomes 2

Xv

~

At these points one

U(z)

z - a_ = A(z) \1=0 v l

which vanishes precisely for z (18) it is clear that for u.

J

u1 ,u2 , ••• ,un. 0

From the differential equations

one has u. is equal to one of the roots J

has the same roots and poles as (20) when u. = 0. Since J

both expressions are integrals

they agree up to a factor, which is 1 by

comparing the asymptotic behavior.

I

Hence we have the identity

~- 2J!L

\I=O z - ~ - A(z) • This formula shows that the F\1 are functions of n1 ,n 2 , ••• ,nn , hence integrals in involution. References [1]

C. Neumann, De problemate quodam mechanico, quod ad primam integralium ultraellipticorum

classem revocatur, Journ. reine Angew. Math.

~,

1859,

183 pp. 46-63. [2]

K. Uhlenbeck, Minimal 2-spheres and tori in Sk (informal preprint, received 1975).

[3]

R. Devaney,

Transversal homoclinic orbits in an integrable system

preprint On the separation of Hamilton-Jacobi equations: [4]

E. Rosochatius,

Uber Bewegungen eines Punktes Dissertation at Univ.

G6ttingen, Druck

von Gebr. Unger,

Berlin, 1877

(available at Library of the Math. Institut, G6ttingen) (5]

P. Stackel, fiber die Integration gleichung Halle

mittelst

1891

der Hamilton-Jacobischen Differential-

Separation der Variablen.

Habilitationsschrift,

(available at Library of the Math. Institut, G6ttingen). 8.

Hill's Equation

(a) In recent years remarkable progress has been achieved in the description of the spectrum of the Hill's equation, including the description and construction of those periodic potentials which belong to a given spectrum, the socalled inverse spectral problem. intriguing

We can enter into this complicated and

subject only to a very limited extent.

We want to show that the

integrable system of differential equations of Section

is intimately

connected with the Hill's equati.on in the case of a finite

gap potential.

We begin with a description of the spectrum Hill's equation and the related inverse problem.

The Hill's equation -- so called because Hill encountered it

in his lunar theory -- is of the form (- D2

(1)

where D

= d/dx

and q(x)

= q(x

+ q (x)) cp (x)

=

A cp (x)

,

+ 1) is called the potential of the operator,

which is assumed to be continuous (at least) •

With this operator one can

associate a number of spectra depending on the domains of definition and t t e

184 boundary condition. Perhaps the most common problem is to consider in a dense linear aani.fold of L 2 {R1 ) in which the operator is essentially self adjoint.

In this case the

spectrum is continuous and consists in general of infinitely many intervals extending to + ""

{"band spectrum").

In exceptional cases one has only

finitely many such intervals, one being infinitely

long.

The end points of the above continuous spectrum are given by the operator

- o2 +

q

considered in a dense linear manifold of L 2 (0,2] of periodic per

functions of period 2 {not 1) with the boundary condition

4» (0) = 4> (1) = 0

(3)

This problem gives rise to a discrete spectrum, with simple eigenvalues p 1 ,1J2 , ••• which lie in the following intervals (j=l,2, ••• ) •

· 1 e to choose the parameters Aj

I

~

). ;

so that e (

9

Im z > 0

into

Re w = e(IA2j-l )

e(IA 2 j)

jTI I so

188 that ~(A 2 jl = 2 cos j~ = 2(-l)j.

Thus we can parametrize the spectrum most

effectively by choosing N positive numbers

Im z > 0

the ur..ique schlicht conformal mapping of from

Im

h 1 ,h 2 , ••• ,hN and define 6(z) as onto the domain obtained

w > 0 by deleting the slits Re

W

= +

j~

I

0 < Im w < + h.

and such that, with z!.;i = :!::. ~ , z:0 = :!::. 6(z!.;i) for j = 1,2, ••• ,N; and 6(0)

:!::. ~j 0.

IXj

6(z:0)

--

= :!::.

j=l,2, ••• N

J·

~j + ihj

Moreover, for large values of z we have

6(z) "' z. This shows that the 2N numbers A1 , ••• ,A 2N cannot be chosen arbitrarily, but depend only on theN parameters h 1 ,h 2 , •••

,~. I

I

I

Given the spectrum, however, also the A1 ,A 2 , ••• ,~ as well as the other eigenvalues A. (j > 2N) are uniquely determined. J

A0

= 0,

Consequently, the spectrum

A1 , ••• A2N determines Aj and all the double eigenvalues, hence the

discriminant ~(A)= 2 cos 6(/r).

This takes care of (A).

(d) Description of the potential in terms of an auxiliary spectrum. We ask for the set of all potentials belonging to a given spectrum of the type (7). To fix a potential one may use another spectrum, e.g.

the eigenvalues belong-

ing to the boundary conditions (3) (actually this gives rise to 2N potentials in general.) In this part we will be sketchy

and refer to the paper (e.g. Trubowitz [6]).

We mentioned that for any potential q(x) = q(x + 1)

the eigenvalues

~ -

J

for (3)

lie in the interval A2 J· 1 22\1

"V=O

All solutions on this torus are

= 1.

defining the potentials

q(x + t). it follows that on this

Since the l 2 j-l interlace the l 2 j

torus all the Fk have positive values. All solutions q, 2\l are hyperelliptic functions of x and so are the corresponding potentials according to (13), both periodic of period 2 or 1. One can determine a particular potential with given simple spectrum l 0 ,l 1 , •.• ,l 2N by setting

~j =

as the "origin" on the torus

l 2 j-l = ~j are roots

of

A2 j-l"

.J!l.)

y 2 (l,l)

(In McKean-Trubowitz

this is choser

Then q (x) is an even function and the while the l 2 j are the roots of y~(l,l).

Therefore for thL choice of the potential q(x) one haS y 2 (l,z)

Yi (l,z) Since the roots of y 2 (l,z), yi(l,z) in A2 j-l obove expression is in fact a rational

N the

function.

It is suggestive to investigate the potentials belonging to other tori, i.e. to different values of the constants

cj

= Fj.

elliptic, but now in general quasi-periodic.

All these solutions are hyperSince very little is known about

the spectral theory of quasi-periodic potentials it would be worthwhile investigating even these special examples

which present themselves through

195 this surprising connection between Hill's equation and the mechanical problem. (This connection was found by E. Trubowi tz and the author.)

However, this

approach has not yet been carried out. References [1]

A. A. Dubrovin,

v. B. Matveev and

S.

P. Novikov, Nonlinear equations

of Korteweg de·vries type, finite zone linear operators varieties, [2]

Russ. Math. Surveys 31 (1976)

v. A. Marcenko and

I.

of Hill's operator, [3]

H. Hochstadt,

v. Ostrovskii,

and Abelian

59-146.

A characterization of the spectrum

Mat. Sbornik 97 (139) 1975, pp. 493-554.

On the determination of Hill's equation from its spectrum,

arch. Rat. Mech. Anal., Vol. 19 (1965) 353-362. [4]

H.P. McKean and P. van Moerbeke, Inventiones Math.

[5]

in the presence of infinitely many branch points, Comm. Pure

Appl. Math. E. Trubowitz,

29 (1976) 14 -226. The inverse problem for periodic potentials, Comm. Pure

Appl. Math. 30 (1977) [7]

217-274.

H. P. McKean and E. Trubowitz, Hill's operator and hyperelliptic function theory

[6]

30 (1975)

The spectrum of Hill's equation,

N. Levinson, (1949) 25-30.

321-337.

The inverse Sturm-Liouville problem, Mat. Tidsskr. B.

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)

HOPF BIFURCATION FOR INVARIANT TORI

A. CHENCINER

HOPF BIFURCATION FOR INVARIANT TORI ( *) by A.Chenciner- I.M.S.P. Mathematics Dept. Pare Valrose - 06034 NICE Cedex -France

I want to describe some joint work with Gerard IOOSS of Nice University connected with the Ruelle and Takens deterministic approach to turbulence. The main references are [R.T.J,[C.I.1.],[C.I.2.]. Navier-Stokes equations can be thought of as a flow in some infinite dimensional Banach space of divergence-free vector-fields in :FI 3 with some Sobolev norm (for regularity properties analogous to those found in the finite dimensional case, even though we-deal with unbounded operators, see [1]. As a certain parameter varies (for ex. the T.aylor number in Couette-Taylor experiment) the flow is at first supposed to undergo a sequence of bifurcations of the Hopf type, eventually leading to a stable (this means here asymptotically stable) invariant r-torus, r nottoo big say r

= 4).

The transitions from a stable equilibrium to a stable closed orbit and from

a

stable closed orbit to a stable invarianc 2-torus are well understood "generic" phenomena (see [R.T.]; here "generic" is in the very strong sense that a finite number of quantities be non zero); on the other hand, the next bifurcations, starting with the transition

from a stable invariant 2-torus

to a stable invariant 3-torus appear to be highly exceptionnal (in any reasonnable sense, measure or category): it is the purpose of what follows to describe, and discuss on an example, sufficient conditions for this last bifurcation to occur (for the general case, see [C.I.1.y. First, as in [R. T] we replace the flo\15 by maps, thus slecreasing by one the dimension of invariant sets; more precisely, suppose that we consider

(*)

Slightly expanded version of a conference given at the Bressanone CIME session on dynamical ~ystems, June 1978.

