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
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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
the inequality
IR
-+
is
?J
f (x) in the U
( x , ~ ) plane is
= 0.
When there are two fixed.points of (
F : 1R U f (x) < x
For
The locus of fixed points of
the parabola x2
Consider the map
-x. 2
and one is unstable
f
for
v
f
x
< 1,
0 <
1 >1
As
p
one is stable
increases further
to 1, the stable fixed point becomes unstable in a flip bifurcation which is described in the next example.
It is worthwhile noting that another sort of
qualitative change in the behavior of orbits near the stable fixed point occurs when 0 < 4
<
u = 1/4.
In this case,
0
f' (0) = 0. U
is fixed and
For
u 2 1/4 , trajectories approach the fixed point montonically, while for u
< 1, trajectories approach the fixed point in an oscillatory manner.
When a trajectory comes close enough to the fixed point, successive points of See Figure 2.3.
the trajectory lie on opposite sides of the fixed point. Example 4:
Let us continue our analysis of
The flip bifurcation.
-
1 has f (x) = (UT) x2 of the preceding example. when y = 1. f P I U a fixed point at 1/2 and f' (1/2) = -1. When p > 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 <"'
<
C
L-1
[C~,C~+~I We.
shall use the following terminology of Milnor and Thurston 1701 :
I. = [cj, c ~ + ~ is ] a 3
9 of
f.
The address of a point
x
is the symbol
I if x E I and not a turning point, and the symbol C if x = c j j' j j' The itinerary of a point x -is the sequence A(x) = { A ~ , A ~ , A ~. ,I of
..
addresses of the points x, of
x
is defined as follows:
creasing on the lap
I j'
numbers) with basis
11,
. .
2 f(x), f (x),..
€(I.) = +1 or -1
if
3
Let V
...,III'
The invariant coordinate f
is increasing or de-
be the vector space (over the rational I .+I
Set
C
=
3in v
The invariant coordinate g(x)
j 2 is the sequence
structed from the itinerary of
x
with
ei
.. I (Al). .. (Ai-lAi.
{go,el, e2,.
= €(AO)
E
We shall speak of the invariant coordinate of a sequence of well as the invariant coordinate of a point.
..
I ~ , .,I~
into the reals so that
I1 < I2 <
E(C = 0. j C v con-
and
€
eddresses as
Order the laps -ding
.-.<
IQ- Then V
acquires
an ordering which induces the lexicographical order on the sequences
'
~(x< ) g(y)
8: -
if there is a
B.(x) < e.(y). 3
j with
Bi(x) = Bi (y) for i ' < j and
A basic property of kneading sequences for continuous maps
3
is the following proposition: Proposition: Proof: -
If x
<
.
then 9 (x) 5 8 (y)
y <
Suppose x
script with em(x) # em(y). the interval
[x,yl
the identity.
.
for
8 (x) # B (y) and
y,
m
is the smallest sub-
We assert inductively that i
5 m.
The case i = 0
Suppose we have proved that x =A
= I
i f
fi
is monotonic on
is trivial since f0
is monotonic on
is
[x,y] for.
fi([x.y]) C I since fi j is monotonic on [x,y]. Since f is monotonic on I it is monotonic on j' i f ( [x,yl ) . Consequently, fi+l is monotonic on [x,y], proving the ini <
t
A
ductive assertion. i < m
for
j'
Notice that
or
fi (x) and
fi (y) cannot be turning points
since turning points are isolated.
increasing or decreasing on = +l
Notice that
-1
Am(y) = Bm(y)
for
.
[x, yl
z E [x,y].
If
fm
If
depending on whether
fm
-$(I) = A m(.)
is
... m-1 (z)
so (z)s1 (z)
is increasing, then
is decreasing,
fm
Next we observe that
E
= Am(x) <
9,(x)
> Am(y) = -Bm(y).
These two cases prove the proposition. The proposition gives us the means of relating the order of the line to order properties of the itineraries of individual points. itineraries of the ineraries A(x)
5 a , determine completely the set of it-
ci,O i'
as x
varies in
the kneading sequences of
In particular, the
f.
I.
The itineraries of the c
i
are called
There are consistency conditions which must be
satisfied by the kneading sequences coming from the proposition abwe.
We
shall see later how these limit the kneading sequences which actually occur. To fix ideas before treating the general case, let us consider maps which
satisfy
I = [0,11, . f (0) = f (1) = 0, and there is a single turning point
which must be a maximum. by
A family of maps with these properties is given
f (x) = 4px(1-x), 11 s (0,l).
u
c
We shall illustrate how the kneading sequence
of the critical point
c of
I. .The critical value
f
determines the itineraries of all points in
f (c) is the largest point in the image of
means that the itinerary of every invariant coordinate of f(c).
f(x)
x
f.
This
must satisfy the condition that the
is no larger than the invariant coordinate of
Conversely, if a prospective itinerary has this property, then there
is indeed a point with this itinerary.
Proposition: (1) if
Let
i > 0
{Bo,B1,B 2...)
and
B # C, i-1
the invariant coordinate associated to the
..
is smaller than the invariant coordinate of
I B ,~ B ~ ,+B ~
sequence
be a sequence of addresses such that
+ ~ .I ,
f (c), and (2) if
Bi-l = C
then the sequence ( B ~ , B ~ + ~ }, B ~is+ ~the itinerary of
.
the critical point Then there is an
f (c)
x E I
such that
IB0,B1 ,B2,..
.}
is the itinerary of
Proof: Denote by & the invariant coordinate associated to {Bo,B1,B2, ...) = B . Let L = {x IIi(x) < &I. We assert that E
It is nonempty unless Bi = IO 0.
Let
x
E L
and let
for all
is not the turning point
A. (x) is not the turning point, then there is a 3
i
5 z,
8 . (z)
x
c
8 (x)< + 4.
for
with
Hypo-
i < j.
If
Ai(z) = Ai(x)
.
Thus L is open if A (x) is not %i j the turning point. If A.(x) is the turning point, then B is not the 3 j turning point, and there are z > x such that A.(z) = B (2). Moreover, the 3 j continuity of f implies that the itinerary of fj+l(z) tends to the itinerary of
implying that
z >
is open.
L
is the itinerary of
j be the smallest integer with
thesis (2) implies that Ai(x)
for
B
i, but then
x.
fj"
3
4
(x) = f (c) in the sense that, given any
A. (fj+l(x)) (provided that one of the addresses are 1
has a maximum at
x,
so
E
~
+= -1 ~
for
hypothesis, the invariant coordinate of
c).
k,
(2)) =
Ai (f"'
Note that
j+l f
z slightly larger than x. (Bj+l,~j+2,...)
By
is smaller than
-0 (f(c)) , =
'j+l
-1
Bj+k # Aj+k (x)
z
>
with c(z) <
U = {xi& < g(x) ]
,
>
this implies that there is a Similarly, if
.
0 with
hence there is a smallest k
x
then U
L
is open.
I
is connected, we conclude that there is an x
Since
&.
Thus
is also open.
with g(x) =
.
Since
This
proves the proposition. Remark: not quite.
The proposition almbst characterizes the itineraries of
k
c: >
points with
as an address, but then end with the itinerary
Ai+k(x) = Ai (f(c)) and
interval containing c exist.
C
A. (x) # c 3
for
j 5 k,
There are examples which show either possibility occurs.
0.
but
We cannot determine without further information whether sequences
occur which do not contain of
f,
not exist.
If some
tends to a stable periodic orbit, then such x
On the other hand, if every interval containing
which contains c
for some
c
will
has forward image
again, then points with the property indicated a b w e will
The quadratic map which has exactly one periodic orbit of each
power of 2 is an example of such a map (see Section 56). The proposition allows us to determine information about the periodic orbits of
f.
Each periodic itinerary is represented by at least one periodic
orbit, but information from itineraries and kneading sequencies cannot specify how many.
This prmpts the definition of monotone equivalence:
points x
and
same itinerary.
y
are monotone equivalent if all points in
two periodic
[x,yl
have the
Counting monotone equivalence classes of periodic orbits can
be accomplished by counting periodic itineraries.
Moreover, information about
periodic orbits gives information about bifurcations. .For a map
f with a single critical point, the maximum possible number
of monotone equivalence classes is such that classes of fixed points.
91
has
2*
The minimum possible number is
monotone equivalence 1.
Consider a one
parameter family which begins with a map having only one periodic orbit, necessarily a fixed point.
If a map
f
v
in this family has many periodic
orbits, then there must have been bifurcations in the family which gave rise to each periodic orbit. order.
These bifurcations cannot occur in an arbitrary
Define an ordering of periodic orbits by calling the larger periodic
orbit the one with the largest .point: with x
>
y
for every y
E
e2.
el
If
g,
5.
A
if there is an x
E
8
1
B are two periodic itless than the invariant co-
then the preceding proposition implies that a map having a
periodic orbit 9 with itinerary g 1 itinerary
. O2
and
ineraries with the invariant coordinate of ordinate of
>
e2 with
also has a periodic orbit
This means that a bifurcation which gave rise to
before the bifurcation which gave rise to the periodic orbit
el.
8
2
occurred
The order-
ing of periodic orbits for a map in the family gives the same ordering for a sequence of bifurcations giving rise to these orbits. Thus we obtain a combinatorial algorithm for deciding the order of the bifurcations giving rise to a pair of periodic orbits:
it is the order of the invariant co-
ordinate of the largest points in these orbits.
Given two periodic it-
ineraries, comparing their invariant coordinates involves canparing sequences whose length is bounded by the product of the periods of the two itineraries. There are two types of bifurcations of periodic orbits which are generic for one parameter families of one dimensional maps:
saddle-nodes and flips.
Hopf bifurcatims cannot occur on a one dimensional space.
A typical bi-
furcation structure m e encounters is illustrated in Figure 5.1.
An initial
saddle-node gives rise to a pair of periodic orbits, one stable and one unstable.
For the stable orbit to persist but lose its stability, it must pass
through a flip bifurcation. twice the period.
This generates a new stable periodic orbit of
This new orbit must also go through a flip if it is to
persist but lose its stability. The result is an infinite sequence of flip bifurcations following the saddle-node. At each flip a new periodic orbit of twice the period of the preceding one is born.
For the quadratic family,
one can prwe that each saddle-node is follawed by a sequence of flips before
bifurcations involving other periodic orbits occur. These results tell us much about the order in which bifurcations occur for families which are qualitatively similar to the family of quadratic maps. They do not give quantitative information about how large the bifuraction set might be.
In particular, the outstanding question in this area is whether
the bifurcation sets of such one parameter fmilies have measure zero (generically or for specific examples). results related to this question.
Jacobson has just announced the first We discuss it more fully at the end of
Section 96. There are three directions in which we would like to extend the theory we have just outlined for maps with a single critical point.
like to allow more critical points. maps as well as maps of an interval.
First, we would
Second, we would like to consider circle Third, we would like to consider maps
which have discontinuities. All three of these considerations are motivated by examples.
Both the first and second are necessary for our analysis of the
forced van der Pol equation.
The third is needed to understand the Lorenz
attractor. We want to describe the periodic orbits of a function having several critical points. sider.
There are two aspects to this question which we need to con-
The first cssumes that a map is given together with the kneading se-
quences of its critical points and then asks for the itineraries of points' in the dcmain of the map.
The second asks for a list of which kneading se-
quences do occur for maps.
Once these two questions are answered, we want
to embed the information in a picture which describes the relationship between the bifurcations of periodic orbits with various itineraries.
Lemma:
Consider a map
I = [co,cQ].
If x
f: I + I which has turning points
has the itinerary
&
and
x
inequality of invariant coordinates is satisfied:
E
L-1 cl,-.*rc
and
I, then the following
Proof: The map as
E (Ii)
f is monotone on
is +1 or -1.
IC~,~,C~I, increasing or decreasing
The lemma follows from the monotonicity of the
Tnvariant coordinates. This simple lemma gives the compatibility conditions which must be satisfied by the itineraries of orbits of f. The itinerary of f(x) obtained from the itinerary of x by dropping the first term. the shift map i.
a
of symbolic dynamics:
a@)
(5) if
=
Bi =
is
This is just
for all i+l . To determine whether a given sequence occurs for a particular point, we
need to apply the lemma to the points in the orbit of x. the address of
A
In particular, if
fj(x) is i, then the inequality
must be satisfied. These inequalities can be evaluated from a knowledge of the kneading sequences of the critical points and the itinerary of .x. The first of our results is a partial converse. Theorem:
Let f: I + I be a map with turning points clr...,c
11-1'
Assume that no c contains a turning point in its itinerary. i Let 5 be a sequence of addresses which satisfies the inequalities E
(Ii)0 (f(ci-l)) < dri)0 (ajtl
) < E (Ii)0 (f(ci)
with i (j) chosen so that
Ii=a If a = C i , assume that oJ(g) is the itinerary of Ci. j' j there is a point x E I such that g is the itinerary of x. Proof: -
Then
The proof proceeds in the same way as the case for a single
critical point.
We argue that the sets L = {x 10 (x) < 0 (a)
U = 1x1 0 (x) a 0 (5))
are open.
The connectedness of the interval I implies
that there is an x with 8(x) = 0 (a), Consider the set U.
and
If x
SO
that
a
is the itinerary of x.
is in U then there is a smallest integer
such that the jth address A (x) of j is a C;t for E < j, AE(x) is not a
x
j
not a
Ao(~)r...A.(~) is open. longing to
U.
borhood of
x
'i (v)
E
It remains to check that if A.(x) = Ci,
U.
still lies in
is a neighborhood of
for which
AE ( y ) = AE (ci) for x
1
5
for
11 <
c E
which lies in
# \(ci). a j+k fi (v) of c i
j.
\(ci)
is not a E
0(W)
In particular,
U.
