Dynamical Systems (C.I.M.E. Summer Schools, 78)

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 978-3-642-13928-4 e-ISBN: 978-3-642-13929-1 DOI:10.1007/978-3-642-13929-1 Springer Heidelberg Dordrecht London New York

©Springer-Verlag Berlin Heidelberg 2010 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

C O N T E N T S

J . GUCKENHEIMER: B i f u r c a t i o n s o f Dynamical Systems M. MISIUREWICZ. : Horseshoes f o r Continuous mappings o f an I n t e r v a l J. MOSER : Various a s p e c t s o f i n t e g r a b l e Hamiltonian systems A. CHENCINER : Hopf B i f u r c a t i o n f o r Invariant Tori S . E . NEWHOUSE : Lectures on Dynamical Systems

Pag. 5 Pag. 125

Pag. 137 Pag. 197 Pag. 209

BIFURCATIONS OF DYNAMICAL SYSTEMS

JOHN GUCKENHEIMER

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

51 -

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

systems.

They are not cmprehensive, 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 1661 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]. intent is pedagogic.

The current state of bifurcation theory is a mixture of

mathematical fact and conjecture. proved is small [ll].

Our style is informal and our

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

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

-.

There are three kinds of systems we examine on a

OD

finite dimensional C

manifold M:

(1) smooth, continuous time flows @: MxR

M

-+

obtained from in-

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

M,

smooth, discrete time flows obtained by iterating a diffeomorphism

(2)

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. A

In some applications, we shall find exafnples of all three.

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 M

At each of

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 ~obustproperty.

This corre-

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

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 than 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,mathematical 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 results

unspecified.

The sophisticated reader desiring a precise accounting of the

state of the art will not find it here. Let us lay out our pl'an of action.

We begin with a description of the

elementary bifurcations. 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

sphere, complicated recurrent behavior does not occur. proceeds much more smoothly than in other situations.

Thus the theory Sotomayor 11041 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

results are those of Dunortier [241.

Takens' concept of normal forms I1113

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 11141 and Bogdanov 1161.

They independently computed the unfolding of

this bifurcation. Holmes and Rand [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 [911. Mathematical models with these properties are appearing in a rapidly growing list of disciplines which now includes chemical kinetics, geodynamics, fluid mechanics, electriccil 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.

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 characterized by.two parameters.

In the case of the van der Pol equation,

o b 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 glob-albifurcations 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 [ 7 4 ] , and the moduli of saddle connections discovered by Palis 1851. Before embarking upon our exposition, it is necessary to set the framework for our discussions. This requires that we provide a minimal background 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 O: MxlR M.

4

M

be a continuous time, smooth flow on a smooth manifold

Recall that this means that C p t = O(.,t)

diffeomorphisms of M. fined by

d X ( x ) = -(O(x,t) dt

a bijection when M

is a one parameter group of

Associated with each flow is its vector field de)

.

The map

O+ X

is a compact manifold.

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

existence and uniqueness theorem for ordinary differential equations. The curves fQ(x,t)

I

t

IR

E

called orbits or trajectories of 8, are the orbits

of the group action of IR smooth submanifold of M

on M.

This implies that each trajectory is a

which is homeomorphic to the quotient of R

closed subgroup. There are three possibilities:

by a

a trajectory can be homeo-

morphic to a point (called a sing ilar or equilibrium point), a circle (called a periodic orbit), or a line. Trajectories which are homeomorphic to lines may sit inside M

in such

a way that their intrinsic topology is not the relative topology induced from

M.

Such trajectories are called (non-trivially) recurrent.

cumulation points of the trajectory through x w and a limit sets of

sequence it.? C R 1

x :y

with ti

is in the w ( a ) +

+ -(-0)

variations on the theme of recurrence.

and

as t +

+

The set of acare called the

limit set of x 8(x,ti) + y.

if there is a

There are many

The concept which has played the

largest role in the study of smooth dynamical systems is that of the wandering set 0, defined as the set of x

E

M with the following property:

if U

is a neighborhood of x, then there is a y

Q (y,t)

E

U.

non-

E

U

and a t

2 1 suchthat

The set $l is easily seen to be closed.

The general aim of the bifurcation theory for dynamical systems is to study how the geomztry of the various kinds of orbits and their limit sets depend upon parameters which enter the definition of a system.

The emphasis is

upon describing those changes which are robust in the sense that they are resistant to perturbation in the proper framework.

It is h a d to make one de-

finition which encompasses all of the interesting phenomena which have been studied. We discuss below the question of strategy which is embraced by bifurcation theory, but the picture upon which the strategy is based turns into a maze of exceptions when examined in detail. Most of the emphasis in the study of bifurcations has been placed upon periodic orbits and singular points.

Since these orbits are compact, they

are easier to deal with than more complicated kinds of recurrence. The topological properties of generic flows near a singular point or periodic orbit are completely described by Hartman's Theorem [ 4 4 ] .

The condition

which must be satisfied for the genericity of a singular point x it be hyperbolic.

This means that the Jacobian matrix

DX(x)

is that

of the vector

field at x have no pure imaginary eigenvalues. Hyperbolicity at x T M I the tangent space to M X

*at

@ ES such that DX (x)

implies

at x, can be deccmposed as a direct sum

leaves each factor invariant, the spectrum of

DX(x)

restricted to EU lies in the right half plane, and the spectrum of

DX(x)

restricted to ES lies in the left half iiane.

theorem implies that ES and EU

The stable manifold

are tangent to smooth submanifolds of MI

called stable and unstable manifolds of x, characterized as the .union of trajectories which tend to x

as t

-+ 2

Hartman's theorem states that

m.

the flow O is topologically equivalent to a linear flow near x.

This

means that there is a.continuous change of coordinates so that trajectories of O become the solutions of a linear system of differential equations. There is a similar definition of hyperbolicity for periodic orbits. Geometrically, this definition is best approached through the Poincare' or return map of a cross section. to y at x y

Given a periodic orbit y, a cross section

is a submanifold NcM

transversally at x.

which has codimension 1 and intersects

The Poincare' map @

:

N -+ N

hood of x by setting @ (y) to be the point

O (y,t) where t is chosen

to be the smallest positive number such that O(y,t) small enough, the map

@ has a fixed point at x.

bolicity is that its derivative @(x) one.

is defined in a neighbor-

E

N.

Provided N

The criterion for hyper-

has no eigenvalue of absolute value

In this case, there is a splitting of the tangent space TX M

EU 0 ES $ E0

with

E0

is

tangent to the flow and

.

into

EU @ ES a direct sum de-

compositio~iof T N invariant under D@ The spectrum of D restricted X U s to E (E ) lies outside (inside) the unit circle in the complex plane. Once

again, there are stable and unstable manifolds intersecting transversally along y

with tangent spaces ES @ EO

and EU @

Hartman's Theorem,implies that the return map @

EO

at x.

A

version of

can be linearized on a

neighborhood of x in N by a continuous change of coordinates. Hyperbolicity for periodic orbits of a discrete time dynamical system is defined similarly. points in its orbit.

A

periodic orbit here is one with a finite number of

If f: M

+

M

is a ma@ eith fn(x) = x, then the de-

finitions and theorems which were applied to to fn and M.

e

and N - above can be applied

Note the shift in dimensions which occurs in this corre-

spondence between periodic orbits for discrete and continuous time flows. Progress in understanding dynamical systems the past decade has been based in large part on the generalization of these hyperbolicity conditions to invariant sets larger than a single orbit.

There is an intimate con-

nection between structural stability and the hyperbolicity of the nonwandering set of a flow.

A

large class of structurally stable flows (no

others are known) have been described in these terms. characterized by two properties: Smale's Axiom versal'i'tyProperty. Axiom

A

A

The flows are

and the Strong Trans-

states that the nonwandering set of a flow is

the closure of its periodic orbits (apart from singular points in the continuous time case) and has a hyperbolic structure. some Axiom

A

The non-wandering sets

flows contain an infinite number of periodic orbits, all of

which are hyperbolic. The number a(n)

of these periodic orbits of period

< n is finite, and a(n) grows exponentially with n. One of the themes -

of bifurcation theory which we pursue in later sections is the changes which occur as a flow with a finite number of periodic orbits is deformed into one which has a hyperbolic invariant set with an infinite number of orbits. The strategy of bifurcation theory is based upon a picture which applies better to algebraic geometry, compact group actions, and the theory of singularities of smooth mappings.

The basic idea is that of a strat-

ification. Let us illustrate the idea with an example.

The singular set of

a.complexalgebraic variety V is a subset of V which is itself an algebraic variety S(V).

The set S(V)

'has lower dimension than V.

One

can iterate this construction, obtaining the chain of subsets 2

V 3 S (V) 3 S (V)= - 3 Si (V)a -a$where si+'(V) is the singular set i i of S (V). The properties which are of interest to us are that each S (V) i i+l is a closed set.and that S (V)-S (V) is a manifold which is open and i' dense in S (V). For our purposes, these will be the distinguishing properties of a stratification, further regularity conditions which are o r dinarily imposed upon stratifications will not be needed. of a set V is a sequence of closed subsets V

-

v1

V*

A

stratification

vi

...

vi vi+l is a manifold and dense in with the property that each stratum i V The dimensions of the V are required to decrease in some reasonable i

.

sense. The second illustrative example is the one which comes from the singularity theory of smooth functions. sider the space c-(M)

Given a compact manifold M, con-

of smooth functions f: M + IR

.

The group

G = Diff(M)x Diff(lR) . acts by composition on the left and right:

.

(h,k)

acting on f gives kofoh. The orbits of this group action can be used to w 6 define a stratification of C (M). Until one reaches V , each stratum is a finite union of orbits of the action of G. functions on M. codimension i.

In a natural way, Vi For values of i

2

V-v1

is the set of Morse

is a submanifold of V of finite

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'allfunctions with a "finite" degree of degeneracy, in principle. Transverse sections to orbits are called universal unfoldings. Their intersection with the various strata show how the functions of the orbit can be approximated by less degenerate functions.

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 o f a flow as a group action of

Z

or

.

JR )

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 intrbduction of weaker notions of equivalence between flows.

Two flows are topologically equivalent if there is a homeo-

morphism of

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

M

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

of its topological equivalence class (with respect to the C spaczeof flows.)

1

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 w e 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 53, we

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

d,imensionalvector fields. Otherwise, efforts in this direction have encountered the counterexamples which have proved the bane of attempts to find generic properties of dynamical systems. Still, there is much of practical use in the theory.

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 acquaintance with the theory of dynamical systems. Nonetheless, I hope that they will prove useful not only to mathematicians but also to workers in a n&er

of disciplines which make substantial use of systems of differential

and difference equations having complicated dynamical behavior.

52 -

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 53 the formulation of invariant expressions which are sufficient for the existence of the nonnal forms.

There are three kinds of bifurcations

called the saddle-node, flip, and ' ~ o pbifurcation f

, 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 x if the Jacobian of a map circle.

f

of a vector field X is Wypefbolic

(DX)x has no pure imaginary eigenvalues, A fixed point x

is hyperbolic if

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 X : M -+ TM be vector field which depends smoothly upon the paraP meter u E IR. Let uo be a value of p and x E M be such that x is

.

a non-hyperbolic singular point of X

The easiest ways for this to come

"0

about are that a single eigenvalue of

(DX

)

is 0 or that a single pair

"0

of eigenvalues of

(DXp Ix

are pure imaginary.

The first of these situations

0

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 : R P

2

2

-+

IR

m e singular points of X are given by solving the equations X (x,y) = lJ lJ 2 (0,O) = (u+x ,+y) . Thus the locus of singular points in (x,y,l~) space is a parabola in the plane while for l~ < 0, X

y = 0. For

l~ > 0, X lJ

has two singular points at

,

X (x,y) is given by

v

the singular points of X

11

and the other is a node.

when

p < 0.

However, when

have coalesced, at

(?fi,O)

.

The Jacobian of

which has real non-zero eigenvalues at

point of X and it is not hyperbolic. o1 ist for l~ < 0

has no singular points,

One of these is a saddle point, p = 0,

there is a single singular

The saddle and the node which ex-

l~ = 0 to give a saddle-node.

For

I.! >

0,

saddle and node have cancelled each other entirely, producing a vector field with no singular points at all. X

lJ

The phase portraits of the vector fields

are illustrated in Figure 2.1.

chosen to be

-.

(The sign in the definition of X

IJ

is

We remark that the saddle-node could have been defined

for a one dimensional vector field. Example 2:

The Hopf bifurcation.

In polar coordinates in

I R ~ con-

sider the vector fields

In Cartesian coordinates

From the polar coordinate representation, one sees that the only singular point is at the origin. From the Cartesian representation one computes the Jacobian

(DXIJ) (0,o)

has eigenvalues whose real parts, have the same sign as l~.

bifurcation occurs at

Of special interest in this bifurcation is

l~ = 0.

the role of periodic orbits.

From the polar coordinate representation of

notice that the vector field X of the plane.

A

X

lJ

is invariant under the group of rotations

lJ Periodic orbits occur on circles defined by

the radial velocity of the vector field is zero. this equation forms a parabolic bowl in

l~-r2 = 0 where

The set of points satisfying

(x,y,l~) space.

See Figure 2.2.

This bowl separates trajectories which spiral away from the origin from those which spiral inward with decreasing

r.

For

toward the origin which is a global attractor.

l~ < 0 ,

all of the orbits spiral

As the bifurcation occurs, the

stable equilibrium behavior of the singular point gives way to a stable periodic behavior. There are many applications of the Hopf bifurcation theorem.

It de-

scribes one of the simplest ways in which a stable equilibrium of a physical system 'can become unstable and lead to non-equilibrium behavior. dynamics provides a number of thoroughly studied examples.

Hydro-

Let us look at

one of these, Couette flow [541. Couette flow consists of the following experiment.

The space between two

concentric vertical cylinders is filled with a fluid and the two cylinders are rotated at different rates.

The friction of the fluid with the bounding

cylinders causes the fluid to rotate as well.

When the relative speed of

rotation of the cylinders is small, the fluid flow follows horizontal circles concentric with the cylinders.

As the relative rotation speed of the cyl-

inders increases, several bifurcations occur. flow pattern to change to,Tayior vortices.

The first of these causes the

The flow remains steady, but no

longer follows horizontal circles. each band being a solid torus.

The fluid separates into a stack of bands,

The streamlines of the flow now follow curves

which spiral around the centerline of each of these solid torii. furcation.of Couette flow is a Hopf bifurcation. main, but they do not remain steady.

The next bi-

The Taylor vortices re-

The tops and bottans of each torus be-

canes wavy; and the entire flow pattern rotates around the cyclinder in a periodic fashion.

Careful experiments provide strong evidence for the regular

periodicity of this flow. Let us briefly indicate how the Hopf bifurcation enters into the mathematical analysis of this experiment. by the Navier'Stokes

The motion of the fluid is described

equations together with appropratie boundary conditions.

These partial differential equations describe the evolution of the fluid in time.

With considerable care, one can represent the Navier-Stokes equations

as a vector field on the appropriate function space.

The relative speed of

rotation of,the cylinders enters the vector field in a dimensionless parameter called the Reynold's number.

As the Reynolds number increases, there are

several bifurcations which occur before the flow becomes irregular and turOne of these is a Hopf bifurcation in which a pair of eigenvalues

bulent.

for the linearized vector field at a stable equilibrim~crossthe imaginary axis.

In a surface tangent to the plane spanned.by the corresponding eigen-

vectors, a Hopf bifurcatim like that of the above exemple takes place.

Ex-

haustive discussions of the Hopf theorem in the context of Couette flow and fluid mechanics can be found in 154,673. Let us turn from singular paints of vector fields to fixed points of maps.

There are three basic examples of bifurcations, the saddle-node, flip,

and Hopf bifurcation.

We shall see that the Hopf bifurcation in this case

is considerably more cauplicated than the other exemples described in this section.

Example 3: given by

The saddle node for maps.

1 fU (x) = (UX)

always satisfied. x =

2

+ fl.

For

+

x

-

p > 0

there are two fixed points given by

P < 0,

1 (pT)

1f

x

1 1, this fixed point beP comes unstable and trajectories starting near the fixed point move away in the map

an oscillating fashion.

To understand the asymptotic behavior of these tra-

jectories, it is necessary to look at

f

u.

e

= 1

n

f

2 1 )'( = 1 , so U

that

f undergoes something like a saddle-node bifurcation. However, the U 2 1 second derivative (f )"(?I is forced to be zero, and the graph of f U

has a higher order tangency with the diagonal. than 1.

f2U

has one fixed point near

x = 1/2,

When

p

is slightly less

and when

p

is slightly

f2 has three fixed points near x = 1/2. Since f has U U only one fixed point near 1/2, two of the fixed points of f2 must form a U periodic orbit of period.2 in the second case. It is not hard to verify that larger than 1,

this periodic orbit is stable. Figure 2.4 gives some of the relevant diagrams. Example 5: Hopf bifurcation: Consider the map depending upon two parameters p,v

given by the following formula in polar coordinates:

In rectanguiar coordinates, 2 2

U," (XVY?

f

(x(u-(x +Y

))

+ (cosv)x-(sinv) y,

As with the Hopf bifurcation for vector fields, the map leaves invariant the circle r2 = p when p >

0.

However, the circle is no longer a single

periodic orbit, and the dynamics of f within the circle must be considered. In this particular example, the map is rotation through the angle v, independent of the parameter p.

Now the study of maps of the circle is a

delicate subject, and this example has features which are atypical. For instance, the generic behavior of a diffeomorphism of the circle with rational rotation number is that there be a finite number of periodic orbits, half of which are stable and half of which are unstable. We discuss this matter much more fully in Section 95. Another difficulty with the example will be evident in the next section when we discuss the reduction of fixed points to normal forms. If v/2n

is a fraction with small denominator ( ~ 5 then ) ~ the

example no longer represents the generic normal form for a Hopf bifurcation with rotation number v.

There are other tens in the normal form which

affect the dynamic behavior at these "resonant" bifurcations. The complete statement of the Hopf theorem for maps, proved by RuelleTakens [92] , follows: Theorem: Let fp: IR2 + lfI2 satisfying the following:

be a 1-parameter family of diffeomorphisms

(a) The origin is a fixed point of f Dfu

at the origin by (b) For

u

= 0,

A1 (u) and .I2(u)

u

.

.

Denote the eigenvalnes of

A1(d and A2(u) are complex numbers of absolute k

value 1. For k 2 5 , Xi # 1. (c)

d

(Ih(u)I)

> 0

Then there-is a u-dependent change of coordinates bringing

f

to the

lJ

following form in polar coordinates: fy (r.~)= ((l+fl(~) r

-

3 f2 (~)r,

G

+

f3(p)

+

fi(y) .r2)

+

terms of

order 5 . If fZ(u) > 0, then there is a continuous one parameter family of invariant

attracting circles of f

e

, one for each

11

E

(0.,6) when 6 > 0 is small

enough. The bifurcations exhibited in these last three examples are found both in discrete time dynamical systems and in continuous ones.

For continuous

systems they appear as the bifurcation behavior of periodic orbits through the construction of cross-sections and return maps.

The Hopf theorem for

maps was first formulated by Ruelle and Takens in developing their theory of the transition to turbulence in fluid flow.

We shall exhibit all of the bi-

furcations of maps discussed in this section later in ecological models. When dealing with one dimensional maps or two dimensional vector fields the Hopf bifurcation for maps cannot occur since there is only one eigenvalue present.

Figure 2.1 Saddle-note bifurcation in two dimensions

4 lJ

riadic orbits

4 y Singular points

Figure 2.2

The mpf Bifurcation

fixed p o i n t s 9

Figure 2.3 Saddle-node for maps

I

fixed points

graph f

P'l

graph f Figure 2.4

The flip bifurcatiar

53 Normal Forms: -

Recall two of the related, but.slightly different, approaches to studying structurally unstable dynamical systems which fail to be structurally stable in a mild way.

In the first approach, one studies parametrized families of

vecitor fields.

Certain structurally unstable systems will be robust or per-

sistent in the sense that they will occur in some open set of families. For example, non-singular vector fields on the torus T~ with irrational rotation numbers are structurally unstable, but they occur robustly in one parameter 2

families of vector fields on T

.

The'second approach picks out a submanifold

of a space of dynamical systems and then studies the vector fields in this subspace. In a typical case, the submanifold is defined by algebraic conditions on the Taylor series (jet) of a vector field at a singularity. For example, saddle-node bifurcations occur at points where the linearization of a veceor field at a singularity has

0

as an eiqenvalue.

This section deals

with this second approach unlike much of the remainder of this work. The submanifolds of the space of dynamical systems which we consider have the property that they are invariant under the group of diffeomorphisms of the underlying space on which the system is defined. The group action here is the one which corresponds to changes of coordinates on the underlying space.

For example, if a discrete system is defined by a map

cp is a diffeomeorphism of

+

M

and

M, then f is transformed by cP to cP ofo*g-l.

The orbit of f beginning at x at p

f: M

-1 is sent to the orbit of pofocP

(x). Vector fields transform by the rule X

I--

D G % X O { '

.

beginning

Assume that

we have identified a submanifold of the space of dynamical systems which is invariant imder the diffeomorphism group.

This means that the submanifold is

a union of orbits of the diffeomorphism group. With luck, from each of the orbits of the diffeomorphism group we can pick out a particular system whose expression is especially simple.

We then say that this particular system is

a normal form for the system. The problem to which this strategy has been employed most systematically is the one of linearizing a vector field at a singular point.

There are two

parts to this problem, one algebraic and one analytic. Here we shall deal with the algebraic problem and only briefly discuss the analytic difficulties. The approach to the problem begins by looking at the Taylor expansion of a vector field which has a singular point at 0. of

lRn

into itself which have

0

Polynomial maps (of degree k)

as fixed point and an invertible linear

part define local diffeomorphisms at 0.

This group can be identified with

the k-jets of diffeomorphisms at 0, and it acts on the Taylor polynomials of degree k of vector fields having singularities at

0.

The orbits of this

algebraic group action define submanifolds of the space of vector fields (by pull back via the projection) which are certainly invariant for the group action.

The nonnal form problem consists of identifying those jets of vector

fields which have the simplest expressions, "simplest" usually interpreted to mean the fewest non-linear terms. The original problem of this sort was'considered by poincarg, Siegel, and Sternberg [97,107,108].

It is the question of whether there is a non-

linear change of coordinates (analytic or smooth) near a singularity of a vector field, so that the vector field is actually linear in the new coordinates.

The answer to this question is complicated, and the analysis in-

volves problems of small divisors.

Necessary conditions for this linear

-

ization problem come from solving for the coefficients of the Taylor series at the singular point of the diffeomorphism defining the change of coordinates. One obtains a recursive system of equations which involve linear combinations of the eigenvalues of the singular point as coefficients.

If the solution is

to exist, all of these linear combinations of eigenvalues must be non-zero. If one of these coefficients is zero, then the corresponding non-linear term in the Taylor series of the vector field cannot be eliminated by a smooth

change of coordinatei. The cases which we are interested in here are ones in which there are zero or pure imaginary eigenvalues, and the litsearization proTo deal with this case we shall follow Takens'

blem is certainly unsolvable.

Lie algebra formulation of the normal form problem [lll]. Let X be vector field on

lRn

with a singularity at

the linear-vector field obtained by linearizing X.

0.

Let

L be

L

We want to think of

as a differential operator acting on the space Hk of vector fields Y whose coefficients are homogeneous polynomials of degree k. given by the Lie derivative

[L,Y].

To see explicitly what this action

looks like, let us consider the case where L and Y mononomial terms:

Then with

[L,Y] = 6im

a L = xi -a x ~Y = X 'i...'n 1 j . . - x li+l

Rx j 1

i

R . 2 1. 3

are given by typical

a

xwhere naxm

... x'j1-1 ...

the Kronecker delta and

The action is

Rn n

If R

R

1

+--.+

II = k. n

I1 ... x n ' a axm &im X1 n

j

a axj

= 0, the first term is

absent. For this computation, we see that L defines a linear map of Hk k itself. Let Hk = B

+

G~ be a direct sum decompositon of

image and a complementary subspace.