200

an invariant 2-torus for a flow and let us make the assumption that the restriction of the flow to this torus admits a cross section (i.e. a transverse closed curve on which the Poincare return map can· be defined); such a cross - section exists for example if the flow on the 2-torus is close to a quasi-periodic one, or if the 2-torus recently bifurcated from a closed orbit. We extend this Poincare-map to get a cross-section of the flow in a neighborhood of the 2-torus; if the restriction to the transverse closed curve of the normal bundle of the 2-torus in ambiant space

is trivial (an assump-

tion which can be easily achieved by eventually adding variables) the flow in the neighborhood of the invariant torus is described by a map F : T1x ~ , T1 x 0 where

'I,·

~ T1 x E , T1 x 0

is a neighborhood of 0 in the Banach space E , and T1 is the circle.

This leads to the following problem (compare with [R.T.J where bifurcation from a closed orbit into a 2-torus is reduced to Hopf bifurcation for maps): Let F

IJ.

: T1 x ~ ~ T1 x E be a Ck mapping.

Suppose that for

1.1.

negative small enough, F admits an invariant circle IJ.

1

getting close to T x 0 (invariant by Fa) when

1.1.

goes toO, and

exponentially stable. Suppose moreover that for 1.1. > 0 there is no stable invariant closed curve near T1 x 0 for F : what is the new attractor (if any) ofF in a neighborhood of T1 x OIJ.for IJ. > 0 small ? 1.1. The following

thec~em

gives sufficient conditions for the appearance for

> 0 small of a stable invariant 2-torus of F , on which F is isotopic IJ. IJ. to identity(so that the attractor for the flow is really a 3-torus). In

1.1.

fact the hypotheses are strang enoUgh to imply from the mere ofT

1

invariance

x 0 under Fa, the existence for ~ small (positive or negative) of a

closed curve invariant under F That such a c.u rve exists for

1.1.

1

IJ.

and getting close to T x 0 as

1.1.

goes to 0.

> 0 does not seem true in general, at least

if the rotation number of Fa!T1 x 0 is rational (see [BJ and [C.Y.J}· To state the theorem, we need some notations and definitions The circle T1 will be identified with ~/Z, and formulas will be 1

1

written in ~ (for ex. a map from T to T will be lifted to a map from :fl to ~ ) • The rotation R

X

is defined by the lifting R (g) W

e+ w

•

201 If (

e I X)

F (e I x) IJ. For any G0

(

eI

g(e)

X)

=

E T1 =

F(e

X

'If

I X I

we write

I

IJ.)

=

(f(e

I X I

IJ.)

I

1

(e, x)E T xE let us .set = (g (e) I Ao (e) X )e T1 X E

f(e,o,o), Ao(e) 1 ~

I

~ce

I X I

IJ.J) E T\ E •

where

~! (e,o,o).'_,

We make the following assumptions : F is. ck I k large enough, g( e) is a Ck-diffeomorphism, t( e I 0

DEFINITION 1 :

I

o)

o.

for J., ~k-1, the J.,-spectrograph of F 0 is the spectrum of the

linear map

defined by

(Go cp) (e)= Ao(g- 1 (eJ) q>(g- 1(eJ) i.e. graph (ao cp)= Go (graph cp) in T1 space of cl maps from T 1 to E • Remark

X

E

if0is a small enough neighborhood of

I

for any cp E c 1 (T\ E) the

o in c1 + 1 (T 1 ;

E), the for-

mula graph (:1 0 cp) = F 0 (graph cp) defines a map :'f'o:

{j....

cl+1(T1; E).

This map is not differentiable at 0, but the composed map

(j~cl+ 1 (T\ E)

c:..._cl(T\ E)

is differentiable, and its derivative at 0 is precisely ao •

DEFINITION 2 :

Suppose that the rotation number Wo of g is irrational; 2·n n 0 an eigenvalue ). 0 =p e ~ of G0 is called "non-real" if for all q E Z , 2(l0 + qwo ~ Z.

Motivation : one can easily show that to such a non-real eigenvalue may be associated a c1 - subbundle r;2 of T1 x E isomo-rphic to T1 x :Fl 2 , invariant under G0 , on which G0 is c1 - conjugate to the map

z),

(e,z) 1----. (g(e), 1. 0 2 where m has been identified with C • Notice that if g is actually C1-conjugate to A

Wo

1

wo irrational, the

1 -spectrograph of F 0 contains the whole circle of center 0 and radius

202

·p as soon as it contains the eigenvalue A. 0 = p e 2i.TTCla •

THEOREM : keeping the above notations we make the following assumptions 1)

2)

g is c 1 -conjugate to the irrational rotation R , t large enough. Wo The t- spectrograph of F 0 may be written cr1U cr2 , where cr1 is contained in a disk of C centered at 0 of radius less than one, and cr2 co!ncides with the unit circle; moreover one assumes that cr2 is "generated" by a single non-real eigenvalue A.o = e 2inno in the sense that the decomposi-

e1e e2 into closed u0 - invariant subspaces relative cr1 and 02 satisfies e2 = rLC~) the subspace of cL-sections of the G0 - invariant subbundle ~;2 of T1x E described after definition 2. tion cL(T 1 ; E)=

3)

(i) rn 0 + qwo , Z

for

r = 1,2~3,4,

q

to

EZ

(ii) 3 e E ]o, [ , C > 0 , such that 'V p E Z, 'V q E Z-0,

I rno

+ qWo-

PI

> ~ '

lql

+~

for r

=1

and r

= 3.

Then one has, in general , the following conclusions :

1)

For small

~

1

there exists a closed curve near T x 0, invariant under

F

, and depending continuously on ~ ; the stability of this invariant IJo curve changes when ~ goes through the value 0 (and depends on the

sign of a certain quantity which we suppose non zero). 2)

For small ~ > 0 or small ~ < 0 (depending on the sign of another quantity which we suppose non zero) there bifurcates a 2-torus invariant under F is the

op ~ osite

~

and depending continously on

~

, whose stability

of the one of the invariant closed curve described

in 1) J the restriction ofF ~

to this 2-torus is isotopic to identity.

Moreover, the distance to T1x 0 of the invariant closed curve 1

(resp.invariant 2-torus) is of order ~ (resp. order ~~~~).

Discussion of the assumptions : (a)

From 1) follows that the restriction of the original flow to the invariant torus whose F 0 is Poincare map defines the same foli a tion (up to diffeomorphism) as a quasi-periodic flow. Notice however that in 3) there is no condition on the appro ximation of wo itself by rational numbers, so that this flow need

not be

203

equivalent up to diffeomorphism to a

quasi-periodic flow.

Anyway, in view of Arnold-Herman theorem (see [HJ) the situation of hypothesis 1) occurs with non zero probability and so is worth studying. b)

The non-resonance assumptions 3) (i) are fundamental (see [C.I.1.J Chapter IIJ for

a:

study of what happens in general in the "strong

res_onance" cases where3) (i) is violated) but it is unclear whether or not it is possible to get rid completely of the two remaining diophantine approximation assumptions 3) (ii) (see [C.I.1.] Chapter V and [C.I.2.Jft) as only truncated normal forms are important this could be expected c)

(?).

Assumption 2) is the most stringent and also the most interesting because of its geometrical meaning: it implies that G0 leaves invariant all the 2-tori of a 1-parameter family foliating the subbundle ~;2 ~ T\ :fl 2 of T1x E and that, on each of these 2-tori , G0 is C·(conjugate to a map of the form (e, 'Ill)~ (e + w0

,

\lr+Oo).

Recall that in the case of Hopf bifurcation for maps (as described in [R.T.J) the existence.of a 1-parameter family of invariant circles for d F 0 ( 0) is automatic from the spectral hypothesis; here i f the spectrum of u0 is as in hypothesis 2) except that cr2 is not generated by an eigenvalue (which is the general case for such a spectrum) . the restriction of G0 to ~ may well have a dense orbit! Some geometrical hypothesis of this kind is definitely needed if one looks for a nice bifurcation theorem: namely, if one starts with a 1-parameter family of maps 1 1 1 1 F : Tx?r, TxO ... TxE, TxO lJ.

for which the conclusions of the theorem hold, the invariant 2-torus which bifurcates being the image of a map from T2 to T1x E of the form ( e' w) .... ( e' x( e' w)) with x( e' w) = IIJolaxo( e' w) +o( h;.r:r), Xo regular, then G0 leaves invariant each 2-torus image of a map · ( 9 , \lr) .... ( 9 , COnstant

.oc

X0(

9 , W)) •

This is not surprising if we view the Hopf bifurcation theorem for maps as an "unfolding" in the IJo direction of a 1-parameter family of invariant circles for d F 0 ( 0) in the plane

(*)

1.1.

=

0.

Note that these assumptions are almost everywhere satisfied in the plane of couples Cwo , noJ

204

linear situation

general situation (the direction of the parabolo!d may be reversed).