We conclude that
such
dIi)0 (f(ci)
such that y
5 j.
be-
we conclude that
Since
theorem is proved when the same argument is applied to Remark:
of x
V
(a)) <
Since r (Ii)0 ( o ' +
Ci.
there is a neighborhood W
neighborhood of
There is a.neighborhood
then AE(y) = AQ (x) = aQ
V,
x
then a neigh-
3
there is a smallest k
that
3
In particular, there is a neighborhood of
3
y
Since no a j' E j. If A.(x) is
E <
for
a
this implies that the set of points whose itineraries begin
=kt
that if
Ck
differs from
W
contains a
is open.
U
implies
The
L.
The theorem leaves aside an analysis of what happens when s m e
image of one critical point is also a critical point.
The theorem remains
valid in these cases, but the proof must be modified to account for the fact that a point
ci
could be an endpoint of one of the sets L
happens only when a turning point which occur for points near
c
i
c i
is periodic.
or
U.
This
The kneading sequences
can be explicitly determined in that case so.
that the conclusion of the theorem can be checked. As with the case of maps with a single critical point, we do not determine whether there are itineraries not containing a point minating with the itinerary of depends upon whether inerary.
f(ci)
f(ci).
cir
but then ter-
Whether or not such itineraries exist
lies in an interval of points with constant it-
Apart from this consideration, the theorem characterizes the it-
ineraries which occur for a given map in terms of its kneading sequences. The next question we consider is a description of the set of kneading sequences which do occur for maps having
E-1 critical points.
It follows from
the theorem we have just proved that a necessary compatibility condition is that the images of the critical points satisfy the inequalities stated there. Inductively, these provide restrictions on the way in which finite portions of kneading sequences might be extended.
Another restriction comes from the
fact that the turning points alternate between local minima and maxima. the map is increasing m 11,
If
this gives the inequalities f ( c . ) < 21
-
These requirements should be sufficient. Theorem:
Suppose we are given
11+1
sequences
q,
yitj E {I1,
...,It'
such that these sequences satisfy the coditions
C1,...,CL} (1) e(12i) <
e(12i+l)
and
(2) If Q , ~= Ikp then < - 1
k+l
(-l)k+lg (a (y*-l) 1 < (-1)
e (a(&))
Then there exists a map
f: i + I with
the ith kneading sequence of
i
(11-1) turning points so that
The assumptions we have made about the se-
are made to ensure that we can determine the order of the points
fj(y.) on the line in a manner that will be consistent with the definition 1
of the map
f.
The invariant coordinates determine an ordering of the se-
quences a3 (q) , j 2 0. map
o
y
Map these sequences into the unit interval by a
that is a homeomorphism with respect to the topology on the space
of formal p m e r series. f (4a
is
f.
We outline the proof briefly. quences y
(oJtl
) =
4 ( a 1y
Define a map )
f on the image of
4
so that
and extend it to a continuous map which is
monotonic on each of the intervals
1 (
y, y
the invariant coordinates guarantees that Such an
i f
The monotonicity of can be found.
Rather than giving a more detailed proof of this theorem, we want to investigate a family of maps which do not have isolated turning points.
Even
thou~hthe maps in the family have intervals of local maxima and minima, their topological behavior resembles maps which do have isolated turning points. Moreover, the family is universal in the sense that any set of kneading sequences allowed by the previous theorem is represented by some map of the family.
The kneading sequences arecalculableto the extent that relative
orders of bifurcations can be determined.
The construction of the family is
motivated by the symbolic dynamics of the kneading sequences. For the sake of exposition, we shall assume that the maps we consider carry the boundary of an interval into itself with a fixed point at the left end. Consider the piecewise linear map f: [0,11 + [0,11 defined by
Represent points of the interval by base n expansions. If x then the coordinate of f(x) while if x
E
E
2i+l IZi, -I 1 n
is given by dropping the first coordinate of x;
[ 2 i , ~ ] , the coordinate of x
is given by dropping the first
coordinate of x and replacing each j in the expansion by (n-1)-j. image of x
If no
is a turning point, this allows the itinerary of x to be
calculated from its base n expansion in such a way that the mth address is determined frcan the first m places in the base n expansion of x. from the behavior at the turning points, the map
Apart
f is topologically con-
jugate to the shift m m on the space of (one-sided) sequences of n symbols in a particularly nice way.
The itinerary of a point is associated with the
same sequence in the symbol space, and the itinerary bears an explicit re-
lationship to the base n expansion of the point. From the map f, we construct an
(n-1) paraneter family of maps by
slicing the graph of f horizontally near each of the turning points
- ,,Inml which are to serve as parameters,
In particular, we pick numbers p I, subject to the condition that
u 2i <
Yi+l. The map
f
f Pl
i/n.
(x)
will be
vl~...ll.'n-l
defined by truncating the graph of f by the line y = turning point
i/n.
2i
near the ith
Precisely,
(x) = I...I?J~-~
1
f (x) otherwise
See Figure 5.2. Our immediate object is to study the dynamics of the family f lJll...rlJn-l
In particular, we want to determine the kneading sequences of the maps
.
f = f For almost all choices of pl,..., these will be P ulr.. ,un n ' eventually periodic. To compute the itineraries of the turning "intervals" •
for the maps
f
, we begin computing the itineraries of the points with
?J
coordinates
pi.
In this computation, we do not consider the corners of the
graph of f as turning points unless interval on which
ui
= 0
or 1.
th f is constant (say the k )
do not, then the subsequent itinerary of
If £'(pi)
but the
lies in an
R
f (ui) 0 2 11 < j
fJ(pi) is the one followed by
p
k'
This forces condition (2) of the theorem quoted above to be true by modifying itineraries which do not satisfy it.
Given a set of itineraries satisfying
the theorem, we can find p = (pl,...,l!n) of the map
f. To do so we no'te that the map
itineraries since all sequences of The values of UlI
SO
-.pn-l
sponding map
(1,
...,n)
that these are the itineraries f has orbits with these occur as itineraries for f.
f at the points with these itineraries give the values
The hypotheses of the theorem then guarantee that the corref ? J -
has the desired properties. Condition (1) guarantees
that the intervals where the graph of f is truncated do not overlap.
Con-
dition (2) guarantees that the subsequent images of each pi do not fall into the interior of the intervals on which f is constant. are the same for f and f
so f
lJ'
.
Such trajectories
will have the desired kneading se-
lJ
quencies
The family f
lJ
allows us to determine the relationship between the bi-
furcations of various periodic orbits. Each periodic orbit of the map characterized by its itinerary. of l~ for which the maps f
u
f is
Corresponding to this itinerary is the set
have a periodic orbit with this itinerary.
n-1 Each such set is a rectangle in R
, determined by
the condition that no
point of the periodic orbit of f lies in one of the intervals on which f
v
is constant.
Thus the values of the pi at thel'smallest"vertex of this rect-
angle are determined by the distances of each turning point of f to the periodic orbit. If x is the closest point of the orbit to the turning point 2i/n,
then the value of
l~~~
1 is min(T,nlx-2i/nl).
the closest point of the orbkt to the turning point value of l
~
1 is ~ max ~ (-, +1-n 1~ x-2i/n 1 ) 2
.
Similarly, if y is (2i+l)/n, then the
The existence of a periodic orbit
for a map with (n-1) turning points with a given itinerary implies the existence of another if the "rectangle of existence" for the first contains the rectangle of existence for the second in the family f
v
.
The family f
lJ
is universal in the sense that any smooth family can be mapped continuously into f,, so that corresponding maps represent the same monotone equivalence class. Let us illustrate this procedure with a brief example. Consider maps of [0,1] which fix both endpoints and have two turning points.
Suppose we are
interested in periodic orbits of periods 3 and 5 with itineraries 323,323
..., respectively.
and 32112,
...
Do there exist two maps within this class such
that the first has the period 3 orbit but not the period 5 orbit, and the second has the period 5 orbit but not the period 3 orbit? We answer this
question by examining the piecewise linear map
f defined by
As explained above, we can find the base 3 expansions of the periodic orbits with the itineraries above. .2122121221.. of
Is
14'
These are
:::
. = - , respectively.
14' 14
and
{E 121' 121'
.210012210012...
=
11 14
The two pkriodic orbits are comprised
iS 2
121' 1211 121
I
respectively. The closest
14
values of the period 3 orbit to the two turning points are the period 3 orbit and 5 -1< - 14
45 < -121 and
1
45 121
79
and
2
-
<
79 121
111 - 12
one but not both periodic orbits. by truncating the graph of whose height is between
for the period 5 orbit.
and
Since-
, there will be two maps which have
A map with the period 3 orbit is obtained
f at the turning point
-
and l1 for 14
14 '
3
by a horizontal line
The map with the period 5 but not
the period 3 orbit is obtained by truncating the graph of point 1/3 by a horizontal line whose height is between
f at the turning
lo7 121
and
l3 ' 14
See
Figure 5.3. There is little difficulty in passing from the bifurcation properties of continuous maps to the bifurcation properties of piecewise continuous, piecewise monotone maps.
If a map
f has a finite number of jump dis-
continuities, then the structure of the set of its itineraries can still be studied by using the theory we have developed this far.
In order to do so,
imagine intervals of infinitesimallength added to the domain of definition of the function around each of the singularities. Extend the map to be con-
tinuous across each of these intervals.
The itineraries for this extended
continuous map can be studied, and the ones which occur for the discontinuous map are those whose invariant coordinates do not fall between the invariant coordinates of the left and right limits of a discontinuity.
See Figure 5.4.
Note that the extension of the discontinuous map at each discontinuity increases the number of laps for the function by
o r 1.
0
We can also incorporate the topologica1,theory of rotation numbers of homecnnorphisms of the circle into the kneading sequence analysis described above.
Recall the definition of the rotation number of an orientation-
1 preserving homeomorphism y :sl + S
r:
tinuous map that
%six
humber a
) =
y (e
a
There is a lifting of
y
to a con-
r (x+l) = I' (x) +
R + R with the properties that
1 and
Independent of x, the map 'l determines a n by the property that 7+ a as n + w The number
e 2nir (X).
.
[O,11
E
.
is called the rotation number of
through the angle
2ra
The map of 'S
has rotation number a
relative order of the points by the rotation number
y.
a.
y
n
.
which is rotation
A basic result is that the
(x) on the circle is essentially determined
~oincare'proved that the rotation number of
is rational if and only if y
y
has a periodic point.
Orientation preserving homeomorphisms of the circle are equivalent to piecewise continuous maps
f: [O,lJ -+ [O,1] with the properties that (1)
f(0) = f (1), (2) f has a single discontinuity
+ f (c) = and y
to
0,
( c , 1]
r:
and
.
c, (3) f-(c) = 1 and
(4) f is strictly increasing on the intervals
Starting with
IR + IR
y:
1 S + sl, the map
and setting
f
f (x) = r(x) (mod 1).
is defined by lifting The kneading theory of
f contains the analysis of the rotation number in terms of intervals then
rn(x)
times that
[O,c] and
[c,l] as
I1
is easily.seen to be
and
8(x) + k(n) k(n)/n.
r.
Label the
If we begin with an x c [O,11,
12.
where k(n)
I2 occurs in the first (n+l)addresses of
the rotation number is the limit of
[O,c)
Since
x. f
is the number of
This implies that
is increasing on
I1
and on 12,
the signs appearing in the invariant coordinates for the map
will all be positive.
f
This means that the-lexicographicordering of it-
ineraries gives the order of the invariant coordinates. The monotonicity of the invariant coordinates then gives us the information about the relative order of points in a trajectory. Asymptotically, all of the itineraries will be the same. Either all or none are periodic.
Since a periodic orbit of f
certainly has a periodic itinerary, the result of ~oincardis recwered. As a concluding remark for this section, we note that the kneading theory applies to maps induced on the interval by all piecewise monotone maps of the circle. The corresponding maps of the interval need only satisfy the conditions that (1) f (0) = f (1) and (2) at each discontinuity c of f,
+
f-(c) = 1 and
T
f (c) = 0.
Thus the kneading theory gives us a way of ex-
tending the topological theory of rotation numbers to all piecewise monotone maps of the circle. Newhouse, Palis, and Takens [77] take a different approach to this question.
Figure 5.1 A typical sequence of flip bifurcations
Figure 5.2
The graph of the map
f
,
,
, p2
v3
p4
Figure 5.3 Periodic orbits with itineraries 3,2,3
and
map with three laps.
3,2,1,1,2,
for a
Figure 5.4
The graph of a 1 dimensional map with discmtinuities and its extension
56 Kneading Sequences, Analysis, and Applications -
In this section, we consider bifurcation problems which involve rotation numbers and the application of the theory of kneading sequences to the examples discussed in Sections 54.
We juxtapose these two issues to emphasize
their relatianship. First, we shall resume our discussion of the van der Pol equation.
Then we consider the Hopf bifurcation theorem for maps in more
detail than in Section 92. Following this, we take up the question of bifurcations and moduli for the Lorenz attractor. The section ends with a brief discussion of some questions concerning one parameter families of maps of an interval with a single critical point.
While the hurried discussion of
kneading theory in the last section was topological and combinatorial, here we shall encounter analytic questicms involving kneading sequences, some of which are delicate. 2 Consider the forced van der Pol equation ? + ~(1-x):
+x
= Acoswt
as
a three parameter family involving the non-linearity E, the amplitude A
of
the forcing term, and the frequency w A, and w
are non-negative.
of the forcing term. We assume E,
The bifurcation structure of this three para-
meter family is inchpletely known.