Ilk

into

into its

The main theorem about normal forms

states that we can eliminate terms which lie in B~ by changing coordinates. k

Let X =

Theorem:

Z Xi + \+l i=l

be the Taylor series of degree k

a vector field defined on

IRn

with a singularity of

space of vector fields on

lRn

whose coefficients are homogeneous poly-

0.

of

Let H~ be the

nomials of degree k, let B~ be the image of the map [x,, . ] : H~ -+ Hk and k in W . If Xi = bi + gi vith let G~ be a subspace complementary to bi p:

Bit gi (lRn ,0)

k

E

-+

Gi, 2

2 i 2 k, then there is a local diffeomorphism

(lRn ,0) such that the Taylor series of degree k

of DPoXOP

-1

Proof:

The proof of this theorem proceeds by induction. We define in-

Pi changes the ith term in

ductively local diffeomorphisms cpi so that the Taylor series of the ith vector field

xi

xi

to

gi while leaving the lower

Then Oko-Q k-lo. so 4 = is the local diffeo1 the morphism sought in the theorem. The induction begins trivially with 1 order terms of

identity.

unchanged.

Suppose that pl,...,

have been constructed and relabel

CPQ

L -1 X=D(Qo---ocpl) X(rpQo-..04Pl) as X. L

2 11 lie in the spaces

the Taylor series of X of degree Y L"'v

be a vector field with the property that

,Jlt be the time t map of the flow of Y. d easy calculation shows that (X t) = dt

-

)

[Y,XtlI .

IS=O

L derivatives of Dqt and

+

R11+2

with

R11+2

II

is X 1

and let

= Y (Xt)

)

-

Xt (Y)

s=O jets of Xt are all the same since the first

Jlt -1 agree with those of the identity at the [X1,Y] = bll+l,we find that

having zero

+ X2 +

X , Y = b

[Y,Xtl- [&(x~)]~

(&+I) jet

[Xt,Y] = bll+l

since the

[Xt,Yl depends only on the linear part of Xt. jet of Xt

Let

GI.

-1 If Xt = (D$~)oxo$~ then an

ds

Moreover, the

origin. From the equation

-

x

= D d

-1

d = ds(D's+tX('s+t) =

Thusweassumethatthetermsof

This implies that the

..-+ XII+l+ bII+l= X1 + g2 + "-

bII+l+ tbll+l. Set t = -1, we find

l.+l = $ 4

L+1 jet of 11+1

+ gQ + gII+l+

is the diffeomorphism re-

quired to complete the induction step. This normal form theorem can be used to systematically study expressions for vector fields which one hopes are structurally unstable in a mild way.

An analogous situation in which these procedures for finding normal forms work much better is the study of singularities of maps which we introduced in Section 51.

Let us briefly recall some of these results so that they

can serve as motivation and comparison with the results about vector fields. One impetus for developments in both of these areas came from Thom's speculations in his Catastrophe Theory where he foresaw the development of

bifurcation theory for vector fields along the lines of the singularity theory for maps.

As we shall see, his hopes were too optimistic, and the bi-

furcation theory lacks the structmal elegance of singularity theory. There is a normal form problem in singularity theory as well.

Let us

concentrate on the local theory for functions, which exmines germs of functions f: lRn of lRn

+

fixing 0

IR

at the origin.

The group of germs of diffeomorphisms

acts on such functions, by right composition. Germs in

the same orbit of this action are called right equivalent.

In each orbit

one would like to find a particularly nice representative, which is again called a normal form. A basic result is that a polynomial representative can be found for each germ of finite codimension. A germ of finite codimension will be equivalent to every function with the same jet of some order R, in particular to the polynomial obtained by truncating its Taylor series at degree

R.

Here

R

can be estimated by the codimension of the orbit.

The

consequence of this fact is that the normal form and classification problems can be reduced to the level of jets.

They become algebraic problems involv-

ing the action of'the finite dimensional groups of jets of diffeomorphisms on the jets of maps (which are just polynomials in n-variables.)

The clas-

sification scheme has been carried amazingly far by Arnold and suggests intriguing relationships between the classification of singularities of functions and the classification of Lie groups. Let us return from this digression about singularity theory to the normal fcrm problem for vector fields.

Here the natural action of the group of

diffeomorphisms does not lead to a nice stratification of the space of vector fields.

Immediately one is confrontea with the fact that the Jordan

canonical form of the linearization at a singular point and the period of a periodic orbit are invariants of the action.

No orbits of this group ac-

tior, on a compact manifold are open, so that one cannot even begin to form a stratification in this way.

It is necessary 'to weaken the equivalence re-

lation between vector fields to even start finding a stratification. The immediate choice f0r.a weakened equivalence relation is topological equivalence: two vector fields are topologically equivalent if there is an orientation preserving homeomorphism sending trajectories of one into trajectories of the other.

The equivalence classes are unions of orbits of the diffeomorphism

group, but they do not form submanifolds of the space of vector fields in an evident way.

For singularity theory, the equivalence classes are determined

by looking at the jets of the maps.

No systematic theory for topological

equivalence of vector fields seems possible here, and this makes the normal form problem harder.. One can only use information like that obtained from the normal forms to guess that certain submanifolds of the space of vector fields will lie in a single equivalence class for the appropriate equivalence.re'lation. With extraordinary luck in addition, there may be-some chance that a reasonable unfo'lding of the submanifold exists as a family of vector fields which is stable with respect to some equivalence relation. All of this sounds dismally vague.

There are examples and counter-

examples which do not seem to lead to anything coherent. Let us review the current state of knowledge about the classification of singular points of vector fields by listing those situations which have been examined. (1) Codimension one.

Here the situation is satisfactory, and the

phenanena have been illustrated in the examples of Section 52.

There are

two principal theorems needed to classify the codimension one singularities: the Hopf bifurcation theorem and an invariant manifold theorem [501 for normally hyperbolic submanifolds.

The specific results on normal hyper-

bolicity are due to Palis and Takens [ 8 5 ] .

These imply that a topological

equivalence of a normally hyperbolic submanifold can be extended to a neighborhood.

The consequence of this result is that one can reduce the study of

bifurcation behavior at a singular point to a submanifold tangent to the eigenspace spanned by pure imaginary eigenvalues.

For the saddle-node

bifurcation this space is one dimensional, and for the Hopf bifurcation it is two dimensional. For the saddle-node, the relevant vector field on the line has a singularity at 0 with vanishing 1-jet and non-vanishing 2-jet. This already determines the topological equivalence class of the orbit in a neighborhood of 0: the origin is an isolated singular point which is the forward limit of the trajectory on one side and the backward limit of the trajectory on the other.

The one parameter family of Section 52 is easily seen to be

transversal to the suhanifold of vector fields having a singularity with a single 0 eigenvalue. The case of the Hopf bifurcation has a long history.

~oincar; con-

sidered the "center" problem for a singularities of a plane analytic vector field.

This question asks for conditions on the Taylor expansion of a plane

vector field at a singularity which allow one to determine whether the singularity is a center, i-e., a neighborhood is filled by periodic orbits. One approach to these questions begins by looking for the normal form for a vector field having a pair of pure imaginary eigenvalues. By an initial linear change of coordinates, we may assume that the linear part of the vector field is given by

a a ax ay

.

X1 = a(-y-+x-)

A

moderate computation shows that

X1

acts transitively on H2, the space of vector fields whose coefficients are homogeneous polynomials of degree 2, but that the image of the action of on H

3

has codimension 2.

2 2 a a (X +Y ) (*y) a x ay and

A

2 2

(x +y

XI

basis for its complement is spanned by

a

(x--$) ay

ax

If the coefficient of the first

of these two terms in a normal form expression is non-zerb, then its sign determines the topological equivalence class of the vector field near zero as being either a (weak) sink or source.

To obtain an unfolding of such a

vector field, one must chose a family so that the real parts of the eigenvalues of the linearization at its singular point cross 0

transversally.

The Hopf bifurcation theorem finally enters the decription. It states that there will be a family of closed orbits to one side of the bifurcation.

A

precise statement follows: Hopf Birfurcation Theorem:

Let X

:

lJ

I R ~+ IFt2

Assume that (1) each X

has a singular point at the origin.

U

l~ = 0, (DXIJ)O has a pair of non-zero pure imaginary eigen-

(2) when

values and the coefficient of Xo

be a 1 parameter family of vector fields.

2 2

( x +y )

a a (-yay)

in the normal form of

is non-zero, and (3) $(W(DX 11

) )

U 0

# 0.

Then there is a smooth surface periodic orbits of X

lJ

.

(

= (x, y , ~ )space) consisting of

There are no other closed orbits of X

U

neighborhood of the origin in with the plane defined by

C C.lR3

lR3

l~ = 0

, and in R

in a

C has a non-degenerate tangency 3

.

(2) Higher codimensions for Two Dimensional Vector Fields. At least three authors, Lefschetz 1581, Seidenberg [94], and Dumortier 1241 have considered, independently, the problem of higher order singularities

in the plane. cases:

Classical theorems of ~oincar4and Bendixon distinguish two

those in which there are trajectories Which approach (or leave) the

singularity with a well defined tangent and those which are centers or focii. In the first of these cases, a neighborhood of the origin is filled by sectors of orbits of the same general type and by the separatrices which bound these sectors. There are three types of sectors, illustrated in Figure 3.1.

These are called fans, hyperbolic sectors, and elliptic sectors.

The procedure used in the study of these singularities involves singular coordinate changes which blow-up the singularity itself into a circle which is invariant for a vector field defined on a new two dimensional manifold.

Repeated applicatidns of this technique leads to vector fields having singularities whose topological rype is simple enough to analyze directly.

Two

variants of the blowing up procedure employ polar coordinate representations of the vector field followed by multiplication with a radial function and monoidal transformations of the sort common in algebraic geometry. Dumortier 1241 gives an explicit classification of two dimensional singularities possessing separatrices through codimension four, but does not attempt to calculate unfoldings. There is one codimension two singularity for which an unfolding has been calculated.

Its analysis indicates some of the difficulties encountered in

dealing with higher codimension bifurcations. The bifurcation is the least degenerate one in the plane for which the linearization at the singularity is nilpotent.

The generalized eigenspace of 0

linearization is given by

a

X = y1 ax

is two dimensional and the

in asuitable coordinate system.

normal form calculations show that the image of X1

a 2a (xrax, y

The

acting on H2 has

za-

a 2a 2 x 5 - x -1. ay' ax a One choice of complementary subspace is the one spanned by xZL and ay x% Takens [Ill] proves that any vector field whose normal form includes a nondimension four with image spanned by

y

zero coefficient for the term x 2 L is topologically equivalent to the ay a The phase portriat for this vector field is vector field y + x2L ax ay illustrated in Figure 3.2. This analysis does not give a complete picture of the bifurcation behavior of this vector field.

For this purpose we want to select an unfolding,

a two parameter family of vector fields which is transverse to the submanifold

of vector fields having singular points with nilpotent parts.

Takens [I141

and Bogdanov [16] have given independent analyses of the unfolding of this singularity.

The result is illustrated in the series of diagrams of Figure

3.3, taken from Arnold 191.

The family is given by

X

XE

lfE2(x,y)

a

y +(E.

ax

+E

1

a -

X+X2+xX) 2 1 - 1 2 a y e The center of Figure 3.3 illustrates the

parameter space

( E ~ , E ~with ) the bifurcation curves indicated.

The phase

portriats of Figure 3.3 correspond to the letters labelled in the regions and along the bifurcation curves of the parameter space.

A notable feature of this example is the presence of the bifurcation labelled

S.

This vector fiela has an orbit which is asymptotic to a saddle

point in forward and backward time. cycles of the vector fields labelled

It occurs as the period of the limit

C

became infinite.

exists only on one side of the bifurcation.

The limit cycle

This bifurcation is different

in character fran the previous bifurcations we,have encountered because of its qlobal character (though the Hopf bifurcation has same aspects of this sort.) Global bifurcations of this sort present a significant obstruction to the systematic construction of topological unfoldings.

Perturbations of

higher order singularities are Likely to contain several singular points, and the intersections of.their stable and unstable manifolds can be quite canplicated.

These features occur even within the class of gradient vector

fields 1371. Nevertheless, locating highly degenerate singularities can be useful'in discovering the bifurcation structure of examples which depend upon several parameters.

Holmes and Rand I511 have found the codimension two singularity

analyzed a b w e in an averaged version of the forced van der Pol equation. This fact, together with results of Gavrilov and Sil'nikov 1321, yields strong analytic evidence for the presence of homoclinic trajectories of the forced van der Pol equatim.

We shall discuss the forced van der Pol equation

later frat a quite different point of view. Another example where this codimension two bifurcation arises canes from the CSTR, the continuous stirred tank reactor.

This is an idealized

model of a chemical reactor in which a single reaction takes place in a well

stirred and continuously cooled reactor.

The equations describing the CSTR

are as follows:

Here u and v

are dimensionless variables representing the proportional

conversion of the reactant inside the reactor and a temperature.

Uppal,

Ray and Poore [I171 investigated the bifurcation of this system of equations in considerable detail and found complicated bifurcation sets. Part of their results are intuitively clearer if one recognizes that this system of equations contains a codimension two bifurcation of the sort illustrated above. To find this bifurcation, one solves the above equations to find equilibrium values for u and v: u = v/$ =

aE (v) l+aE (v)

.

The codimension two .bi-

furcation occurs when the trace and determinant of the linearization at equilibrium vanish.

- aE + aB(1-u)E/6

These give the equations 1

+

aE

- a6 (1-u)

= 0 =

- 1-1/6

. . There will be one solution of this system of four

equations for each value of v which yields a positive value of a:

We leave to the reader the exercise of computing the second order coefficient in the normal form for the singularity of the CSTR and verifying that it is not zero.

The conclusion of this exercise is that the bifurcation

diagram illustrated in Figure 3.3 occurs in the analysis of the bifurcations

of the CSTR.

Compare this treatment, based on the unfolding of the co-

dimension two singularity, with the analysis of Uppal, Ray, and Poore 11171. Note that the unfolding of the codimension two singularity does not portray all of the bifurcation phenomena which occurs in this problem due to the presence of a third singular point.

Aris 1123 has noticed that the coin-

cidence of the three singular points leads to a surface of equilibria like that found in the "cusp catastrophe" 11161.

e l l i p t i c region

Figure 3 - 1 A highly degenerate singular point

Figure 3.2

Phase portrait of

a +

y-ax

a

x2

ay

Figure 3 . 3

The unfolding of

a

+ x2E

ay

54 -

Homoclinic Orbits and Three Dimensional Vector Fields:

All of-the phenomena which have been considered to this point involve plane vector 5elds and avoid questions dealing with limit sets having a complicated topological structure.

Here we begin to confront these matters and find that

new kinds of bifurcation questions arise.

To fix ideas, we shall discuss two

examples, the forced van der Pol equation [20] and the Larenz attractor [641, at some length before considering more general'matters in the next section. The van der Pol equation is the seccmd order differential equation

which is equivalent to the plane vector field defined by

When attracting.

E 3

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

The only initial condition whose trajectory does not lead to the

limit cycle is the singular point at the origin.

As c

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.

E,

the limit cycle is il-

Approximately, the limit cycle trajectory alter3

nately follows sections of the curve y2 = yl /3

- yl

and Worizontal 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:

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

R 2 x S1

with the equations

9, =

E(y2

-

3

(yl /3 -yl))

$2 = -y /E + a cos 8 1

where

8

is a variable along the unit circle

S

.

1

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

Cartwright and Littlewood 1201 made the first ob-

servations which indicated that basically new dynamical phenanena are present. Further analysis of this system by them 162,631 and of a similar system by Levinson 1601 prwided basic motivation for the development of Smale's ideas about dynamical systems [102].

All of this work has yet to yield a complete

analysis of the dynamical behavior of the forced v m der Pol equation.

It

remains a good model system to study in terms of dlpamical system theory. Since the speed of the forced van der Pol vector field is constant in the 0

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

which are multiples of 2n/o

.

Solutions of period

2n1r/w with n > 1 are

called subharmonic. The fundamental observation of Cartwright and Littlewood was that there are parameter values for which two different stable subharmonic solutions were present with different periods.

For the proper parameters,

different initial conditions can lead either to a stable subharmonic of period (Zn-l)21~/o or

(Zn+l)Za/w

.

This behavior is robust, as it m u s t >e

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 prwed 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 51. 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 B = constant (say 0 ) .

Observe that the time

2r/o

map of the flow

maps this plarie into itself, and thus defines a diffemorphism 0: R2 4

The asymptotic behavior of flow, and we may study

cD

+

R

2

.

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 prablems that we discuss later. about qualitative properties of

4.

We start with observations

Consider an annulus A of moderate size

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

0 will map

A

annulus which is much thinner. parameter

E

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

0

0.

by a map with

an image of 0 image.

thickness; i.e., by a map Y of rank 1 with a one dimensional

We assume that an approximating

Y

can be found with the property

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

@

is homotopic to the identity,map of A

in which it is embedded.) he core of A two "kinks".

once.

This implies that the image of

winds around

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

projection n of A ro'f'on-l: C + C

Y

@

However, numerical computations show that the image has

Y to establish a coordinate system on A

and

(through the flcw

as S1 x I and a corresponding

onto one of its boundary components C.

The map

is well defined since the inverse images of points for

are the same. The map

-1 f = noYos

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

T

is a map of the circle which is

.

It is this circle map

f 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: jr = -lox

4; = 5 where

U

+

ax -y

- -8z/3

10y

- xz + xy.

is near 28.

This system of differential equations was studied by

Lorenz in the 1960's 1641 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 dynamical systems theory [38,42,55,90,118]. We shall summarize the setting in which the equations (5) arise and some of the results about them.

In Section 56, we consider their dynamics in more detail.

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 bottan, so there is a buoyant force pushing the lighter fluid upward.

If the density

gradient 5s sufficiently large, then this force actually causes the fluid to mwe.

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 became 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 Larenz 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 solutions 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 (51, the solutions correspond to fluid =+ion which is irregular. The fluid rotates in alternate directions for varying lengths of time.

The time intervals between successive

changes of direction vary with changes in the initial condition. Within certain 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 fran a geometric description of the flow of the Looeenz equations.

There are three singular points of the

Lorenz system, all of which are in the plane y = x .

One is the origin, and

it is a saddle point with t w dimensional stable manifold containing the z-axis and one dinensional unstable manifold tangent to the x-y plane.

The

other two singularities lie in the intersection of the planes y = x z = 0-1.

and

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

mensional unstable manifold. away from the singular points.

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

y = x,

then one

sees a.vertica1 barrier directing points left and right away from the origin as they descend 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 apposite side of the barrier.

See Figure 4.4.

Williams 11181 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 con-

siderably 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 Loren2 attractor via a cross-section in the

An-

horizontal plane of the singular points.

This allows us to

See Figure 4.6.

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 y which do not return to the cross section at all. y

of points

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 y

.

The points which started to the left of

y

return to a similar set.

Thus the return map would a homeomorphism of the cross-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 carefylly chosen. We want to obtain a one-dimensional map from the cross-section.

To do

so, we assume that there is an invariant foliation of the cross-section which

contains the curve y.

In more prosaic language, we assume that coordinates

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

should tend to

0

uniformly

With such a representation, many of the pro@

g.

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

matters in Section 16 after taking up the properties of one dimensional maps in Section 05.

Figure 4.1 The van der P o l l i m i t cycle for large

E

Figure 4.2

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

@

Figure 4.3

The singular points of the Loren2 system viewed in the plane

y = x

Figure 4.4

The unstable manifold of the saddle point in the Lorenz attractor

Singular point

Figure 4 . 5

The branched manifold f o r the Lorenz attractor

Figure 4.6

A cross section to the Lorenz attractor

05 -

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 applicatim which provides impetus for the study of one dimensional maps lies in population dynamics, discussed in Section 57.

One dimensional

maps are subject to severe constraints which come fran the order praperties of the line.

In this section, we shall consider these constraints with

particular attentim 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

f (x) = 1-1 1-I

f: I + I will be a continuous hap of the in-

Throughout this section terval

[cO,c ]

II

This means that

- x2.

with a finite number of turning points f

C1

is monotonic on each of the intervals

< C2 problem for the two parameter family of diffeomorphisms of S1 depending upon We ask for a determination of the region in the

(E,w)

.

(~,w) plane for which there

will be periodic orbit with a given rational rotation number.

For a generic

family, this region will be wedge shaped with the tip of the wedge along the line

E =

0. On the line

E

= 0 we assume that the family is given by

actual rotations gith rotation number to zero rapidly with of a line of constant all values of

w.

The width of these wedges tends

in such a way that the measure of the intersection

E

E

with all the wedges tends to 0 with E.

For

E, the complement has positive measure, indicating that the

width of most wedges is very small indeed. When E

is large, the flow of the forced van der Pol equation carries

an annulus into a thin annulus as described in Section 54.

We approximated

this map by a rank 1 map which then induces a map on the circle. While one obtains a diffeomorphism of the circle for small E, when map of the circle;as "kinks."

E

is large, the

Because of the symmetry properties of the van

der Pol equation, the map will have four critical points rather than two. The maps are such that there are large regions in parameter space in which 2r are stable. Thereare open sets in which periodic orbits of period (2n-l)w two of these regions overlap. When this happens, there are also an infinite number of periodic homoclinic orbits.

The knegding theory can be used to

estimate how many periodic homoclinic orbits of a given period occur. 1 1 For concreteness, let us illustrate with a specific map g: S + S which seems to have qualitative properties corresponding to values of the parameters for which the forced van der Pol equation has stable limit cycles of periods 6r

and

lor

.

The graph of the map g is shown in Figure 6.1.

We assume that it has the following properties.

There are four critical

points of g, all of which are assumed to lie in stable periodic orbits. (This is a non-generic property, but it simplifies our descriptionwithout changing the nonwandering set of g.)

There are.two stable periodic orbits of

periods 5 and 3, whose points we label al'a2,a3,a4,a5

and bl,b21b3 re-

spectively. We assume that the ordering of these points on the circle is

a b a b a a b a a with 1 1 2 2 3 4 3 5 1

g(ai) =ai+l

for

1

~

i

~ g(a5=q,g(bl) 4 ,

= b2, g (b2) = b3 and g(b 3) = b 1' The critical points are b2,a3 ,b3, and a 5'

These assumptions determine the nonwandering set of of symbolic dynamics.

through the use

Symbolic dynamics keeps track of which images of

intervals intersect which other intervals as we iterate circle into eight intervals: I4 ' [b2,a31 , Is

g

g.

Partition the

I1 = [al,blll I2 = [blra2, I3 = [a2,b2],

[a31a411I6 = [a4,b31, I, = [b3rasll and

Our assmptions about

g

imply that the images of

9(11) = Ijl 9(12) = 14, 4(13) = Is g(16) = I8 I J Ill g(17) = Ill and I7 have reversed orientation.

u Is,

11,...,~8

I8 = [aslal]. are given by

g(14) = 16# g(15) = I6

g(18) = I~ (1

I2.

U 17,

The images of

I4

and

This information about images can be em-

bodied in the transition matrix.

o if I which is characterized by

a

j

g ( ~ ~ )

ij = { 1

The transition matrix

d

A

if

1

j

kg(Ii)

.

allows us to calculate the image gk (I.) of 1

Ii

for the kth iterate of

g.

Indeed, the transition matrix of

gk

is

A ~ . This is seen easily from the formula for Ak : k (A )ij =

2

i 1 . i .

k-1

ai i a i2ai3... ai;,lik =1 0 1

with

Each term is zero unless all factors are 1, and then 0

2

5 k. This implies I

j

k

c g (1~1. Therefore,

i

0

= i

I, (14;fj

and

ik = j.

c g(1,)

for

is k e number of

k times that g (Ii) covers the interval I j' The number of unstable fixed points of gk Since gk

k is given by the trace of A k is contracting at a fixed endpoint. each time g (Ii) covers Ii.

there is at least one fixed point in the intepior of

Our

Ii.

.

assumption that

g has only two stable periodic orbits implies that there will be exactly one fixed point in the inteirior of

Ii.