Picture in Fl 2 with assumption that F (o) = o.

'"" The word "in general" in the theorem means that conclusion holds provided two quantities are non zero: as these are best understood

iv)

after some changes of variables, I refer to [c. I.1. J for an explicit statement; let me just say that they correspond to the Ruelle-Takens assumptions that eigenvalues of, d F 0 ( 0) cross the unit circle transversely and that 0 be vaguely attractive for F 0 • A warning is necessary at this point: thanks to the first conclusion of the theorem, we may suppose that F (T 1x 0) cT1x 0 for 1.1. small 1.1. (after a change of coordinates eventually) so that a linear map G can be defined in the same way as was

IJo·

Go • The hypothesis do not imply that

the spectrum of GIJo contains an eigenvalue. In fact, the cr2 - part of the spectrum of G is expected to explode into an annulus at values of 1.1. 1.1o for which f( e, 0 , ~J,) has rational rotation number, but the hypothesis can be shown to imply that it stays away from the unit circle as soon as

1.1.

is non zero (and small).

Quick sketch of the proof : 1)

The steps are as in [R.T.J. Reducing to a problem in T1x Fl 2 by center-manifold technique : the spectral assumptions on fio and the irrationality of the rotation number

205

of g are used to prove a center-manifold theorem which, when applied to of T\

~o~o) ...

(F (a ' x)' IJ.) exhibits for IJ. small a submani fold 'II , diffeomorphic t~ T\ JR 2 , which contains all the local

the mapping (a ,

X ,

1

recurrence of F in the neighborhood of T x 0 • IJ.

More precisely one starts with a change of coordinates eliminating terms of order

1.1.

in ~(a, x , ~o~o) so that the theorem could be applied

(see [c. I. 2. J).

2)

Finding truncated normal forms for F approximately) equations of the form

one has to solve (resp.to solve 1.1.

est , which account for the diophantine approximation assumptions. 3)

Using classical fixed point theory in Banach space: the truncated normal form allows one to find a good approximation to the seeked invariant set, from which standard iteration technique works to get the result. One prove in this way

1)

The persistence of an invariant circle near T1x 0 for F

this is

IJ.

more complicated that in the Hopf case where the existence of a fixed point near 0 for F follows, via the implicit . function theorem, from the fact that F 0 (0J = O, and d F 0 (0) does not admit 1 as an eigenvalue. 1 The invariant circle one finds is the graph of a mapping from T to E

of the form

2)

The change in the stability of this invariant circle when

1.1.

crosses

the value 0 (here diophantine approximation assumptions can be avoided and hypothesis 2) can be weakened, see [C.I.1.J chapter v).

3)

The bifurcation result : the invariant 2-torus one finds for the map 1

F may be written (after identification of ; 2 to T x lR t~e set of ( a , r e 2i TT such that

'iJ

2

= T1x

C) as

206

Question :

do a persistence and a bifurcation result still hold if one replaces the assumption on the existence of an eigenvalue for u0 by the more geometrical assumption that there exists an invariant subbundle ~;2 for G0 , foliated by G0 -invariant 2-tori ? Of course the restriction of G0 to these tori is not necessarily supposed to be

isotopic to identity so that, for the flow, we do not . expect an invariant 3-torus to appear, but instead a non-trivial fiber space over r 1 with fiber~. For some partial results, see [C.I.1~] Chapter V. An example: the following example, withE= JR, is highly non generic but one expects generic examples of this kind to occur for dim E > 1 • Let F

IJ.

T1 x :fl -+ T1x :fl be defined by

FIJ.(e·, x} = (e-+:.wo, (1 where a(e)>O is

s 1

c•

+~J.)

a(e)(x-x3 )),

(or analytic),

!-og a( e) d e= 0 ' Wo E :fl . Q •

0

It is easy to show that for any J, the t -spectrograph of F is the IJ. circle of center 0 and radius 1 + IJ. : from this it follows that, in a neighborhood of T1x 0 , F is topologically conjugate (for IJ. small IJ. non zero) to the mapping G defined by IJ.

GIJ.(e, x) = (e+w•• (1

+~J.)

a(e) x)

(see [H.P.S. J).

1) 2)

Now let V be the interval [O,CJ in :fl. If max a( e) -min a( e) is not too large, iT. is possible to find a C such that the restriction of F to T1x V is one-to-one. 1

1

IJ.

F (T x v) c: T x [o,c[ for ~mall. IJ. The intersection n~ F:(r\ JO,CJ)contains then, for IJ.>D small, a compact set K invariant under F and having no intersection with 1 IJ. IJo T x 0 (we use here the fact that, for~> 0 and n large enough, G:(T1x [fl,aJ) contains T1x (0, a] for any a ). Let us try to analyse K : if, as in the discussion of hypothesis 2) IJo of the theorem, we assume that K is the graph of a function IJo e ... IJ.Q' Xo (e) + 0 ( !~J.I Q'), Xo regular' then log Xo ( e) is easily seen to be solution of the difference equation.

(*)

207

On the other hand, if w0 is too well approximated qy rational numbers) (say, if w is a Liouville number) it is possible to choose a( e) 0

analytic such that the above equation had even no measurable solution. Notice that

(*)

having a regular solution amountsto the existence of a

change of variables for F0 transformin~ a(e) into 1, so that all the circles T1x est are invariant under G0 after this change of variable~. If

(*)

has no continuous solution, it was shown by Hedlund that G0 has a dense orbit in T1x JR+ (resp. in T1x Fl-) •

Question: what can one say on K in this case ? ~

IJ.

: if in the formula for FIJ. one replaces 6 + Wo by 6 + Wo + 1J.

UJot ,

the spectrograph of FIJ. explodes at rational values of e + Wo + ~ Wot and is no more on one side of the unit circle. What happens then ?

BIBLIOGRAPHY [BJ R.Bowen, A model for Couette flow data, in Berkeley turbulence seminar, Lect.Notes in Math §12, Springer Verlag, Berlin 1977. [C.I.1.J A.Chenciner, G.Iooss, · Bifurcations de tores invariants, to .appear in Archives of rational mechanics and analysis,1979. [c.I.2.J A.Chenciner,G.Iooss, invariants, to appear.

Persistence et Bifurcation de tores

[C.Y.] J.H.Curry, J.A.Yorke, A transition fro~Hopf bifurcation to chaos: computer experiments with maps on Fl • Preprint 1978, [H] M.R.Herman, Mesure de Lebesgue et nombre de rotation, in Lect. Notes in Math, §@2 1 Springer Verlag, Berlin 1977,p.271,293. [H.P.S.] M.W.Hirsch, C.C.Pugh, M.Shub, Invariant manifolds, Notes in Math. ~. Springer Verlag, Berlin 1977,

Lect.

[I.JG.Iooss, Sur la deuxieme bifurcation d'une solution stationnaire de systemesdu type Navier-Stokes. Arch.Rat.Mech.Anal.~,4, p. 339-369 ( 1977) • [R.T.J O.Ruelle, F.Takens, On the nature of turbulence.Comm.Math. Phys.20,p.167-192 (1971).

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)

LECTURES ON DYNAMICAL SYSTEMS

SHELDON E. NEWHOUSE

C. I. M. E. Summer session in Dynamical Systems, Bressanone, Italy, June 19-27, 1978.

Partially supported by N. S.F. Grant MCS76-05854.

Contents '~'

Introduction

213

1.

Periodic points, flows, diffeomorphism&, and generic properties

215

2.

Hyperbolic sets and homoclinic points

223

3.

Homoclinic classes, shadowing lemma and hyperbolic basic sets

234

4.

Hyperbolic limit sets

245

5.

Attr~ctors--topology

254

6.

Attractors--ergodic theory

1'64

7.

The measure

275

8.

Diffeomorphism& with infinitely many attractors

288

References

soB

~A

213

Introduction. A basic question in the theory of dynamical systems is to study the asymptotic behaviour of orbits.

This has led to the development of many

different subjects in mathematics.

To name a few, we have ergodic theory,

hamiltonian mechanics, and the qualitative theory of differential equations. A particularly baffling and interesting problem is to describe systems with non-trivial recurrence. diffeomorphism

f

For example, consider a smooth area preserving

n2 •

of the two dimensional disk

Poincare recurrence theorem, almost all points in

on

x

infinitely often.

x

accumulates

On the other hand, if we consider the

on the two-dimensional torus

g

That

are recurrent.

However, except for simple cases, we have no global

model that describes all the motion. mapping

D2

off a set of Lebesgue measure zero, the orbit of

x

is, for

the

According to

r2

induced by the matrix

(2

tl

~·

However, in this case not only

then again almost all points are recurrent.

do we have a fairly good picture of the total motion, but this picture persists for any

g'

which is

c1

close to

g.

This example gives an indica-

tion of some remarkable progress which has been made in describing non-trivial recurrence during the last twenty years. The main feature possessed by the toral mapping shared by

f

is what is called

hyperbolicity.

g

above which is not

In its present form, this

concept arose in the work of Anosov on geodesic flows on negatively curved Riemannian manifolds

[ 2 ].