A
thorough exposition of what is known
requires an extraordinary amount of the bifurcation theory of dynamical systems. This example remains an excellent test case for the study of non-linear oscillations. The theory which has been developed thus far splits neatly into the small E large
E
and large E
case in Section 54
cases. We have already discussed the
as a motivation for studying maps of the circle
which possess critical points, so let us begin here with a review of the small E
case. When
E = 0, the van der Pol equation is linear.
given explicitly by trigonometric functions. (w # 11, each solution has the form c1cost
The solutions are
In the non-resonant case A + c2sint + 7 coswt. In the 1-w
resonant case, the term which depends on w case w # 1.
becmes
A 2
tsint.
Consider the
Interpreted as trajectories in R2 x S1, the solutions lie on
tori and the motion corresponds to a translation invariant flow on each torus. If o/2n
is rational, the trajectories are all periodic, and the numerator
and denominator give the winding numbers with respect to the two generators of the fundamental group of the torus.
If w/2n
is irrational, then the re-
turn map of a cross-section is an irrational rotation of the circle. Thus the bifurcation behavior of the limiting case
E
is that of rotations of the circle. Each w
= 0
for the van der Pol equation
in an interval gives a topo-
locially different flow. If
E
persists.
> 0,
then only one of the tori which were invariant for
E
= 0
Much o r the classical theory of non-linear oscillations deals with
phenmena of this sort. There are various techniques, such as perturbation theory and the method of averaging, for analyzing the question of which tori persist when
E
is small.
Still there remains the question of describing
the flow on the invariant torus which does exist when
E
>
0.
This is a
question which involves rotation numbers in a fundamental way. Restricting our attention to the invariant tori which persist for
E
> 0
we obtain a family of non singular vector fields on T~ with
and fixing
A,
parameters
(E,w). When
number depending upon w.
E
= 0, the vectorfields are linear with rotation
Passing to return maps, the small
E
> 0 problem
is one of describing the bifurcations of a family which is a deformation of the family defined by
y (8) = 9+w
w
1 on S
.
This problem has a long history
which is closely intertwined with questions about the structure of Hamiltonian vector fields near elliptic periodic orbits.
Questions of small divisors and
number theoretic properties of the rotation numbers play a decisive role in the theory. Some of the first results in this context are due to Arnold [51.
He
established that 'analyticone parameter families obtained by fixing E small have the property that there is a set of positive measure for which the vector field will be topologically equivalent to an irrational rotation. The proof of his results is based upon a "hard" implicit function theorem. Herman I 4 9 1 has made striking progress on these questions in the past few years, removing the necessity for placing undertermined bounds on the size of
r.
In any family of diffeomrophisms of the torus with varying rotation
number, there will be a set of positive measure in the parameter space for which the diffemorphisms have irrational rotation number.
This leads to
the following picture of the structure of a one parameter family of diffeomorphisms of the circle. The rotation number varies ccntinuously as a function of the psrameter.
This function is differentiable at most
irrational values, but need not be at rational values.
The generic behavior
of families is to assume each rational value on a whole parameter interval. When the rotation number is rational there will be typically a finite number of periodic orbits, but the number of these orbits will be difficult to predict.
When the rotation number is irrational, the classical theorem
of Denjoy implies that the diffeomorphism is topologically conjugate to rotation by its rotation number.
Herman proves that there is a set of rotation
numbers of full measure for which the conjugacy is smooth. We are left then with the conclusion that for a generic family an open-dense set of parameters lead to diffeomorphisms.with rational rotation numbers.
Each of these
diffeomorphisms has a finite number of periodic orbits, stable and unstable ones alternating. The complementary set of parameter values is nonetheless large.
In particular, it has positive measure 1481.
There is another interesting way of looking at the bifurcatio:> 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 <
In-
+ f c.
The endpoints are the only fixed points of the map and they are stable. The next bifurcation occurs when
+
f-(c) = c = f (c).
The symmetry
guarantees that these two equations will be satisfied simultaneously. This
bifurcation generates a behavior which Kaplan and Yorke describe as preturbulence. Because
f' (x) +
as x + c, when
w
f-(c) > c >
with the differences small, there will be fixed points pc with p- < c < p+.
near and p
+
of
<
f-(c).
I1 = [p;,c]
Since
On the interval and
f+(c)
and p+
of f
+
f' (x) is.large near c, f (c) <
p-
[p-,p+l, f is expanding and the images
I2 = [c,p+] each cover
implies that there is an invariant set A
in
[p-,p+l completely.
[p-,p+l which is topologically
conjugate to the one-sided shift on two symbols. The set A containing all the orbits which remain in
This
is a Cantor set
[p-,p+l. In the language of
kneading sequences, the statement that it is topologically conjugate to the shift on two symbols means that there is a unique point in erary is any given sequence of
I 1
and
I*.
A
whose itin-
Yorke calls this phenomenon
preturbulence because most orbits which start in
[p-,p+] behave in an
erratic fashion as long as they remain inside the interval. After each iteration there is a small probability that an orbit will be mapped outside [p-,p+l. If it does, then it quickly tends to one of the attracting fixed points at the ends of As
-
I.
+
a is increased more, a value is reached when p- = f (c) and
p+ = f (c).
This marks a qualitative change in that the interval
[f+(c), f-(c) ]
is now mapped into itself.
The invariant set A
the whole interval and behaves like an attractor.
+
A = [f (c), f- (c)1
becomes
After the bifurcation
is the attractor and the larger interval
its region of attraction. Points outside
[p-,p+l =
[p-,p+l
is
[p-,p+l are still attracted to
the endpoints of the interval, so that the asymptotic behavior of typical trajectories depends upon the region where they start. that
+ [f (c),p]
and
[~,f-(c)} cover
itineraries of the points
+ f-(c)
[f+(c), f-(c)]
It is no longer true completely. The
form the kneading sequences which determine
the topological equivalence class of f.
In the symmetric case we are con-
sidering, these two kneading sequences are complementary.
Each of the
addresses of
f+(c)
and
+
differ unless fn(f (c)) = c = fn(f-(c)).
f-(c)
This is the region of the Lorenz attractor to which the results of 1421 apply. Notice that the attractor has lower entropy (fewer itineraries) as the physical size of the attractor grows. As I.
a
increases still more, a bifurcation occurs at the endpoints of
The points p-
and p+
approach the endpoints of
of f becomes larger than 1 there.
m3
for the flow.
I as the derivative
This represents a Hopf bifurcation in
It is a Hopf bifurcation in which an unstable periodic
orbit, represented by
p- or p+, collapses onto a stable singular point as
the singular point becomes unstable.
This Hopf bifurcation occurs after the
Lorenz attractor has formed and does not involve it directly.
The only global
effect on the Lorsnz attractor is that after the'Hopf bifurcation, the complement of its domain of attraction has measure zero. This analysis of the bifurcations of the Lorenz attractor can be extended to the non-symmetric case. family of maps
Two parameters are needed to describe a
f which encompass the behavior encountered.
It appears
that the main difference between non-symmetric maps and symmetric ones is that the bifurcation which creates preturbulent behavior does not create an invariant set conjugate to a full shift.
This one bifurcation splits into a
Cantor set of bifurcations much like those encountered in the bifurcation theory of the quadratic family.
A
complete analysis of this two parameter
family of bifurcations has not been given, but the kneading theory should contain the tools to do so. One conclusion of this analysis is that the topological type of an Lorenz attractor can be specified by two parameters which determine the kneading sequences of
8(c).
The general kneading theory allows us to
characterize which sequences occur for some map of the interval. The two dimensional family of vector fields which we obtain is a universal unfolding
for the Lorenz attractor in the weak sense that any perturbation of one member of the family will have an attractor topologically equivalent to the attractor some member of the family.
In 1421, we indicate further how these
topological equivalences extend to neighborhoods of the attractors.
This
stability of codimension 2 is manifest in spite of the variation of the topological types of the attractors within the family 138,1181. Let us end the section with a few comments about analytic questions connected with the kneading theory.
The kneading theory is topological.
It requires only that maps be continuous (or piecewise continuous), and gives no information of an analytic character. There are unanswered bifurcation questions which do depend upon the smoothness properties of maps.
Their
answers are likely to require considerably more sophisticated tools of analysis than currently exist for the kneading theory.
The situation is
superficially reminiscent of the theory of rotation numbers. There, proving that a diffeomorphism of
s1
is smoothly conjugate to a rotation
(Herman's Theorem) is much harder than proving that it is topologically conjugate to a rotation (Dejoy's Theorem), and this is harder than showing that the diffeomorphism is in the same monotone equivalence class as a rotation (~oincar6'sTheory).
There are two questions about maps with a single
critical point which we leave as challenges to the reader. The first question is based upon computations and heuristic arguments of Feigenbaum 1261.
It is suggested by the quantitative behavior Feigenbaum
observes for the sequences of flips which one encounter as successive bifurcations. He finds that the distance between successive flips decrease geometrically at a rate which is the same for all such sequences as long as the map has a non-degenerate critical point.
This behavior is incorporated
into the map which one obtains at the end of a sequence of flips. Consider a map
f: I
+
I which has a single critical point that is
non-degenerate and a single periodic orbit of period 2i for each i. is a quadratic map which has these properties.
The kneading sequence v
.
of f is 1011101010111011~~~ The sequence v i perty that if j < 2 zi-1 and 0.)
-
1, then the terms j
i+l2 1 are different.
+
.
is determined by the proi
2
agree while the terms
(Recall that itineraries start with index
The sequence formed from the odd terms of v
sequence to v
There
This implies that f and
form the complementary
fL represent the same mono-
tone equivalence class up to an orientation reversing homeanorphism of the interval.
If the map
f has no stable periodic orbits (this is true of the
quadratic example), then f and
fL will be topologically conjugate by
an orientation reversing homeomorphism h. fixed point at the critical point of c.
The conjugacy h will have a The conjecture we pose is that h
is differentiable at c with a derivative independent of which
f we con-
sider satisfying the conditions at the beginning of this paragraph. h' (c) = u
with
2i la 1 < 1, then the sequence f (c)
+
If
c at the geometric
rate a. The second question we pose concerns the measure of the bifurcation set in a one parameter family. Does it usually have measure zero? For the quadratic family, we know that if there is a stable periodic orbit, then the critical point tends to it.
In this case the derivatives of the functions
fn at the image of the critical point tend to zero at a geometric rate.
quantity
h = lim
nfor f [91].
1 n
The
i C In f'(f (cj) is called the characteristic exponent i=l
It is negative if the critical point tends to a stable periodic
orbit. Thus, one can try to measure the size of the bifurcation set by calculating characteristic exponents.
Calculations of Rob Shaw 1951 for the
quadratic family suggest that there is a set of parameter yalues of positive measure for which characteristic exponents are positive.
Jacobson has
announced results which substiante this calculation.
He asserts that there
is a set of parameters of positive measure for which the map
f has an
invariant measure, absolutely continuous with respect to Lebesgue measure. His results rely on the fact that the map be the restriction of a complex analytic map in the plane.
Are they more general? The calculation of
characteristic exponents for the quadratic family indicates how prevalent "chaos" is when numerically solving dynamical systems.
Figure 6.1
The graph of the map
g
periodic orbits
with i t s stable
fixed points
Locus of fixed points and periodic orbits
Figure 6.2
The period 3 resonant Hopf bifurcation
07. -
Population Models:
As an illustration of the use of bifurcation theory, we shall discuss briefly some dynamical systems which arise in the study of ecology.
The
basic problem we shall consider is the construction of mathematical models which reflect the abundance of animal populations. pects.of these models:
We deal with three as-
the basic balance laws for a population [80,811,
the effects on a model of competition within a species, and the interaction between different species. We then select three examples which illustrate that complicated dynamics do occur in the models and examine their bifurcations. We will not attempt to give a precise definition of a population. Think of a population as a group of animals of the same species located in a given area (for example, the area may be an island, a country, or a laboratory cage).
The population size is the number of individuals in the group,
which we assume to be a continuous rather than discrete variable.
The basic
balance law for the population states that the change in the population size is entirely due to births, deaths, and migrations of individuals into or out of the area considered.
This is not a sufficient description for our
purposes, however, because the vital parameters may depend upon other factors than the population size.
For example, birth and death rates are likely to be
age dependent, and may depend upon other factors such as size, genes, and nutritional state of individuals. Let us example a single population in the absence of migration.
To ob-
tain a self-contained model, we need to subdivide the population into subgroups with the property that the population dynamics expected for a subgroup can be explicitly calculated from the current sizes of the other populations and subgroups entering the model.
The division into subgroups
can be done either continuously or discretel9.
The most frequent variable
which is used to distinguish subgroups is age.
The equations for a single
age structured population have been extensively studied by demographers (in For a continuous age variable, the
the case that the system is linear).
basic equation can be represented either as a functional differential equation, or as a partial differential equation with boundary conditions given by an integral expression.
Denote by
individuals in a population: there are is between al and a2 at time t.
is the per capita death rate.
p
to be a function of
the age density of
J aa2n (t,a)da
individuals whose age 1 Each cohort ages at the same rate time
passes with an attrition due to death.
where
n(t,a)
This yields the equation
In a linear model, 11
is allowed
(t,a), but not of n.. This equation tells us the rate
of change in the size of the population due to already living individuals. To this equation must be appended an equation describing births:
with b(a)
the age-specific per capita birth rate.
may depend upon t
In linear models, b
as well; we shall allow it to depend upon n when we
examine a non-linear model.
The initial population distribution n(o,a)
must be specified as the initial conditions for predicting the future population size. If age is regarded as a discrete variable, it is convenient to regard time as discrete with the same basic intervals. We normalize these intervals to be
1.
The current state of the population can then be represented by a
vector N(t) = (n(t,l), of age
....n(t,k))
where n(t,i)
is the number of individuals
i at time t and k is the maximum age an individual can attain.
common convention in human demography is to take five year age classes so
A
that n(t,l)
is the number of individuals aged
number of individuals aged 5 to 9, etc.