Table 1 gives the number of unstable fixed

for k 5 15.

points of gk

1 2 3

k # unstable fixed points gk

0 0

# unstable periodic 0 orbits, period k

3 16. 5 . 3

0 1

7

4 5 6

4

8

9 1 0

11

12

13

14

15

28 64 39 45 176 307 260 392 1028

1 0

4

6

4

4

16

24

20

26

68

Table 1 The (one-sided) subshift of finite type E with transition matrix is a topological space (again denoted E

9,

m

R+1

= 1

for all

2.

A

1

OD

The

with the property that

distance function can be defined by

d(L.1) =

with

=

.

This metric is complete on

E

.

1 if i R # jR

R=O The map o:E

-+

E

is defined by

o(L) =

i

equivalent to the map g

jR = iR+1'

with

indices on a sequence, omitting the first term.

set.

.

C

0 if i, = j, 6,2-111

together with a map o

= {iR l~i,~81 ,,iQ=O

space consists of all sequences a i i

)

A

The map o

It shifts the is topologically

restricted to the unstable part of its nonwandering

This means that there is a homeomorphism h

part of the nonwandering set of g

from

such that ho = gh.

Z

to the unstable

Put somewhat dif-

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

5;

for each sequence

L

with transition matrix the nap - g.

A

Note that 'A

has a dense orbit in

I C g(Ii ) , a ) 0. Thus, the subshift i a+1 a allows us to characterize the nonwandering set of

with

has only positive entries.

This implies that

a

C.

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. 1361. forced van der Pol equatiw for large hard to extract [ 621.

E

Information about the

based upon rigorous analysis is

Much more has been proved with less effort by

Levinson [60] and Levi 1591 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 nioderate 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

bifurcations of the van der Pol equation.

E

An invariant curve, homeomorphic to , ' s map as the bifurcation occurs.

appears around a fixed point of the

The behavior of the map on the invariant

curve depends on its rotation number.

In a typical family, this rotation

number will not be constant, and a set of parameter values of positive measure will yield invariant curves with irrational rotation numbers.

When

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.rationa1with a small denaminator.

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.

Gicen a map

f:

mn

-+

lRn

with a fixed point at

local change of coordinates ation such a

4

4: mn

-+

IRn

which linearizes f. The equ-

(or its inverse) should satisfy is $oL = £04

the linearization of f at the origin. problem was to expand

f and

4

where L

is

The approach of ~oincar6to this

in Taylor series, thereby obtaining a re-

cursive system of equations for the coefficients of eigenvalues of

0, we want to find a

.

If sane of the

L 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 1~' we want to consider is rotation by is L(x,y) =

2.rr 3'

, and

the linear map

L

The expression of the linear map

1 6 6 1 (-7 + p, - 7 - 3). Writing

the quadratic terms of

2 2 2 2 2 2 2 2 (al(x +Y ) + a2 (X -Y ) + a3(2xy), b1 (x +y ) + b2 (X -y )

+

b3 (2xy))

4

as

and those

o

f

2 2 2 2 2 2 (al(X +y + a2(x -y ) + a3 ( 2 ~ ~ 1B1,(x +y 1 +

as

B2 (x2-y2) + B3 (2xy)

leads to the following system of equations for the a 's and b.*s. i 1

The last two pairs of these equations have rank 1, so the coefficients a 2 and B3 cannot both be eliminated and similarly with f3 and a Thus the 2 3' nonnal form of f can be taken as

Perturbation of this map will not have a stable periodic orbit of period 3 near the origin. function

One way of studying the behavior of this map is to notice that the

Ilf (P)II

2

- llP112

has a singularity at 0 which is an "elliptic

umbilic" [lid for almost all choices of pressing the map the usual way. where

wLI

(a,B).

Another approach involves ex-

f in complex terms by identifying the plane with C

The expression for f becanes f (z) =

= 1 and y

depends upon a and 3

one can easily canpute that f (z) = z

-3

6

.

wz

+

in

y(z2 + r2)

In this representation

~ ~ +0 higher ~ 2 ~ order terms. The

map

f3 (z) -z has an elliptic, corank 2 singularity at 0

of perturbations is given by

fE(2) = (W+E)Z + y(z2-z

glecting terms which are quadratic in

E

-2

,

)

[ A

E

1.

One family

real.

Ne-

-

and cubic in

(z,z) gives

3

In looking.for solutions of f (z) = z near 0 the dominant terms lead to the approximate equations size has order

E.

i2 =

caz which has 3 non zero solutions whose

There is no condition of the sign of

E, illustrating

the qualitative difference of this bifurcation from the non-resonant Hopf bifurcations.

Instead of pairs of periodic orbits emerging from one side of

the bifurcation, a single periodic orbit exists on both sides of the bifurcation. See Figure 6.2 Arnold [ll] and Takens 11141 study the resonant Hopf bifurcations via equivariant vector fields on Let us return now to the Lorenz attractor.

.

In Section 54, we saw that

by taking a cross-section and projecting along an (assumed) invariant foliation, the Lorenz attractor determines a map of an interval to itself. This map of the interval has a discontinuity which is due to the presence of a singular point inside the attractor with a two dimensional stable manifold. If the cross-section is large enough to include the other two singular points of the vector field, then the interval map has fixed points at both endpoints of the interval. Thus we are led to study maps

f: I + I of the

interval which satisfy the following conditions: (1)

f fixes both endpoints

(2)

f has a single discontinuity c

(3)

f' (x) exists and is positive if x # c

(4)

lim f' (x) =

as x + c

from the right or left

This last condition requires that the positive eigenvalue at the singular point in the attractor have larger absolute value than one of the negative eigenvalues.

The Lorenz equations do have this property, and it plays an

important role in some of their bifurcation properties. Denote the two limit values of f (x) at c by divide the interval I at eraries of the map

f-(c)

f.

fi , then

and

c.

f-(c)

and

f+(c), and

The kneading theory implies that the itin-

+

f (c) determine the monotone equivalence class of

If we regard these as parameters, and

f'(x)

is always at least

they are the moduli which determine the topological type and the

topological equivalence class of the attractor.

The constructions which lead

to an interval map can be reversed by taking inverse limits and suspending. The attractor as a topological space with a flow is reconstructed from the map in this way.

This process is described in [ 4 2 ] .

Here we shall examine

the bifurcations of f and compare these with the development of the Lorenz attractor in terms of the parameter a Kaplan and Yorke [ 5 5 ] .

of equations 4-5 as described by

To begin with, we describe the "symmetric" case in

which the vector fields are invariant under rotation by When the parameter a

s around the z-axis.

is small in the Lorenz equations, there is a

single singular point at the origin which is globally stable.

This under-

goes a bifurcation giving rise to two new stable singular points.

This bi-

furcation is not generic among unrestricted vector fields but only among those invariant under the symmetry which rotates When

a

m3

by

n

around the z-axis.

is increased further, a pair of the eigenvalues of each new singular

point become complex.

After this occurs, the map

f can be defined.

itially it has slope less than one at the endpoints and

f-(c) < c
p

, then

The loop we consider will

It will behave like a limit cycle which on one side < 0

or repels if y + X > 0

which we want to avoid, is that y + X = 0

.

.

The indeterminate case,

This argument shows that the

topological equivalence of a vector field with a loop depends upon the sign of the sum of the two eigenvalues at the saddle. Let us now embed a vectorfield with a loop in a one

family.

To assure non-degeneracy of the family, a transversality condition must be formulated. Let X IJ

be the vectorfield with a loop formed from a

separatrix S y

to S.

p = po.

of the saddle p, when

Choose a transverse section

Saddles and their stable and unstable manifolds vary smoothly with

perturbatians, so that the stable and unstable manifolds of the saddle for X intersect y

when

p

- po

the parmeter interval into

is small. y

Near

po,

lJ

we have two smooth maps of

which give the intersections of the stable

and unstable manifolds of the saddle p of X which is near p. The P lJ transversality condition we want is that the graphs of these two maps intersect transversally.

In addition, we continue to require that the sum of the

eigenvalues at the saddle is non-zero. These two transversality conditions are enough to determine the one parameter family as unfolding of the loop bifurcation for a plane vector field. The situation can be understood in terms of the one dimensional poincarg maps

a

lJ

of a transversal section to the loop.

Ccnsider the case where the sum

of the eigenvalues at the saddle is smaller than zero. obtained from this one by reversing time).

(The other case is

If the section y

is chosen

small enough, then the derivative of the poincard map is smaller than one. This implies that the vector field the loop of

X

.

X

u

has at most one periodic orbit near

If there is a periodic orbit, then it is stable.

The trans-

versality condition on the stable and unstable manifolds of the saddle relative to the parameter imply that there will be a periodic orbit near the loop when

lies on one side of

but not the other.

po

ing the graph of a paincard map or

This can be seen by look-

as a function of the parameter

y

p.

These are illustrated in Figure 8.1 in a p-dependent coordinate system on

y

in which the origin is located at the intersection of the stable manifold of the saddle with

y.

In these coordinates, the fixed points of

periodic orbits passing through

y.

a

I!

locate

Phase partraits of the corresponding

vector fields are also illustrated in Figure 8.1.

Note that this bifurcation

gives yet another mechanism by which a periodic orbit can disappear. case, the period of the orbit tends to

m

In this

as the periodic orbit vanishes by

becoming a loop. Example:

Moduli of Saddle Connections [84]

The preceding example illustrates a situation in which the relative sizes of eigenvalues at a saddle affect the topological equivalence class of a system with a saddle connection. The condition on the eigenvalues in that example was persistent under perturbations of the eigenvalue since it was defined by an inequality.

In the next example we give, the topological equi-

valence class will depend upon the exact value of a certain expressions inThere is

volving eigenvalues. This expression is an example of a modulus. a continuous family of systems, no two of which are equivalent.

This type

of phenomenon was encountered earlier for the Lorenz attractor,-wherea pair of parameters were necessary to specify the topological equivalence class of the attractor. The systems we shall study are diffeomorpWisms f: R

2 -F

R2

which

have a pair of fixed points which are saddles and have a single orbit of tanqencies between their stable and unstable manifolds. We assume that there are saddle points p qencies between'

wS

and

wU (q).

and q with an orbit of tan-

The eigenvalues which will be espe-

A

cially important are the expanding eigenvalue tracting eigenvalue

p

of q.

wS tq) ,

See Figure 8.2.

We assume h

wU (q)

3

of p

and the con-

1 > p > 0.

wS (p)

at a point r u of tangency of their stable and unstable manifolds, and yg to W (p). cross sections yl to

y,

to

and

Pick three

From this situation we want to derive a topological means of determining the number

logp/logA.

The argument we use relies upon the A-lemma of Palis [821. In this example the A-lemma states that the forward iterates of y2

will tend uni-

u formly (in the cl-sense) to W (q), and the backward iterates will tend uniformly to wS(p).

This implies that there is an integer N

with the pro-

p e r t y that if

n > N

then

fn(y2)

closest point to

r

on

on

y2

y2 with

relative order of the points

y3

Ei IiLN.

Thus we can define sequences is the closest point to r

n

5

(qili2N

ji(qi)

E

n

y,

are n m empty.

E

yl.

yj

and that

is the i We want to show that the q

determine the quantity

qi

E1

with the properties that

i f (Si)

with

and

i

-n f (y,)

and

log~/logh

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

and

q.

is linear at the saddle

This is not a severe restriction since there are w-

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

y1

and

y

Moreover, we may

lie in these neighborhoods.

3

With these assumptions, we can easily describe how the sequences 5 and q i i tend to r. Measuring Si/Si+l

Near

Si

(and rl ) by arc length £ran r i

.

v

+

p, the coordinate along w ~ ( ~ is ) multiplied by

Similarly

-1 p

geometric rate

ni+l/rli

.

+ X

and the sequence

along y2,

The sequence

qi +

r

ai+l/ai = a

relative order of the a

i

Bi+l/Bi

and

Bj

and

>

> am+i+l '

'j+n

B

j+" = Bn, and

' j

Then

m+l > a

am+'

.

j'

{ai)

r

at the

a

{Pj1

and

.

with

ai

>

.

loga/logB

.

0. > ai+l and 3

a

Now

i

This gives the estimate

m+l = a

m-1 a ,

t+l

m-1 ,On m+l a > a .

i'

Taking logarithms (and remembering that logB/loga
am+i+l j

i+l

am+i+l =

Ei

determines the quantity

a

a m+i

6.

=

i, j, m, n with the properties that

Pick integers

this implies that

at the geometric rate h

Consider now two gemetrLc sequences of numbers the property that

P.

m+l

estimate for

.

By picking

logB/loga

in the two sequences.

n

loga < 0). we obtain


, h ( i )

*)=

Jn

P r o o f . C o r o l l a r y f o l l o w s from t h e f a c t , t h a t t h e r e i s

a t l e a s t one f i x e d p o i n t of

f kn

i n every element of

.

m e folloviing c o r o l l a q answers a q u e s t i o n r a i s e d i n [I]. C o r o l l a r y 3.

If

h(f)

>0

, then

p e r i o d i c p o i n t which p e r i o d i s n o t a power of

, 3

3

has a p e r i o d i c p o i n t of p e r i o d

t h e p e r i o d of t h i s p o i n t w i t h r e s p e c t t o by

.

2

It f o l l o w s e a s i l y from Theorem t h a t f o r some

Proof. n

has a

f

f

.

But then

is d i v i s i b l e

I Row we s t a r t t o prove Theorem. We s h a l l u s e t h e following

lemma ( c f . Lelnma 0 of l2] ): Leraal. f ( J ) n K f pj that

Let

, then

f(L)=f(J)nK

3 , K

besubintervalsof

there e x i s t s an interval

.

LC I

I

.If such

Proof.

f(J)nK

be its endpoints. For some

,

f(z) = x

t

and assume

.

may

.

an a p p r o p r i a t e (i.e.

L

z

,

' ini*(k8,+-)nf-1(y))

We t a k e as

,

J

Define:

z8 = sup ((--,t]nf-'(x))

t

y

from t h e c l o s u r e of

1. Without l o s s of g e n e r a l i t y we

J

t

e ,
/ h ( f , ~ )

r(a,b,n))

.

l i m sup n --+m

;l o g y ( a ,

!be t r a n a f o r m a t i o n

a p e r i o d i c p o i n t . Denote i t by

ni

,

can

hence

.

)o

lp ( a ) en) 3 h(f9*)

has t o have

, and by

a.

p e r i o d , R o t i c e that

f o r any

u

But

i s f i n i t e , f o r some t r a n s f o r m a t i o n

we have

aCE

>/

K(a.b,n))

i = O,l,...,m-X

.

merefore

m

- its

$ log

l i m au2 n-00

a

a

n

,

.

A

'I'he r e v e r s e i n e q u a l i t y follows from the obvious

inequslity ( 2 ) holds.

J'-(ao?aoTn)

Assume now t h a t (if

)

a fixed

n

I

.

h(f)> 0 ~

h(f)=+o.

-

>n+2

(if

and

, there

A

, such

B

, then

h ( f ) = +oc

kn

P

.

B

).For

By Lemma 4

e x i s t : an i n t e r v a l

a.

that

such that A

of

, applied and

, such t h a t

,m

1 10s r(ao,a,,$) mn

,

r = l

).Then we t a k e a f i n i t e p a r t i t i o n

a positive integer

Jn = a0

we s e t

We t a k e a n i n t e g e r ,then

i n t o subintervals, f i n e r than

to

, then

h(f) = 0

If

we take a f i n i t e open cover

h ( f r , ~ )3 h(?) h(f,B)

Hence, t h e e q u a l i t y

I

Proof of Theorem.

P

< Cmd

9 r*mn

. We s e t :

-

~(F,A)

,

- such a p a r t i t i o n of

Dn

i n t o s u b i n t e r v a l s , t h a t f o r every

d E Dn

there exists

a unique element

e € E

(fr)mn ( e ) 3 a.

( i n such a way we o b t a i n t h e o o n d i t i o n ( b ) ) .

We have

r;;;

l o g Card D ;

f o r which

a,

-q 1

e C d

and

l o g F(ao,ao9m,) 3

l i m i n f -1- l o g n-okn

then

- $)

3 l i m (n+2 n+m

n

m e inequality

l i m 1 ( h ( p ) -;) 2 n+mr

= h(f)

,

-

lim i n f 1 l o g Card D , ), n - e - kn

. I n both c a s e s we have

=+-

1 l i r n inf f ;- l o g Card Dn

n+w

'

, then

h( f) = + 00

and i f

>

Card3

.

3 h(f)

h ( f) $ l i m sup L l o g Card Dll n 3 0 0 n'

follows from ( b ) as i n t h e proof o f C o r o l l a r y 1. Hence, t h e property (a) i s a l s o s a t i s f i e d . Remark.

, Theorem

A s i n [3]

continuous mappings c l o s e d s u b s e t of

a where e i t h e r

f : X-X

I

value) property, o r

and

is also valid for

f

X

is a

h a s t h e Darboux ( i n t e r m e d i a t e

is a c i r c l e . I n t h e l a s t case, a l l

X

corollaries a r e also valid. Bef e r e n c e s

.

[l] R. Bowen, -J. Franks

-

d i s k and t h e i n t e r v a l [2]

T. L i , J. Yorke

The p e r i o d i c p o i n t s of maps o f t h e

- Topology 15 (19761,

1. M i ~ i u r e w i c z ,1. Szlenk

-

992

- E n t r o ~ yof p i e c e r i s e monotone

v e r s i o n w i l l appear i n S t u d i a Math. 67) I n s t i t u t e of Mathematics, Warsaw University PKii? UI p. 00-901 Varszawa

Poland

- 742

- P e r i o d t h r e e i m p l i e s chaos - Amer.

b t h . Monthly 8 2 (19751,. 985

[3]

337

C ENTRO INTERNAZIONALE MATEMATICO ESTIVO (c.I.M.E.

)

VARIOUS ASPECTS OF INTEGRABLE HAMILTONIAN SYSTEMS

J e

MOSER

J. Moser ~okan~ t n s t i t u t eof Mathematical Sciences, New York University

Contents Integrable System, Basic Facts and Examples Examples of Integrable Systems, Isospectral

Deformations

Reduction of Hamiltonian Systems with Symmetries The Inverse Square Potential Extension of Geodesic Flow Geodesics on an Ellipsoid An Integrable System on the Sphere

H i l l 's Equation 1. Integrable Hamiltonian Systems

(a) In these informal lecture notes we discuss a nunber of integrable Hamiltonian systems tions.

which have surfaced recently i n very d i f f e r e n t connec-

I t i s our goal to discuss various aspects

i t y of a system l i k e t h a t of group representation, and geometrical considerations.

underlying the integrabilisospectral deformation

Since this subject is s t i l l f a r from being

understood o r being systematic we discuss a number of examples which a r e seemingly disconnected.

I n f a c t , there are some rather unexpected connections

( * ) p a r t i a l l y supported by National Science Foundation Grant MCS 77-01986.

l i k e between the inverse square p o t e n t i a l of Calogero (Section 4) and the Korteweg de Vries equation.

Here we show a surprising new connection between

the geodesics on an e l l i p s o i d and H i l l ' s equation with f i n i t e gap potential. (b) The d i f f e r e n t i a l equations of mechanics can be written i n Hamiltonian

aH

(1)

when x = (xlr..

..xn

n

) E R

,

Yk=--

a~

.,yn) E

y = (ylI..

phase space R~~ o r an open subset of R

(k=1,2,.. .n) Rn , a r e coordinates i n the

Thus ~ . a function H defines a vector

~

f i e l d ]k defined by

For any function F

i s antisymmetric

the expression

i n F, H.

I t is called the ~ o i s s o nbracket of F and H.

The

Hamiltonian systems form a Lie algebra and [X

X I = -

HI G

A

nonconstant

I n p a r t i c u l a r , H is an integral.

A

~

~

,

~

~

function F is c a l l e d an i n t e g r a l of XH i f X~IF ='

i n t e g r a l of

X

IF,H) = o

.

I f F is an i n t e g r a l of

then H is an

5.

s e t of functions F1,F2,

...,Fr

are s a i d t o be "in involution" o r t o

,...,

f o r k , j = 1,2 r. k I This implies c l e a r l y t h a t the vector f i e l d s X comute. Fk I f $ = $(El, ,Er); $ = $(El ,Sr) then {F ,F.) = 0

...

Thus, i f F1,

,...

...,Fr are i n involution,

so are any functions of F1,F2,

. . . IFr'

2n DEFINITION 1. A Hamiltonian system (1), def ined i n an open domain D C R

i s c a l l e d "integrable" i f t h e r e exist n i n t e g r a l s F1,F2,.

..,Fn

in

involution

. i n D we have (iii) dFl, ...,dF l i n e a r l y independent. n

with l i n e a r l y independent gradients, i .e (i){ H,F

.I

3

= 0;

Example 1: H Fk =

2

%+

Example 2: with Fk =

i

-r1 1

F F

k 2

%(\ +

2 yk (k = 1,2

= 0

3

yi)

,

defines an integrable system i n FtZn with

,..,,n ) .

I f H = H(y) is independent of x

then the system is integrable

Yk' any system is integrable; i n

Locally, t h a t i s near any p o i n t where dH f 0

which i s a

f a c t , i n appropriate canonical coordinates H agrees with yl s p e c i a l case of Example 2.

Generally it makes sense t o speak o f a system being i n t e g r a b l e i n a domain which

E

i n v a r i a n t under the flow generated by

5.

I t i s highly exceptionai! f o r a Hamiltonian system t o be i n t e g r a b l e gZobaZZy

i n an i n v a r i a n t open domain

dH = 0).

However

-- o r even

l o c a l l y n e a r a s t a t i o n a r y p o i n t (where

many systems occurring i n a p p l i c a t i o n a r e c l o s e l y approxi-

mated by integrable systems.

For example, t h e n-body problem becomes

i n t e g r a b l e i n the l i m i t when a l l b u t one mass tends t o zero.

The r e s u l t i n g

system is a decoupled system o f Kepler problems. Another example is a system near a s t a t i o n a r y p o i n t , say dH(0) = 0.

x = y = 0 where

Assume t h a t t h e Taylor expansion o f H begins w i t h

with r e a l numbers al,a2

I . . .,an.

By a theorem of G. D. Birkhoff one can

approximate t h i s system l o c a l l y by an i n t e g r a b l e one (see 131, 141.): Given any l a r g e i n t e g e r N , assume t h a t j l a l +

-< N

, jk

- - - + jnun=O,

i n t e g e r s implies t h a t jl = j2 =

c a l variables

xi,.

.., x i , Yl,...,Y,*

...

--

ljll

+

lj21

+

+

ljnl

jn = 0; then t h e r e e x i s t canoni-

such t h a t

H = c b (F1,F2,.-.,Fn)

+

%+1

where F j

%+1 vanishes

= x . ~ + y;2. I

with derivatives of order

Thus i f we drop the term

5N

a t the origin and

the system i s integrable.

(To be precise, one has t o excise the h s e r p l a n e s

,

where e

3.

= (0,0,1) and

A > 0. Then

it follows from

- X3Y2 = 0 - 1y 3 = 0

X2Y3

xyl

X

t h a t x3 = y3 = 0 and the problem i s reduced t o one i n R2 x R2, where we have the quadric

x1y2

- x2yl

= 1.

This problem i s s t i l l rotation invariant under

SO(2) and we may use polar coordinates r , $ and conjugate variables p

r' '4.

They can be defined by the canonical transformation

with W = r(ylcos @

+

~ = r c o s $ = W y1 x2 = r s i n @ = W y2 y2sin $1. Then pr = Wr , p@= W$ gives

$

4 sin $ v2 = p s i n 4 + 3 cos $ r r Y 1 = prcos

and xly2

-

x2y1 = p4 = A.

Thus p

4

is the i n t e g r a l and H independent of $.

The reduced Hamiltonian i s

and the $-dependence i s obtained separately from

This type of reduction was known t o Jacobi who "reduced" the 3 body problem i n R~ by using the invariance of t h i s system under the G a l i l e i contains the rotation group SO(3) as a subgroup.

group which

Eliminating the integrals

of the center of mass and the angular momentum one can reduce t h i s system of 9 degrees

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

space from 18 dimensio~st o 8 dimensions.

Using, i n addition, the conserva-

tion of energy

one has a vector f i e l d on a seven dimensional manifold.