It was subsequently realized by Smale that hyper-

bolicity could be used to describe other systems with non-trivial recurrence, and this led him to define Axiom A diffeomorphisms.

The recurrent orbits

in Axiom A diffeomorphisms lie in certa in sets which Smale called hyperbolic

214

~~·

These sets have been studied by many authors.

Perhaps the most

significant result about hyperbolic basic sets is that .t hey can be modeled by certain symbolic spaces (called subshifts of finite type), and this gives one very precise information about their orbit structures. If one thinks about the structure of hyperbolic basic sets a bit, one realizes that they are special cases of certain ·sets which we call h-closures which exist for many diffeomorphisms.

At the present time we have relatively

little information about the fine structure of non-hyperbolic h-closures. but hopefully we will understand more about them in the future. Our goal in these lectures is to introduce the reader to some of the results in this fascinating area of mathematics.

Considerations of time and

space have forced us to choose a rather limited set of topics to present here . Our intention has been to describe a variety of results with a special emphasis on the theory of attractors.

While many references are given in the

ensuing sections, we recommend that the reader consult the recent survey of Bowen [ 9l, and the lectures of Ruelle !49]

for different perspectives .

215

1.

Periodic points, flows, diffeomorphisms, and generic properties .

In this section, we shall begin to motivate the concept of hyperbolicity. First, let us consider the relationship between flows and diffeomorphisms . Let

Ck mapping from

a

for each

x

€

M.

always assume ¢> : lR ·>
·

: lRxM +M

[s + t], f[s+tJx}}

with

216 where

[s + t)

Examples:

1.

is the greatest integer in Let

0

as

IR/Z




(actually defined for forward time only so we

should say semi-flow) whose orbits come in at the boundary. could make this .a global flow (via differentiable change coordinates)

s 1 x D2

embedding

extending the vector field of

4>

in

We

of

s3, the 3-sphere,

to all of

s3,

and taking

the flow of this extended vector field. If

4>

is the suspension of

qualitative features, and As we shall see, with

~his

f

f,

then they have essentially the same

gives us the advantage of one less dimension.

enables us to describe interesting 3-dimensional flows

2-dimensional diffeomorphisms. For most of these lectures, we shall be concerned with diffeomorphisms.

We begin by describing certain generic properties of diffeomorphisms. an integer We give

ck

k ~ 1,

DiffkM

and let

be the set of

a topology as follows.

diffeomorphism from

space

DiffkM

1Rm, m =dim M.

Ui

diffeomorphisms of

Fix a finite covering of

onto a bounded open set

Choose the pairs

ck

Vi

Fix

M by

in the Euclidean

M.

217

a= (a1 , • •• ,ar)

If

tinuous.

JaJ "'a 1+ . • . +ar.

set

For

are uniformly con-

k

partial derivatives of order less than or equal to

is a multi-index of non-negative integers , we

f,g

€

let

DiffkM,

:5:. i

s n, lsjsn,JaJ

$

k,

and

x·e: v.} J

where

D~' (O

e > 0,

let

is a partial derivative at B}(e)

= {g

€

Diff~ : ~(g,f)

neighborhood base for a topology on

1; of order

of

X

The sets

< £}.

JaJ. Given k Bf(E) form a

Diff~ called the uniform ck· topology.

This topology is independent of the choices of the charts

{(Ui,~i)}.

A point

Let us consider the local structure near a periodic point. p

€

M is periodic for

The least such period

n

is called the period of p.

is hyperboZioff the derivative

n

n

if there is an integer

f

Note that

eigenvalues of absolute value 1.

fn

of

Tpfn : TPM

automorphism, and any two local representativ~s of Thus, the eigenvalues of

automorphisms.

fn

p

at +

TPM

Theorem (1.1). point of a of

p

hfh- 1

T fn p

Suppose

ck diff eomorphism f ·: M + M •

in M and a homeomorphism

h : U

+

p

has no is a linear

TPM with

E.m •

i s a hyper bolic fi xed

The r e i s a neighborhood lRm

suoh t hat h(p)

U

= 0 and

= Tpf wher e both sides are defined. Thus, via a continuous change of coordinates,

mapp i ng

T f. p

The structure of

T f

lR m

= Es

a di rect sum decomposit ion that

of

are well-defined.

as a linear isomorphism of

(Hartman and Grohman ).

= p.

define conjugate

For convenience of notation, we will frequently identify 1R m and think of

p

A periodic point

T fn p

fn(p)

so that

2: 1

TpfJ Es

f

looks like the linear

is given by l i near a l gebra .

p

$

Eu

Ther e is

into two invari ant subspaces so

has eigenvalue s of norm less than one while

T f!Eu p

has

218 eigenvalues of

no~

For some norm on

greater than one.

lRm

'

Some hyperbolic linear automorphisms are sketched in the next figure.

=

f(x,y) f(x,y) • ('l!x,>.y)

0


is

i ,f z)

~x,f

E:

~~(x)

ho~

oo

+ d(f ~y

x

A(f)

+

o0

~

i i i i ) + d(g y ,f ~Yn ) + d(f ~y ,f z) nk k nk nk

i

e:

2

By the

i.

i,

But, for all

Thus, we have proved that for each E

k

as

x

i i i i i i d(f ~x,f z) ~ d(f ~x,g x) + d(g x,g y ~

A(g), there is a

E

be a subsequence such that

(yn )

Let

k.

E

x

is not continuous at

~

for . all

Let us prove that

is unique with this property.

E:, ~(x)

continuous.

2E:

g -> h

g

Then,

Clearly,

is continuous.

f

which has a local product structure and has

fiA

has a dense orbit is called a hyperbolic

basic set. Such sets have been studied a great deal.

Their orbit structures can

be modeled very well by certain generalizations of the 2-shift described above. Let

J

= {l, ... ,N}, and let

of the elements of shift on

EN

N-shift.

If

J

EN= J 2

be the set of hi-infinite sequences

with the compact open topology.

as before by

c:r(~)

is an

(i) =

~(i

+

1)

for

One defines the

~ E

EN.

N x N matrix whose entries are

This is the full O's and

l's,

243

we may consider the subset

LA c LN A

~(i)

defined by

LA

Thus a sequence ( ... a_ 1a 0 a 1 , ... ) is in yields

aiai+l

iL

for all

if, and only i f each of its 2-blocks

A.

when used as indices for the matrix

1

a-invariant set and

a closed

1

,:!!_(i+l)

oiLA (or sometimes

LA

The set

LA is

itself) is called a

subshift of finite type. :meorem (3. 8).

Let

be a hyperbo lia basia set for a c1

A

Then there are a matrix A of

f .

tinuous surjection

~

and l's

O's

and a finite- to-one con-

: LA+ A s o t hat the following diagram aommutes

This important result was proved by Sinai [53] by Bowen [ 4] for genera l bas i c sets. of

diffeomorphism

when

A

= M,

and later

The proof involves special coverings This is treated

A by local product sets called Markov partitions.

nicely in [ 7 ] • The mapping of

~

-1

matrix

[ 9].

The space

and the cardinality

on many points in

is

for each

is bounded by

(x) A

~

LA

x

is the order of the

where

f

codes the action of

A in a very

on

comprehensible way , and can be used to prove many facts about instance the minimal sets in

For

A are zero-dimensional [ 5 ], [22], and one

can compute the number of periodic points of period each

fi A.

n

of

f [ 68 ]

for

n .::_ 1.

Remark:

1.

Theorem (3 , 8) holds for hyperbolic sets with local product struc-

tures (i.e. with1:Jut .as s uming

·f iA ha s a>O so that for g

in

f

a homeomorphism from

f

Say that

and no cycles.

0-stable

f

must satisfy Axiom A and must

The analogous result holds for absolute structural stability

Another characterization of Axiom A and strong transversality is

Franks' time-dependent stability [66]. The main importance of the Axiom A and no cycle diffeomorphisms is that, at present, they give the largest open set of diffeomorphisms whose orbi t structures are well understood.

254

Attractors - topology

5. Let

: M-+- M be a

f

c1 diffeomorphism. A closed £-invariant set U of

A is an attractor if there is a compact neighborhood f(U) c int U, r-\fn(U) ~A,

that every

x

ri~

U, w(x)

~

has a dense orbit.

f!A

is the w-limit

w(x)

A where

c

and

A such

set of

Thus, for The open

x.

uf-~ is called the basin of n

r

for

i

that

0,

X

(z'' w')

z

€

2:

sl.

0,

€

U and

V are

fn(U n A) n V n A

Sl,

E

>'

.p.

x

o2

Toward this

N

there is an integer

(z,w)

E

v n A.

1le: (( z, w))

for small

e:,

g-nl(z)

such that

fnl({zl} x D2 )c V.

u n A and a small

e:

>

0

>

= sl.

gn(I)

o2 _,. sl be the projection. Let

zl

s1

open sets in

Note that

n

n

is an interval in

such that Let

has a dense orbit using proposition (3 . 1)

to show that if

then for some

g(z)

(a)

f!A

0

Taking

so we may choose an integer Now pick

such that Wu((z',w')) c U n A. e:

rWu (( Z' , W 1 ) ) is an interval in sl, we can find an e: n u rf 2We:((z',w')) = sl. Then, there is a point p in

n2

>

0

such

256

f f

n2

2 n {z 1} x D.