0 to 4 ,
n(t,2) is the
The population equations can now be
expressed in a matrix form ~-(t+l= L(t)N(t)
with
Here b represents the per capita birth rate of the ith age class, and i is the survivorship of the ith age class which is defined 'to be the i
'
proportion of the individuals in the ith age class at time t which survive to time t+l L(t)
(when they then fall into the
(i+l) age class).
The matrix
is called the Leslie matrix for the population model, which we shall
call a Leslie model.
The Leslie model can be regarded as a finite difference
approximation of the continuous time and age model [go]. If a Leslie matrix L(t) = L is constant, then the predicted population dynamics can be analyzed in terms of linear algebra.
In pasticGar, the
matrix L is nor;-negative and the theorem of Frobenius [31] can be applied. The matrix L will have a single positive eigenvalue, and the absolute value of
X is the maximum absolute value among all the eigenvalues of L.
Pick an initial vector N(0) # 0 with non-negative components. Then the t direction of N(t) = L (N(o)) of h
(in some cases t will have to be taken to have the form mt' for an
integer m < k) geometric rate X.
X
will tend to the direction of the eigenvector
= 1, and
and the length of N(t)
will grow asymptotically at the
The three qualitatively different cases are
X < 1,
X > 1 corresponding to extinction for the population, approach
to equilibrium, and unbounded exponential population growth.
These pos-
sibilities do not describe the dynamics of all isolated populatims, so we shall examine nonlinear models which exhibit a wider range of asymptotic behavior. Consider the simplest sort of population experiment in which one grows an isolated'population in a laboratory under carefully controlled conditions. Such a popuhtim cannot g r m indefinitely. One of its resources, such as food or space, will ultimately limit the growth of the population.
It is
reasonable to expect that such a population will then maintain the largest equilibrium population which the available resources will support. While this is often the case, it is not universally so.
The delays inherent in
the maturation process from the time an egg is laid until the individual lays her first eggs can give rise to instabilities in the population dynamics. This behavior is clearly illustrated by the classical experiments with blowflies conducted by A.J. Nicholson during the 1950's in Australia 1791.
There
are a variety of ways of incorporating the effects of competition into the population model described above.
The one which we shall employ in a later
example makes the vital parameters of the model dependent on the totnl population.
In ecological terms, the model becomes density dependent.
Natural populations are seldom isolated from the effects of other species. They may be predators or prey or both, and they compete with each other in various ways.
The result of these interactions is that the dynamics of in-
eracting populations are coupled.
Only a system of equations which expresses
the effect of each population upon the others is likely to yield good predictions. Lotka and Volterra developed a classical theory of two interacting populations during the 1920's on the mathematical foundation of plane vector fields.
They dealt with two different ecological situations: two competing
species and a predator-prey interaction. In both situations, the differential equations describing the populations were based on the assumption that per capita growth rates were linear
98
functions of the abundance of the two species:
x
1
and
x2
being the two population sizes.
Biological assumptions led to
restrictions on the coefficients in these equations. two important conclusions.
Their analysis led to
The first was the competitive exclusion principle:
Two species utilizing exactly the same resources in an environment cannot coexist.
The other conclusion was that these models provided a satisfactory
explanation for fluctuations of predator and prey which were observed historically.
The sets of data which are usually cited came fran fur trapping
records of the Hudson Bay Company, fishing catches in the Adriatic Sea, and moose-wolf studies on Isle Royale in Mi-chigan. This is not the place to assess the ecological validity of these conclusions or even to discuss these models in detail.
We do want to stressthat
prior to 1970 there was little awareness among ecologists that simple p o p ulation models could have complicated dynamics.
The Lotka-Volterra equations
for a predator-prey interaction are integrable and all solutions are periodic. The fact that this situation is non-generic was not recognized.
The in-
tegrability of these equations prompted attempts to make "satistical mechanical" theories of population behavior based upon hamiltonian dynamics. The theory was artificial because there was no underlying biological basis for the integrability of the original equations.
Modelling population dynamics
is not easy and the limits of our ability to do so are still uncertain. Let us turn to three examples which illustrate the ease with which complicated dynamics are to be found in population models.
The first model we
consider is one in which there is density dependence and discrete generations. Think of an insect which has one generation a year.
A
common life history for
moths or butterflies is that eggs are laid in the fall and over winter in a dormant state.
In the spring they hatch and development of one generation
proceeds through the summer until the next batch of eggs is laid.
It is
reasonable to assume that a critical resource which will affect the size of the succeeding generation is the amount of food available to the larvae or caterpillars.
If per capita resources are abundant, then the mortality of
the population during development is low.
On the other hand, if too 'inany
larvae are present in the spring, few will be able to obtain the food necessary for maturation and the resulting adult population will be small.
This leads
to a simple population model of the form Nt+l = f (Nt)
the pop-
ulation size of generation t and
f: ' R l
+
lR+
with Nt
a function which is
specified. The function f cannot be arbitrary. Plausible assumptions that f should satisfy are that f(0) = 0 and that f has a single maximum.
The
first assumption asserts that if there are no individuals in the population, none will be born the next generation. The second assumption is based on the idea that competitive effects grow with population size and that these place an absolute limit on the maximum attainable population size.
Functions f
satisfying these hypotheses fall into the class we studied in the first part of Section 95.
They can have large numbers of periodic orbits, and these
periodic orbits appear in a specific order as the function is allowed to depend upon a parameter.
The model is too crude to be realistic, but it does
show that even very simple ecological systems may be capable of complicated dynamics.
The actual causes of outbreaks for many forest insects are poorly
understood, but we do not need to look far for possible mechanisms. Little is known about the asymptotic behavior of solutions to the partial differential equations for an age structured population when the co-
in [411, and the answer is complicated.
Let us sunnnarize the findings of
that numerical study.
F) must lie on the line
Equilibrium populations (fixed points of x1 = x2.
If
fixed point. 1 5
b <
.
b
is sufficiently small, then the origin will be the only 2bxe-2ax = x
This requires b >
For
-21
to have no positive solutim, or
there will be a positive equilibrium population.
Its
stability is determined by the eignvalues of the matrix
at the equilibrium x 2 = ~ 1 = 2 b x-2ax e 1. 1
When
3 ax = 3 / 2 o r b = e / 2 1
bifurcation occurs with complex eigenvalues which are cube roots of
a
1.
"Hopf bifurcation" occurs there,.but the bifurcation is a resonant one.
A
The
generic picture of this resonant bifurcation involves a loss of stability for the equilibrium without a stable invariant curve of period 3 orbit appearing near the bifurcation.
Instead there are saddle type period 3 orbits near.the
equilibrium for parameter values
-
b
3 both larger and smaller than e /2.
These period 3 orbits originate at a saddle-node bifurcation for scme value of
b
less than
furcation and
e"/2.
e3/2
In the
b
interval between the saddle-node bi-
there are two stable periodic orbits for
F.
The
asymptotic behavior of the populations then depends upon initial conditions. When the parameter
b
increases further, the stable period 3 orbit goes
through a sequence of flips generating new stable periodic orbits whose periods are 3 . ~for~ increasing values of
k.
As
k
becomes large, one can no
longer follow the details of the dynamics numerically. map
It appears that the
F has a strange attractor in the sense that the points of a typical
trajectory seem to fill out a set which looks locally like the nroduct of an interval and a Cantor set.
This set first appears to have six components
efficients are allowed to depend upcm the current population size. There are two general rules of thumb based upon a modest amount of calculation. The first is that the dynamics tend toward instability if there is a long delay in the time of first reproducti~nfor an individual. This is the smallest
a for which b(a) f 0 .
The second is that uniform reproductive rates as a
function of age tend to produce stability: to produce.unstabledynamics.
irregular functions b(a)
tend
In relation to Nicholson's experiments cited
earlier, it is interesting to note that female blowflies lay their eggs in batches of a couple hundred separated by intervals of three to four days. Irregular functions for b(a)
need not be unrealistic.
These rules of thumb have an impact when one looks at the discrete models of an age structured population.
If there are few age classes, then the
severe discretization introduces an intrinsic irregularity into the continuous version of the equations. The resulting equations lead to canplieated dynamics. On the other hand, it is far easier to visualize the dynamics of models which do have few age classes. With the warning that caution is needed in viewing such a model as typical of the continuous age dependent one, let us consider a model for a population with two age classes. Imagine a population which is divided into two age classes, young ones and old ones.
Xssume that all young ones survive to be old ones and that all
old ones die at the next time step. We assume further that both young and old are equally fecund and that the per capita birth rates are exponentially decreasing functions of the total population size.
If x(t) = (xl(t) ,x2(t)) are
the population sizes of young and old, then we assume that the population dynamics are described by the map F(xl,xZ) = (b(x +x )e-a(xl"2),~1) 1 2
which sends
the positive quadrant of the plane into itself. The parameter a can be normalized by a change of scale.
The question we want to address is now the pop-
ulation dynamics change with increasing b (the reproductive rate of individuals insmall populations). The answer to this question has been explored numerically
which then join to .form three components. As b is increased still more the three ccmponents grow in size, join, and then coallesce to a stable period 4 orbit.
This period 4 orbit goes through the same sort of sequence of bi-
furcations as the period 3 orbit, eventually yielding an apparent strange attractor with 4 components. 5 orbit.
A Q ~so on.
These join and then coallesce around a period
The reader is encouraged to do his own numerical ex-
perimentation with this example to understand its canplexity. Simple models for interacting populations can behave in a complicated fashioh as well. There is a class of vector fields which embody plausible assumptions about models for competing species. Smale [lo31 proved that any asymptotic behavior displayed by a smooth flow is exhibited by one which canes from a vector field in this restricted class.
The assumptions place no con-
straints on the qualitative properties of the dynamics which occur.
The
prod of the theorem is geometrically explicit about how one realizes a system with a given asymptotic behavior without giving actual equations. However, there are an explicit discrete models for interacting populaticns whose dynamics are canplicated.
For example, a model of a host-parasite
interaction studied numerically by Beddington, Free and Lawton [13], shows a Hopf bifurcation with stable invariant curves clearly.
The equations are
given by the following plane map with parameters a, r and K:
Let me end this section by expressing some opinions about population models of this sort. Even with laboratory populations, 'collecting good long term data is a horrendous task.
Laboratory ecosystems with several species
have a notorious habit of not surviving, and single species systems with small organisms have a tendency to evolve.
The problems of data collection mean
that questions about the asymptotic properties of population models have
limited relevance for ecology.
They may be relevant if one tries to study
evolutionary trends, but the simplest models which incorporate genetic variation are formidable.
Moreover, there has been an unfortunate tendency
in ecology for theoretical studies to ignore the data that does exist.
I
believe that further progress in understanding the.dynamics of natural p o p ulations will depend upon careful, detailed and prolonged examination of particular systems.
This is not a question of large scale "biome" studies
which try to explain everything in terms of models based upon myriad unknown parameters and functions, but one of identifying environments in which the essential influences on the population dynamics can be identified and measured.
The two age class model described above leaves one with the im-
pression that non-linear population models can be expected to behave in complicated and counter-intuitive ways.
The utility of these models is more
likely to be limited by issues other than ones of mathematical intractibility.
58. -
Bifurcations of Global Behavior The preceding sections have dealt with bifurcation behavior connected
wiyh changes in a dynamical system that occur near a single equilibrium or periodic orbit.
Here we want to briefly consider bifurcations that arise
through the tangency of stable and unstable manifolds of periodic orbits. This is a more recent area of investigation in which a number of delicate phenomena have been discovered.
We shall concern ourselves m9re with
diffeomorphisms rather than flows since the theory is.more fully developed in this case.
A m m o n feature of these results is that they involve the
analysis of how the eigenvalues of the linearization of a system at hyperbolic periodic orbits affect the dynamics along stable and unstable manifolds. We shall proceed by describing a number of examples illustrating the different phenomena which have been discovered.
Example:
Plane Loops [lo41
Let us first examine flows in the plane (or two dimensional sphere) where limit sets are simple.
Recall that a saddle point p is an equilibrium
whose linearization has one positive eigenvalue and one negative eigenvalue. There are two orbits asymptotic to p to p
as t +
4.
as t +
0,
and two orbits asymptotic
. . All of these orbits are called separatrices of p.
Together with p, the orbits asymptotic to p
as t +
(t +
smooth curve called the stable (unstable) manifold of p
.
-95)
form a
A necessary
condition for the structural stability of a plane vectorfield is that there are no saddle connections, non-equilibrium frajectories which are in both stable and unstable manifolds of saddle points.
A saddle connection can
either join a pair of distinct saddle points or it can be a tory which is asymptotic to the same equilibrium as t
.+ +w
a, a trajec.
Loops have already been encountered in our analysis of the two dimensional unfolding of an equilibrium with nilpotent linear part in Section 1 3 .
There
are aspects of the bifurcation associated with a loop which depend upon the relative size of the eigenvalues at the saddle which forms the corner of the loop. To understand this phenonemon, regard the loop as being like a periodic orbit and pick a transverse section y.
The return map will not be
defined on the loop itself, but it will be defined on one side of the loop (the side opposite to that of the other separatrices of p). passing through points of y
Trajectories
near the loop will also pass near p
the behavior of the flow near p
,.
and
is decisive in determining the qualitative
properties of the return map. Assume that the eigenvalues of the linearized vectorfield at the saddle point p are X < 0 < y
.
The trajectories near p will behave much as if The linear vector field defined by (kl,k2) =
the vector field is linear.
(Ax1, yx2) has trajectories X(t) = (xl(o)eXt curves of the function x x ' form x = E~
and x = E~
-'.
1
= a(x2) + 0 along
a (x2) + 0 along R2 a'(x 2)
+
II
2
.
attracts if y
+X
and
R2
of the
and the map a which sends a point of kl to
.