(b) The moment map. abstract form.

W e describe the generalization of this reduction i n

W e wnsider a manifold M with a one-form 8 for which w = d8

i s nondegenerate, so that (M,u) i s a symplectic manifold. refers t o ( M , e )

as an "exact symplectic" manifold.

One sometimes

An example i s the cotan-

gent bundle M = T*N of a manifold N with the natural 1-form. I f G i s a Lie group we speak of a symplectic gmup action

(4

g E G t h e n exists a diffeomrphism @ : M + M such t h a t 9 g.h E G. (id = id. and

4

9

is symplectic, i.e.

4)

= w.

9

=

i f for every

mgh

for

I f , moreover.

4i8 = 8 we speak of an exact symplectic group action. For G

R t h i s concept i s the same as that

of symplectic flow.

Every such

flow is generated by a Hamiltonian vector f i e l d , say X for which X J

w is closed.

W e w i l l assume t h a t t h i s one form is even exact and s e t

For example, i n the exact sym-

F being a Hamiltonian for t h i s vector field. plectic case we can define F by X J 8 P P .

I f A i s the Lie,algebra of G I and a E A, expta = g = g ( t ) E G function on M

-- the

then f o r any

relation d = Xf x f(@g)1 t=*

defines a symplectic vector f i e l d on M. a E A.

If

bg , hence

X

This vector f i e l d depends linearly on

are exact symplectic

then we can define the cones-

ponding Hamiltonian F = F(p,a) by X J 8 = F . This Hamiltonian depends linearly on a E A $ E A* i n the dual of the Lie algebra via

and therefore defines an element

The mapping

9:

M

+

A*: s o defined is called the moment map (of Souriau).

These concepts a r e e a s i l y i l l u s t r a t e d and motivated by Example 2, where G = S0(3),

A

= R3 with [a,bl = a Ab.

3

1

space (M,O) = (R6,

j=l

y .dx.) 3 3

The group action on the exact symplectic

i s given by

$g(~,~ = ) (Rx,RY)

,

R E SO(3)

and the cofresponding vector f i e l d s f o r R = em, A x = a Ax

x

= a (x a 1 2 X3

i s obtained by

- x ax2 ) + cyclic permutation

This vector f i e l d i s Hamiltonian with Hamiltonian 3 F = a F F~ = x g 3 x3y2 (and cyclic perm). j=l j j ' The moment map 9: R6 + R3 takes p = (x,y) i n t o the angular moment vector

-

1

(F1tF2,F3) If H

0

i s invariant under the group action $ g

COrresponding flow

mot

, i.e.

H

0

o$

g

= Ho then the

generated by Ho leaves @ invariant, i.e.

c0 This generalizes the f a c t t h a t the angular momentum vector i s an integral. The proof of the aboire statement i s immediate (see [3]). An important group action,

(d) The coadjoint representation of a group.

which is.independent of symplectic s t r u c t u r e , i s the adjoint representation of a Lie group G

which is given by

4y Here L

R

denote l e f t

-1 x + g x g

= LgR -lX g and r i g h t multiplication. This mapping takes the.

g ' g u n i t element e E G i n t o i t s e l f and the linearized map a t x = e i s defined as

I t maps

A

+

A and s a t i s f i e d Ad(glg2)

= Ad(gl)Ad(g2).

-

I t is the adjoint

1 representation of G. The induced mapping ~ d * ( g ) on A*, the dual of A,

called the coadjoint

is

representation.

For a fixed p E A* one defines the o r b i t s O ( p ) of the coadjoint representa-

=

t

(Ad*s)v

I t i s a basic r e s u l t due t o K i r i l l o v

a symplectic structure.

I.

I

g E G

and Kostant t h a t these o r b i t s 0(p) carry

We i l l u s t r a t e t h i s f a c t with one example, referring

f o r the f u l l discussion t o [51. Let G = GL(n,R), then one sees a t once t h a t A ~ ( T ) : A + T A T-l where

T E G;

A E

A.

We can identify A* with A by representing any l i n e a r functional on A by tr(AB) with B E A.

representation.,

Thus we can identify the adjoint and coadjoint

I n t h i s case the o r b i t s O(p) a r e the matrices similar t o p. d i s t i n c t eigenvalues

I f p has

th;e o r b i t consists precisely of the isospectral matrices.

The symplectic structure i s defined by giving the values of the two-form on the tangent space of

IT A

$-'(A)

T",

T E G}

The tangent space a t A i s given by T $"(A) A

The

{IB,AI

=

I

B E A).

d i f f e r e n t i a l form i s given by U( [B1,Al

,[B2,AI)

= t r ( A [B1,BZI)

One v e r i f i e s t h a t t h i s form i s nondegenerate, skew symmetric defining a symplectic structure on O ( A )

-

Consider any two group actions Q M g." map T: M + M equivariant i f

. +

M and

..

Qg = QgT

........, : M M of

4

g

+

G.

and closed, thus

One c a l l s a

.

is an exact symplectic group action then the moment map $: M + A i s g M + M and the coadjoint representation of A. equivariant with respect t o 4 9: (This theorem holds under more general assumptions, t h a t is not only j.' ; i s If

exact symplectic)

9

.

Thus the image of {Q (p) g

I

g E G}

f o r fixed p E M

under the moment map

$

i s given by the o r b i t '

I

bid* ( g ) ~ g E

GI

where

IJ

=

$(PI

.

The isotropy group G of l.j is defined a s t h e subgroup lJ

G = 1g.E G lJ

I

~ d * ( g ) l l , =lJI

o f G. (e) The reduced phase space.

I$ : M g

+

M

Given an exact symplectic group action

we construct the moment map $: A

+

A

and f o r a fixed p E A* the

set -1 $J (1-1) = { P E M

I

$(PI = l J ) .

This i s the subset of M of fixed values of the i n t e g r a l .

I n the example 2 it

consists of those s t a t e s f o r which the angular momentum vector i s fixed. t h i s s e t the group G a c t s , which i n t h i s example lJ

On

consists of a l l r o t a t i o n s

leaving the angular momentum vector l~ fixed. I f lJfO t h i s i s a one-dimensional r o t a t i o n group SO(2) quotient s e t

.

(p)/G

lJ

To eliminate t h i s angle o f r o t a t i o n we consider the which corresponds t o elimination of the "ignorable"

angle of rotation.

-1( l ~ )i s a manifold,

Under appropriate assumptions (e.g. t h a t $ that G

?J

a c t s on

(p)

fixed point f r e e )

G

lJ

is compact,

we introduce the "reduced phase-

space"

i = f 1(P)/G, Thus

M

.

i s the base space of the bundle IT: *-'(lJ)

This reduced phase space

+

i.

i s a symplectic manifold again and the symplectic If

s t r u c t u r e i s given by a two form G; which i s defined as follows: j: $-l(lJ)

+-l(p).

+

M

i s the i n j e c t i o n map then j * denotes ~ the pullback

This form is i n v a r i a n t under G and therefore defines 1-I IT*; = j * ~

I n other words, f o r V ,V E T$ 1 2 .

-

-1 ( l ~ ) l e t Vk, = ( ~ I T ~ ),Vthen ~

of

w via

U

to

Moreover, i f H i s a Hamiltonian invariant under $ Ls given by a Hamiltonian

H

g

defined by

-

= HOj

HOT

In ~ t h e words r H O j , the r e s t r i c t i o n of H t o $'l(p) therefore defines a function

then the "reduced flow"

on

i s invariant under G

v

c.

and

I f H = H($ ) is invariant under the group action $ then g 9 (under appropriate assumptions) the phase space can be reduced t o We summarize:

i=

c1(p)/G,

which is a s y q l e c t i c manifold and the Hamiltonian reduces to

i

defined by r e s t r i c t i o n . We do not go i n t o the precise specification of fha ass.mptions since t h e i r v a l i d i t y can be e a s i l y established i n the examples which we discuss. (f) Examples.

and 2. which Example 3:

To i l l u s t r a t e t h i s reduction one may consider the Examples 1

we leave t o the reader.

Consider the group action of G = R (x,y)

on R

~ which ~ , preserves 8 =

=

{ ( x , ~ )E R

* n-1 which i s the cotangent bundle T S the one form < y , W

of minimal distance i n the = 0.

~ 1x1 ~ ,= 1, *,y>

Thus = 0)

of the unit sphere.

projects i n t o the natural 1-form of

One v e r i f i e s t h a t

* n-1 . TS

A s an example we consi'der the Hamiltonian

which is invariant under the above group action. field

;=

lx12y

-

<x,Y>X

The corresponding vector

r e s t r i c t s on T*s"-~ t o

which i s clearly the geodesic flow on the sphere. 1 appears as the reduced system t o H = 2 (1x1

-

Thus the geodesic flow

- QcIy>2).

1 1

Its o r b i t s are

c i r c l e s on the sphere. Example 4:

The above Hamiltonian (*)

is invariant

(x,y)* (ax+by, c x + d y l of ,511 (2 ,R)

.

,

under the group action

ad-bc-1

Its Lie algebra is generated by the Hamiltonians

Ixi 2 , 1y 1

I

<x,y> and we ask f o r the reduced space corresponding t o 1 x 1=~ 1 which defines $-l(1.1)

<xpy>= 0

i n t h i s case.

,

lYl2

= 1

,

We need the isotropy group G which is

P

easily determined t o be given by

Note t h a t $-l(p) is the tangent bundle of c i s e l y the geodesics i n the sphere.

sn'l

and the o r b i t s of G

a r e pre1.I Thus ,the reduced space i s the o r b i t mani-

fold of geodesics on the unit tangent bundle. The reduced flow is constant i n t h i s case. In the next section we w i l l describe a more complicated example of a reduced phase space under the coadjoint action of the unitary group extended t o the cotangent bundle of t h e Lie algebra.

This w i l l lead

f o r appropriate choice

of 1.1 t o the inverse square potential of Calogero. Actually similar constructions are applicable t o the Tode l a t t i c e (the group being given by the upper triangular matrices) and the Korteweg-de Vries equat i o n as was shown recently by M. Adler 151.

References [l]

V. I. Arnold, Mathematical Methods i n Classical Mechanics, Moscow, 1974,

(to appear i n English t r a n s l a t i o n ) , i n p a r t i c u l a r Appendix 5.

[2]

133

J. Marsden

and A. Weinstein,

Reduction of symplectic manifolds with

symmetries,

Reports on Math. Physics 5, 1974, 121-130.

J. Marsden,

Applications t o Global Analysis i n Mathematical Physics,

Publish o r Perish, Inc, 1974, i n p a r t i c u l a r Chap. 6.

143

J. M. Souriau, Structure des syst&nes dynamiques, Dunod, Paris, 1970.

151

A. A.

Kirillov,

Elements of the Theory of Representations, Springer,

1976. [6]

M. Adler,

On a trace functional f o r formal pseudodifferential

and symplectic structure Univ. Wisc., 1978.

operators

of the Korteweg-de Vries equation,. preprint,

The Inverse Square Potential

4.

(a) I n connection with h i s quantum theoretical work Calogero [ 4 ] was led t o He was led t o the conjecture t h a t the

consider the n-body problem on the l i n e .

corresponding c l a s s i c a l problem was integrable which was subsequently proven Recently these r e s u l t s were derived most elegantly from a reduction pro-

( 151).

cess as it was discussed above applied t o the coadjoint representation of the unitary group [ I ] . We formulate the r e s u l t s .

Consider n d i s t i n c t points on the r e a l l i n e with

coordinates X

and define the

1


0

poly-

we consider the cotangent bundle of the unitary

We represents i t s eiement i n t h e form of a p a i r (XIY)

of Hermitian matrices and with the 1-form

The

symplectic structure i s defined by the two form

w = t r (dY

A

dX)

In t h i s symplectic space the Hamiltonian system associated with a function H = H(X,Y) 'is given by

x=$

$ = - H ~

t

where HT is the .matrix with M e e n t r i e s

( a ~ / kaj)~ e t c .

The Poisson bracket i s

given by ( F,G) = tr (FXGy

- FyGX) .

Therefore it i s c l e a r t h a t any two functions F of X are i n involution.

1

= F (Y) 1

, F2

= F (Y) independent 2

Hamiltonian is integrable.

Thus any such

.

I

= t r (yP) P P w i l l be relevant.The system belonging t o G2 i s even linear &has G

(6)

(d) A l l these systems defined by G

P

For us

the solution

are invariant under the group action

The corresponding infinitesimal mapping is (X,Y)

where A is H e d t i a n , i.e.

+ i([X,Al,

[Y,AI)

i A belongs t o the Lie algebra of U(n). This vector

field

'has

the Hamiltonian

GA = i . t r

(A[Y,XI)= t r ( ( i ~ )[ ~ x , ~ Y I )

This defines the moment map as $: (X,Y) t IY,Xl

the right-hand side being interpreted as an element of the dual of the Lie algebra.

The elements of [Y,XI represent the relevant integrals; [Y,Xl i s the

analogue of the angular momentum vector. -1 To construct the reduced space $ ( p ) / E we choose

and Zternberg

-- with

Kazhdan, Kostant

the element p as the matrix

-1 so t h a t $ (p) i s given by the p a i r s of Hermitian matrices (X,Y)

T1E

for which

complete s e t of solutions i s given by the following.

PFOPOSIT~ON 1.

The most general solution of (9) with p given by (8) i s of the

form (10)

x

=

where x ,x 1 2

.

v - ~diag (5, .. ,xn)v

,...,rn

Y =

;

v-1L ( X , Y ) V

are d i s t i n c t zeal numbers, y = (yl

,...,yn)

E

cn

and V a

unitary matrix satisfying

V e =

X e

[i]

where

e = Moreover, the group of the unitary matrices V agrees with the isotropy group G

1-I

o f p .

COROUARY.

The quotient space $ (p) /G

Y = (Y lt...,yk)

and t o any (X,Y) E

1.1

i s parametrized by x = (xl,.

$(u)

..,%)

,

i s by (10) associated a unique p a i r of

matrices ( ~ ( x,) L(x,y))

where

K(x) = diag (xl

... xn)

The corresponding d i f f e r e n t i a l form i s computed as

Proof of the proposition:

where v = (vl

We consider the nwre general equation

... vn) i s any complex vector # 0.

The second term i s determined

s o t h a t the t r a c e of the right-hand side vanishes. goes over i n t o ( 9 ) .

For vk = 1 the equation

We choose a unitary transformation U which diagonalizes X. X + U-~XU

v

+

U-lv.

,Y

+ U-'W

Note t h a t under

the right-hand s i d e goes over i n t o a similar form where

Therefore we can analyze t h e equation f o r a diagonal matrix

X = diag (xl

... xn) = K(x).

From

[ K ( x ) . Y ( x ) I=~o~ = i ( l v k l

2

- -n1 1v12)

2 we conclude t h a t lvk12 i s independent of k. I f r e dspecialize t o lvl = n we i 6. g e t vkI =, 1, o r vk = e Applying the unitary transformation, i0, ie2 e .e I...Ieien]

I

'.

we achieve t h a t v = e and the diagonal form of X is not destroyed. Thus we have reduced our system t o the case X = K(x); v = e. From [ K ( X ) , Y =] ~(xk ~

- xR)Yk

= i

for

k # R

we conclude t h a t the xk are d i s t i n c t and Yk

I f we s e t Ye

=

i for =5c - "a

k # R.

= yk we have the solution

f o r the system ( 9 ) . To find the most general solution we have t o investigate those unitary transformations U which leave the equation invariant. dl(v

k

; ) u R

dwk

;d

Since

-

w = U 1v

where

and since t h i s matrix determines the vector w only up t o a f a c t o r A on we see t h a t the s e t

of U comuting with (vk

GR)

I hl

a

1

is given by those unitary

This proves the remaining assertions of the prop-

matrices f o r which Uv = Av. osition. (e) Thus t o

-1( l ~ )we

any (X,Y) E J,

can associate a unique p a i r

n(X,Y) = ( ~ ( x ) ~, ( x , y ) )= (u-'XU, and t o any function H ( X , Y )

U-~YU) ;

on J , - l ( l ~ ) we can associate a function

h(x,y) = V*H= H ( K ( x ) ,

L(x,y))

This projection takes the symplectic s t r u c t u r e of J , - ~ ( ~ J ) / G i n t o the standard

u

In p a r t i c u l a r , the functions (6) which are i n involution go over i n t o

which

a r e a l s o i n involution.

I f we observe t h a t

1 2 1 t r L (x,y) = 2 ly12 2

-

-

+

U(X)

i s the Hamiltonian o f . t h e inverse square potential

t h e n . i t i s c l e a r t h a t the

F are r a t i o n a l integrals i n involution of t h i s system, proving one part of P our theorem. The solutions of the G2-flow are

The other p a r t i s equally e a s i l y derived. given by (7) which we r e s t r i c t now t o exist

JI-1(p).

By the proposition

there

unitary matrices V such t h a t t (V~K~V;',

where we s e t K

t

vtLtvil)

=

= ( ~ ( t )L) = , L

(vO(KO + ty )

tLO)v;',

t = (KO

v~L~v-~)

Hence with Ut=

+

v;hO

we h a w

tLO,LO)

The equality of t h e second component shows again t h a t t r )L(:

= tr LoP are

i n t e g r a l s , the equality of the f i r s t component shows t h a t the x k ( t ) are eigenvalues of KO

+

tLo

, proving

the theorem.

( f ) O u r derivation was based on t h e f a c t t h a t the functions tr'Y are mapped n tr KP = t o F = t r LP(x,y). Similarly, t r'X is mapped i n t o P k=l %,' b u t

1

here it i s evident t h a t these functions a r e i n involution.

Nevertheless, it

i s useful t o consider the map (X,Y)

+ (-Y,X)

which c a r r i e s one s e t of functions i n t o the other. -1 symplectic, preserving $ (p).

t h a t there e x i s t s a U E G

P

This mapping i s clearly

Projecting t h i s mapping t o

such t h a t

(u-'K(x)u,

u-?L(x,Y)u)

=

(-

L ( s , ~ ) , K(s))

9-1(p)/G Fr we see

where (x,y) t (St?$

is the induced mapping.

symplectic algebraic mapping of { x < x < 1 2

IS1
= 0. Thus we have i n (2) a H d l t o n i a n system i n R

2n+2

which a f t e r r e s t r i c t i o n t o

the tangent bundle becomes the geodesic flow on the sphere. (b) We want t o generalize

t h i s construction t o a r b i t r a r y n-dimensional n surfaces i s o m t r i c a l l y embedded i n t o with the usual metric ds2 = dS2. P O n+l Let f (x) be a smooth function i n R which a t f i r s t is assumed t o be

1

s t r i c t l y convex, and such t h a t f (3)

-t OD

as 1x1

-t m.

F(x,y) = mint f ( x

+

We define ty)

,

and consider t h e Hani ltonian system

f o r x,y E

W e have the obvious LEMMA 1.

The system (4) possesses t h e i n t e g r a l s ly12

and

F(x,y)

.

W e may r e s t r i c t ourselves t o solutions of (4) with lyl = 1 and F = 0. We

w i l l i n t e r p r e t a p o i n t (x,y) i n phase space with n+l

the p o i n t x E R

with d i r e c t i o n

X

a l i n e through

= 1 as

Thus (x,y) i s represented by an W e w i l l assume

with a distinguished p o i n t x on it.

oriented l i n e L = L(x,y) t h a t the gradient f

y.

lyl

does not vanish i f f = 0, s o t h a t f = 0 represents a

smooth manifold. THEOREM 1.

I f x = x ( t ) , y = y ( t ) i s a solution of (4) with

ly[

= 1, F = 0

then t h e l i n e L = L ( x ( t ),y (t)) i s tangent t o the surface f (x) = 0 and the p o i n t of tangency

c(t)

of L ( x ( t ) ,y (t)) and { f = 0) moves on a

geodesic of the manifold { E = 0). Proof:

Let s = s ( x , y ) be determined such t h a t F(x,y) = f ( x

i-e. that f ( x

+

sy)

,

+ t y ) has i t s minimum a t t = s. Then one has c l e a r l y < FX,y> = < f x ( c ),y> = 0 with 5 = x + s ( x , y ) y

(5) This implies

Fx =

f x ( c ) + < f x ( c ) ,y'sX

F = sfX(c) Y

= fx(c)

+ < f x ( c ) ,y>sY = s (x,Y)

fx(c)

Hence t h e d i f f e r e n t i a l equation (4) become

;= For t h e p o i n t

6

= x

+

sy

s fX(S)

,

y =

we g e t the equation

-

fX(S)

.

.

.

I f we use s = s ( x ( t ) , k ( t ) ) a s independent variable we f i n d

This is the d i f f e r e n t i a l equation f o r a geodesYc of f = const since the second derivat

of

E i s normal t o the surface.

I f we r e s t r i c t ourselves t o F = 0 then f (

= 0.

Thus

i1 (0) /Go 2 M

and we a r r i v e a t t h e r e s u l t :

I f we reduce (4) by the group action (11)we obtain up t o parametrization the geodesic

fZow

on f (x)

the unit tangent bmdZe of

= const. (f = 0

Moreover,

F = 0

gives r i s e t o the f7.w on

;.

Conversely we can view (4) a s an extzr;&?9 Hamiltonian system of t h e geodesic

flow

--

i n the sense t h a t the l a t t e r i s obtained from (4) by a reduction v i a a

synplectic group action. Clearly, there a r e many such extensions.

The use of such an extension is t o

be compared with the use of horiwgeneous coordinates t o describe points on the sphere instead of spherical coordinates.

Of course i n homogeneous coordinates

one has to take account of the i d e n t i f i c a t i o n of x and Ax f o r

A >

0. Similarly,

in the Hamiltonian one has to eliminate the redundancy of the group action. ( f ) For the f o l l w i n g we have t o generalize the above considerations somewhat. I t is, of course, not necessary t h a t f ( x ) i s a convex function. A l l t h a t matters is t h a t f o r some point

5

= x

-

t*(x,y)y

on the l i n e x

+

ty

Then $(x,y) = c is the equation f o r the tangents of f (x) = c. Since we are not interested i n the parameter dependence of t h e - ~ a m i l t o n i a n system we could take any function H(x,y,c) defining the tangent t o f ( x ) = c by

as a Hamiltonian, as long as

a

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

Indeed since @(x,y) i s obtained by solving

we have Hx + Hc$x = 0

and therefore

,

H

Y

+ H $ - 0 C Y

which i s , up t o parametrization, the above system. (g) We consider two surfaces f (x) = 0, g(x) = 0 with nonvanishing gradient and l e t @(x,y), $(x,y) l i n e flows. PROPOSITION.

respectively be the Hamiltonians of the corresponding

We inquire when the corresponding flows commute. I f f o r any l i n e tangent t o both f = 0 and g = 0 one has

Ern

f o r the points of contact

where f(S) = 0, g(n) = 0. then

{$,$I=

0

on

$ = $ = O

,

Thus i f fhe moments of f = 0, g =O a r e perpendicular

i .e. both flows commute.

then the flows commute. Proof:

I f x +' t y describes the l i n e one has '$(Spy) = f ( S )

a t the point

6

= x

$,$I

+

$ x ( S r ~ )=

t

t * of ~ contact.

= < $ ,$

x Y 6.

>

-

(1)

be a positive d e f i n i t e quadratic form i n R"+' O


= 1

For z < .ao these are e l l i p s ~ i d ,but f o r a. < 'z < they are hyperboloids;

4

or

i n f a c t one has n+l d i f f e r e n t types

...

.

of such quadrics corresponding t o the i n t e r v a l s ( 4 , ao) , (ao, all (an-l an) n xv # 0 pass n+l such confocal quadratics Through every point x E R ~ " with 0 (2). Indeed f o r a given ,such x t h e r a t i o n a l function 1-QZ(x) has n+l d i s t i n c t roots, one i n each of the above i n t e r v a l s and one can write

The equation z = u

v

defines the desired n+l hyperquadrics intersecting a t x;

it i s well known t h a t they i n t e r s e c t orthog~?zazZy. Actually they possess a

stronger orthogonality property

which is l e s s well known

-- and which w i l l

be of i n t e r e s t t o us. I f L is a s t r a i g h t l i n e which touches two confocal quadric Q

THEOREM.