W~((z', w'))

nl+n2

(U n A) n V n A

f

Hence,

-n

2

£UnA

p

and

f

nl

p

E

so

as required.

~ ~

The construction of the solenoid (as a hyperbolic attractor) to Smale

V n A,

is due

-[ 56 ] •

It leads to a general construction of one dimensional hyperbolic To describe this construction we need

[ 61 ] •

attractors due to Williams some definitions. Let

and

~1 : 1R +1R

be defined by

1R+:JR

~2

1

cpl (u)

=e

-~

u

~Pz (u)

0

u

~

u

=0

=e u

u

>

0

0

u

~

0

0 1 --"7

S be the set of functions a • cp 1, B • cp 2 where B varies

Let

through the real numbers. The graphs of elements of {(u, v)

£

1R 2

have infinite contact with

(0, 0).

at

v • 0}

S

A compact branched !-manifold

K is compact Hausdorff topological

space satisfying the following property. such that

each point

x

in

There is a finite subset

Bc K

B has a neighborhood which is homeomorphic to

a finite union of graphs of elements of

S,

and each

x

E

K - B has a

neighborhood which is homeomorphic to a real open interval. Typical pictures of branched !-manifolds are in Figure 5-2.

Figure 5.2

257

The set

B in

K is called the branch set

There is a finite open covering there is a homeomorphism CL Yi • {(u,

¢>~(u))

of

{U } CL

¢>a : Ua -

Ya

e lR 2 : ¢>iCL

€

{ (U '¢> ) }

defines a

of

K.

K such that for each

Ua

CL

and

CL

Ya • Y1 u ••. u Yn

where

S},i cl, ••• ,n • ~1he open intervals correspond to

¢>~ = 0. The family

CL

CL

as usual by saying that a function and for each V

of

CL

a,

¢> (U ) CL

f to

CL

o ;

lR •

1

differentiable structure on f : K + lR

extends to a

CL

is

Cr

Since the graphs of two elements of

One defines

Riemann metrics and

branched !-manifold can be

Cr

embedded in

We now always assume that with a fixed Riemann metric.

I· I

Let

0, >.

>

1

so that

For example, if

sl n s2 - {p},

c1

r

Cr Cr

0.

~

branched !-manifold TK.

so that there are constants

ITx(gn)(v)l ~ c >.nlvf for all xeK, n2:0, and veTxi

K has a well-defined

Every compact

JR3

K is a compact

S have

Crmaps between

branched !-manifolds and other manifolds as usual.

c

if it is continuous

Cr function- from neighborhood

infinite order contact at any point where they meet, tangent bundle.

K

= zn,

n

>

0

on

K • s1,

then

g

is expanding.

K to be a wedge of two circles, say

and let

g

be the map

Cf)., sl

g

described

sl

>

sl + s2 + sl

s2

7

sl + s2

s1 u s 2

in figure 5-3.

Figure 5.3

With the indicated orientations, lengths on

s 1 , and

g(p) •

p.

g

doubles lengths on

s2 ,

triples

This expanding map is intimately related

258

(i i)·

to the Anosov diffeomorphism induced by If

g

K+K

the set

K

= { (a0 ,

is a map, we define the inverse limit of a 1 , ... ) ; a 1 E K and

One frequently writes

g

K+ K

~

• lim K

K

defined by

+

~(~)

= a0 •

J

a 1 , a 2 , a 3 , etc.

with inverse

Now, one can prove that

2 g (z) .. z ,

g

....

g : sl

if

is topologically equivalent to

sl

fjA

n2

r: s 1 x

the projection, then

rfr -1

t n (S 1 x D),

an(z) • rf get that

-n

for

z

n:O 2 Let {h(a}}=n fn({a}xD). Then h isalsocontinuous,and ho~=id., o h • idsl•

"

n

n2:0

~

As

~f

•

g~,

we see that

is a topological conjugacy.

~

Williams has given general theorems of this type.

Theorem 5.1 (Willi'ama

[ 61 )J.

Suppose that

f

is a

c1 diffeomorphism

naving a 1-dimensional hyper150Ua attraator A UJith splitting

TAM .. Es e Eu

of a branahed 1-manifold

and dim Eu .. 1

Then there is an expanding

K suah that

is topologiaally aonjugate to the inverse limit map

In [ 61),

fjA

77tzp

g

Williams assumed that the stable manifold foliation was

It is well-known now that this assumption can be removed as follows. U be a small fundamental neighborhood of

A.

g.

c1 .

Let

Approximate ·

259

f

by

f1

so

th~t

is

£1

topologically conjugate to

c2 . f!A. f1

stable manifold foliation of There is a converse to

Theorem (5 •.2)

and

~

K+K

g

every point of

(b)

eaah point in

K

be an expanding map of a

K

is non-ux:zndering

has a neighborhood whose image by _ a power

is an aro:

Then thel'e is a diffeomorphism f . : 54

+

54

on whiah it is topologiaally aonjugate to

whiah has a hyperbolia attraator g.

The idea of the proof of theorem (5,2) is as follows, K in of

1R 3 .

4>(K) .

via

4>: K + 1R

and let

4>(K)

First embed

N be a "tubular neighb-orhood''

This is a 2-disk bundle over

branch points of

the

suah that

(a)

gm of g

(5.1).

Let

[ 17 ],

c1 •

U is

on

f 1 Qf~(U) is

(6.5) in

By theorem

the~rem

(Williams [ 61] ).

branahed 1-manifold K

By theorem (3.72,

4>(K)

where the corners at

have been rounded off to look like

pants legs as

in figure 5.4.

Figure 5.4

One may write

N as a union of

{D }, x € 4>(K), in which

2-d i sks

a t most two 2-disks have a point in common.

X

If

one forms the quo tient

260

space by identifying to a point any two 2'-disks which intersect, one gets a homeomorphic to

K1

space

The map the map

g1

~ o

K.

Let

g : K ~ N may be approximated by an embedding 11_ : K~ N , and

~-l : ~(K) ~ N extends· to a diffeomorphism

o

and

its interior so that each x e:

be the identification map.

n : N ~ K1

g 2 from

g 2 contracts

D

X

N into for .

~(K).

g2

We picture part of the image of

in figure

5.5.

Figure 5.5 Then x, y

Q g2 (N) ,. A(g2 ) €

N, then

ng2x"'

is easy to see tl"at g3

is a hyperbolic set for ng2y,

so

We have to know when

g

if

TIX

~

Kl.

g3 : Kl

g2jA(~2).

That

g3

s: K +

and

rs .. gm.

diagrams.

It

requires more work. g

Kl

for

is also topologically

and

g

and

K

L

K

two expanding maps and we wish

are topologically conjugate.

Williams shows that

a sufficient condition is that there exist continuous mappings and

= ny

is an expanding map, and that the inverse limit

g3

K1 ~ K1 g3

induces. a map

g2

is topologically conjugate to

conjugate to

Also,

g2.

and an integer

m

~

The conditions can be

1

such that expre~sed

r : Kl

...

K

gr = rg3, g3s = sg , sr

as the following commutative

m g3,

261

g3

K~

r1

Kl

lr

K~ K

g

't '(

g3

K __&_> ·K

K

g3

is conjugate to

Since

K_1_,. Kl 1

g 2 : N + 1R 3

we choose an embedding a diffeomorphism o~

s4

an

g, g3 ,

In the case of our maps so

m

g

g~

K.._L.> K

'"Ill

K _L_., K

\:I '\i

m

See [ 61 ]

~

1

~'

Kl

r, s

and maps

for more details.

is homotopic to the inclusion

1jl •

1R 3

+

can be found,

s 4 , then

1jl o

g2

o

i : N + 1R 3 ,

ljl-llljl (N)

if

extends to

by standara techniques in differential topology.

Let us give one more example of a 1-dimensionalattractor. variant of an example due to Plykin [43].

This is a

His was the first example of a

!-dimensional hyperbolic attractor in the two dimensional disk. Let

1R 2

D be a disk in

with three holes foliated as in figure 5.6.

Figure 5.6

We define a diffeomorphism foliation and have

f(D)

f

from

D into its interior to preserve the

as in figure 5.7.

262

Figure 5.7

The branched manifold is a union of 3 circles and the map (on homology) is

A+ C- A

A

~

B

~A

C .-.- B

Remarks 1.

One can use this example to show that non-trivial hyperbolic

attractors (for flows) appear in arbitrarily small perturbations of constant vector fields

on

tori

Tn ·of dimension greater than

2 [36).

As a consequence,

hyperbolic attractors appear in perturbations of three or more coupled harmonic oscillators, or three or more coupled relaxation oscillators. explicit, tecall that a harmonic oscillator has equation

X ==

where

v

m

and

k

are positive constants.

To be more

=0

mx + kx

If we have

n

or

such

mv • -kx oscillators, we obtain the system (1)

on

lR 2n.

lie on

There is a stable equilibrium at the origin, and all other orbits

n-dimensional invariant tori.