As x + 0 along R1 2
, then
The crucial point is that the rate at which
depends upon the sign of X
0 while if -1 < y
behave similarly.
which are level
Consider two lines R1
the intersection of its trajectory with R2 x
, x2 (0)epf
, then
oS(x2) -+
m
.
+
p
.
If -X > 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 <
-+
Then we assert that the
B +n > 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
<
sufficiently large, we obtain as good an
as we desire fran the relative order of the points
This argument can be applied to the sequences
{Ei)
and
{nil
that their relative order on y2 determines the quantity logu/logh.
to show The
relative order of the points will be preserved under topological conjugacies, so that quantity class of f.
logp/log)i is an invariant af the topological equivalence
Melo 1691
has investigated this situation further. He has
considered the question of whether there are additional moduli of the topological equivalence of two dimensional diffeomorphisms which are MorseSmale I821 except for the presence of one cubit of non-degenerate tangencies between the stable and unstable manifolds of a pair of saddles p and q. The answer depends upon whether there are other stable and unstable manifolds of saddles which intersect those of p and q.
This gives an example (the
first?) in which the behavior of a diffeomorphism far away from the degeneracy which causes it to be unstable has a large impact on the topological equivalence class. Example:
The Newhouse Phenomenon 1743
The final example we consider in this section is the diffeomorphism counterpart of the loop for plane vector fields. diffeomorphisms f: lR2 + lR2
Our concern will be with
having a saddle point whose stable and un-
stable manifolds have a point of tangency and with perturbations of these. Saddle points are often embedded in a larger set of homoclinic orbits [741. Examples with this property have been extensively studied by Newhouse [72]. He has used them to provide ccunter examples to a number of different genericity conjectures for dynamical systems: notably, the finiteness of the number of attracting periodic orbits of a system [72] and the manifold structure of stable and unstable sets 1741. The geometric phenomenon underlying these examples involves pr~perties of Cantor sets of the line, so we must take a brief detour to discuss the
thickness of Cantor sets. A Cantor set C
is a closed subset of a closed
interval F which has no interior and no isolated points.
The complement
of a Cantor set is a countable collection of disjoint open intervals, called the
gaps of C. We are primarily interested in conditions which guarantee
that a pair of Cantor sets have a non-empty intersection. While all of the Cantor sets we deal with have zero Lebesque measure, they nonetheless can have a thickness (defined below) which prevents two of them from being disjoint. Take F C.
Let
to be the smallest closed inte'rval containing the Cantor set
{uiJi,
be an ordering of the open intervals which comprise F-C.
-
U Ui form a defining sequence of closed sets whose iLj intersection is C. The thickness of C is defined in terms of the sequence
m e sets F = F
5
F Denote the length of an interval I by k(1). Each F is a finite j' j union of closed intervals and Fj+l is obtained from F by deleting the j interval U from one of the intervals I of F Suppose that I = j+l - j j j 1 I O U u. U I and that r = mink^.^, 1 U j Define j 3 j j 3 3 T({F.}) 3
to be the i n f ~ and the thickness T(C) j
of C to be the
13
supremum of T({F.})
taken over all defining sequences F . j' U t us examine this rather complicated definition for a particular ex3
ample.
Suppose C is constructed as a decreasing sequence of closed sets
obtained from C by deleting from each component I of j the middle interval of length a-k(1). This means that for any defining
Cj with Cj+l C j
sequence we will have
g(1.O) = R(1. 3
3
1
)
1-a
a
(T).
k(I.1 3
-
and
k(U.1 = aSk(U.). 3
3
1-a Therefore, all of the T 's are the same number 2u ' and this is the j thickness of the Cantor set. The fundamental result of Newhouse about the thickness of Cantor sets in the following:
Proposition:
Suppose C1
and C2 are two Cantor sets of the line with
the following properties: (1) C1 and C2 are contained in disjoint closed intervals (2) C1
is not contained in the closure bf a gap of C2, and C 2
is not contained in the closure of a gap of (3)
T(C1)
-
T(C2)
>
Cl,
1.
4.
Then C1 U C2 #
The proof of the proposition proceeds by picking defining sequences {F~} and
(Gi}
for Cl and C2 such that r({Fi))
then proves inductively that intFi n Gi #
>
1. m e
For i = 0, this is the
The inductive step is completed by investigating two com-
first hypothesis. ponents
4.
r((Gi})
I and J of Fi and G which intersect and such that I is i
not contained in a gap of C2 and J
is not contained in a gap of C1.
The thickness assumption then implies that if Fi+l and Gi+l are obtained by removing gaps from I and J, then the entire intersection of I and J will not be removed.
The argument 'is the following. See Figure 8.3.
least one endpoint of I or J lies in I of I.
If we delete gap G of C1
I is still in J.
n J,
At
say the right endpoint
from I, then the right hand portion of
Delete now a gap H
from J.
Since I is not contained
in H, (I-G) n (J-H) can be empty only if the right hand component I-G is in H and the left hand component JI of J-H is in G. requires that %(I2)
%&
(GI
- L'(H) J1'
to assumption.
<
< %(H)
and
~ ( J I )< %(GI
1 which implies
Therefore
r ((~~1)
of This
Then T ((~~1)<
1 contrary
(I-G) n (J-H) contains a non-trivial closed in-
terval, and the induction can 'proceed. This result on thickness of Cantor sets is used by Newhouse to prove that there are open regions in the space of diffeomorphisms of the plane
where each diffeomorphism contains a
wild hyperbolic
invariant set: one
having tangencies between its stable and unstable manifolds.
Indeed, he
proves that any diffeomrophisms for which there is a saddle having a nondegenerate tangency between its stable and unstable manifolds can be perturbed into such a region. The result shows that such a perturbation can be found
in any one parameter family which is transverse to the hypersurface of diffeomorphisms with a tangency of the sort just described. such a family has is given by the map fp(x,y) = (y,-EX maps have been studied numerically by Iienon 1471.
+
An example of
These
py (1-y)).
Earlier Newhouse had
proved the existence of a residual subset of two dimensional systems with wild hyperbolic sets which each have an infinite number of attracting periodic orbits. All of this shows that the structure of bifurcations associated with a non-degenerate tangency of stable and unstable manifolds of a saddle point is extremely complicated.
It also raises puzzling questions.
example can be regarded as a function of the two parameters When
E
=
The Henon p
and
E
.
0, the map has rank 1 and is essentially the same as the quad-
It is known
ratic family of one dimensional maps discussed in Section 55.
that a quadratic one dimensional system can have at most a single attracting periodic orbit, 1251
.
Thus, many qualitative differences in the bi-
furcation structure of the Henon example seem to exist when E
> 0.
E
= 0
and
Can these differences be resolved? If attention is focussed upon
a particular periodic orbit, nothing dramatic occurs.
Somehow there must be
many changes in the relative order of bifurcations with respect to p E
+ 0.
as
This prompts the question of how much of the universal order of bi-
furcations for one dimensional maps carries over to families of two dimensional diffeomorphisms which are perturbations of these.
One would like
to be optimistic and think that some features are robust in the families studied by Newhouse.
The three examples described in this section are hardly a comprehensive survey of bifurcation results for the global behaviors of dynamical systems. Notable omissions are the work of Newhouse-Palis [75,76] on describing the bifurcation behavior at the boundary.of the set of Morse-Smale systems, the work of Newhouse-Palis-Takens 1771 on continuity properties of topological equivalence and families and the work of ~il'nikovon saddle-foci 1981, I appologize to the reader for not doing so and meekly refer him to the bibliography. pidly.
The bifurcation theory of dynamical systems is growing ra-
An attempt to capture a canplete picture of its current state would
be transitory, but I have tried to make the bibliography comprehensive.
graph
a
phase portrait
?J
phase portrait
0
phase portrait - p+
?J-
Figure 8.1 The unfolding of a saddle loop in
IRL
Figure 8.2
A saddle connection for a dif feomorphism of
lRL
Figure 8 . 3
The gaps of a Cantor .set
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CENTRO INTERNAZIONALE MATEMATICO ESTIVO
( c . I . M . E .)
HORSESHOES FOR CONTINUOUS MAPPINGS OF AN INTERVAL
Me MISIUREWICZ
Horseshoes f o r continuous mappings o f an i n t e r v a l by 36. Misiurewicz
Let f :X
be a compact Hausdorff space and l e t
X
---+X
be a continuous mapping. Let u s r e c a l l t h e
n o t i o n of t o p o l o g i c a l entroyy. For a cover put
h(f,A)
N ( A ) = n i n {card B : B
= lim
1
n+-
of
i s a subcover of
A]
we
X
,
( t h e l i a i t zlways e x i s t s and i s
log N ( A ~ )
n o t g r e a t e r ttan
A
l o g E(A)
1.
The t o p o l o g i c a l e n t r o p y of.
i s defined as (f) =
3
{ h ( f : A
i s an open cover of
i}
The main p r o p e r t i e s of t o p o l o g i c s l entropy a r e a s ' 1 ) If the Ciagram-
Y
2 X
compact ZausZorfi znd
g
----+ 1 commutes ( h e r e
f g
, f ,t
X
,Y
are
a r e continuous), then
f
.
a) if t is a surjection, then
h(f) ,< h(g)
,
b) if t is an injection, then
h(f) 2 h(g)
,
c) if t is a bijection, then
h ( f ) = h(g)
2)
.
h(fn) = n*h(f) Mow let
I
.
f : I --+ I
be a closed interval and let
be a continuous mapping. It was shown in
that if
is
f
also piecewise monotone, then we can observe a phenomenon which is (provided
h(f)70
) similar to the Smale's
horseshoe effect. Here we obtain a similar result without an assumption of piecewise mo~otonicity: Theorem.
Let
f : I --+I
be a continuous mapping
of a closed interval into itself. Then there exist sequences: (1) (ii)
(~n)n?-l
I:),@
of subintervals of
I
,
of ?artitions of corresponding
Jn' s
into subintervals, (iii)
(kn)E1
of positive integers,
such that: (a) (b)
lim L log Card Dn = h(f) , n+- kn fkn (d)D Jn for any d €Dn
Our Theorem has some important corollaries:
1.
Corollary 1. (cf. Theorem 5 of entropy, regarded as a function
h
:
~'(1,I)
The topological
----+ ~
-0) ,
+ {+ v
is lower semi-continuous. Proof.
Similarly as for the classical horseshoe, the
existence of an interval
J1,
and its partition
into
s u b i n t e r v a l s such t h a t (b) h o l d s , i m p l i e s h(f) )
?
If
N
r D~ Dn
? yu
f o r some i n t e r v a l
N
and
is sufficiently close t o k
i n ~ ~ - t o ~ o then l o ~a ~l s o,
f d
.
l o g Card Dn
(@) 27,
s l i g h t l y s h o r t e r than
Jn
with c a r d i n a l i t y g r e a t e r o r e q u a l than . l i m i n f h ( g ) 3 l i m 1 l o g (Card Dn l3-f n+- n
Thus,
( c f . C o r o l l a r y 2 of
Corollary 2 l i m sup n +eo
for a l l
$ l o g Card
{=,I
:
-
13)
Card Dn -2.
2) =
).
.
I]>, 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 ,<
,t
z
and
( i f it i s p o s s i b l e , we take
f(t) = y
from
x
is an i n t e r v a l . Let
i n t e r v a l w i t h endpoints
c l o s e d , open, half-open)
t'
and
z '
•
F u r t h e r c o n s i d e r a t i o n s a r e similar t o t h o s e from [3] Let
be a f i n i t e p a r t i t i o n of
A
For any i n t e r v a l
J
~ a r d ( a € ~ a : nJ#g
Let
E
(a t A : l i m sup n 3-
Clearly, t h e family fmllies
2'
En
a
€
E
E
(provided c(e,a)
f( c ( e , a ) )
I
$ l o g Card
i s non-empty. I
f( e ) n a #
( ~ ~ 1 , =) h ( f , ~ ) ]
(n =. 1 , 2 , .
e
..
:
6 En
and
choose some o(e,a) c e
and
( i t i s p o s s i b l e by Lemma 1).
i n ( e ) f la :
e &En
,
aEE
, fn(e)na
o ( e , a ) i s c o n t a i n e d i n some element of
and i n a n y (non-empty) a unique element o f
element of En+=
E~"
. .
We d e f i n e by i n d u c t i o n
$
such t h a t
En+1 = {c(e,a)
n o t i c e that any
a \ ~ f $ ] , C2
fs a l r e a d y d e f i n e d , then f o r
interval
Put
and
of s u b i n t e r v a l s of
En
If
into intervals.
we have:
(1)
x
I
it i s contained
.
We shall a l s o u s e t h e f o l l o w i n g lemma:
#
$1.
' 9
lemma 2.
~ e t ( O L ~ ) ~ aTn 8~
be two
( tn1n20
sequences. of r e a l non-negative numbers and l e t
Then
(k-0n xexp
limsup$log 11400
Let us f i x a n a r b i t r a r y
Proof.
e x i s t two numbers
-nP
OCn
and
k E (0,.
and hence
l i m sup n -00
cLk
. 1f
( dk
+
(ps
n
3p
lim sup ;1l o g n -+a* Proof.
-
+ nu))
1 lh sup ;;logcard n- = ,
(E,I
d o=
Denote: I
l o g Card (En
la
)
.
u
n
2'
k-0
3 0 0
Card (En
T
la
= 1
u
>t
Since'
I For any
aCE
we
. , we
have
.