Z.

= 1,

Qz = 1 a t the points I"), I K ( ~ ) then the normal of the quadrics a t i e s e 2 points of tangency are perpendicular t o each other, provided zl = z2. Proof:

Let t h e points of L be given by x

x'j) = x

+

t.y. 3

The condition

+

ty where t varies over R, and l e t

of tangency of L a t x ( j ) i s given by

We want t o show t h a t

vanishes.

For t h i s purpose we use t h e i d e n t i t y

Therefore, i f QZ(x,y)

i s the symmetric b i l i n e a r form belonging t o QZ(x) we

have f o r (5)

-

-1

tfr

(Q,

(x"') + ct21 which vanishes on account of (4). = 4

z 2

Q,

This ,remarkable property implies t h a t a l i n e confocal qbadrics

-- has

- pz2

(x(l),y) 1

(X

-- which

(2)

generally touches n

n mutually perpendicular normals associated with it.

These normals a r e c l e a r l y a l s o perpendicular t o the l i n e , s o t h a t we have an orthogonal (n+l) frame associated with each such l i n e . (c) TO determine the d i f f e r e n t i a l equations of the geodesics of the e l l i p -

s o i d we generalize t h i s question i n two ways: (i)We replace t h e e l l i p s o i d by any of the confocal quadrics Q (x) = m and (ii)determine the extended " l i n e flow" of the previous section. For t h i s purpose l e t u . (x) = z be any of these quadrics and consider a l l 3 l i n e s x + t y which are tangential to it. Since f o r the given value z we have Q, (x) = m the condition f o r x

t h a t f o r the

+

point of contact

ty t o

5

Qz(S)

=

represent a tangent to Q (x) = m i s z

x + t*y

= m ,

Q,(2 whose gradient vanishes H0 =

F(ly12+ 0 there i s no stable i n v a r i a n t closed curve

Suppose moreover t h a t f o r

1

near T

x 0 for F

1

borhood of T

*

: what i s the new a t t r a c t o r (i any) f of F i n a neighP

x 0 f o r ~r,> 0 small 2

The f o l l o w i n g thecrem gives s u f f i c i e n t conditions f o r the appearance for on which F i s i s o t o p i c > 0 small of a stable i n v a r i a n t 2-torus o f F P ' CI t o i d e n t i t y ( s o t h a t the a t t r a c t o r f o r the flow i s r e a l l y a 3-torus). I n

p

f a c t the hypotheses are s t w n g e n o k h t o imply from t h e mere invariance 1 x 0 under F, the existence f o r p small ( p o s i t i v e o r negative) o f a

of T

a

closed curve i n v a r i a n t under F

and g e t t i n g close to

if the r o t a t i o n number of FoIT'

x 0 i s r a t i o n a l (see [B] and [C.Y.])

x 0 as p goes t o 0. 1.1 That such a curve e x i s t s for u, > 0 does not seem t r u e i n general, a t l e a s t TI

.

To s t a t e the theorem, we need some notations and d e f i n i t i o n s :

1 and formulas w i l l be The . c i r c l e T w i l l be i d e n t i f i e d w i t h R/Z 1 1 w r i t t e n i n R ( f o r ex. a map f r o m T t o T w i l l be l i f t e d t o a map from

,

R t o R ) . The r o t a t i o n R

p

i s defined by the l i f t i n g R

(a)

tu

= g+,

.

1

€ T xV

~f ( e , x )

FJB,

= F(B,

XI

X,

, we

write

PI = (f(0,

X.

PI,

G(B,

X,

PI)

E

T'X E

1 For any (.e x)E T xE l e t us .set 1 ~ ~ ( x)8 =, (g(B) A,(B) x ) E T x E

,

g(e) =

Where , f ( e , o,oJ , el'= a (e,o,oI-"

W e make the following assumptions : k k F is C k l a r g e enough, g(e) i s a C -diffeomrphism, '

,

4(9, 0 , 0) = 0.

DEFINIlTON 1 : f o r ~ l k - I , the 1-spectrograph o f Fo i s the spectrum of the l i n e a r map

G, : c'(T';E)

+ cJ(T';E]

defined by

(to v~ (0) = A.(~-'(B)) p(gd (01) (a, cp)= G o (graph cp) i n T1 x E

i. e. graph

space o f 'C Remark : i f

0i s

1 maps from T t o E

.

, for

E

any

1 C ~ [ T; E) the

(TI ;E],

a small enough neighborhood of 0 i n C"

the for-

mula graph (9, cpI = ~ , ( g r a ~ cp) h defines a map 9,

:

O+c'+'(T'

i E).

This map i s not d i f f e r e n t i a b l e a t 0, but the composed map

O.&CJ+

'(TI; E LCL(TI;

E

i s d i f f e r e n t i a b l e , and i t s d e r i v a t i v e a t 0 i s precisely Ci,

.

DEFINIlTON 2 : Suppose t h a t the r o t a t i o n number wo of g i s i r r a t i o n a l ; an eigenvalue ho=p q

E Z

, 2n0+ quo fz Z

.

no o f

Go i s c a l l e d "non-real"

i f for a l l

Motivation : one can e a s i l y show t h a t t o such a non-real eigenvalue may be 1 1 2 associated a Ca- subbundle t2 o f T x E isonmphic t o T x R

-

i n v a r i a n t under G o (e,zI where

m2

, on

,

which G o i s CR- conjugate t o the map (g(eI

, 10z ) ,

has been i d e n t i f i e d w i t h B:

.

Notice t h a t i f g i s a c t u a l l y Ca-conjugate t o R wo

, wo

i r r a t i o n a l , the

&-spectrograph o f Fo contains the whole c i r c l e o f center 0 and radius

.p as soon as i t contains the eigenvalue h a = p e

.

awn,

THEOREM : keeping the above notations we make the f o l l o w i n g assumptions : g i s caeonjugate t o the i r r a t i o n a l r o t a t i o n

R Wo

,Q

l a r g e enough.

The Q -spectrograph of Fa may be w r i t t e n alU 02, where al i s contained i n a disk o f C centered a t 0 of radius l e s s than one, .and s2 corncides

I

w i t h the u n i t c i r c l e ; moreover one assumes t h a t o2 i s "generated" by a 2inQ s i n g l e non-real eigenvalue A, = e i n the sense t h a t the decomposi1 t i o n C'(T i E) = el@ i n t o closed C&- i n v a r i a n t subspaces r e l a t i v e to

%

q

and

9 satisfies

C2 = fl(k)lthe

Go- i n v a r i a n t subbundle

Q

subspace o f c'-sections

9 z f o r r = 1,2;3,4,' 10, [ , C > 0 , such t h a t

(i rRo )+ v o (ii 3) E E

o f the

of T X E described a f t e r d e f i n i t i o n 2. q

E Z

V p E Z , VqEZ-0,

I rn, + qwo - p l

C >:p,

Then one has, i n general

f o r r = I and r = 3 .

, the

following conclusions :

1 For small p there e x i s t s a closed curve near T x 0, i n v a r i a n t under F and depending continuously on p i the s t a b i l i t y of t h i s i n v a r i a n t P Y curve changes when p goes through the value 0 (and depends on the sign o f a c e r t a i n quantity which we suppose non zero). For small p

> 0 o r small

< 0 (depending on the sign of another

quantity which we suppose non zero) there bifurcates a 2-torus

,

i n v a r i a n t under F and depending continously on p whose s t a b i l i t y FL i s the opcosite of the one o f the i n v a r i a n t closed curve described i n 1) 1 the r e s t r i c t i o n o f F

t o t h i s 2-torus i s i s o t o p i c t o i d e n t i t y .

lFL

Moreover, the distance t o T x 0 of the i n v a r i a n t closed curve (resp.invariant

2-torus)

i s of order p (respa order

lFL 19 ).

Discussion o f the assumptions : (a)

From 1) follows t h a t t h e r e s t r i c t i o n of the o r i g i n a l o lfw

t o the

i n v a r i a n t torus whose Fa i s Poincare map defines the same f o l i a t i o n (up t o d i f f e ~ m r ~ h i s m as ) a quasi-periodic flow. Notice however t h a t i n 3) there i s no condition on the approximation of wo i t s e l f by r a t i o n a l numbers, so t h a t t h i s flow need

not be

equivalent up t o diffeomorphism

to

a

quasi-periodic flow.

Anyway, i n view o f Arnold-Herman theorem (see

PI) the

s i t u a t i o n of

hypothesis I]occurs with non zero p r o b a b i l i t y and so i s worth studying. b)

The non-resonance assumptions 3) (i are )fundamental (see [C.I.

.]

1

Chapter.IV f o r a' ~ t u d yo f what happens i n general i n the "strong resonance" cases where31 (i i] s violated) but i t i s unclear whether o r not i t i s possible t o get r i d completely o f the two remaining diophant i h e approximation assumptions 3) (ii (see ) [C.I.I.]

[c. 1.2.

Chapter V and

]f'as only truncated normal forms are important t h i s could be

expected (?). c]

Assumption 2) i s the most s t r i n g e n t and also the most i n t e r e s t i n g because o f i t s geometrical meaning: i t implies t h a t G o leaves i n v a r i a n t a l l the 2 - t o r i o f a I-parameter family f o l i a t i n g the subbundle 1 2 1 €2 :2 T x R o f T x E and that, on each o f these 2 - t o r i Go i s

,

cJ-conjugate t o a map of the form ( 0 ,

9)b ( e

+

w,

, %+ Q,).

Recall t h a t i n the case o f Hopf b i f u r c a t i o n f o r maps (as described i n [R.T.])

the existence.of a I-parameter family o f i n v a r i a n t c i r c l e s

for d ~ ~ ( 0 i s ) automatic from the spectral hypothesis; here i f the spectrum of Go i s as i n hypothesis 2) except t h a t o2 i s not generated by an eigenvalue (which i s the general case f o r such a spectrum] .the r e s t r i c t i o n o f Go t o

52

may w e l l have a-dense o r b i t !

Some geometrical hypothesis o f t h i s k i n d i s d e f i n i t e l y needed i f one looks f o r a nice b i f u r c a t i o n theorem: namely, i f one

starts with a

I-parameter family of maps

f o r which thk conclusions o f the theorem hold, the i n v a r i a n t 2-torus 2 '1 t o T x E of the

which bifurcates being the image o f a map f r o m T form

(e, d

H

(e,

x(e,

d) w i t h

x(e,

@I=

Jpl'*,(e,

@ I + o ( ~ p I " l , x0

regular, then G o leaves i n v a r i a n t each 2-torus image of a map

'

This i s not surprising if we view t h e kbpf b i f u r c a t i o n theorem f o r maps as an "unfolding" i n the p d i r e c t i o n o f a l-parameter family of i n v a r i a n t c i r c l e s for d ~ ~ ( 0 i n ) t h e plane p = 0.

(*)

Note t h a t these assumptions are almost everywhere s a t i s f i e d i n the plane of couples (cue no)

,

linear situation

eneral s i t u a t i o n the d i r e c t i o n o f the paraboloyd may be reversed).

Picture i n

iv)

m2

w i t h assumption t h a t F (0) = 0.

P

The word " i n general" ,in

the theorem means t h a t conclusion holds

provided two q u a n t i t i e s are non zero: as these are best understood .

I refer to

a f t e r some changes o f variables,

[C.I.I.]

f o r an e x p l i c i t

statement; l e t me j u s t say t h a t they correspond t o the Ruelle-Takens assumptions t h a t eigenvalues o f ' d F,(o)

cross the u n i t c i r c l e transver-

sely and t h a t 0 be vaguely a t t r a c t i v e f o r F.,

A warning is necessary a t t h i s point: thanks t o the f i r s t conclusion 1 1 o f the theorem, we may suppose t h a t F (T x 0 ) c T x 0 for small

P

( a f t e r a change o f coordinates eventually) so t h a t a l i n e a r map be defined i n the same way as was

Ci,

.

R e hypothesis-do not imply t h a t

the spectrum o f O contains an eigenvalue. I n fact, the spectrum o f

p f o r which

f(e

w

G P

G can CI..

the

o2 - p a r t o f

i s expected t o explode i n t o an annulus a t values o f

,0 , u) has r a t i o n a l r o t a t i o n number,

but the hypothesis

can be shown t o imply t h a t it stays away f r o m the u n i t c i r c l e as soon as Y i s non zero (and small).

Quick sketch o f the proof : The steps are as i n [R.T.]. I]

1 Reducing t o a problem i n T x s p e c t r a l assumptions on

Ci,

m2

by center-manifold technique : the

and the i r r a t i o n a l i t y o f t h e r o t a t i o n number

o f g are used t o prove a center-manifold theorem which, when applied t o

*

,

t h e mapping ( 0 , x y) (F ( 8 , x ) , ~ ) e x h i b i t s f o r II small a submanifold 1 lJ 1 o f T x 'lf diffeomorphic t o T x FI2, which contains a l l t h e l o c a l 1 recurrence o f F i n t h e neighborhood of T x 0 FL

,

.

More p r e c i s e l y one s t a r t s w i t h a changd o f coordinates e l i m i n a t i n g terms o f . o r d e r p i n (see

2)

@ ( e , x , y)

so t h a t t h e theorem could be applied

[c.I.2.7)

Fin'ding truncated normal forms f o r F approximately) equations of t h e f o h

II

: one has t o s o l v e (resp.to

solve

which account f o r the diophantine approximation assumptions.

3)

Using c l a s s i c a l f i x e d p o i n t theory i n Banach space: t h e truncated normal form a l l o w s one t o f i n d a good approximation t o t h e seeked i n v a r i a n t set, from which standard i t e r a t i o n technique works t o g e t the result. One prove i n t h i s way

1)

1 The persistance o f an i n v a r i a n t c i r c l e near T x 0 f o r F : t h i s i s P more complicated t h a t i n t h e Hopf case where t h e existence o f a f i x e d p o i n t near 0 f o r F

follows,

v i a t h e i m p l i c i t . f u n c t i o n theorem, f r o m

t h e f a c t t h a t Fo(o'f = 0, and d

F,(o)

does n o t admit 1 as an eigenvalue.

1

The i n v a r i a n t c i r c l e one f i n d s i s t h e graph o f a mapping from T

to E

o f t h e form

e* 2)

px,(e)+

o(l~I1.

The change i n t h e s t a b i l i t y o f t h i s i n v a r i a n t c i r c l e when p crosses t h e value 0 (here diophantine approximation assumptions can be avoided and hypothesis 2) can be weakened, see [C.I.I.]

3)

chapter

v).

The b i f u r c a t i o n r e s u l t : t h e i n v a r i a n t 2-torus one f i n d s f o r t h e map 1 1 F may be w r i t t e n ( a f t e r i d e n t i f i c a t i o n o f t o T x TI2= T x C) as

v

an

the set o f ( 0 , r e

IE)

5

such t h a t

Question : do a persistence and a bifurcatiori r e s u l t s t i l l hold i f one replaces the assumption on the existence of an eigenvalue for Go by the more geometrical assumption t h a t there e x i s t s an i n v a r i a n t subbundle

5

for G o

, foliated

by G o - i n v a r i a n t 2 - t o r i

r e s t r i c t i o n of G o t o these tori

? O f course t h e

i s not necessarily supposed t o be

i s o t o p i c t o i d e n t i t y so that, f o r the flow, we do not.expect an i n variant 3-torus to appear, but instead a n o n - t r i v i a l f i b e r space over 1 T with f i b e r For some p a r t i a l results, see [C.I.l;] Chapter V.

P.

An example : the following example, w i t h E = R , i s h i g h l y non generic but one expects generic examples o f t h i s k i n d to occur f o r d i m E Let F

w

1

=. 1

.

1

: T x R + T x R be defined by

, (1 +PI a(e)(x

XI

~ ~ ( 8 . . = (etm. where a(e) >O i s C' ~ ' ~ a(8) o g dB=

o

-x31)

,

(or analytic),

, w0

ER 0

.

0

I t i s easy t o show t h a t f o r any Q the &-spectrograph of F i s the P c i r c l e o f center 0 and radius 1+w : from t h i s i t follows that, i n e 1 neighborhood o f T x 0 , F i s topologically conjugate (for CI, small P non zero) to the mapping G defined by IJJ

(see

V.P.S.]). i n R. If max a(e9 -min ace) i s n o t too

Now l e t V be the i n t e r v a l [O,C]

13

I

2

large, i t i s possible to f i n d a C such t h a t 1 t h e r e s t r i c t i o n o f F t o T x V i s one-todne. 1 1 w F (T X V ) C T x [O,C[ f o r p m a l l . Lb

1 n& F"(T x ]o,c]) P compact s e t K i n v a r i a n t under F 1 II P T x 0 (we use here t h e f a c t that, 1 1 G"(T x [a,@]) contains T x [O a] P . L e t us try to analyse K : i f , as The i n t e r s e c t i o n

,

LL

contains then, for p >0 small, a and having no i n t e r s e c t i o n w i t h f o r IL > 0 and n l a r g e enough, for any

cr ).

i n the discussion o f hypothesis 2)

i s the graph of a function P regular, then l o g xo[e) i s e a s i l y seen to

o f t h e theorem, we assume t h a t K

eH

~ ~ ( +0 o(lpla), )

x,

be s o l u t i o n o f the difference

I*)

l o g x0(o+mo)

equation.

- l o g xoCe)

-

l o g aC0) *

On t h e o t h e r hand, i f (say, i f

0

(o

4

i s too w e l l approximated by r a t i o n a l numbers)

i s a L i o u v i l l e number) i t i s possible t o choose a( e)

a n a l y t i c such t h a t t h e above equation had even no measurable solution. Notice t h a t

(*) having a r e g u l a r s o l u t i o n amountsto t h e existence of a

change o f v a r i a b l e s f o r Fo transforming a(e) i n t o 1, so t h a t a l l t h e 1 c i r c l e s T x c s t are i n v a r i a n t under Go a f t e r t h i s change o f variables. If

{*) has no, continuous s o l u t i o n , i t was shown by Hedlund t h a t

1 1 has a dense o r b i t i n T x X3+ (resp. i n T x R-]

.

Go

i n t h i s case ? CL Remark : i f i n t h e formula f o r F one replaces Q + w,, by g + wo + p P t h e spectrograph o f F explodes a t r a t i o n a l values o f e + % + r c u l CL i s no more on one s i d e o f t h e u n i t c i r c l e . What happens then ? Question: what can one say on K

, and

BIBLIOGRAPHY [B] R.Bowen, seminar,

A model f o r Couette flow.data, i n Berkeley turbulence Lect.Notes i n Math 615, Springer Verlag, B e r l i n 1977.

E.I. 1.1

A. Chenciner, G. Iooss, ' if u r c a t i o n s de t o r e s i n v a r i a n t s , to'appear i n Archives o f r a t i o n a l mechanics and analysis,1979.

1C.I.2.1 A.Chenciner,G.Iooss, invariants, t o appear.

Persistance e t B i f u r c a t i o n de t o r e s

.

[C.Y. ] J.H.Curry, J. A.Yorke, A t r a n s i t i o n fro%Hopf b i f u r c a t i o n t o chaos: computer experiments w i t h maps on R P r e p r i n t 1978.

m] M.R.Herman,

Mesure de Lebesgue e t nombre de r o t a t i o n , i n Lect. Notes i n Math, 597, Springer Verlag, B e r l i n 1977,p.271,293.

W.P.S. ] M.W.Hirsch, C.C.Pugh, M.Shub, I n v a r i a n t manifolds, Notes i n Math. 583, Springer Verlag, B e r l i n 1977.

Lect.

Sur l a deuxigme b i f u r c a t i o n d'une s o l u t i o n s t a t i o n n a i r e [~.]G.Iooss, de systbmesdu type NavierStokes. Arch.Rat.Mech.Anal.E,4, p. 3394369 ( 1977). m.T.1 D.Ruelle, F.Takens, On t h e nature of turbulence.Comm.Math. Phys.20,~. 167-192 (1971).

CENYRO 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 Periodic points, flows, diffeomorphisms, and generic properties Hyperbolic sets and homoclinic points Homoclinic classes, shadowing lemma and hyperbolic basic sets Hyperbolic limit sets

A t t r a c ~ o r ~ r g o d itheory c The measure

p,

Diffeomorphisms with infinitely many attractors References

Introduction. A b a s i c question i n t h e theory of dynamical systems i s t o study t h e

asymptotic behaviour of o r b i t s .

This has l e d t o t h e development of many

d i f f e r e n t s u b j e c t s i n mathematics.

To name a few, we have ergodic theory,

hamiltonian mechanics, and t h e q u a l i t a t i v e theory of d i f f e r e n t i a l equations. A p a r t i c u l a r l y b a f f l i n g and i n t e r e s t i n g problem i s t o d e s c r i b e systems

with n o n - t r i v i a l recurrence. diffeomorphism

f

For example, consider a smooth a r e a preserving

of t h e two dimensional d i s k

D

2

.

Poincare recurrence theorem, almost a l l p o i n t s i n

is, for on

x

x

x

That

accumulates

However, except f o r simple cases, we have no global

model t h a t describes a l l t h e motion. g

the

a r e recurrent.

D~

o f f a s e t of Lebesgue measure zero, t h e o r b i t of

i n f i n i t e l y often.

mapping

According t o

On t h e o t h e r hand, i f we consider the

on t h e two-dimensional t o r u s

T~ induced by t h e matrix

then again almost a l l p o i n t s a r e r e c u r r e n t .

[: :] *

However, i n t h i s c a s e not only

do we have a f a i r l y good p i c t u r e of t h e t o t a l motion, b u t t h i s p i c t u r e per-

sists f o r any

g'

which is

c1

close t o

g.

Thid example g i v e s an indica-

t i o n of some remarkable progress which has been made i n d e s c r i b i n g non-trivial recurrence during t h e last twenty years. The main f e a t u r e possessed by t h e t o r a l mapping shared by

f

i s what i s c a l l e d

hyperbolicity.

g

above which is not

I n i t s p r e s e n t form, t h i s

concept a r o s e i n t h e work of Anosov on geodesic flows on n e g a t i v e l y curved Riemannian manifolds

[2

1.

It was subsequently r e a l i z e d by Smale t h a t hyper-

b o l i c i t y could b e used t o describe o t h e r systems with n o n - t r i v i a l recurrence, and t h i s l e d him t o d e f i n e Axiom A diffeomorphisms.

The r e c u r r e n t o r b i t s

i n Axiom A diffeomorphisms l i e i n c e r t a i n s e t s which b a l e c a l l e d hyperbolic

basic s e t s . These s e t s have been studied by many authors. --

Perhaps t h e most

s i g n i f i c a n t r e s u l t about hyperbolic basic s e t s i s t h a t they can be modeled by c e r t a i n symbolic spaces ( c a l l e d s u b s h i f t s of f i n i t e type), and t h i s gives one very p r e c i s e information about t h e i r o r b i t s t r u c t u r e s . If one thinks about t h e s t r u c t u r e of hyperbolic b a s i c s e t s a b i t , one

r e a l i z e s t h a t they a r e s p e c i a l cases of c e r t a i n sets which we c a l l h-closures which e x i s t f o r many diffeomorphisms.

A t t h e present time we have r e l a t i v e l y

l i t t l e information about t h e f i n e s t r u c t u r e of non-hyperbolic h-closures.

but

hopefully we w i l l understand more about them i n t h e f u t u r e . Our goal i n these l e c t u r e s i s t o introduce t h e reader t o some of t h e r e s u l t s i n t h i s f a s c i n a t i n g a r e a of mathematics.

Considerations of time and

space have forced us t o choose a r a t h e r limited s e t of t o p i c s t o present here. Our i n t e n t i o n has been t o d e s c r i b e a v a r i e t y of r e s u l t s w i t h a s p e c i a l emphasis on the theory of a t t r a c t o r s .

While many references a r e given i n t h e

ensuing s e c t i o n s , we recommend t h a t t h e reader consult t h e recent survey of Bowen [ 9 l, and t h e l e c t u r e s of Ruelle 1-49]

f o r d i f f e r e n t perspectives.