263 A relaxation oscillator is a differential equation 'Of the form X+ f(x)x + x

and

f(x)

=0

0

>

for

where, for some constant

I xi

2: k.

For example, i f

f(x) 0,

f(x)

= p(x2

for

- 1),

l'x1

p

~

k

> 0,

one has Vander Pol's equation which comes up~n Vacuum tube circuits (see e.g. [ 59]). in [ 15]),

Under certain conditions on. f

the system

~ = v-

Jxf(u) du

(as in theorem 10.2

has a single asymptotically

v = -x stable periodic solution.

If one has

n

such Aystems,

o~e

(2) 1

on

1R 2n.

If

This n

2:

3,

syst~ffi

has a unique invariant attracting

gets the system

s i s n n-torus.

there are small perturbations of both systems (1)

which possess non-trivial hyperbolic attractors. to us, a recent paper in Science [ 12 ]

As

and

J. Ford pointed out

gives related experimental results.

In particular, the broad band noise spectrum in figure 2B of

[ 12]

may be

due to a non-trivial hyperbolic attractor. 2.

Williams has extended theorems (5.1) and (5.2) to higher dimensional

"expanding" attractors

[ 63 ] ,

(2)

and has given general conditions for

topological equivalence of one dimensional attractors

{62 ].

264

Attractors -- ergodic theory

6.

X.

be a homeomorphism of the compact metric space invariant Borel probability measures

on X.

p

p(f

-1

in

for every Borel set

B) • p(B)

X.

v

The measure

~

: X

This means that

Such a function

R.

+

~

real-valued functions on

X, and we set

~ €

space. M(f)

f(B}

= B,

converges to

M(f)

C(X).

then

~.

is ergodic if and

~(x} = J~dp

for p-almost

MCf)

as a

is the space of continuous

.I

p( ~)

ll ~

$clp

for

With this topology,

oy

(~

if and only i f

M(f)

C(X).

~ €

This

C(X) * ,

is the point mass at

11. (~) + 11(~) ~

becomes a compact metrizable

M(f)

are the ergodic invariant measures of

subsequence of

•

is constant almost everywhere.

C(X)

where

It is also a convex subset of

empty because if

p

~of

M(f) (called the weak or vague topology) so that a

gives a topology on

for each

8 and

Riesz representation theorem, one may think of

ccx>*

€

Then

f.

1 in L (}1)

subset of the dual space

lli

~

R is invariant if

+

~of= ~.

From the

sequence

: X

is constant on orbits of

~

That is, i f x.

B

That is, any invariant p-measurable set has measure

only if any invariant function

all

and

An equivalent condition can be given in terms of real valued

zero or one. functions

= 1,

~(x)

is called ergodia if whenever

p(B) • 0 or 1.

we have

p ~ M(f) is a

That is,

B be the a-field of Borel sets

Let

B.

be the set of

Let M(f)

regular Borel non-negative measure on X such that

f : X +X

Let

We begin with some notions from ergodic theory.

and the extreme points of f.

y,

Note that

M(f)

is non-

then any weak limit c f a

M(f).

265

The most basic result of ergodic theory is the following theorem.

x. and let

thP- compact metric space there is a set lim n....,.

n-1

I·

1 n

and define

= JM

Let

m(~) XE·w

=

JM~·w

where

for XE

any~

L 1 (~).

€

A•

= li~!

n-1

l

n....,. n kaQ

¢(fkx)

= J$dp

E

J$d~ n-1 L ~ 1

n

k=O

almost everyk ¢(f x) along

J¢dp.

M is not orientable, let

Take the measure

= m(11~ E).

m on

w be a nowhere vanishing n-form, n =dimM,

€ C(M).

That

i~ define

is the characteristic function of a Borel set E:

XE(x)

1T*m(E)

$(x}

approach the space average

measure which can be defined.

If

€

¢

M is an orientable compact manifold, there is a natural Borel

If

m(E)

x

the time-averages

x,

Thus, for IJ-almost all x

For any

in theorem (6.1) is ergodic, then the functi9n $

ll

must be constant p-almost everywhere,. so

the orbit of

'•'

Moreover, if we set ¢(x)

-exists.

If the measure

where.

M(f).

€

X: of IJ-measUPe 1 such that for

.A c

¢(fkx)

k=O

p

be a homeomorphism of

Let f

Theorem (6.1) (Birkhoff ergodic theorem).

=

0ifx¢E lifx€E

M+

~:

M be an orientable 2-to-1 covering.

M and let 1T*m be the measure on M defined by

Any measure

m on

will be called Lebesgue measure on

M.

M induced by ann-form on Dividing by

M or

m(M), we will assume

m(M) = 1. Ruelle has proved the following theorem. p

€

M(f)

containing

is the set of points x, IJ(U)>O.

x

€

The support of a measure

M such that for every open set

U

-M

266 Theorem 6.2([48], [ 7 J) . a

Let

be Lebesgue mea8U1'e on

m

and let

M,

f

be

c2 diffeomorphism having a hyperbo l ic attractor A. There is an ergodic

f-in~~riant

probability measure

There is a subset A c ~(A)

property.

~

x e W8 (A)-A and

x

A with the following

with m(A) • 0 such that if

is any continuous function on M,

Thus, form-almost all orbit of

supported on

pA

x

in

then

~(A), the time average of

converges to a definite limit.

~

along the

This resul t is quite remarkable.

For, in a natural sense, Lebesgue measure zero agrees with our intuitive feeling of what is exceptional (or avoidable) in smooth systems. ~

If we think of

as an observable physical quantity evolving along an orbit, then, with

probability one, we can compute its expected value. the Anosov case, hyperbolic attractors for Lebesgue measure zero .

c2

Furthermore, except in

diffeomorphism& have

Therefore, it is surprising that one can say anything

about time averages of points in sets of positive Lebesgue measure near these at tractors. If Ws(A)

PA

is the measure in theorem (6 . 2), and

such that

tl

A(C£1J) •

n-1 k lim.! Xu(f x) • PA. n n-+1

c.eu

with

For

Since

e;

v

c

n>O

is arbitrary, and

m(A) • 0.

•..i'

and

and: x

= ~A(U)

n k=O

Given e;>Q ,

v be an

and let

J.lA (U)z

and

u

if (A)

•

Before proceeding to the proof of theorem (6.2), let us note that it implies a celebrated theorem of Anosov.

Theorem (6.3).

Let

be a c2 topologiaaZly transitive Anosov

f

diffeomorphism and suppose aontinuous with respeat Proof.

Let

4>

f

to

C(M).

preserves a measure v whiah is absolutely

f

Lebesgue measure m. Let

A c M be such that

I

k 1 n-1 lim$(f x) =

n+= n k=O

Since

v

also.

Thus,

Then

J$df.l

A

is absolutely continuous with respect to ~

$(x)

1

n-1

n-+ d~A •

dv •

Remarks. being

k=-0

4>

of

k

dv ..

Hence, and

-

v

By the bounded convergence

v is ergodic.

Theorem (6.2) and (6.3) hold under weaker assumptions than

1.

f

c 2 • The proof given here works just as well (with straightforward

changes) i f

Ha

is

f

C

,

O 0 E

E·>O

suah that for any x E A and n>O, m(~3 ~(x,n)) ~ C p E

~

A

(~(x,n)) E

We defer the proof of proposition (6.4).

Proof of theorem (6.2). E>O

so that

Let

d(~(x), ~(y))O,

f~dpj>~} A

set

..

for infinitely many

n}

00

We first claim m(~(A) n E(~, 3~)) • 0 • E

(1)

Assume (1) is proved for the moment. f

As

E(~.

36)

is £-invariant, and

preserves sets of m-measure zero, we have m(f-n(Ws(A)) n E(~, 36)) = 0

for each Letting But if

n~O.

But

36 • -1

m

A(~) •

Ws(A) ..

for

U m~l

m~l,

U

n~O

E

f""V(A),

so

E

we get that

m(~(A) n E(O.

M.

is

c2 by

Hence, the maps

This implies that

~u

274

Note that, ,.u(fkx}

1

'f

=

n- ·

7T

k=O

k

lji(f x}

-1

so,

Thus,

contracts the

measures how much

wfnx - volume. Now, we can construct f

of

in

A of period

2 p,;:Per(n)

zn

e

% 1

Let

~E.

1.

fiA.

for

such that

0

= T(m) = 2N(E) + n

Let

n

0 be arbitrary and let

use specification to give a q(z)

> 0 such that S .pU(x) E ~A(W:(x,n)) ~bEen

be an expansive constant for

0

>

>

>

b

0, there is a constant

for any

q(z)

and

E

Ws(x,n). E

z

and q(w)

Also, by

w get at get at least

By lemmas (7.3) and (7.5), we have

(-2N(E)-2P) II

zn =

Recall that

be an expansive constant for

such that

c1d(f j y,fj z)

2E.