,
?Q= l o g
is
exp ( d
) 4 Card
) (h(f,~)
to
a
p
1)
Card (En ) = h ( f , ~ ) la
Since
n-k
. Therefore
< p s + nu
Lemma 3. ( o f . Lemma 2 of C3] ). have
,m
n32p
tpk <
l i m sup 1 l o g (
a r b i t r a r y , we g e t
,
n
k
men t h e r e
such t h a t :
foran3
(n-p)u
+$
l o g (n+l)
. p
,( p s
kt0
>t
n
then e i t h e r
n
u
forall
. +tWk+
..,n j
)),
>0
p
and
$ p n < a
$ log ( C e m
,( l i m (
new
)/ u
and
-n1 % , < a and
8
pn-k
(dk
g n = l o g Card ( P I a ) = (
G
btA\E
Card
(~~1~)) T
n = 2
,
. It i s easy t o s e e t h a t
.
s m a l l e s t number f o r which the image of a given element of
~~l~
under
For any
i s contained i n a ' n element of
r e have
b e A\ E
< h ( f , ~ ) , a n d hence the d e f i n i t i o n of 1 i m sup n 3 -
l i m sup n 3
l i m sup
n+m
l i m sup
n 4-
+ xn3 ~
Lim sup
(2)
n
Proof.
p
.
> log 3
h(i.~)
implies
,b
Lem 3
.
E E
From
$ log
thus, by Lema 2,
:
Assume t h a t
1.
Then t h e r e e x i s t s an
such t h a t
. a 6 E
r(ao,ao,n) = h ( i , ~ )
.
3 0 4 4
a E E
.
and a r e a l number
Suppose t h a t t h e r e
log 3
Card (E,+~I,)
t h a t then
.
.E a
Fix a s e t
such t h a t
( A ~b)) <
h(f,A)
, and
l o g Card ( ~ ~ 1 , =) h ( f ,A)
Lemma 4 (ef. Lemma 3 of [3]
such t h a t
;f n<
).
it follows t h a t
E
Denote f o r a r b i t r a r y
h(f,A)
$ l o g Card
A\B
l i m sup n3 0 0
<3
Card (E,
$ l o g Card
Therefore we have:
la
(En\ ) a
)
\C u
.
n exf sts
It i s easy t o s e e
, which
contradicts
(3)
f o r every number n )p
p
such that
c a r d (En+1
),1
t h e r e e x i s t s an i n t e g e r 1 ;, l o g Card (Enl ) u and
) 3 Card (En
>
8
I1 a
f ( e ) i s an i n t e r v a l , and Enla t h e r e f o r e i n view o f (1) i f it meets r elements o f E , Pix a Bet
t h e n it c o n t a i n s a t l e a s t t h e d e f i n i t i o n of Hence,
En+l
card(b€E:
b6 E
-2
o f them. But i n view of
(Ell+l
r = Card
we have
P ( e ) ~ b ]2, Card
e 6 Enla
Summing over
r
e
)
le)
-2
.
we o b t a i n :
r ( a , b , n ) ) Card (En+l
la)
-
(En\a'
I n view o f (3) we conclude t h a t l i m sup n 3 -
$ log
(
b €E
b e chosen a r b i t r a r i l y c l o s e t o l i m sup n Since
E
)o : E +E f o r any
C
log (
be E
.
u
, and
h(f,A)
>/ 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
<
=
Yi= 0 where the dF j
are l i n e a r l y dependent. ) On the other hand one can show t h a t i n general, even i f the q,a2,...,an are rationally independent
t h e system is not infegrable i n any neighborhood
of the origin. (c) *e
is p a r t i c u l a r l y simple. Given
s t r u c t u r e of the integrable system
the i n t e g r a l s one considers the manifolds Nn defined by
with appropriate constants
clrc2,
...,cn.
These manifolds are invariant
not
,
only under X
H
(because of (i) of definition 1) but also under XF
.
.
, ..,X
-J
(because
Since these span t h e tangent space of Nn. Fn vector f i e l d s comute each component of N is topologically a cylinder n of ( i i ) ) Thus XF ,Xp2 1
--
and i n case i t i s compact, a torus.
Thus i n the l a t t e r case D i s f o l i a t e d by
n-dimensional t o r i . By a theorem due t o Arnold [ I ] , [21 and J o s t e n t of Nn H = H (yl
one can near a compact compon-
introduce canonical coordinates, called x , y again, such t h a t
,...,yn) . t h a t
y = 0 corresponds t o Nn and t h a t points (x,y) , (G,y)
with the i n t e g e r (x 2 . ) (271)-1 correspond t o the same points i n D. The j 3 yk, 5 a r e called the "action-angle" variables, respectively. In other words,
-
the example 2
m
is typical.
Example 1 these t o r i are given by
The flaws generated by the commuting XF
52 + y2k = % i f
,...,X .
the c a r e positive. k
are the n rotations i n
Fn
the s t y k plane. In action-angle variables the d i f f e r e n t i a l equations become
Thus the d i f f e r e n t i a l equations a r e linear on No,
If the frequencies
are rationally independent then t h e o r b i t s are dense on Nn. Yn I f one solution on N i s periodic then a l l are. This occurs i f and only i f n
H ,...,Xi Y1
H /jk = p is independent of k with some integers jl,j2, ...,jn. Thus f o r an Yk integrable system the periodic Solutions forma-1 dimensional families. The proof t h a t Hamiltonian systems generally a r e ?tot integrable i s based on the f a c t t h a t generically the periodic solutions on a fixed energy surface are isolated. References [I]
V. I. Arnold and A. Avez, ~roblzmesErgodiquies de l a ~e'canique
Classique, [2]
Gauthier-Villars, P a r i s , 1967.
V. I. Arnold,
(Russian).
Mathematical Methods i n Classical Mechanics, Moscow, 1974
English translation to appear.
[31 C. L. Siege1 and J. Moser, Lectures on C e l e s t i a l Mechanics, Springer,
1971. [41 J. Moser, Stable and random motions i n dynamical systems, Ann. Math. Studies, 77, 1973. Examples of Integrable Systems, Isospectral
2.
I n s p i t e of t h e i r exceptional character a n-er
Deformations
of integrable Hamiltonian
s y s t e m have been discovered recently for.which the underlying symmetries are highly unexpected.
The most i n t e r e s t i n g ones are given by p a r t i a l differen-
t i a l equations and hence of i n f i n i t e degrees of freedom.
For lack of space
they w i l l not be discussed here, but some of the following systems can be viewed as discretized versions of those of i n f i n i t e degrees of freedom. (a) Toda l a t t i c e . XlrX2t
..
tX
n
Consider n mass points on the l i n e with coordinates
and s a t i s f y i n g the d i f f e r e n t i a l equations
where
This system possesses n integrals i n involution F1,F ZI...,F rational i n yk = then by Flaschka. form:
e
Xk
constants
;rk and e ~ ~ i $ - ~ + ~This ) .
n
which are
f a c t was discovered by Henon and
The solutions of t h i s system can be expressed in rational
are rational functions of e
,...e Int
with some distinct
\.
The original Toda l a t t i c e refers t o infinitely many particles, but we w i l l r e s t r i c t ourselves to n < m (see [1]- 143). (b) The inverse square potential of Calogero is also given by (1)but (2)
i s replaced by
This system possesses n rational integrals i n involution 153 (see Section 4 ) . The solutions ~ ( tare ) algebraic functions of t, i n f a c t they are the eigenvalues of a matrix depending linearly on t. (c) The oldest example
with different axis.
ellipsoid i n Section 6).
of t h i s type is the geodesic flow on an n dimensional This f a c t was discovered by Jacobi (see
He showed, moreover, t h a t a particle mving under the influence
of the radial
force ax on the ellipsoid i s integrable.
geodesic flow as a special case.
For a = 0 one gets the
The solutions are expressible i n ternts of
hyperelliptic fmctions. (dl A related problem is t h a t of the motion of a mass point on the n dimensional sphere 1x1 = 1, x E R ~ + ' V(x)
under the influence of a quadratic potential
with d i s t i n c t eigenvalues. The solutions are also expressible i n terms of
hyperelliptic functions (Section 7). How Qes one recognize the integrable character of these systems and the underlying hidden symmetries?
There is no systematic approach.
different reasons underlying the existence of the integrals.
W e w i l l find
In case
(a) and
(b) the integrals a r e
found as eigenvalues
of some classes of matrix, and
t h e . d i f f e r e n t i a 1 equations correspond t o deformations of these matrices leaving t h e spectrum fixed (isospectral
deformations).
In case (b) we w i l l see t h a t coadjoint
rep&sentation of the unitary group
i s the ultimate reason for the symmetries (see also [ l o ] ) .
In cases ( c ) and (d) we w i l l use a geometrical f a c t about confocal quadrics t o find rational integrals i n involution. ing:
I f a l i n e touches n confocal quadrics i n
then the normals of the quadrics lar
The geometrical f a c t is the follow-
.
a t points Pl,PZ,.
a t these points P are mutually perpendicuj
I t i s l i k e l y t h a t a conumn reason underlies a l l these phenomena.
ment i n t h i s direction i s t h a t a l l these examples de Vries equation.
...Pn One argu-
are related t o the Korteweg-
For (a) and (b) t h i s i s w e l l known [6-91 b u t we w i l l show
such a connection of (d) with H i l l ' s equation
(which i s closely related t o
the Korteweg-de Vries equation) i n Section 8.
This observation seems new.
We conclude t h i s section with a description of the irrteg-rals f o r problem ( a ) , following Flaschka.
He constructed the Jacobi matrices
He showed t h a t the d i f f e r e n t i a l equations (11, (2) are equivalent t o the system d L= dt
[B,L]
i f we s e t
From t h i s one deduces L3at the eigenvalues of L a r e out t o be i n involution. function o r
t h e traces
integrals
-- which
Thus any functions of t k s e , e.g. a symmetric
turn
tr L'
,
p=1,2,
...n
are also i n t e g r a l s i n involution. A similar construction succeeds i n (b)
b u t we w i l l describe a d i f f e r e n t
approach i n Section 4. References M. Toda, Wave propagation i n
a n h a ~ w n i cl a t t i c e s , Jour. Phys. Soc. Japan
23 (1967)
501-506.
M. Henon,
Phys. Rev. B9, 1974, 1921-1923
H. Flaschka,
The Toda l a t t i c e I, Phys. Rev. B9, (1974) 1924-1925.
J. Moser, F i n i t e l y many mass points on t h e l i n e under the influence of an
exponential p o t e n t i a l
-- An
integrable system,
Lecture Notes i n Physics
38, Springer, 1975, 467-497. J. Moser, Three integrable Hamiltonian systems connected with i s o s p e c t r a l
deformation, Adv. Math. 16 (1975)
197-220.
H. A i r a u l t , H. P. McKean and J. Moser, Rational and e l l i p t i c solutions of
the Korteweg-de Vries equation and a r e l a t e d many body problem, C o p . Pure Appl. Math. 30 (1977)
95-148.
M. Adler and J. Moser, On a c l a s s of polynomials connected with the
Korteweg de Vries equation, Comm. Math. Phys. (1978) 1-30. F. Calogero,
Motion of poles and zeroes of s p e c i a l solutions of non-
l i n e a r and l i n e a r d i f f e r e n t i a l equations and r e l a t e d "solvable" many-body problems, p r e p r i n t , Univ. di Roma, 1977. D. V. Choodnovsky, and G. V. Chodnovsky,
p a r t i a l d i f f e r e n t i a l equations, I1
NWVO
Pole expansions of nonlinear Cimento, 40 B, 2, 1977.
M. A. Olshanetsky and A. M. Perelomov, Completely integrable Hanultonian
systems connected with semisimple
Lie algebras, Inv. Math. 37 (1976)
1111 F. Calogero, Exactly solvable one-dimensional many body problems, Lettres
a1 Nuovo Cimento, 13, No. 11 (1975) 3.
411-416.
Reduction of a Hamiltonian System with Synnnetries
(a) Integrals of a system are i n close r e l a t i o n t o invariance groups of a system,
i s Abelian
also
For integrable systems the relevant group action
a5 i s well known.
-- b u t
following we w i l l study more generally arbitrary
i n the
noncommutative group actions.
--
A good example i s the rotation group
i n R3 x R3
invariant under the rotation
SO(3)
and a Hamiltonian system
group
can be reduced t o a system i n the r a d i a l variable only, i.e.
to a
system of one degree of freedom. How t h i s reduction can be effected is the topic of t h i s section.
- .
excellent presentations of t h i s reduction, see [ l ] 141 the approach informally
without
proof
There are
Therefore we describe
and s t r e s s simple examples i l l u s t r a -
t i n g the approach. 2n Example 1: Let H(x,y) be a Hamiltonian i n R
+
t i o n xk +
S,
yk
+
, i.e.
yk
invariant under the transla-
satisfying k=l
Then, clearly
a
Xk
8 - 0 .
i s an i n t e g r a l generating t h e above group action and
This system can be reduced t o a system of n-1 degrees of freedom, e ,g., by introducing 'relative
coordinates :'
E&=%-xn '
F., =
Xn
,
'lksYk 11, = Yl + Y2 +
for
k = 1,2,. ..,n-1;
.-.
+ Yn
This i s a canonical transformation and the integral becomes TIn new Hamiltonian
, so
t h a t the
i s independent of
5n'
Thus i f we f i x On = c the system
can be solved f o r qk = c.
4 - r "k
, ik=-rG
(k 5 n-1)
Afterwards one can solve the equation f o r
Thus one i n t e g r a l allows us to reduce the phase-space by 2 dimensions: One dimension is l o s t by fixing the i n t e g r a l F = c and the second by ignoring the variable
En
= xn along the o r b i t of the group action.
This example i s rather t y p i c a l f o r integrals i n involution: I f a system admits
r i n t e g r a l s i n involution, one can reduce t h e system by 2r dimensions
t o a system of n-.r degrees of freedom.
However, i f t h e i n t e g r a l s are not i n
involution the reduction i s more ccunplicated. Example 2:
Let
H(x,y) =
1 ly 2
1 + V( 1x1)
be a Hamiltonian i n R
6
invariant
under the orthogonal group SO (3) : x + ~ x , y + ~ ywhere
RESO(3)
.