1. Periodic points, flows, diffeomorphisms, and generic properties. In t h i s s e c t i o n , we s h a l l begin t o motivate the concept of hyperbolicity. F i r s t , l e t us consider the r e l a t i o n s h i p between flows and diffeomorphisms.

a

Let M be a compact cW manifold. A G~ vector f i e l d X on M is k C mapping from M i n t o t h e tangent bundle TM of M so t h a t X(x)s T M X

x

f o r each

Here

M.

E

i s t h e tangent space t o M

TxM

always assume

k r 1. The vector f i e l d

$ : R-r M + M

where

X

at k

induces a

C

We s h a l l

x.

mapping x E M

R i s t h e r e a l l i n e such t h a t f o r each

and

t , s E IR we have

(11

6(0,x) = x

(2)

$(t

+

a4

(31

s,x) = 4(t,$b,x))

(t,x2 = X($Ct,x))

Conditions (1) and (2) imply t h a t t h e mapping morphism from

The mapping X.

The mapping It is

9

+ 1.

follows. by

f a c t t h a t every

t = 1, and tl = 0,

MI/"

"

= M

ck

a

d i f feoinorphisms of

ck

diffeomorphism

flow

X

M,

of

a manifold

z

on a manifold f

with

dim

M.

x = f(xl),

x = xl

and

t = 0, and

or

t = tl

tl = 1.

+

t

-

[s

+ tl,

arises

=

-

on

M1

xl = f (x),

The quotient space

i n h e r i t s a d i f f e r e n t i a b l e s t r u c t u r e and a flow ) : Ix $(s. { ( t , x ) I ) = ( ( s

M

and i s defined a s

and d e f i n e t h e equivalence r e l a t i o n

i f and only i f or

4

f

i s c a l l e d t h e suspension of

$

M1 = [0,1]

(t,x) " (tl,xl)

ck

homo-

i s c a l l e d t h e time-one map of t h e flow $.

x w eC1,x)

The flow

Let

of

,y(tf>~) is a

is c a l l e d the flow o r one parameter group generated by

$

a s t h e time-one map of dim M

iff%

IR t o the group

t M

f

[ s+tJ ,) )

+%with

where

[s

Examples:

+

t]

1.

s

i s the g r e a t e s t i n t e g e r i n Let as

s1

0 < a < 1 and l e t IR/Z

f (x) = x 2

with

b e the c i r h e which we think of

1R t h e r e a l s and

+ a mod

T~ = IR. /z2

+ t.

and

Let

i s diffeomorphic t o the two torus

fi

1. Then

the integers.

Z

may be thought of a s the flow induced

$

;=

by the d i f f e r e n t i a l equations

1 on

R2.

? = a

2.

Let

2 D = I (x,y)}

and let

f :D

2

-+

B 2 : x2

E

D2 b e a

+ y2

S

1)

be the unit disk i n B

diffeomorphism from

ck

2

D2 i n t o

i t s i n t e r i o r which preserves o r i e n t a t i o n . The suspension contruction applied t o on

1 S

x

f

gives us a flow

$

( a c t u a l l y defined f o r forward time only so we

D2

We

should say semi-flow) whose o r b i t s come i n a t the boundary. could make t h i s . a g l o b a l flow (via d i f f e r e n t i a b l e change coordinates)

1 S x D2

embedding

extending the v e c t o r f i e l d of

$

in

of

.S3, t h e 3-sphere,

t o a l l of

s3,

and taking

t h e flow of t h i s extended vector f i e l d . If

$

is t h e suspension of

q u a l i t a t i t r e f e a t u r e s , and

f

f,

then they have e s s e n t i a l l y the same

g i v e s us the advantage of one l e s s dimension.

A s we s h a l l see, c h i s enables us t o describe i n t e r e s t i n g 3-dimensional flows

with

2-dimensional diffeomorphisms. For most of t h e s e l e c t u r e s , we s h a l l be concerned w i t h diffeomorphisms.

We begin by describing c e r t a i n g e n e r i c p r o p e r t i e s of diffeomorphisms. an i n t e g e r We g i v e

k 2 1, and l e t

diffeomorphism from

space

be t h e s e t of

~ i f f ha topology as follows.

open coordinate c h a r t s

ck

s iff^^

lRm, m = dim M.

!U1,$l),....,(Un,$n) Ui

ck

M with

onto a bounded open s e t

Choose t h e p a i r s

d i f f eomorphisms of

Fix a f i n i t e covering of of

(Ui,$ti)

Fix

$i : U i +

M

M.

by IRm

a

i n the Euclidean -1 . s o t h a t $ i 0 $j and i t s Vi

p a r t i a l d e r i v a t i v e s of order l e s s than o r equal t o tinuous. set

If

$(g,f)

For

supll~:((~g(;'-

f ,g

E

(if(j

iff^^,

let

-1 dl : 1S i

> 0,

let

B:(E)

= lg

E

5

n, 1 S j

x of

D$'(E) i s a p a r t i a l d e r i v a t i v e a t

where E

-

..-Car.

a r e uniformly con-

i s a multi-index of non-negative i n t e g e r s , we

a = .(al,...,ar)

la1 = al+.

k

DiffkM

a

The s e t s

la1

.

k

ck'

c a l l e d the uniform

E

is p e r i o d i c f o r

M

The l e a s t such period

n

n

f

i f there is an integer

is c a l l e d the period of p.

A point

n r 1 so t h a t

A periodic p o i n t

automorphisms

.

Thus, the eigenvalues of

T f P

fn

and think of

Theorem (1.1).

hfh'l

p

T fn P

a r e well-defined.

a s a l i n e a r isomorphism of

(Hartman and Grobman). &pposs

~k diffemorcphism

f : M -+ M

i n M and a homeomorphism h : U = T f P

-+

.

p

%re

TpM with

R ~ .

i s a hyperbozic fixed i s a neighborhood

I R ~such t h a t

U

h(p) = 0 and

where both sides are defined.

Thus, v i a a continuous change of coordinates, T f . The s t r u c t u r e of T f P P a d i r e c t sum decomposition lRm = Es

f

look's l i k e t h e l i n e a r

i s given by l i n e a r algebra.

mapping

that

of

d e f i n e conjugate

For convenience of notation, we w i l l frequently i d e n t i f y

of

p

T f n of f n a t p has no P Note t h a t T f n : TpM + T M i s a l i n e a r P P

automorphism, and any two l o c a l r e p r e s e n t a t i v ? ~of

point of a

fn(p) = p.

is hyperbozicif the d e r i v a t s v e

eigenvalues of absolute value 1.

IR

topology.

i(Ui,(i)).

Let us consider t h e l o c a l s t r u c t u r e near a periodic point. p

Given form a

Bf(e)

This topology i s independent of t h e choices of the c h a r t s

and

k,

5

5 of order

DiffkM : $(g,f) c c } .

neighborhood base f o r a topology on

S

(B

There i s

EU i n t o two i n v a r i a n t subspaces so

has eigenvalues of norm l e s s than one while ~ ~ ES£ 1

1

T f E~ P

has

For some norm on

eigenvalues of n o w g r e a t e r than one.

1 1Tpf 1 E' I 1

11

< 1 and

-1 u Tpf E

1

1

Some hyperbolic l i n e a r automorphisms

< 1.

'

IR.~,

a r e sketclied i n t h e next f i g u r e .

f(x,~)

I

(ax

+

By,-Bx

+ ay)

o < ~ ~ + B ~ < I

= (ax + By,-Bx 2 2 O < a + B < 1 , A > 1

f(x,y,z)

+ ay,Xz)

Figure 1.1 The map

h

i n theorem

l i n e a r i z a t i o n theorems,

r

a hyperbolic p e r i o d i c p o i n t

is called

(1.1) 2

One may assume

p

of period

f : I R -+ ~ I R ~and

Cr

~~f~ 'near

which even works i n Banach spaces i s The i d e a is a s follows.

f ( 0 ) = 0.

Let

L = Tof.

With a

(a function which is one on a neighbor-

0 and zero o f f a s l i g h t l y bigger neighborhood), one may replace

hood of af

a

looks l i k e

n > 1, fn

[40] and Pugh [45].

s u i t a b l e choice of bump f u n c t i o n

by

For

1, and ocher normal forms, s e e [58], [60]. For

An e l e g a n t proof of theorem (1.1)

due independently t o P a l i s

linearization.

CO

+

(1

- a)L

and assume t h a t for

x o u t s i d e some neighborhood

(1)

f ( x ) = Lx

(2)

t h e L i p s c h l t z conatant of

Then one t r i e s t o f i n d

h = id

u i s a bounded continuous function.

f-L

+u

of

0

i s small. where

id

i s t h e i d e n t i t y map and

f

Consider (L

(3) with and

+

+

42)

+2 L i p s c h i t z f u n c t i o n s w i t h small

$ 1 and

+ u.

h = id

$1) (h) = (h) (L

H : u r---.+ u

u = IT1~-l($,

-

- L-'U(L

+ u))

$l(id

= $(u)

+ 8 ),

+l)h = h (L

we have

.

For

f = L

h+2,

+

0

then

7

,0

h

,

$1- 2

.

+2

Then

Also,

h+l, 42 = i d , s o

he

and

SO

= id

Similarly,

is i n v e r t i b l e , so we g e t

+$2)

t r a c t i o n , s o (3) has a unique s o l u t i o n

+

and L i p s c h i t z s i z e s ,

We g e t

The operator

s o l v e (L

CO

small,

$

i s a con-

h4 ,+ and i d 1 1 h+2,12 = i d . Since

by uniqueness.

h

$1 $2

i s a homeomorphism.

If

i s t h e r e q u i r e d l i n e a r i z a t i o n f o r theorem

(1.1).

I

Now l e t Given a p o i n t let

d

b e a d i s t a n c e f u n c t i o n on M x

E

M y l e t p ( x , f ) = (y

wU(x,f) = p ( x , f - l ) .

wU(x,f)

One c a l l s

t h e u n s t a b l e set of

x.

E

induced by a Riemann metric.

M : d(f"x,fny)

WS(x,f)

-+

0

as

the s t a b l e set of

n x

-+

-1, and

and

The next r e s u l t shows t h a t f o r a hyperbolic

period p o i n t t h e s e sets have n i c e s t r u c t u r e .

(Stable manifoZd theorem for a point .I

.morem (1.2)

ck diffeomorphism

a hyperbolic periodic point of a

a eu. Then

T M = E;

P

P s = dim

with

ad

Hirsch

and Pugh

t o Irwin

[19].

P

eS

and

P

Poincare

and then by

The o u t l i n e we give h e r e i s based on t h e treatment of

[17].

A proof based on t h e i m p l i c i t f u n c t i o n theorem i s due

Related and important r e s u l t s a r e i n Hirsch-Pugh-Shub

The b a s i c i d e a of the proof of theorem (1.2) s = dim

s copy of 1

i s t a r t g h t a t p to E'.

p(p,f)

Early v e r s i o n s of t h i s theorem were given by Hadamard and Perron.

with splitting

f

ck injectiveiy inmersed

is a

P(p,f)

Szlppose p i s

u

= dim EU. P

Replace

f

i s as follows.

by a power of

f

[18],.

Let

s o we may assume

£(PI = P. Choose a neighborhood such t h a t

4 (p)

we i d e n t i f y L

+

S X

B

p

= IR'

*

To(IRS x IRU) with

Ji where L is a l i n e a r

B = B1

maps

= 0 , To ((E;)

of

U

and a diffeomorphism {.O},

and

IRS x lRU.

To4(~:)

For

hyperbolic map and

8;

b e t h e product of t h e u n i t b a l l s i n

as

i n the next f i g u r e .

U

Jl is IRs

4 :U

= {O} x

small, C'

-f

small. and

lRu.

+

RS

R u.

R

Here

~ f @ -=l Let Now

I-'

-f -l(B

Also,

s t r i p i n the a

ck

n F I B ) n B n ?-'B.

lRU-direction.

= F~B, n

FIBn

ry

transfo?mation

ck

mapping from

g,

r-n(g)

0;

nLO B whose forward o r b i t s

Actually, i t tmrns out b e t t e r t o consider the

defined by graph (rF(g)) =

BU

to

BS

details,

-g

so t h a t r-(i) f

W e warn t h e r e a d e r t h a t

see [17],,[18].

i n any s u i t a b l e metric.

F1 (graph

g)

of L i p s c h i t z c o n s t a n t 51.

converges t o a unique

f

7"~ is

It i s reasonable and provable t h a t

manifold, and equals t h e s e t of p o i n t s i n

remain t h e r e and approach

is a thinner

B

~ ~ ~ ~ ~= ()'Ip ,(graph f )

Now,

is a

g

Then, f o r any such and

=

r-f

where

is

F)

not

T

g=

0

0.

For

a contraction

is c a l l e d . t h e l o c a l

= uf-n$o,(p,f) is an n2O expanding union of immersed d i s k s , and hence i t is an i n j e c t i v e l y immersed s t a b l e n i f o l d of

p

for

f.

Then,

I$(p,f)

Euclidean space. Applying theorem (1.2) t o $(p,f)

and

wU(p,f)

f-I

gives a similar structure t o

~ ~ ( ~ , f ) .

a r e c a l l e d t h e s t a b l e and unstable manifolds of

p.

It was r e a l i z e d long ago t h a t i t is impossible t o d e s c r i b e the o r b i t

s t r u c t u r e s of - a l l d i f f e r e r e n t i a l e q u a t i o n s o r a l l d i f f eomorphisms.

~oincarg

and Birkhoff emphasized t h e concept of t y p i c a l o r general systems i n which c e r t a i n exceptional o r r a r e phenomena were t o be excluded.

There a r e

many p r e c i s e notions of t y p i c a l i t y which can b e introduced i n t h e space Dif fkM.

One of t h e most f r e q u e n t l y used i s t h e n o t i o n of r e s i d u a l i t y .

subset 8 c

D i f f k ~ is called

s e c t i o n of dense open sets.

residuaZ i f i t contains a countable i n t e r Residual sets a r e dense, and a countable i n t e r -

s e c t i o n of r e s i d u a l sets is again r e s i d u a l . residual s e t s a r e called

A

P r o p e r t i e s which a r e t r u e f o r

g~neric* It i s t o b e hoped t h a t one day we w i l l

be a b l e t o understand t h e o r b i t s t r u c t u r e s of elements i n a r e s i d u a l s e t i n DiffkM.

A t p r e s e n t , we a r e f a r from t h i s goal.

The next two r e s u l t s de-

s c r i b e some u s e f u l g e n e r i c p r o p e r t i e s . Let

gl : N

1

.+

M

and

$2 : N2 + M

b e two immersions.

We s a y t h a t ,

i s transverse to

(bl

x E N~ , Y

E

(or t h a t

$*

That is, the tangent spaces t o M

at

@ (x)

$2 (y)

1 with t h e i r images and j u s t say t h a t

N2

N1

.

N1

at

and

for

N~

any

+

at

,+,2(y)

Sometimes we i d e n t i f y and

N1 a r e transverse..

N2

Theorem ( 1 . 3 ) (ICupka-Shale). There i s a residual s e t 8 each f

if

= Tx$l (TxN1)

1

Wan the tangent Space t o and

@2 a r e transverse)

(XI= $ (Y) We have T4 (,)M

N2 , such t h a t

TY$2 (TyN 2 )

and

c

if f k ~so that

i n 8 has only hyperbolic perioao points and the stable and

unstable manifolds of the periodic points are transverse. A point

of

x

M

is non-wandering f o r

t h e r e is an integer

x

wandering points i s denoted

n > 0

such t h a t

for

f

E

8,

( pugh

[ 44

1)

fnu n U +

f

U

The s e t of non-

9.

f-invariant s e t , and it

mere i s a residual s e t

the periodic points of

8 c ~ i f f lso~ that

are dense i n n ( f )

.

holds i n ~iLffkM, k > 1.

It does

M = sl, 2s tke c i r c l e ,

Proofs of theorem (1.3)

(1.4)

f o r every neighborhood

f.

It i s s t i l l unknown i f theorem (1.4)

hold i f

if

~(f). It i s a closed

contains a l l t h e recurrent behavior of

Theorem ( I . 4 )

f

is i n

[46].

are i n

1371, [41].

The b e s t proof of theorem

Hyperbolic S e t s and Homoclinic Points.

2.

I n t h i s s e c t i o n we extend the condition of hyperbolicity t o l a r g e r s e t s than a s i n g l e o r b i t .

This extended condition i s f a i r l y r e s t r i c t i v e , but it

permits us t o understand many complicated o r b i t s t r u c t u r e s .

W e have already defined hyperbolic p e r i o d i c points, and we have estabIt is c l e a r how t o define a hyperbolic non-

l i s h e d some of t h e i r properties. periodic o r b i t

o x .

s

One should r e q u i r e a s p l i t t i n g

u Txf(Ex) = Efxy Txf(Ex) = Efx, S

U

expansion f o r

y

o E:

and some s o r t of contraction f o r

so that

Tf lES

and

S u i t a b l e d e f i n i t i o n s of t h e contraction and expansion

Tf lEU.

1.1

a r e t h a t , i n some Riemann norm that for

TxM = E:

TM, t h e r e i s a constant

on

1< A

so

o(x),

E

and

If

is a closed i n v a r i a n t s e t , i.e.

A c M

hyperbolic i f a l l of the o r b i t s i n

A

f (A)

I I

on each

Definition 2.1.

E A,

(2)

T

A Riemann metric on

f-invariant s e t

A

T M = ES o EU for each

x

X

X

a constant

~ A

V

is called

TM

induces

We w i l l c a l l t h i s . a Riemann norm.

X

A closed

splitting with x

T M.

A

a r e hyperbolic i n a uniform way.

More precisely, we have the following. a norm

= A, then

X

i s hyperbolic i f there i s a E A,

X > 1, and a Riemann norm

o

v c ,:E

and

which varies continuousZy

I .I

such t h a t

l ~ ~ f ( v t) ~ j l v for l

v c E .:

I t can be proved t h a t t h e bundles

conditions (2.1.1)

and (2.1.2).

Follows from (2.1.1)

adapted t o

Es -and EU a r e unique subject t o

The continuity of x -+

and (2.1.2).

The norm

u

and x ->E

E'

X

X

also

i n d e f i n i t i o n (2.1) i s c a l l e d

1.1

A.

A d e f i n i t i o n of hyperbolicity which i s independent of any p a r t i c u l a r

Riemann metric involves replacing (2.1.2) with

1

(2.1.2)'

I ~ ~ f " ( v ) ~ A - ~ l v lv,

E

E,:

and I ~ ~ f " ( v )L lC f o r any

n 2 0

-T n A

I"(,

and some c o n s t a n t s

changing the norm merely changes

u v e Ex C > 0 , A > 1 independent of

C

and

A.

n.

Then

We w i l l always use an adapted

norm. Definition (2.1) has t h e d e f e c t t h a t i t is hard t o e s t a b l i s h i t s exisThere is an equivalent formulation of hyperbolicity which

tence i n examples.

i s e a s i e r t o use. Let

be a Riemannian manifold with norm

M

T M = E lx e E2x

for

X

on

Define t h e

M.

x E M.

Let

E(x)-sector

on

1'1

TM, and l e t

~ ( x ) be a p o s i t i v e real-valued function S,

(Elx,E2x)

of

(Elx~E2x) by

When t h e s p l i t t i n g TxM = Elx e E2x

is understood, we w r i t e

S,(x) (Elx,EZx)

- Sc(XI

is a

say Txf lA

Theorem ( 2 . 2 ) . manifold

M,

bolic for x

E

and

E

: I\

for

A-expansion i f

Let

f

axd l e t A

be a c

c1

TxM

IT Xf (v) I ->

If lvl

+

A > 1 and for all

A c T M, we X

v e A.

diffeonwrphism o f the cornpact Riemannian

be a ctosed f-invariant s e t .

i f and.onZy i f there are a s p l i t t i n g

f

A, an integer

tion

S'E (x)

for

m > 0, a constant

Then A i s hyper-

TxM = E

lx

E2x for

A > 1, and a positive real-valued func-

R satisfying the following conditions.

(2)

F o r each

x

E

A, ue have

( a ) T x f m l s E ( ~ ) sc(f04)

are A-eccpansions

and both

TXfml

and ~ ~ f - ~ l s : ( ~ ~

.

Theorem 2.2 says t h a t t o e s t a b l i s h hyperbolicity, one only needs t o find

a f i e l d of cones and f o r some fields

x >-

Cx

in

TM X

for

m > 0, TXfm expands C

X

x

such t h a t

A

E

and

Cx,

Txf

maps

T X f l expands

do not even have t o be continuous.

to

Cx

TxM

- Cx.

Cfx,

The

In most applltcations,

however, they a r e piecewise continuous. For a proof of theorem (2.2), s e e [33], [34]. Let us give a well-known example of a hyperbolic set--the

Smale horse-

shoe diffeomorphism. Let

Q

be a square i n the plane

and described i n f i g u r e 2.1.

The map

R 2 ' and f

define

f

f i r s t squeezes

then s t r e t c h e s i t v e r t i c a l l y , and f i n a l l y wraps the top of the figure.

Write

f(A) = A ' ,

f(B) = B ' ,

etc.

from Q

Q

into

horizontally, Q

around a s i n

R

2

Q Figure 2.1 Label the two components ,of

and

[ ] -a

Txf = where

0 < a
0

defined by

and t h i s implies t h e same f a c t f o r I n general, i f

U

U whenever

€

i

and

f ]A

A.

such t h a t

that

hf = oh, or

From t h i s i t follows t h a t t h e periodic o r b i t s of

= o.

For .a sequence

lil

so we may d e f i n e a sequence

A2,

u

by

C2

E

x

a > 0 fc(x,y) =

One e a s i l y hhecks t h a t

i s t h e inverse of

fc

80

f C is

The images of h o r i z o n t a l l i n e s a r e v e r t i c a l , and

t h e images of v e r t i c a l l i n e s a r e parabolas.

Using theorem 2.2,

t h e reader may

prove t h a t there i s a

c(a,b) > 0

invariant s e t f o r

is a s e t

lent to

0

on

fc

so that for

on which

A(fc)

Q

g

is

c1

near

is a l s o hyperbolic f o r

only choose

m = 1,-X =

coordinate l i n e s a s valent t o

fc

is topologically equiva-

Z2.

Let u s . r e t u r n t o our horseshoe map 2.2 t h a t i f

c 2 c ( a , b ) , the only bounded

g.

-1

a 2 ,

and

Elx

It follows e a s i l y from theorem

f.

f , thenthe l a r g e s t g-invariant subset For t h i s note t h a t e(x) = 1 f o r a l l

E

Also,

2x'

A(g) =

A(g)

of

f ) g n ~ ,and we

need ne Z x, and the tangents t o the

g ( A(g) remains topologically equi-

(a,C2). i n trying t o

Horseshoe type mappings were 'discovered by Smale [55]

geometrically describe a v a r i a n t ofVan d e r Pol's equation studied by Levinson They a r i s e i n many physical s i t u a t i o n s near what a r e c a l l e d homoclinic

[20]. points. If point

p

is a hyperbolic p e r i o d i c point of

x r w ~ ( n~ w) ' ( ~ )

s e c t i o n of

w ~ ( ~and )

- {p?

w'(~)

c1

diffeomorphism

i s c a l l e d a homoclinic point.

at

x

f , then a

I f the i n t e r -

is transverse, the homoclinic point i s

c a l l e d transverse.

Theorem (2.31, (Sttale homocZimc theorem [55]). phism with a hyperbolic periodic point

set

There is an integer

x.

point A

containing

x and

p

so that

be a

f

c1

diffeomor-

having a transverse homoclinic

n > 0 such that

t o the s h i f t mrtonorphism ( , C 2 . Corollary ( 2 . 4 ) .

p

Let

fn(A

fn

has a closed invariant

i s topologically equivalent

Moreover, A i s a hype~bolics e t for

fn.

Each transverse homoclinic point of a diffeomrphism f

i n the closure of the hyperbolic periodic p i n t s of

f.

Let us sketch a proof of t h e homoclinic theorem. Let placing

p f

be a hyperbolic p e r i o d i c p o i n t of the diffeomo~phism f . by a power of

f , we assume

f ( p ) -= p.

Let

x

Re-

be a transverse

is

homoclinic i n t e r s e c t i o n of u = dim wU(p).