L

esncpu(p)

Let 2E > 0

pe:Per(n)

Per(n), there is 0 s s Thus, We:(p 1 ,n) n We:(p 2 ,n) = e!. So, p1

If

~

p2

in

I m(Ws(p,n)) > c-1z by the volume lemma (7.4). - e: n E pe:Per(n) hand, by specification (lemma (7.1)), for each x e: A, and n

m(M) ~

a p e: Per(n + P(E))

such that

'¢u .

l;u, and, hence, it gives (b) for

Again, as above, this gives (b) for

Proof of lemma (7. 5).

~

x e: Ws(p,n). e:

~

j < n

On the other ~

1, there is

Thus

and

for

This gives

n > P(e:)

and (a)

clearly follows. To prove (b), we first prove (c)

there is a constant

DE

>

0

so that for

have

D~r

r nl i=

zn .-P ~zn.+ . . . +nr ~ D~ ,J..

i

r z +P n i=l ni

n1

>

P(~)

P, we

286

For (f

0

p

r

Per(

€

o+nl+•+ni-l

L ni)

p,ni).

For convenience , we set

(z 1 (p), ... ,zr(p)). for some

i r

if

T

i=l

p1 ~ p 2 , then for some

If i

and

= L ni'

1 ~ i ~ r, let

and

i=l

i

k,d(f zk (p 1 ) ,f zk (p 2))

and

>

3 , E

zi(p) e: Per(ni + P) n0 j

= 0. >

and

3E

Then let ~(p)

-specify

=

0, d(fjp 1 ,fjp 2)

>

-

Thus,

z(p 1 )

~

-

z(p 2).

e:, so

K is as in lemma (7.3),

zT = I

pe:Per T

Simiiarly, if ~

= (z 1 , ... ,zr)'

there is a f

+·+n

n 0

p(~) e: PerT

zi e: Per(ni- P), and

such that

i-lp(~)

Hence, r

f l z i I n

i=l

i

-P

=

and (c) is established .

Now, we claim Zn+P

~

De:Zn

for all n

>

1.

Z +P > (l+a)D Z with a suitable small a. n e: n ~ > D-kzk Th en, f or k ::_ 1 ' -k (n+2P) - e: n+P > -

D-k(l+a)kDk Zk E

E D

If not, let n

be such that

287

So 1

.k(n+2P) log zk(n+2P)

>

1

p+2P(log(l+a) + log Zn] .

By (a), the term on the left approaches zero as tradiction.

Similarly, we get

follows se~ting

cl =

n;.

z

> D- 1z n-P- e: n

k

~ ~

for all n

which gives a con>

P.

Then (b)

288

Diffeomorphisms with infinitely many attractors.

8.

Given a

We have seen that hyperbolic attractors have a rich structure. diffeomorphism

f, one would like to know what kinds of attractors

In particular, are there finitely many?

sesses.

f

pos-

Do almost all points, say in

the sense of Lebesgue measure, approach attractors?

For even simple diffeb-

morphisms which arise in practice, these questions are very difficult. On the other hand,if

Cl Hp (f)

By theorem

(3.8~

Cl H (f)

H

p

(f)

in

M

p

has a dense orbit and a local product is a finite-to--one quotient of a

fjCl H (f) p

Also, by theorem (3.7) there are neighborhoods

subshift of finite type.

Cl

N of

and

f

such that for

in

is topologically conjugate to

gl ngn (U) n

the structure· of

f, and

is a hyperbolic periodic point of

is hyperbolic, then

structure.

of

p

fic.t H (f). p

g

E

U

N,

Thus, we understand

fiCl H (f) very well and this structure persists when p

f

is

perturbed. Now there are many diffeomorphisms

f

for which

Cl Hp (f)

is not hyper-

bolic for some hypqrbolic periodic point p, and we would like to understand these.

At present our knowledge of these diffeomorphisms is quite incomplete,

and here we shall merely focus on a few typical examples and some of their properties. It follows from proposition (4.2) that a diffeomorphism with a hyperbolic limit set has only finitely many attractors. faces

Cl H (f) p

We will see below that on sur-

not being hyperbolic frequently leads to the existence of

infinitely many periodic attractors. To begin with, let us try to imagine the simplest way we might have

Cl Hp not hyperbolic.

Clearly~

we should try to find a non-transverse

289 homoclinic point

q

of

However, the Kupka-Smale theorem (1.3) tells

o(p).

us that generically all homoclinic points are transverse, so even if we found such a

we could perturb it away.

q

If we want a persistent lack of trans-

versality, it is natural to replace and try to arrange for sistent way.

Wu(A)

and

o (p)

by ·an

Ws(A)

infinite hyperbolic set

to be non-transverse in a per-

This turns out to be easy to do in dimension larger than

For irtstance, return to Plykin's example in section 5. feomorphism

f

of a subset

~ f~ = A1 (f)

n2:0

fixed point of

x E W~(p), w:(x)

D c 1R 2

in

into its interior such that

u

A1 , and consider

is an interval, and

foliation of a neighborhood of

p

We(p)

for some small

{W:(x)}xEWu(p)

D.

in

Let

Let

p

A > 1, and let

g(x)

D x IR + D x IR.

is a hyperbolic saddle point for

and

dim Wu«p,o) ,f x g)

=

2.

z 1 E W~((p,o),f xg) - {(p,o)}

Let

Consider the mapping

= Ax

f x g f x g

be a curve joining two points

y

and

a

gives a !-dimensional

£

1R •

(p,o)

~e

For each

e.

be a linear expansion on the line Clearly,

2.

This is a dif-

is a one-dimensional hyperbolic attractor. f

s z 2 E WE((p,o),fxg)- {(p,o)}

as in

figure 8.1. y

Figure 8.1

Let

N be a small tubular neighborhood of · y, and let

feomorphism such that N n Wu((p,o),f x g) £

¢(z) = z

A

for

to a curved disk

z i N and

~

~'which meets

~

maps a disk

be a dif~

in

N n Ws((p,o),f x g) £

290

in a circle as in figure 8.2

Figure 8.2

Let

I •

~o(fx g).

I,

is hyperbolic for gencies with x,y

A'(f1 )

E

and for any near

Ws(A 1 (f1),f1) such that

Wu(x)

in this case, if of

f1

near

A'= Al x {0}

N is small, one may check that

If

c1

f1

near

(p,o)

f

has tan-

That is, there are points

61 •

is tangent to near

I, Wu~Xf 1 ),f 1 ) Ws(y)

Moreover,

61•

near

is the hyperbolic fixed point

and

is not hyperbolic.

This shows that the Kupka-Smale theorem fails if we try to replace periodic points by hyperbolic sets.

It also shows that Axiom A

diffeomorp~

or more generally diffeomorphisms with hyperbolic limit sets are not dense in

Diff~ for any M with dim M > 2.

It is less obvious that hyperbolic

sets with persistently tangent stable and unstable manifolds exist in dimension two.

We will see that they do in the

DiffrM with

r

>

2.

A hyperbolic basic set

if there is a neighborhood there are points somewhere.

x

r > 2, but it

M is a compact two-dimensional manifold and

Now let us assume that E

topology with

r = 1.

is still not known if

f

Cr

and

y

A for

f

is called wi l d

N of f in DiffrM such that for any g E

A(g)

such that· Wu(x)

and

W5 (y)

E

N,

are tangent

2 91

We will omit the word basic and call such sets A periodic point

dissipative if of

f.

T fn

p

of period

det T fn

That is, if

p

E

of a diffeomorphism will be called

1.

Let

s (f)

denote the set of periodic sinks

S(f)

and

fn(p)

= p,


0 D(v0 )

n > 0

as in Figure 8.4.

Figure 8.4

yu(g1) aud

=

corresponds to

u. ir(u) r - o. u->0 juj2

{(u.v} :

so that

lui

D(v ) 0

Ve may choose

~ e:r lv-vo[ ~ e:2}

and

g~(v0 )

look

293

D(v )

Of course

0

may intersect

ys' and

may be below

g~(v0 )

The important thing is that by translating

0

up or down via a family s,t € (-tS,tS)

(-tS,tS), tS >'"0, we can find

in some interval

t

with

{4>t}

D(v ).

t such that . n (1) g D(v ) n D(v0 ) "' 0 s n n (2) gtD(vo) n D(v0 ) "' Al u A2 and the index of gt on -n gt Ai is +1 for i = 1,2. n (3) g\1 has no fixed points on ClD(v0 ) and the index of n on D(v ) is 0 for \1€(-tS,tS). g\1 0 s

with

~

.

See figure 8.5.

g"n(v)=r\, s

0

Now for some a between point

in

D(v0 )

< 1

for

z

det Twgan

with eigenvalue w

~

D(v ) 0

are near the dis sipative orbit be

1

and

).1

where 1)1 1

Now there is a to the eigenspace of

and

s

Cr

t, i t follows that 1.

D(v )

If

0

o (p).

has a fixed

is very near

since most of the iterates

x, then

gj D(v ) , 0 < j -

0

Thus, the eigenvalues of

n Tzg


o f,

then

z1

1.

If

is a periodic

294

g.

sink for

Cr diffeomorphism

Then

tangent at some point.

Proof.