This group i s three dimensional and generated by the vector f i e l d s
where
The vector f i e l d s a r e Hamiltonian with the Hamiltonian
where a = (al,a2 ,a3) an'd x A y
i s the vector product of x and y. Thus the
components
are i n t e g r a l s of the motion, defining the angular momentun vector. Since the number of i n t e g r a l s is 3,
one may expect t h a t one can reduce the
phase space by 6 dimensions but, i n f a c t , i n t h i s case the integrals are not i n involution and the reduced phase space i s of 2 dimensions. follows:
One proceeds as
We f i x the angular momentum vector to, say, xAY=p
where p # 0.
We may assume t h a t p = Ae
3
>,
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 =
Taking p = G = R.
line x
+
1
sy
I t has the Hamiltonian
-1 the manifold $ (p) i s the u n i t sphere and the isotropy group i s
-1(u)/G
To describe $
+
(x + SY,Y)
to lie at s $-~(F()/G
we choose the point
3
0, i.e. we choose *,y>
=
{ ( 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
<
X
2
<
...<
X
'
n
potential
-
We consider the Hamiltonian system with the Hamiltonian
This system i s integrable and the solutions are algebraic functions i n a sense t o be defined. We write the d i f f e r e n t i a l equations i n the form
. ..
\=
Yk
yk=% (1)
= - b U
<=
2b
2
5
1
- a \
-3
(\-xj)
jZk
Using an interesting observation due t o Perel-v
2
- a % .
(*Ione can reduce t h i s system
t o the case a = 0, b = 1 by the following elementary transformation.
\ (t) =
(2)
c cos Cat)
3 (T)
where
Indeed, one finds 2 \ + a \ = where
'
c a
T = tan(at) 2
(cos a t )
3
G
denotes d i f f e r e n t i a t i o n with respect t o T.
(*I I m e t h i s information t o Calogero.
.
5 satisfies
Hence, i f
2 % + a x
a
the above equation with a = 0, b = 1 we have c a
=
.L
at)
(COS
3
2
1
(%-xj)
-3
=
4 2 c a 2 1 (\-xj)
-3
jfk
j#k
Thus with b = a2c4 the transformation (2) takes the system (1) i n t o t h a t with a = 0, b = 1. This may lead t o complex values of c
-- which
need not disturb
us since a l l solutions have complex continuation. (b) In the following we w i l l r e s t r i c t ourselves to the case a = 0, b = 1 and s h w t h a t t h i s system possesses n i n t e g r a l s in.involution, and t h a t f o r the solutions \(t) are algebraic functions of t. To formulate the r e s u l t s it i s good t o introduce the matrix
where zk = (xk
-
L(x,Y) = diag (yl,...,yk) + i(zkR) -i xR) f o r k # R , zkk = 0, and the 5 always d i s t i n c t .
We w i l l prove the. THEOREM.
The Hamiltonian system (1) with a = 0, b = 0 has the n rational
integrals
which are an involution.
The solutions
5
= ~ ( ta r) e the n d i s t i n c t eigen-
values of the matrix diag (x, (01
(5)
... a(01) + t L(X(O)
(01)
I
hence dlgebraic functions. COFOLLARY.
The symmetric functions
are polynomials of degree p i n t.
Hence, using the transformation ( 2 ) one finds
finds t h a t f o r a r b i t r a r y a,b the above expression is a trigonometric nomial i n e
fiat
of degree p.
periodic of period 2ra
a l l solutions are
-1
.
(c) To prove these r e s u l t s
group U(n).
I n p a r t i c u l a r , for a > 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<
c2< ...<
on , 'l
Notice t h a t t h i s mapping i s a
... <
xn
,y
cn3 onto
E
en}.
I t takes our Harniltonian
1 y - tr
2 L (x,y) i n t q
1 t r K~ (5) =
k2
I.
k=l and therefore t h e d i f f e r e n t i a l equations i n t o the l i n e a r ones
5
.
= 0, 'l =
-5.
(g) Scattering map. I f we observe t h a t i n our problem t h e p a r t i c l e s repel each other
one sees t h a t the solution runs a p a r t and has asymptotic behavior,
+
~ ( t= ) %t
(12)
yk(t) where the.
''%
are d i s t i n c t .
B* + o ( t - 5
+ O ( t
for
..,an a r e t h e
+
03
1
I f we i n s e r t t h i s estimate i n L (x (t),y ( t )) and
r e c a l l t h a t i t s eigenvalues a r e independent of t
?,a2,.
t +
-1
then we see t h a t
eigenvalues of L ( x ( t ) , y ( t ) ) , i n p a r t i c u l a r of L(x,y)
where x = x(O), y = ~ ( 0 ) . Thus i n (11) we can i d e n t i f y the
5, 3
with the
asymptotic v e l o c i t i e s y . ( ~ ) Similarly, . by i n s e r t i n g the flows i n the second 3
equation of (11) we g e t
from which we read off . t h a t -'l. = B 3
j'
Thus with the mapping (x,y)
-t
(5,q)
defined by (11) we have for t where %,yk
-+
-
+m
are the i n i t i a l values. Thus (11) is the s c a t t e r i n g mapping of
t h e i n i t i a l values (x,y) For t
+
03
i n t o the asymptotic v e l o c i t i e s
5
and.phases -0.
one obtains similarly asymptotic formulas, say
%
=
-
<+
~(t-l)
for t
-+
-
m.
Then it follows a t once from the r e v e r s i b i l i t y of the system t h a t
Sk
= Lk+l
'
'lk = 'l;-k+l
showing t h a t the s c a t t e r i n g of t h i s system is t h e same a s f o r e l a s t i c reflections.
Exercise 1. Show t h a t f o r any 2 function f j = 12
2 2 t r f . (X +Y ) 3
are i n involution.
.
The same holds for6 tr f (X Y ) 3
Exercise 2.
the functions
j
.
Show t h a t under the above reduction the Hamiltonian
goes over i n t o the Hamiltonian
(5- x,) -2 k<j The tiw dependent acanonical transformation ( ~ , x , Y ) -+ (T,x,Y) 1
T
Exercise 3.
*
x
-
1
1Y12
2 2 a 1x1
= cos T X(tan r) -1
Y = ( c o s , ~ ) Y(tan T)
takes the Hamiltonian system with Hint:
_-
+
Ho =
- s i n T X(tan T)
T1 t r
Y
2
1 -2 - 2 i n t o H = , - t r (Y + X ) 2
.
Note t h a t the solutions ~ ( t= )X ( O ) + ~ Y ( o ) , ~ ( t= )Y ( O ) go i n t o
Exercise 4.
=
COS T
Z(O) +
sin T
Y(o)
=
- sin T
Z(O) +
cos T
?(o)
The above transformation of Exercise 3 s a t i s f i e s [:,?I
= [X,Yl
Therefore reduction of the systems v i a (8), (9)
gives r i s e t o the transforma-
t i o n (2) f o r a = 1 of Perelornov. (Exercises 2, 3, 4 are personal conrmunications by M. Adler.) References [l]
D. Kazhdan, B. Kostant and S. Sternberg, Hamiltonian group actions and
dynamical systems of Calogero type, t o appear, Corn. Pure Appl. Math 1978. [2]
M. Adler, Completely Integrable Systems and Symplectic Actions, MRC Report
Report # 1830, Univ. Wisconsin, 1978. 131 M. Adler, On a trace functional f o r formal pseudo-differential operators and t h e grrmplectic s t r u c t u r e of the Korteweg-de Vries equation, preprint, t o be published.
[41
F. Calogero, Solutions of the one dimensional n-body problems with
quadratic and/or inversely quadratic.pair potentials, Jour. Math. Phys.
151 J. Moser,
Three integrable Hamiltonian systems
isospectral deformations, 5.
connected with iso-
Adv. Math. 16, 1975, 197-220.
Extension of the Geodesic Flow
(a) The geodesic flow o n . a sphere
sn: 1x1 = 1
where x = (xo,xl,.
..,xn
) E R
~
+i s ~described by the d i f f e r e n t i a l equation
x=xx;
is determined such t h a t 1x1 = 1 i s compatible
here the Lagrange parameter
with the d i f f e r e n t i a l equation, i . e . < ~ , i > = 0 , < a , ; >+
1;12
= 0.
Hence
taking the inner product with x gives
x=-
1;j2
and the above equation becomes
;;=
(1)
-
1;l2X
.
This system can be considered as a Hamiltonian system
-
x = aH ay
(2
y = - -a H
ax
with H =
(2')
1 1 x 1 ly12 ~ 2
t o be r e s t r i c t e d t o the tangent bundle 1x1 = 1, Q,y> = 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 (<) = f (x the p o i n t of tangency of the l i n e x
+
+
s ( x y )y) = 0 and
5
is
t y t o f = 0. This proves the assertion.
Thus we can visualize the solutions of (4) on H = 0 as t h e motion of lines: The l i n e s move i n such a away a s t o be tangent t o one and the same geodesic on f = 0.
The distinghished
t h i s l i n e i n t h i s process.
p o i n t x on the l i n e moves perpendicular t o W e w i l l r e f e r t o t h i s flow a s the " l i n e flow"
associated with f ( x ) = 0. (c) This shows t h a t the system (4) i s r e l a t e d t o the geodesic flow on f = 0.
We make t h i s more formal and consider the submanifold of R
(6)
I
x,yER
M =
2n+2
;
ly(
2
2n+2
01
given by
=l,
=
This i s a symplectic submanifold, a s i n general any submanifold given by F 1 = 0 , F 2 = 0 ,
(7)
F2r = 0
i s symplectic provided det { F ,F ) # 0 k j where { ) are the Poisson brackets. I n our case (8)
{lY12,
(X)Y}
=
- *C 0
since we assumed f t o be s t r i c t l y convex.
We r e s t r i c t H t o t h i s manifold and
M denote t h i s r e s t r i c t i o n by H ~ . Then H defines a Hamiltonian vector f i e l d X
M
tangential t o M. W e consider i n general the question of describing the Hamiltonian vector f i e l d XM on a symplectic nanifold M defined by (7) and the a function H t o M.
r e s t r i c t i o n H~ of
This fa:ztor f i e l d can be obtained from a Hamiltonian H
the embedding space which on M agrees with H, but generally has d i f f e r e n t
0
in
derivatives.
Indeed the vector f i e l d defined by H need not be tangential t o M.
We s e t
where the A,
I-'
are determined such t h a t
o
(10)
= {HO,F~} =
{~,d - I AJ 2r
which assures t h a t the flow defined by H
F~,F,,I P=l i s tangential t o M. This vector
f i e l d agrees with XM. We apply this
and the
approach t o the syoaplectic manifold
Hamiltonian H = F(x,y) = min f (x
and since { K , F ~=I 0 one has h2 = 0.
+
ty)
.
Then
Moreover
The flow XM given by
r e s t r i c t e d t o HO = 0 can be simplified by introducing a new parameter T b y
Then the d i f f e r e n t i a l equations become dx
dT
-
52 = A-lHo d~
On the energy surface HO = 0
1 x
we may replace the Hamiltonian simply by
K=-A;$O=-A
-1 1B + F ~
We observe t h a t on M we have min f ( x + t y ) = f ( x ) as < fxIy> = 0, so t h a t t Hx(xIy) = fx(x) On M Hy (x,Y) = 0 I
.
Thus
K (x,y) =
Y
and the d i f f e r e n t i a l equations f o r
a F =Y ay 1
%
on M
reduce t o
which again are the d i f f e r e n t i a l equations f o r the geodesics.
Hence, we have sham that the geodesic f t m on f
= 0
i s obtained from
res.tricting the EamiZtonian (3) t o the synp7Zectic submdfozd
(4)
given by
M,
by (6).
This hoZds up t o repm?ametrizatim of t. I f f (x) =
(d) Example. n
sphere S
.
1
- 1) then we a r e dealing with
(1x1
the unit
We f i n d 2
~ ( x , y )= min f ( x + t y ) =
- l1
Next we r e s t r i c t F t o
so that FM = 2 [lX121y12
and f i n a l l y , s i n c e
~ h u son FO =
E $, 1 y 12)
I$,<X,~>}
= 0,
- l]
= 0 on M we have
o FO=
;a
;= -F:
=
I X I 2Y = Y - 1 ~ =1 -x~ ~
d e f i n e s t h e geodesics on t h e sphere. ( e ) We observe t h a t t h e manifold (6) is obtained by "reducing t h e phase space ~ by~ t h e+ i n~t e g r a l F1 =
R
T1 ( 1y 1 21
( i n t h e sense of S e c t i o n 3 )
. Indeed Fl
i s an i n t e g r a l which belongs t o t h e one dimensional symplectic group a c t i o n (x,y)
-+
(X + ty, Y)
,
Thus $ = F is t h e moment map 1 2n+2 $-'(o)={x,yER , Iyl=l}.
with t h e group G being R. (11) The group Go l e a v i n g
F
1
= 0 i n v a r i a n t i s e v i d e n t l y t h e whole group G = R.
-1 To form t h e q u o t i e n t manifold $ (0)/Go we s i n g l e o u t t h e p o i n t x on t h e l i n e
x
+
t y f o r which
< f X Y, >
= 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.
>
-<4
Y
fx(E)
$ y ( S , ~ )= t*fx(S)
Hence
,J' > X
= (s*- t*) = 0
Geodesics on an Ellipsoid
(a) I t was Jacobi who established the geodesic flow on an e l l i p s o i d as .an integrable one.
He used separation of variables on the Hamiltonian Jacobi This deri-
equation a f t e r introducing e l l i p t i c coordinates and various t r i c k s .
vation can be found i n various places i n the l i t e r a t u r e , s t i l l Jacobi's derivat i o n may be j u s t as well.