DU b e a

let (DU

-

Let

DS

be an

and

wS(p)

s-disk .in

u-disk i n wU(p) with

p

E

I n s u i t a b l e coordinates about

aDU).

t u b u l a r neighborhood .of 6 > 0

wU(p)

is small.

DS

as

N:

.

Write

wS(p) DU

-

s~= dim wS(p)

with

aDU

{p,x) c DS

and

x

E

f(DU

and

- a~',

and

-

-

aDU)

D , ~ ,we may t h i n k of a small

= DSx6DU =

(c,60) ; 6

E

Ds,

q E D

u

1 where

Thus we have t h e following f i g u r e .

Figure 2 . 2

Let us use set

C(z,F)

t o denote t h e connected component of a p o i n t

z

in a

F.

Notice t h a t i f along

wU(p)

6 > 0

is s m a l l , then i t e r a t e s

fnN;

tend t o accumulate

a s i n figure 2.3.

Figure 2 . 3 Let us s e t

Any' 1

t h e boundary of

= c(p,f%"N;

DU

N:),

and

A";

= c(x,f%:

appropriately, then f o r large

n

n N:).

I f we a d j u s t

and s m a l l 6 ,

A;'~

and

and A;'~

look somewhat like the Al

and A2 of the horseshoe diffeomorphism.

In figure 2.4, we indicate several possibilities for f % :

w ~ ( ~ ) need

and

.

Note that

w~(~)

not be transverse everywhere.

Figure 2.4

Let ns : N 6

DS and n : N6 + 6DU be the projections. For F c N u 6' -1 -1 set dU(P) = sup {dim nS (2) n PI and ds(F) = sup {dim n ( z ) n P I . u zeDs ze6DU Assume n is large and 6 is small. Then a little thought shows that -+

,

there are constants c > 0, h > 1 such that for any finite sequence (il,.

..,im)

(1)

with

n

= 1

or

2,

n 6 fnk (A. * ) is a disk homeomorphic to DS Ozkm 4(

x

DU and

Similarly, (2)

n

f?~?'~) -mzkzO %

n

fnk(~n'6) -mzkzO ik

n

with

nk n,6 f (Ai ) are shaded in figure 2.5a and typical Ozkzpl k are shaded in figure 2.Sb.

Some typical sets sets

is a disk homeomorphic t,o D S x

Figure 2.5a

Figure 2.5b

Now i t follows t h a t i f

C

1

Hence, f o r

then

c ~ A where - ~

f - n k ( ~ n ' 6 ) i s a d i s k of diameter l e s s than ik -m r 2

-

(r2). r2,

wU(r2). Thus, then

j.

be non-empty open s u b s e t s of

Cl ~ ~ ( f )We. must show

t h a t t h e forward o ~ b i of t

meets

U1.

t o prove, s o assume t h e r e a r e p o i n t s

ql

such t h a t

get

faql

If E

U

1

q

-

> q2 -

f jaql > q2

-

q1 > q2 > ql.

and

> ql.

Then,

f

2a

ql >

PPZ

f o r each j 2 1. L e t t i n g

and a point

a

> f q1 > j

, 'there q2

E

is nothing

U2 n Hp(f)

r ,r 1

2

,... of

"

q2 E o(q ) 2

> ql.

be t h e period of

fJri

fnu2 # 0 n >O

meets

U1

r

i

ri

is i n

such

Continuing, we ql

transverse homoclinic p o i n t s of

i, and some f i x e d k , we have

ward o r b i t of

F2

-

gives

Looking a t f i g u r e (3.1), we s e e t h a t t h i s implies t h a t

t h e homoclinic theorem t h e s e homoclinic points

U1

n H (f) P

a

q2, t h e r e a r e an i n t e g e r

a l i m i t of a sequence

Forlarge

Hp(f) = o(p)

o(ql) # o(q2).

Since that

U2

Y2 ql.

is By

a r e i n Cl H (f) = Cl B (f). 91 P k f U2. Clearly, t h e for-

for arbitrarily large

j

.

Thus,

a s required.

Figure 3.1

Let us c a l l t h e closure

CL Hp(f) of an h-class of f an h - c l o s ~ . The h-

c l o s u r e s f o r a diffeomorphism form s e t s with dense o r b i t s and periodic p o i n t s dense.

I n general, very l i t t l e is known about t h e i r f i n e s t r u c t u r e .

instance, i t i s not known when they have p o s i t i v e Lebesgue measure. case where

f

i s a generic

c1

For I n the

a r e a preserving diffeomorphism of a compact

two manifold, each h-closure h a s Hausdorff dimension 2

[31].

We w i l l s e e

t h a t when an h-closure i s hyperbolic, then i t has a r i c h s t r u c t u r e .

Also,

we w i l l examine open s e t s of diffeomorphisms which have non-hyperbolic h-closures

.

Our next theorem s t a t e s t h a t t h e o r b i t s forward and backward asymptotic t o a hyperbolic s e t hehave nicely. by a Riemann metric.

For a point

w

~ = )w ( ~ , ) .One c a l l s .E

of

and a n w b e r

M

> 0, l e t

E

u

0

of a compact metric space so t h a t

d(fnx,fny) 5

E

i s c a l l e d an expansive constant f o r If

A

i s a hyperbolic s e t for

X

is c a l l e d

for a l l

n, implies

f.

f, then

f ( A i s expan-

The s p l i t t i n g

Proof. wS(x) E at

is a disk tangent t o :E

y r $(XI, y r W:(X)

and i f n w:(x)

d(fnx,fny) y)- r

n

The o r b i t

i d ( f x,xi)

be i n t e g e r s o r

is a sequence

f

E

a =

for a l l

W;(X) n A x W:(X)

independent of or

-m

(xf )a 0 such

E

has a

A

The name comes from the f a c t t h a t i f t h e condition

x r A, then, f o r small

a < b

5 o for n

d(fnx,fny)

i , y E A, then one says t h a t

which is homeomorphic t o t h e product

Let

o,

is a disk tangent t o :E

W;(X)

has t h e property t h a t t h e r e is an

A

local pmduct structure.

i.

x, and

for

for a l l

W;(X) n w:(~) c A

holds and

is continuous, and f o r small

EU

= {XI.

I f a hyperbolic s e t

A

at

O

u s Wg(x) n WE(x) = {XI. But, i f

Thus,

7:.

that

TAM = ES

b =

such t h a t

of

Ux

n A.

x

in

Of course,

x.

+w.

For , G > 0, a

&pseudo-

d ( f ~ ~ , x ~5+6 ~ f) o r a l l

E-shadows t h e pseudo-orbit

{xi]

if

i. Most of t h e properties of hyperbolic s e t s may

be proved from theorem (3.3) and t h e following . r e s u l t .

Theorem (3.5) (Shadowing Lemma).

a ZocaZ product s t h c t u r e .

Suppose A i s a hyperbolic s e t for

For every

every 6-pseudo-02 bit i n A

can be

E

>0

f

with

there i s a 6 > 0 so that

E-shadmd by an o r b i t i n A.

With a s l i g h t l y d i f f e r e n t formulation theorem (3.5) was f i r s t proved by Anosov [ 2 1 . Proof. properties. (a)

(b) and (c)

The formulation we give i s due t o Bowen 1 7 ?here a r e constants If

0 < o1
0

and

1.

X > 1 with the following

E ~ ,then

u f o r x , r~ A, WE (XI n W: (y) c A i s a t most one point, 1 1 y E W: (x) implies d(f-ny,f-nx) y)- X-"ol f o r n 2 0, 1. y r W:

(x) 1

implies

d(fny ,fnx) ji n c l

for

n

t 0.

c > 1 such t h a t

From the l o c a l produci s t r u c t u r e we may f i n d a constant if

z

wU

E

wS

(x) n

€0

€0

d(z,x) 5 cd(x,y) and

( y ) , then

Pick an i n t e g e r .N > 0

such t h a t

t o shadow f N pseudo-orbits. pick

such t h a t 6 ;0 1

Choose

Then choose

is a has d(f

Ni


0.

E E

Let

6 > 0

A and

i s a unique point.

We work with pseudo-orbits indexed on a l l of Let

is a r b i t r a r y ,

>0

(E.

Assume

E.

E

6-pseudo-orbit

be a point i n

Then f o r

E

f N can be

f N pseudo-orbits can be shadowed.

imply

d(x,y) < 26

E 2'

pseudo-orbit f o r

{xi}

d ( f XNi'%i+j)

We now show be such t h a t

sl-

so that i f

61-pseudo-orbit

d(f j x,f j y)


xi

for a l l

61 7. x.

If

A(£) such t h a t

E

i

+ d(fg x,ggix) + Let

$(x)

E

A(f)

besuch

i d (f ( (x) ,xi)

that

.f

for a l l

i

i i i m e n , d(f (x,g x) ( d ( f (x,xi)

choice of continuous.

If

a sequence all z

E

k.

yk

( E

+ d(xi,g ix) ( TE

i s unique w i t h t h i s property.

((x)

E,

.

i s not continuous a t

A(g)

such t h a t

yk

x

-+

and l e t t i n g

i

k +

contradicts the choice of

gives

k

and

-+

A(f)

x --+$(x)

is continuous.

ho$ = i d

A(g)

gh = hf.

+

A(g)

and

Interchanging

such t h a t

,

(oh = i d

A(f)

'

A

for

the a d d i t i o n a l property t h a t

f

x

E

E

a

6

0

is

> 0 and

d($x,$yk) 'Go

for

converges t o a point

f o r each

E

$

i.

his

2

so

t h e r e i s a unique

A(g)

of t h e g-orbit of

f

and

d(gihx,fix)

g

5:

g ->

h

for a l l

g

x , and

gives a continuous funcThen,

i.

i s a homeomorphism.

h

We let the reader v e r i f y t h a t

A hyperbolic s e t

BY the

E.

whose f-orbit s t a y s within

h i A(f)

5

d (f i(x,f iz)

Thus, we have proved t h a t f o r each

tion

as

(y ) be a subsequence such t h a t (y nk "k Then, d((x,z) 2 &O. But, f o r a l l i,

A(f).

E

, there is

A(g)

E

i.

Let us prove t h a t

Let

Fixing

$(x)

x

for.all

Clearly,

is continuous.

f

which has a l o c a l product s t r u c t u r e and has

IA

has a dense o r b i t i s c a l l e d a hyperbolic

basic set. Such s e t s have been studied a g r e a t deal.

Their o r b i t s t r u c t u r e s can

be modeled very w e l l by c e r t a i n g e n e r a l i z a t i o n s of t h e 2-shift described above. Let

J = (1,.

of the elements of s h i f t on N-shift.

ZN

If

..,N), J

and l e t

ZN = 'J

be the g e t of b i - i n f i n i t e sequences

with t h e compact open topology.

a s before by b (a) ( i ) = ~ A = (A .) i~

i s an

N x N

( + i1 )

for

5

One defines the E

EN.

This i s the f u l l

matrix whose e n t r i e s a r e

0's and

l's,

we may consider the subset

CA =

{a

yields

i i+l

a closed

A

c EN

defined by

,g(i+l) = 1 f o r a l l

Z~ :

a a ...I 1 o 1)

Thus a sequence (...aa a

C

is i n

CA

i).

i f i and only i f each of i t s 2-block

1 when used a s indices f o r the matrix

O

so that

N 0. 1 >

has a well-defined

K

Every compact

R3

i s a compact

K

S have

crmaps between

branched 1-manifolds and other manifolds a s usual.

r

K

i f i t is continuous

i n f i n i t e order contact a t any point where they meet,

with

a Y1 u

Ua

0. The family

Va

{Ua)

of

S1 w S2

i n f i g u r e 5-3.

Figure 5.3

With t h e indicated o r i e n t a t i o n s , lengths on

S1, and

g(p) = p.

g

doubles lengths on

S2, t r i p l e s

This expanding map i s i n t i m a t e l y r e l a t e d

[' 1' .

t o the Anosov diffeomorphism induced by If

i s a map, we. d e f i n e t h e inverse l i m i t of

g : K+ K

..)

A

the s e t

K =

I (aO, al,.

; ai

closed s e t i n t h e product of give

..

K

a.

A

k

K defined by

+

*

g : K + K defined by

...))

g(z) = z

2

,

g

solenoid map above.

rf-"z

an(z) get t h a t

Let @ o

for

{h(a)}

K

K.

is a

K

a = (a,

of

alp a2, ag, e t c .

g : K + K induces a homeomorphism

The map

n (2) = ao.

al,..)

T h e n is a n a t u r a l p r a J e c t t m

a2,.

..).

A

I n a p r e c i s e sense,

rfr

g

g : s1 +

if

unfolds

s1

f

IA

r: S1

x

is t o p o l o g i c a l l y equivalent t o

-1

with inverse

= (gag, ao, al...)

g

into

i s the map

where D2 +

D2i,

is the

f

s1

is

z r b=nfn(si x set n2O Then, l e t t i n g @ ( z ) = (ao(=), al(z) .), we = g,

If

,..

n r 0.

i s a continuous map.

0: A +

we have t h a t

= 1

To s e e t h i s , n o t e t h a t i f

the p r o j e c t i o n , then

Then

One thinks of a p o i n t

Now, one can prove t h a t

then

.

with i t s e l f countably many times and we

K

g((ao, al...))

= (al,

a homeomorphism.

= ail

6

*

r ((ao, al,

g(ai+l)

t o be

A

together with a choice of pre-images

K

E

Qne frequently w r i t e s n

K and

t h e r e l a t i v e topology.

K a s a pofnt

.. :K

E

g

Since

f

l l i ~x D~

is a contraction,

nfn({ a d D2) is a s i n g l e point f o r any = = (aO, al,. .) c sl. n20 2 @=idA, =n fn({an} D ). Then h i s a l s o continuous, and h n20

h = idil.

x

o

x

As

@ f = g#,

we s e e t h a t

@

i s a topological conjugacy.

Williams has given general theorems of t h i s type.

Theorem 5.1 ( W i l l i m . [ 61 11.

Suppose that f

is a

c1 diffeomorphiwn

k v i n g a 1-dimensiml hyper&lic attractor A with aplttting and dim E~ = 1

.

EU

TAM = ES

Then there i s an expanding map g of a branched I-manifold A

K

such that In

1: 611,

f l ~ i s topologically conjugate t o the inverse l i m i t map Williams assumed t h a t t h e s t a b l e manifold f o l i a t i o n was

It is well-knowr~ now t h a t t h i s assumption can be removed a s follows.

U be a small fundamental neighborhood of A.

Then n f % = A . -0

g.

.

1 C

Let

Approximate

f

by

fl

s o th.at

fl

is

By theorem (3.72,

c2.

topologically conjugate t o

f

IA.

s t a b l e manifold f o l i a t i o n of

fl

(6.5) - i n [ 17 I ,

By theorem on

.

is the

,

C1

is

U

flQfy(u)

There i s . a converse t o theorem (5.1).

(GFiZZiams [ 6 1 I

Theorem ,(5.2)

branched I-manifoZd

)

.

Let

g : K+K

be an expanding map of a

K such that

( a ) every point df

K

is non-wrmdering

( b ) each point i n K has a neighborhood whose image by. a power

and

g

m

of

i s an are;

g

Then t h e ~ ei s a diffeomorphism f : S4

+

s4

which has a hyperbozic attractor

,.

on whioh it i s topoZogicaZly conjugate t o g.

The idea of t h e proof of theorem (5.2) i s a s follows. K in

of

R

cp(K)

3

. .

via

cp: K + IR

and l e t

N be a "tubular neighb-orhood"

This i s a 2-disk bundle over

branch p o i n t s of

+(K)

F i r s t embed

$(KJ

where the corners a t

have been rounded off t o look l i k e

pants l e g s a s

i n f i g u r e 5.4.

Figure 5.4

One may w r i t e

N

a s a union of

2-disks

a t most two 2-disks have a point i n cqmuon.

( ~ ~ x1 E, $ (K) If

i n which

one forms t h e quotient

space by i d e n t i f y i n g t o a p o i n t any two 2-disks which i n t e r s e c t , one g e t s a space

homeomorphic t o

Kl

+og

The map the map

gl

0

:K + N

each x

E

:N

B

-,

K

1

be t h e i d e n t i f i c a t i o n map.

may be approximated by an embedding gl : K + N

+-I : @(K)+ N

its interior so that

Let

K.

e x t e n d s t o a. diffeomorphism

g 2 ( ~ x )c D

and

g

2

from

N

Dx

for.

g2 c o n t r a c t s

, and

into

+ogo+-l(x)

+ (K) .

We p i c t u r e p a r t of the image of

g2

5.5.

i n figure

\

Figure 5.5 Then x, y

r\ g2(N) n>O

= A (g2)

N, then

ng2x =

E

is easy t o s e e tl-at g3

is a hyperbolic s e t f o r rg2y,

so

g

W e have

K

t o know when

g

Also,

induces. a map

g3 : K1

if +

3 Kl

,.

I

g2JA(b2).

That

,.

g3

and

rs = gm.

diagrams.

1'

It

i s a l s o topologically

3

and

g

and

K

K

two expanding maps and we wish

a r e topologically conjugate.

Williams shows t h a t

a s u f f i c i e n t condition is t h a t t h e r e e x i s t continuous mappings

s: K + K1

K

for

r e q u i r e s more work.

1

and

nx = ny

i s an expanding map, and t h a t the inverse l i m i t

g3

i s t o p o l o g i c a l l y conjugate t o

conjugate t o

g2

g2.

and an i n t e g e r

m 2 1 such t h a t

r : K1

-+

K

g r = rg3, g3s = sg, sr = ,g;

The conditions can be expressed a s t h e following commutative

g ?

K-K

In the case of our maps g, g3, an m

2

1 and maps r, s can be found,

so g3 is conjugate to g3 See [ 61 1 for more details. Since g2 : N

-+

we choose an embedding

a diffeomorphism of

is homotopic to the inclusion i : N

lR3

s4

) :

4 lR3 + S , then $

0

g2

-1

0

)

I$(N)

+

R

3

,

if

extends to

by standard techniques in differential topology.

Let us give one more example of a 1-dimensionalattractor.

This is a

variant of an example due to Plykin [ 4 3 ] . His was the first example of a 1-dimensional hyperbolic attractor in the two dimensional disk. Let D be a disk in lR2 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.

Figure 5.7

The branched manifold is a union of 3 circles and the map(on homology) is AC--+A+C-A B

h

A

C

W

B

RernurEs 1. One can use this example to show that non-trivial hyperbolic attractors (for flows) appear in arbitrarily small perturbations of constant n vector fields on tori T .of dimension greater than 2 [ 3 6 ] . As a consequence, hyperbolic attractors appear in perturbations of three or more coupled harmonic oscillators, or three or more coupled relaxation oscillators. To be more explicit, xecall that a harmonic oscillator has equation I&

. = v-kx

3

=

where m

+ kx

= 0

and k are positive constants. If we have n

or

such

mv

oscillators, we obtain the system

on 1 ~ ~ " . There is a stable equilibrium at the origin, and all other orbits lie on n-dimensional invariant tori.

A relaxation oscillator is a differential equation vf the form x

+

+x=

f(x)i

0 where, for some constant k > 0, f(x)

and f(x) > 0 for 1x1

2

k. For example, if f(x)

-

p(x2

.O.

x

E

M

The support of a measure

such t h a t f o r every open s e t

U

%

Theorem 6 . 2 ( [ 4 8 1 , L 7 1).

Let

cL diffeomorphism having a hyperbolic attractor

a

and l e t

be Lebesgue measure on M,

m

f

be

l'here i s an ergodic

A.

f-inczriant probability measure uA supported on A with the foZZaving l'here i s a subset A c #(A).

property. x

E

wS (A)-A

with m(A) = 0 such that i f

and $ i s any continuous function on M,

Thus, for m-almost all x orbit of x

in

ws (A),

then

the time average of

converges to a definite limit.

$

along the

This result 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 Furthermore, except in

probability one, we can compute its expected value.

the Anosov case, hyperbolic attractors for c2 diffeomorphisms have Lebesgue measure zero.

Therefore, it is surprising that one can say anything

about time averages of points in sets of positive Lebesgue measure near these attractors.

If uA is the measure in theorem

ws (A)

such that

+I

= p ,(V),

,(C&J)

f k ) )

{x, fx,.

..

n-1 ,f x } n U

densitaes.

in

ws (A),

the average number of points in

for x

E

A

Thus, any open set U

dimensiondl measure zero for all x )I,,(C~ = p,,CUI.

is an open set in

is the characteristic function of U-

approaches pA(U).

unstable manifolds wU(x)

c1

xu

Here

k-0

Thus, for almost all x,

U

then for m-almost all x

1 l:n 1 n-

( 6 . 2 ) , and

One can project CQ

p,

onto the

get conditional measures with

such that

aU n

(x)

has the property that

has u-

n-1

k xU(f X ) t o

;1

The convergence of

k=O

pA(U) f o r such s e t s

is

U

proved with f a m i l i a r methods of measure theory. Let ldt

F

be t h e above subset. of

A

be a closed subset of

open s e t in. $(A) Let

* xy.

42

n-1

;; I

Thus, iim inf nSimilarly,

k=O

l i m sup

rrw Since

E

c

V

wiFh

m(A) = 0.

$or

Given E > O

pA(U)

6 > 0, there i s an

then

such that

Choose poines x,y

"k

-+

d(fjx,fjy) 5

for

E

Given any

Ij 1

5 N(6),

E

y as k

+-

* d(f jxn,fJyn) 5E

for

(j1 5 n and d(xn,yn) ) 60-

"

and subsequences (x ) , (y

A

-.

)

so that x

' jf x,fJ y) 5 e: Then d(x,y) 2 60, but d(4

This contradicts the choice 0-f

(xn) ,

> 0 and there are sequences

If not, there is some

in A

and y

> 0 so t h a t i f

f A.

5 6.

d(x,y)

Proof. (yn)

N(6)

1

be an expansive constant for

0

"k

-+

x

for all j.

E.

We need three more lemmas. We will defer their proofs to the end of this section.

Lema (7.3). x,y

There i s a constant K > 0 such t h a t i f

A with

y

c



0 there i s a constant

and n > 1.

There i s a constant

C;'~Z~~C,

C1 > 0 such t h a t

forall

n

~

and

f b ) for any i n t e g e r s

.

nl,.. ,n > 0, r

> 0 i s srnaZZ and

- s~$~(Y)1 5 K.

1, then

For any

E

l

C

E

> 0

We now move t o t h e proofs of (6.6) and (6.7)

In view of t h e volume lemma (7.41, (a)

Let

x

E

> 0, t h e r e i s a constant

f o r any

x

E

A, l e t

E

Let

E~

E

> 0 be a r b i t r a r y and l e t

d(x,y)

5 E.

Let

use s p e c i f i c a t i o n t o give a

expansiveness f o r least

-

E

2.

a p a r t somewhere.

I l$UI I

2 bee

2 1.

n

£ 1 ~ . By lemma (7.21, t h e r e

for

P e r ( 2 ~ ( € )+ n

+ IU + 2P.

and

ij

1

5 N(E) and y r Fbr any

(z ,a)

E

j !

q (w)

q(z)

w:(x,n). z

and

and q(w)

A

Per(m) n A,

E

that E

1 P = P(-j)

with delay

Per(m) n A, t h e o r b i t s of

That is, q (z)

z

+ m + ~ P ( E ) ) such

Then, ( f N ( € l q ( z ) )

a p a r t somewhere, s o t h e o r b i t s of

E,

Let

q(z)

z # w in

such t h a t Sneu(x)

> 0

m 2 1 be an i n t e g e r .

( f - N ( E ) x , 2 ~ ( ~+) n)

-specifies

r = ~ ( m )= ZN(E) + n

Let

.

d ( f J x , f f y ) 2 el

N(E) > 0 such t h a t

€1

E

> 0 be an expansive constant f o r

implies t h a t

q(r)

b

n 2 1, uA(w:(x,n))

and any

.

is an

(6.6) follows from

Yor any

A

.

.

Also, by w

get a t

get a t l e a s t

.