If

n

B 2 (x) • €

~(o(p))

~d.

in a set

z

C(x,Ws(x)n B (x))

•

€

x

is an interval about

yu €

Ws(x), and, for

in

x

y8 e:

and

are

Here, as usual, d is the distance fUnction

is small enough, then

E

is an interval about €

€

€

{y EM: d(x,y)

€

and

Wu(o(p))

yu • C(x,Wu(x) n B (x))

small, set

E > 0

i s f y

and

denote the connected component of a point

C(z,F)

where B (x)

and

rrrzy be cr perturbed to g such that o(p)

f

x be the point at which

Let

Let

tantent.

M.

are

have a tangency arbitrarily near p.

Ws(o(q),g)

on

Wu(o(p)) and ~(o(p))

Suppose

o(p).

are hyperbolic periodic orbits of g, and Wu(o(q),g)

and o(q)

F. For

q;

with

f

h-related hyperbolic periodic points of a

be

Let p and q

Lemna (8.J).

in Wu(x),

fiy~

i / 0,

0.

We will produce sequences of intervals

such

J 1 ,J2 , •••

and

I 1 ,I2 , .•.

that (a)

Ii c Wu(o(q))

(b)

Ii

(c)

for

Ji c Ws(o(q))

and

~ y: and Ji ~ y: in the c1 topologies n

~

1

i,

and large

Suppose these sequences have been found. ~

tees we may find a diffeomorphism is tangent to g

= ~of,

Ji

near

X

and

Hn>

s Ji c W (o(q) ,g). orbit under

g

will pass near

produce the sequences

and

T

be such that

be a common period of u W (p1 )

and

s W (p 2 )

for

I f we then set

11 ; B (x) . €

Hii) c Wu(o(q),g) Hii)

and

and

is near

Ji

~(Ii)

x, its

Thus, to prove the lemma we only need to

p.

the analogous construction of the Let

=n

Since the tangency of

i, (b) guaran-

near the identity such that

Cr

then (a) and (c) guarantee that

Then for large

(Ji). J 's i

p

and

I 's

We will produce the

i

and leave

to the reader. q.

Let

are tangent at

p1 x.

E

o(p)

Let

and

p2

E

o(p)

D be an interval

295

U

such that {p 1 ,x} c: D - aD and xI. f-jD, and let s lSjST an interval in W (p 2 ) such that p·2 e: D' - aD' and x t D'. in

Wu(p 1)

Since

p

and

intersection of

q

are h-related, we may choose a point

Wu(o(q))

and

Ws(pz)

so that

and

for

~

n

fi

small tubular neighborhood

N

0 and

neighborhood

z e: D'- aD'.

be

of transverse Because

= tJ, so we may choose a small

o(q) n o(p)-= tJ, we have Wu(o(q)) n Wu(o(p)) e: > 0

z

D'

of

D such that

f-nz n N

= r)

we may take a small tubular n B (x) ..

N'

e: Osjs-r -r = 1. Observe that

gives a typical situation when

tJ.

Figure (8.6)

D may meet D'

in

several places

B (x)

e:

Figure 8.6

It

I

is a small interval about

z

in

Wu(o(q))

and

e: is small,

then

0 for

(d)

n

>

0

and (e)

if

k

>

0

is the smallest non-negative integer such that

fki n B (x) 1< e:

for

0, then fii

0 s i < k.

c:

v

lSjST

f-jN u

v

OSj

-n

x

€

u

We:(f

-n

g

n

=~

for

n > 1.

f

and Wu(o(p) ,g)

x.

U n B2 e: (x) ':"

A and

~(z 2 ) at

x.

fnx

z1 ) c U and

~. Let Pick

n

0

>

0

Ws(fnz 2) c U. . e:

€

< n •

-

0

Let· yu

Then be

x.

s Similarly, there is an interval. ys c W (z 2)

~ 1.

Then

0 be small enough so that if

Wu{z 1 ) is tangent to f

and

about

x · such

Since the orbits of the homoclinic points

of homoclinic A, there are sequences -n 0 z1 and i ~ ""· For o(p) such that ri ~ f -n w2u e:(ri) is near wu2e:(f 0 z1 ), so there are intervals

are dense in.

points of large

i,

I1 c

W~e:(ri)

topology.

c

Wu(o(p))

For large

such that

fn°Ii

~

in the

yu

c1

~(w)

some large (a)

= w,

for

~

= ~.

As

be an appropriate function

Cr

w i B0 (x), and we set

g

=

~og,

i, is tangent to

Cr)

=~

Ji c W~e:(si) c Ws(o(p)) n-n 0 Ji n B (x) i ~~.and, for large i and all n ~ 1, f 0 proof of lemma (8.3), if we let

(actually

for n ~ 1. Similarly, -n such that f 0 J 1 ~ ys as

f-n+nori n B0 (x)

i, one has

there are intervals

with

e:

suah that Wu(A)

f

(O,e:)

€

fnys n B6 (x)

that

u

y •

1, as required.

so that fjB 6 (x) n B0 (x) = ~ for -n n 0 0 u z 1 )n B0 {x) containing a small interval in f We:{f choose

at\

near

Let p be. a periodia point in A.

x.

have a tangenay arbitrarily near

A be such that

€

=~

n B (x)

Using corollary (3.6), let

Proof. U "' {y

is

I,N)

c1

is an interval

Let A be a hyperbolic basia set for

Lemma (8.4). Ws(A)

Ti

Ii • C(fTii,N) n Be:(x), then (d), (e), and the construction of

If we let

and N'

C(fTii,N) n B (x) e:

D, so

interval

C(f

(2.5) insures that

~-lemma

near

x

in the near id

then we have, for

297

n

(c)

g.

is a hyperbolic periodic point for

p

(b)

~(f

0

u

Ii) c W (o(p),g)

and

This proves lemma (8.4). Now we can prove theorem (8.1).

N be a neighborhood of f so that if g is in N,

p. Let

periodic point

s W (A(g))J

is tangent somewhere to n

For

f containing the dissipative

A be a wild hyperbolic set for

Let

~

be the set of hyperbolic periodic points of

Per(n,g)

1, let

g.

Hn be the set of diffeomorphisms

Finally, let

N such that for each q

g

be the set of all hyper-

Per(g)

n, and let

of period less than or equal to belie periodic orbits of

is dissipative.

p(g)

and

Per(n,g) n Hp(g), there is a periodic 1 Clearly, H is an open subsink s (q) of g such that d(s(q) ,q) < -n n set of N. Using lemmas ( 8. 2), (8. 3), and (8.4), we can prove Hn is dense g

in

E

.

in

N.

To see this let

g

E

By lemma (8.4) we may perturb to is tangent somewhere to

(8. 3) again enables us to perturb

s(q)


O 1

The

F.'s

2i

Fi

components of length

easy to check that

in the example are not a defining sequence for

T(F(8))

1-6

28

Observe that the Lebesgue measure of each Fi

= 2i a i

Finally, set

1

F~8), be~ause we have taken out several gaps of

of

ai.

.

So

F at each stage.

T(F(8)) F(8)

is

+a:>

0

as

It is

8 + 0.

since the measure

The reason for thickness is the following.

300

If F and G are Cantor sets UJith

Lemma 8.6.

-r(F) ·-r(G)

neither is aontained in a gap of the other, then Proof.

{Fi}

Let

and

{Gi}

-r({Fi})·-r({Gi})

in a gap of the other,

F

If

(*)

c

Once (*)

0

Fi+l

1.

>

1

and

F

and

G

nor

G is

~.

F

Since neither

We prove the following statement.

n G ; ~.

G, then

which meets

Fi

is a component of

a component of

1

be defining sequences of

respectively such that

0

F n G

>

which meets

contains

c

G.

has been proved, we have that each

F meets

so

G,

Fi meets

Let us prove (*).

G.

c

Suppose

also a component of is obtained from ponents of

c - ui

Fi

F-gap

by removing the

/~ )

UF c c. i

Let

is Fi+l be the com-

c£., c r

\]

Figure 8.8

r

and

Assume by way of contradicition that

cr

c

If

as in figure 8.8.

c

and

~.

Fi+l, there is nothing to prove, so· assume that

(

both

c n G;

such that

Fi

is a component of

are in G-gaps .

both

If

c

do not meet

r

c

and

G.

Thus,

were in unbounded

r

G-gaps, then we would have one of the FOllowing situations. (a)

G

lies to the left of

(b)

G

lies to the right of

(c)

G c u~

0 0

c

r

0

Now (a) and (b) contradict the assumption that G

in the

c n G

f

~.

and (c) would put

F-gap

Hence, at least one of

and

c

r

is in a bounded G-gap.

Suppos e

301

c.f. , is in the bounded G-gap

G

Uj.

The argument is similar

if

cr

is in a

bounded G-gap.

'

Let

cr

be the components of

~Gj+l

adjacent to

UG as in j

figure 8.9.

r

Figure 8.9

Now

c

r

cannot be in

Then we have,

UG

for this would give

j

c c UjG

contrary to hypothesis.

ui

Case 1:

c~

c

Case 2:

c

n c1

or r

r

;

~.

In case 1, we have figure 8.10.

cl

[

E I~