= ~ u ~ { l $ ~11. ( c ) By lemmas (7.3)

and (7.51, we have

I

I f we s e t Now l e t t i n g

i + m

I / $UI I - 2 ~ ) , we have

= c-%x~((-zN(E)-u)) 1

mi

be such t h a t

T (mi)

+

= ni = ~ N ( E ) 2P

gives

UP:(x,n)

sn$"(x)

2 bEe

which is (a).

snoU(x)

Tws(x,n) E

+ n + mi,

2 bEe

and l e t t i n g

To prove (6.7), i . e . t h a t t h e r e i s a constant

(b)

and

i s ergodic, we f i r s t prove

uA

C > 0

such t h a t f o r any.Bore1 s e t s

B,

Once (b) i s established, (6.7) follows e a s i l y . v a r i a n t and

-A

0 < uA(A) < 1. Then, M < 1. But from (b) , we g e t

0 < u,(M-A)

A

For suppose

is in-

A

is a l s o i n v a r i a n t and

0 = pA(A n (M-A)) 2 CpA(A)-p,(M-A)

Q

>

a contradiction. To prove (b), it s u f f i c e s t o show t h a t i f of

A, and

6 > 0, then

(c)

l i m i n f u,(B,(A) n-

where

B6(A) = {y : d ( y , ~ )5 6 )

Heri,

uA(A) > 0

62 > 0

and

- A1)

B (A ) c . U 1

and

1

a r e compact subsets

6.

0 < 6

1

a s before.

uA(A) o r

pA(B) = 0.

If

< 1 be a r b i t r a r y , and choose

s o small t h a t

uA(U1

6

B

B6 (B) = {y : d(y,B) 5 61

and

uA(B) > 0, l e t

Then pick compact s e t s that

A, .By and

Then (b) is obvious i f

For suppose (c) holds. both

and

n f a ( ~ 6(B)) 2 Cy,,(A) .u,(B)

is independent of

C

A

< 62

A c A, B c B 1 1

and

-

u,(V1

B (B ) c V1. 6 1

and open s e t s

B1)

v,(B, (al)

> uA(B6(A1) -

By (c) and t h e choices of

Next take

3

A, V

1

6 > 0

3

such

B

so t h a t

Then,

-

-> Y , ( B ~ ( A ~n )

-

< 62.

U1

n

f-%&(

+

- A) - uA(v1

-

.

we have

-

~ , ( f - % ~ f-%))

-

~ ~ 1 L1,(ul )

n f - n ~ 6 ( ~ 1 ) ) 262

62, A1, B1,

- A)

(p,(ul

- B)

was a r b i t r a r y , we g e t (b).

As

Let

6 > 0, and l e t

Given any l a r g e i n t e g e r s

n , r , s > 0, l e t

To prove ( c ) , we again use s p e c i f i c a t i o n . be compact s e t s i n

B

z

1

A n Per(n), z2 E ~ e r ( r )n A, z

E

El P = P(-) , let 3

let

delay implies

-

-

z = w

and

wi

since i f

get

d(f [n121q(~),zl)

£-n-r-2P

Bg (B)

E

cl

z

i

B n ~ e r ( n ) ,and and let

T

q(T) E Per(r) n A.

$ wi

.

and Now,

z

= 2n

4

E

Per(s) n A.

+ r + s + 4P.

,(z2,r) , (f-[n12'z3,n) , and f o r some

a p a r t somewhere.

Note t h a t

Also, i f I:[

f n + r t [n/2]+2P q(Y)

E

(z4 ,s)

q(;)

Then,

We with

=\(;)

1 I i I 4, then t h e o r b i t s of

5 6 , and d(f n+r+[n/2]+2P q(z2 - ,z3)

f'n121q(~) E B6(A) n

(f-[n1211,n)

by a p e r i o d i c p o i n t

P

3

= (zl,z2,z3,z4),

1 3 -specify t h e p a i r s

zi

A.

and

A

B,(B),

.(

,N(6), 6. so

then

Thus, f [ n 1 2 1 q ~E B6(A)

with

depending only on

C

through

ni)

E

and

1

s

Letting

(so t h a t

+

T

runs

gives

r -+

Then, l e t t i n g

we get

w,

l i m i n f u,(B6 (A) n ~-'B,(B)) )CY~(A) .Y~(B) j-

Finally, l e t t i n g n = ni

and

i

gives us (c).

-+

,

We w i l l now give t h e proofs of lemma (7.3)

P m o f of Z e m (7.3). b o l i c i t y of Let

E

x

(7.5).

X > ,1 be a s i n t h e d e f i n i t i o n of hyper-

A. x,y

be small enough s o t h a t f o r each

0

is .at most one poilit. each

Let

, and

(7.4)

Then choose

E

1

E

A , W'

(x) n 2Eo

0

I

if

y

B

E

that

(x) n A , t h e r e a r e p o i n t s

E

s

{y) = WE (zl)

z2

and

y

such t h a t

r

(x)

, and

co

{Y}

Now suppose there is a

d(x,y)

x,y

wU

w E

- wS

c(d(x,zl)

l

2co

E

(2,)

(fn-lx)

-< E

Similarly, i f z

2

E

wSXl-nE

j

.I

(x) n

wU

(y)

.

zl e

2E 0

Now, l e t

+

WE (x)

c > 0

"

p-1y

5

E

for

s

WE (wl)).

0

< n, so

z1 =- fl-nwl

5 j < n.

Then

NOW,

0.

for

such

0

E

wzE0 s (y)nwU

n

w2€u

(z2).

(x) and

-n < j

5 0 , t h e r e i s a point for

z2rwS

0

Hence

Ij 1

< n

implies

(x)

suchthat

0

d(x,y) 5 c(d(x,zl)

+

d(x.z2))

5 2ch1-n~0-

0

L > 0

and

Oi).

X1-"E-

Thus, d ( f j x,f j y) 5

wU X1-='E

(zl)

E

2E0

0

V8

z2

independent of u d(xyz2)) whenever zl E WEn(x),

d ( f J x , f Y)

o 2j

for

d(f x , f y) 5 c1

there are points IY) =

That is,

wU2Eo.(z2).

such t h a t

€0

( w l y n l x )

n

satisfy

A

W : (x) and

E

There i s a constant

0

0

x

zl

0

u n WE (z2).

0

( 0 , ~) small enough s o t h a t f o r

E

A , BE (x)nA i s contained i n an co-product neighborhood.

E

wlE(y)

0 < a < 1 b= such t h a t

d($Ux,$Uy)

5 Ld(x,yIn

0

for

x,y E A, and suppose

WS (x,n)

y

.

Then

I

Proof of (7.4). Let (us$)

be a

flattens

wU(x).

+

:U

W:(XI

-+

of

x

(5,n)

c2

x

That is, i f

E

with

B'

and

be coordinates on If

= {Z E 1~'

E

small, x

z

E

in(=) 1) n $(x,n)

x

E

A, and

x which contains :

(21

5 I), u

c2 d i f feomorphism such t h a t

(01) c w;€(x)

= ( ( z , ( .

+-1(Bu x

wS(x,n)

coordinate c h a r t about

is a

BS

c (-'(B~

Let z

BU

Consider

=

n . 2 1.

wS(x,n) E

then

S,U,

+(x) = (0,O)

and

,

Ip2E(x,n) c O.

B~ x ' B S , and f o r let

D;(z,d)

containing

z.

z

E

U, w r i t e

be t h e connected component We d e p i c t t h i s i n figure 7.1.

Now each

fjD:(z,n)

metric induces a If

y

E

C

1

0

5 j < n , is a c2 u-disk i n M y and t h e Riemann

u-dimensional volume form w(j ,z) = w(j ,z,n)

fjDU(x,n), then

w(j ,x)

E

coincides with t h e form

Y

w Y

y # f j ~ ~ ( x , n t)h,i s . w i l l n o t generally be t r u e .

$U, but i f

w i l l be t h a t they come from a smooth coordinate

w(j ,z,n)

system, ana l a t e r w e w i l l be a b l e t o use Fubini's theorem. t h e d e f i n i t i o n of

Then, we l e t

au

$u,

y

E

(f-n) w(o,z)

Y

= e

sn?(y)

As before, i f

w(n,z)

D:(z,n),

y

E

by t h e equation

D;(z,n)

,z

E

,

f(x)

fny

Now we claim t h e r e i s a constant y

Proceeding a s i n

u f 3DE(.z,n), we d e f i n e F(y)

(y) = -log T(y).

*

then

if

used t o define

The advantage

E

of using t h e forms

on i t .

K > 0

S

such t h a t i f

z E W (x) E

and

then

(a)

IsnGU(2) -

Sn$

(b)

IsnTU(y) -

Sn$ (z> 5 K .

(XI I 5 K

-U

and

I

Suppose (a) and (b) hold f o r t h e moment.

It i s evident t h a t , f o r

small,

Let

m

s

be s-dimensional volume, and

Clearly, t h e r e i s a constant

C1,E

> 0

m b e u-dimensional volume. u

s o that

By Fubini's theorem and (c) and ( d ) , t h e volume lemma (7.4) w i l l be proved i f we can f i n d a constant

CzSE > 0

such t h a t f o r

z

E

W;(X),

E

To prove (e), we f i r s t use t h e change of v a r i a b l e s formula f o r multiple integrals t o get

verge t o

ff~;(x,n)

fY(x,n)

i n the

topology a s

C'

= w:(fnd.

f j DEu (z,n)

1-lemma (25) , t h e d i s k s

By an (estimate s i m i l a r t o the

j

increases.

Furthermore,

Thus, t h e r e i s a constant

that

-

C-'3,s < t(f%:(z,n))

5

con-

C

3,c

> 0 such

c ~ ., ~

By ( a ) , (b), and ( f ) ,

So, (e) follows taking

2K

C 2,s = C3 , E e

'

Now t h e proofs of estimates (a) and (b) a r e very s i m i l a r t o t h a t of lemma (7.3) except t h a t we r e p l a c e u-dimensional planes over y r fjDU(z,n) E

Tu becomes

M.

M by t h e Grassmann bundle

Since

f

is

c2,

i f we i d e n t i f y points

with t h e i r u-dimensional tangent planes, T f'Dr(r,n),

a C'

function

cu

m t h e d i s j o i n t union

0

Ojjfn

Also, t h e r e i s a constant 4 > 0 independent of j, e, x , and

Let

d

f o r some

-

be t h e metric on

C > 0 and

z = T DU(z,n) Z E

G U ( ~ ) of

and

G ~ ( I ~ ) I. f

1 > 1 by a

u x = Ex

gives

~~(fj%(x,n)). n

such t h a t

z r W;(X), then

1-lemma type estimate.

then

Setting

This implies (a) f o r

guy s o (a) follows f o r

0 5 j < n , .each disk

On t h e o t h e r hand, f o r

of a

c1

function

qj

to

f r n W;(f%)

q 's a r e a l l uniformly bounded. j

of t h e

TU . is t h e graph

fjDU(z,n) E

W;(f3x).

Monover, t h e

y

Thisqmplies t h a t i f

6

sizes

C'

~r(z,n),

then dU(Tfjyf

~

j

-< C2h- (n-j 1

f o r some c o n s t a n t s

cU,

Again, a s Above, t h i s gives (b) f o r

Proof of Zema (7.5).

Recall t h a t

1

be an expansive .constant f o r

f A

We f i r s t prove p a r t (a).

If

such t h a t

. d(f 3pl,fJp2)

n(M)2

1

j

z T fnJ z f j ~ r ( z , n ) ) 5 Cld(f y ,f z) > 0

.

Zn =

1

pr Per (n)

sn4u(~) e

.

S > 2 ~ . Thus, WE(pl,n)

+ ~ ( c ) ) such

x

that

G

Let 2~ > 0

# p2 i n Per(n), t h e r e is 0 5 s

n WE(p2,n) =

w:(P,~).

j < n

So,

0.

m ( ~ , 8 ( ~ , n ) ) ~ ~ ; 'bZy~t h e v o l m e l e m m a ( 7 . 4 ) . pePer(n) hand, by s p e c i f i c a t i o n (lemma (7.1)), f o r each x r A, and n a p r Per(n

TU.

and, hence, it g i v e s (b) fok

. pl

C1,C2

Ontheother 2

1, t h e r e is

Thus

U

W ~ ~ N = ~ W ~ ( X ) C $E(~.n) xe A pr Per (n+P (€1 ) and

This gives

-

-1 CEm(M) 2 Zn 2 C2€ e PIICIIm(~:(A))

for

n > P(E)

and (a)

c l e a r l y follows. To prove (b), we f i r s t prove (c)

t h e r e i s a constant have

DE > 0

SO

that for

n i > P)(:

= P, we

For (f

n +n +-+ni,l

,...,

1

f o r some if

7

-

ni) i=l

p,ni).

O

z

1

p r Per(

z

r

12 i 5 r , l e t

For convenience, we s e t

i i E k,d(f -zk(pl) ,f zk(p2)) > 3

r

f

ni, i=l

and

K

with

DE =

z = (zl,..

and

z

exm[

.

,Z

r

n

0

e Per (ni

= 0.

E -specify + P) 3

Then l e t

+ P I I $uI I).

), t h e r e is a

-

(y) #

# w implies

.

and

# y(p2).

=

>

E,

so

Thus,

is a s i n lelaa (7.31,

< exp ( @ + P I IO~ -

-

zi(p)

If p1 # p2, then f o r some j > 0 , d(fjpl,ffp2)

and

i

and

1 I)r )

Z,

simi<arly, i f p(3

r n e x p (sn + ~ $ U ( Z ~ ) ' ) i i=l

zi E per(ni

- PI,

and

such t h a t

Per P

E

1 ,.*.z-

p(G).

Hence,

and (c) i s e s t a b l i s h e d .

Now, we claim

Zn+p

DEZn

f o r a l l n 2 1. I f n o t , l e t n

Zn+p 7 (l+a)DEZn with a stxitable small a.

Then, for

2

' Zk (n+2p) 3

> D

-k k Zn+p

-k

> D E

(l*)

k k k DEZn

.

be such t h a t

So 1 k (n+ZP) log 'k(n+2~)

-.

>- I

n+2P

[log(l+a)

+ log 61 .

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

tradiction.

Similarly, we get

2 follows seCting C1 = DE.

> D Zn-p -

+

for a

-

which gives a con-

n >

.

Then (b)

8.

Diffeomorphisms with i n f i n i t e l y many a t t r a c t o r s . Given a

We have seen t h a t hyperbolic a t t r a c t o r s have a r i c h s t r u c t u r e . diffeomorphism sesses.

f , one would l i k e t o know what kinds of a t t r a c t o r s

f

pos-

I n p a r t i c u l a r , a r e t h e r e f i n i t e l y many? Do almost a l l points, say i n

the sense of Lebesgue measure, approach a t t r a c t o r s ?

For even simple diffeb-

morphisms which a r i s e i n p r a c t i c e , these questions a r e very d i f f i c u l t . On t h e other hand,if

c l HP ( f )

i s hyperbolic, then

structure.

By theorem (3.81,

s u b s h i f t of f i n i t e type. of

p

C l H (f)

ngn

P (U)

in

M

i s a hyperbolic periodic point of

C l H (f)

P f l C l Hp(f)

f , ahd

has a dense o r b i t and a l o c a l product

is a finite-to-one

quotient of a

Also, by theorem (3.7) t h e r e a r e neighborhoods

and

N

of

f

in

~ i f f bsuch t h a t f o r

g

E

U

N,

.

is topologically conjugate t o f 1 C l Hp(f) Thus, we understand n the s t r u c t u r e ' o f flC& H ( f ) very w e l l and t h i s s t r u c t u r e p e r s i s t s when f i s P perturbed. g1

CL H ( f ) is not hyperP b o l i c f o r some hypnrbolic p e r i o d i c p o i n t p, and we would l i k e t o understand

Now t h e r e a r e many diffeomorphisms

these.

f

f o r which

A t present our knowledge of t h e s e ,diffeomorphisms is q u i t e incomplete,

and here we s h a l l merely focus on a few t y p i c a l examples and some of t h e i r properties. It follows from proposition (4.2)

t h a t a diffeomorphism with a hyperbolic

l i m i t s e t has only f i n i t e l y many a t t r a c t o r s .

W e w i l l see below t h a t on sur-

faces

CL H ( f ) not being hyperbolic frequently l e a d s t o t h e existence of P i n f i n i t e l y many p e r i o d i c a t t r a c t o r s . To begin with, l e t us t r y t o imagine t h e simplest way we might have

Cl H

P

not hyperbolic.

Clearly, we should t r y t o f i n d a non-transverse

homoclinic point

af

q

o (p>

.

However, t h e Kupka-Smale theorem (1.3) t e l l s

us t h a t generically a l l homoclinic pointe a r e transverse, s o even i f we found such a

I f we want a p e r s i s t e n t l a c k of trans-

we could.perturb i t away.

q

v e r s a l i t y , i t d s n a t u r a l t o replace and t r y t o arrange f o t s i s t e n t way.

o(p)

s

w ~ ( A )and

W (A)

by a n

i n f i n i t e hyperbolic set

t o be non-transverse i n a per-

This t u r n s out t o be easy t o do i n dimension l a r g e r than

For ifistance, r e t u r n t o Plykin's example i n s e c t i o n 5. Ieomorphism

f

-

r)

f% Al(f) n2O fixed point of

of a subset

D c 3R2

a c wU(p), ~ ' ( 1 0 E

in

A1,

and consider

is an i n t e r v a l , and

f o l i a t i o n of a neighborhood of

This is a d i f -

w:(~)

f o r some small

{W;(X) )xEw;(p)

E

.

b e a l i n e a r expansion on the l i n e

3R.

D x IR -+ D x IR.

i s a hyperbolic saddle p o i n t f o r

and

dim

wUcp, o) ,f x g)

z1 c ~ r ( ~ , o ) , xg) f

-

= 2.

{(p,o)}

(p,o) Let

y

and

For each

gives a 1-dimensional

in

D.

Let

p .be a

Let

p

Clearly,

2.

i n t o i t s i n t e r i o r such t h a t

i s a one-dimensional hyperbolic a t t i a c t o r . f

A > 1, and l e t

Consider t h e mapping

g(x) = Ax f x g : f x g

b e a curve joining two p o i n t s z2 c w;((~,o).f x g)

-

( ( ~ ~ 10 )a s i n

f i g u r e 8.1.

Figure 8.1

Let

N

be a small tubular neighborhood o f ' y , and l e t

feomorphism such t h a t .$(z) = z

N n ~ ~ ( ( p , o ) ,xf g)

for

t o a curved d i s k

A

z

4 N and

$

A ' which meets

@

maps a d i s k

be a difA

in

N n ~ : ( ( ~ , o ) , f x g)

2 90

in a c i r c l e a s i n f i g u r e 8.2

Let

= 90 (f

x

g)

.

If

-f , and

is hyperbolic f o r

f o r any

gencies w i t h

ws(gt (fl) ,fl)

x,y E ~ ' ( f , )

such t h a t

i n t h i s case, i f of

fl

is small, one may check t h a t

N

near

wuh'(f l) ,f l)

f

x ( 0J

has tan-

That is, t h e r e a r e points

A'.

near

C'

f,

i s tangent t o w ' ( ~ ) near

wU(x)

is

C1 near

f

A' z A1

and

pl

Moreover,

A'.

i s the hyperbolic fixed point

(fl) is not hyperbolic. p1 This shows t h a t the Kupka-Smale theorem f a i l s i f we t r y t o replace

fl

near

(p,o)

then

Cl H

It a l s o shows t h a t Axiom A diffeomorphisms

periodic p o i n t s by hyperbolic sets.

o r more generally diffeomorphisms with hyperbolic l i m i t sets a r e not dense in

~ i f f % l f o r any

M with

dim M > 2.

It i s less 'obvious t h a t hyperbolic

s e t s with p e r s i s t e n t l y tangent s t a b l e and unstable manifolds e x i s t i n diraens i o n two.

W e w i l l see t h a t they do i n t h e

is s t i l l n o t known i f

iff^^

with

r 2 2.

M

r

2, b u t it

is a compact two-dimensional manifold and

A hyperbolic basic s e t

i f t h e r e is a neighborhood

N

there a r e points

E A(g)

somewhere.

topology with

r = 1.

Now l e t us assume t h a t f .E

cr

x and y

of

f

in

iff^^

A

for

f

is c a l l e d wiid

such t h a t f o r any

such t h a t ' wU(x)

and

g E N,

w ' ( ~ ) a r e tangent

We w i l l omit t h e word basic and c a l l such s e t s A periodic point

p

of period

n

A, wiZd hyperbolic sets.

of a diffeomorphism w i l l be c a l l e d

dissipative i f

S(f)

of

fn(p) = p, then a l l eigenvalues of

det T f n < 1. Let P That is, i f p E S ( f ) and

f.

T fn P

denote the s e t of periodic sinks

haveanom l e s s than one.

Theorem (8.1). and

f

E

iff^^^,

has a wild hyperbolic s e t

r r 2,

contains a dissipative periodic point

A

N of

Suppose

iff^^^

in

f

then CC H (g)

c

P

C l S (g)

mere are a neighborhood

p.

and a residual subset

B

c

A,

N

such that i f

.

E

B

,

0

Thus under the conditions of theorem (8.1) each point of the h-closure

Cl H (g) is a

l i m i t of i n f i n i t e l y many periodic sinks. I f A i s a hyperP b o l i c basic s e t f o r f and p i s a p e r i o d i c point i n A , then A c Cl H ( f ) , P s o generically, each wild hyperbolic s e t with a d i s s i p a t i v e periodic point

i s i n the closure of the periodic s i n k s .

This gives an i n f i n i t e number of

i n v a r i a n t open s e t s which accumulate on

A , and is the reason why we c a l l

A ' s wild.

such

The f i r s t p a r t of the proof of theorem (8.1) i s the next lemma.

Lemma 8.2.

Suppose p i s a dissipative periodic point for

i s tangent t o W'(O(~)) a t a point M and

x.

and

W~(O(~))

U i s any neighborhood of

If

N i s any neighborhood of f i n

f

iff r ~ then , there i s a

g

E

x

in

N which

has a periodic sink i n U. Proof. at

x.

Let

l y such t h a t of

ys

or

neighborhood

Let ys y

and S

yU.

U

i s tangent t o w ~ ( ~ ~ )

p1,p2 c ~ ( p ) be such t h a t yU be small curves i n w

and

y

Since of

x

u x

a r e tangent a t

~ ) ( and ~ 1

x , and

x

i s not i n the boundary

cannot be a fixed point of

such t h a t

-1 f U

r;

U =

0 and

w ~ ( respective~ ~ )

f , choose a small

ys u yU c U.

Our

perturbations of

f

the i d e n t i t y and

@(z) = z

= 0 and

for

z

f-nyu n U = 0 f o r n > 0 .

@(pl.g)

yg

w i l l be of t h e form

while

+(yU) c

We f i r s t choose

4

g = @of where For

U.

.

near

small enough, fny n u

s

Let us write

s o t h a t t h e curves

degenerate second order contact a t

cr

is

i s p e r i o d i c f o r any such g

Thus, p wU(p2.g)

U

@

ys

and

Y u (g)

-

((yu)

and

.

y (g )

have non-

fu,v)

s o that. x

u

1

x.

This gives us f i g u r e 8 . 3 .

I

Figure 8.3

Assume we have corresponds t o

(0,0),

{(u,v) : v = -ab2 near

v O

x

and

U

i n a coordinate system

ys c ( v = O),

+ r(U))

where

yU(g1)

corresponds to

u * 0 a ~ dlfm- r a-I@ lu12 D ( V =~ ((u,~? : la[ (cl,

a > 0

0 , a s u i t a b l e small disk

,, ,,,, lv-vol

(

E

is small and an i n t e g e r 1 a s i n Figure 8 . 4 .

where

and

n r 0

Figure 8 . 4

so that

D ~ V ~a )d

gpfvo)

~ look

~

3

or

course

D ( V ~ )may be below

gp(v0)

is, and

The important thing is t h a t by t r a n s l a t i n g { @ t ) with

s # t

with

i n some i n t e r v a l

t

such t h a t

(1) 'g>(vo) (2)

g>(v0)

n ~(v.1

-

gp(v0)

up o r d m v i a a family

s,t

6 >