Here we give a d i f f e r e n t derivation using the exten-
sion of the geodesic flow t o a "flow of lines" i n the embedding space as discussed i n the. previous section. (b) The -1
Q(x) = < a X,X>
(1)
be a positive d e f i n i t e quadratic form i n R"+' O<
a < a < 0 1
... <
an.
We can assume t h a t a = diag (ao,al,. Q =
(1')
with d i s t i n c t eigenvalues,
-1 2 avxv.
v=o
, and
Then Q(x) = 1 defines an e l l i p s o i d i n
..,an)
and
we w i l l study the geodesic
flow on t h i s ellipsoid. For t h i s purpose one introduces t h e family of confocal quadrics, defined by Qz(x) = < Ca
(2) f o r r e a l z # a,,. a < 'z < a2 1
etc.
-
z)
-1
X,X>
= 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,(<,Y)
=
o
o r , i f Q (y) f 0 we g e t from the second r e l a t i o n t* = I n s e r t i n g t h i s i n t o the f i r s t
.
- Q,(x,Y)/Q,(Y)
we see t h a t x
+
t y i s tangential to Q ~ ( x = ) m
i f and only i f
-
@ z ( ~ ,=~ ~) Q ~ ( Y ) ( Q ~ ( x Q) ~ ( Y )
- Q2~ ( x , Y ) )
vanishes, always under the assumption Q (y) f 0 . Thus we can take (PZ(x,y) a s the Hamiltonian f o r the desired flow r e s t r i c t e d t o @ =O. A s a consequence of t h e theorem and the proposition of t h e previous section
we have
i.e.
the flows commute.
For large values of m we have
m
2
z
and i t follows t h a t the zeroes o f QZ a r e d i s t i n c t i f m is large enough. For with d i s t i n c t zeroes one can show now
\
t h a t the above .relation
( 0 ,@ l = o z1 z2 holds without f u r t h e r r e s t r i c t i o n , i . e . t h e $Z,r$zq are i n involution (see A
6
This holds f o r l a r g e m, b u t t h e above expression being
Exercises 1, 2).
quadratic i n m, it holds i d e n t i c a l l y f o r all m, i n p a r t i c u l a r f o r m = 1. If we express
4-
( f o r m = 1) i n p a r t i a l functions
one computes t h e F,S as residues to be
1 'xvy,l -
2 FV = YV +
(7) Being functions of
\ ,...,@ 0
n
U#V %f o r i + l d i s t i n c t zo,zl,. {F
(8)
2 xpyv)
..,zn we a l s o have
F ) = O .
v* lJ
Thus the f a c t t h a t the F~ a r e i n involution is a r e f l e c t i o n of the geometric a l proposition of Section 5 .
Of course, t h i s f a c t can
also
be
v e r i f i e d by d i r e c t calculation (Exercise 3). The Hamiltonian of the l i n e flow f o r t h e e l l i p s o i d QO(x) = 1 can be
The above %-flow is i n t e g r a b l e and possesses t h e r a t i o n a l i n t e g r a l s
THEOREM. F
v
given by (7) which are i n involution.
COROLLARY.
The geodesic flow on t h e e l l i p s o i d i s i n t e g r a b l e and has the
i n t e g r a l s Fv,.
COROLLARY.
(*O,l,.
The energy takes t h e form
The r e a l components of t h e algebraic manifold M: Fv = cV
..n) is
a torus, i f dFv are l i n e a r l y independent of M.
Exercise 1. Define t h e polynomials pz(x,y) = l y l - 2 ~ ( z ) ~ where z
i(z) =
N $ TT (r\, -
2)
v=o
Show t h a t
using t h e i d e n t i t y
Exercise 2.
Let P (z,x,y) = z
n
+
n-1 pl (x,y) z
+. ..+
pn(x,y)
be a polynomial
with r a t i o n a l c o e f f i c i e n t s . I f P(z,x,y) has d i s t i n c t zeroes somewhere then ( p Z ,PZ = 0 1 2
implies t h a t
Hint: Factorize P =
n
Pz,
J.
P
2
-
{ p z ,Pz I = 0 1 2 u,,(x,y)) i n some neighborhood and show t h a t a t
uy(g,f),
Z
Thus show t h a t t h e zeroes u
Pz, f o r a l l zl,
= 0 andPZ = 0
.
(z v=o 0 0 a p o i n t (x.y) and f o r zl =
w i t h c # 0.
if
= u
0
0
one has
x y # z
a r e i n involution, hence a l s o
z2,x,y.
L
Note t h a t t h e a s s e r t i o n of Exercise 2 does n o t hold i f double r o o t s occur I f
then f o r P
=P
Exercise 3.
Show t h e functions (7) a r e i n i n v o l u t i o n b y d i r e c t c a l c u l a t i o n .
Exercise 4.
Show t h a t t h e n+2 functions (XJ~ xv~v)2 GV = and v+ "v alJ.
2-
2-
= 0 we always have
C
-
-
2 lyl
a r e i n involution. Therefore they must b e f u n c t i o n a l l y r e l a t e d ; indeed
Exercise 5.
Any l i n e tangent to two confocal cones,
o r tangential t o such a cone and t h e sphere 1x1 = 1 has perpendicular normals a t the points of contact. (This f a c t leads t o the expressions G
v of Exercise 4.)
Exercise 6.
The tangents of any geodesics on an e l l i p s o i d are tangent t o the
same s e t o,f confocal quadrics, i.e. geodesic (see [51)
independently of the point on the given
.
With minor changes one can show t h a t the motion on an. e l l i p s o i d under the influence of a potential lxI2 i s a l s o integrable.
This was shown already by
Jacobi [ I ] .
[l]
C. Jacobi
,
Vorlesungen Uber Dynamik, Gesammelte Werke ,. Supplement band,
Berlin, 1884. [2]
H. S c h U ~ ,
S t a b i l i t a t von periodischen Geoatischen auf n-dimensionalen
Ellipsoiden, Dissertation, 'Bonn, 1972. [3]
A. Thim,
1976.
I n t e g r a b i l i t a t bien
geodatischen Fluss , Diplomarbeit, Bonn,
(In this paper ( i n Theorem 4.1) it i s shown t h a t the geodesic
flow on the
e l l i p s o i d admits nglobaln i n t e g r a l s i n involution. This f a c t
i s evident f r ~ mour representation of the integrals.) [4]
K. Weierstrass,
Math.
[5]
D. H i l b e r t and Cohn
7.
Werke I , pp. 257-266.
Vossen, Auschauliche
An Integrable System on the Sphere
(a) A point moving on the sphere quadratic p o t e n t i a l U (x) = system.
For n = 2
Geowtrie, Dover, 1955, p. 197.
sn:
T1,
1x1 = 1 under t h e influence of a
a = diag (ao,.
.., an)
t h i s was shown by C. Neumann i n 1859 [ l ]
i s an integrable
using Jacobi's
approach of separating variables i n the Hamilton Jacobi equations.
F i r s t we
proceed d i f f e r e n t l y and show t h a t t h i s system is obtained by reduction of
another
integrable system i n R~ (n+ll.
Then we w i l l apply separation of
variables. The equations of motion a r e xv = - a Vx v + A x v
(1)
where h i s determined i n such a way t h a t the p a r t i c l e says on the sphere. This leads t o (2) I n s e r t i n g (2) i n t o (1) we obtain the desired nonlinear system of d i f f e r e n t i a l equations which we want t o e s t a b l i s h a s an integrable
one-
(b) We compare the above system with the Hamiltonian system (3)
with (4)
1
1 +( 2
H = r
1 ~ 1 ~ l- < y xl , ~ y>
2
)
.
Note t h a t t h i s system has the i n t e g r a l / x i 2 since it is i n v a r i a n t under the s p p l e c t i c action (x,y)
+
(x,y+Zxs)
2 generated by 1x1
.
which we f i x a t 1x1 = 1.
sensible t o reduce the above system by t h i s i n t e g r a l , The isotropy group
Therefore it i s
is G = R and given by t h e action (x,y)
+
(x,y+2xs).
We
characterize the quotient manifold
{
I
Y
1x1 = l}/G
by picking the p o i n t of minimal distance i n the l i n e y
+
2xs, obtaining a s
reduced manifold
To determine the reduced flow
we determine
0 H = H
i n such a way t h a t
-
(1x1
2
- 1)
The f i r s t condition i s , of course, s a t i s f i e d a s 1x1 hence of HO, but the second y i e l d s
an6 therefore
,,= 1-< ax,x> 2
is an i n t e g r a l of H, hence
H0 =
=
1
-< ax,%> 2
T1
1
2 ( ~ x ~ 2 ~ -)
$< a ~ , x > l x -l ~< a x , ~ ( ~ x 1 ~ -+171) ( ~ x [ ~ l<xty>') ~ l ~ - .
Dropping the term<x,y>2 whose gradient vanishes H0 =
F(ly12+
1x12
on M we obtain
- < cix,10.(1xI 2 -
1)
and the reduced d i f f e r e n t i a l equations (on M)
which i s precisely the system (11, (2). The system (11, (2) i s obtained from (31, (4) by reduc-
Thus we have shown: t i o n t o the manifold M.
(c) The system ( 3 1 , (4) i s integrable, .and possesses t h e r a t i o n a l i n t e g r a l s
Fv(y,x) =
(6)
xt
+
'xvy,,
1
-
x,,yv) 2 a - a
llzv v u These are the functions of the previous section with x,y exchanged
-- and
therefore are i n involution. To prove t h i s assertion it suffices t o show t h a t H of (4) is a function of
...,Fn .
FO,
But it i s readily v e r i f i e d t h a t H =
-1
%F~(Y,XI -0
Thus t h i s system and the geodesic flow related
-- although not equivalent.
tion" (x,y) + (y ,-XI
and the
e l l i p s o i d are closely
I f one makes the "hodograph transfoxma-
then the i n t e g r a l s of one go i n t o those of the other.
One is governed by the Hamiltonian
the other by
(8) Although t h i s problem was t r e a t e d already i n 1859 by C. Neumann arrive a t the algebraic integrals. Devancy [31
a few years ago.
he did not
They were found by K. Uhlenbeck [2] and
We show how t h i s problem was solved by
(dl Hamilton Jacalji equations.
C. Neumann following the pattern of Jacobi, who used separation of variables
This technique requires an appropriate
of the Hamilton Jacobi equation. choice of variables. Coordinates"
In t h i s case these variables are " e l l i p t i c a l spherical
( e l l i p t i s c h e Kugel koordinaten)
For given a < al < u4 =
...
C
a
n
which are defined as follaws: n , xV # 0 define the and x = (xO.xlr.. , X n
7
.
as the solutions of the equation
U+ (x)
where the u. i n t e r l a c e the a 3
a s follows:
According t o (9) we have f o r z = u. (x) 3
9
Thus the u . (x) can be viewed as "coordinates on the sphere".
z = u
defines j the intersection of the sphere with one of a family of confocal cones. 3
From (9) one can express the xL rationally i n terms of the u
v
--
j
e.g. by
computing the residue X
2 = - u(it) A1(av)
where
V
These formulae express the x Similarly
U(z) =
-
on S" i n terms of u
n
(2
j=l u2t . . . I
- u.) . 3
n'
l i k e the
of coordinates.
e l l i p t i c a l coordinates the u form an orthogonal system j Indeed, one computes n ds2= (dxJ2= g . du2 V=O j=l 3 j
!
I
(11)
gj
=
- L-.u l ( u ) 4
~(u.1 3
TO prove t h i s we compute the coefficient of
duj d\ du
X
The coefficient of
du. d\ 3
i s found t o be
i n ds2. Since by (10)
if j # k
equal t o
and f o r j = k X
2
v
gj
v=o ( u j
=
- a,,)
which establishes (11)
X
- -i-d
2
v
- ---
=
=
j
-- and the orthogonal character of
I
3
j
these coordinates.
we use the variational principle
To describe the d i f f e r e n t i a l equation
6
U'(U.)
-1 2 4 A(u.)
(T
-v)
=
-1
dt-o
where, bY (11),
1 V = T
I a,,
V=o
2
1"
v=o
-2 i n (9). I f we The second equality follows by comparing the coefficients of z introduce the canonically conjugate variables v. by 3
the Hamilton function becomes (12)
H = T
+
Y =
-1 j 2= l
where we dropped the unessential constant
-1 2 (gj v j 1
-
U.1
3
1 av.
Thus the equations of
m t i o n become
and the Hamilton Jacobi equations
as
H(u,
= const.
More precisely, we ask f o r a solution S = S(u,q) depending on
n
= (n1,q2,.
..'0,)
of the equation
(13)
H(u,
Then the canonical mapping (u,v)
H
v
j
(Ern)
= -as
a ~ +
I
J
as
=
n1
.
defined by
5.I
as
=-
all; J
takes the Hamiltonian i n t o H = q, and the d i f f e r e n t i a l equation
into
This i s t h e s t a n d a r d use of t h e Hamilton-Jacobi equations. (el Separation of v a r i a b l e s .
The success of t h i s approach depends on the
p o s s i b i l i t y of s o l v i n g t h e above equation (13) which takes t h e form
is given i n (11). j To solve t h i s equation by s e p a r a t i o n we employ an i d e n t i t y frequently used
where g
by Jacobi : I f P ( z ) = qlz
n-1
+
n-2 q2z
+
... + qn is a polynomial
then
(17) To prove t h i s note t h a t t h e left-hand s i d e of t h e f i r s t r e l a t i o n i s t h e sum o f the residue
of
which makes t h e r e l a t i o n evident. The second can be proven s i m i l a r l y . We r e w r i t e (16)
using t h e expression (11) f o r g . and B .= 3 3
Using t h e i d e n t i t i e s (17)
-
2
4 A ( u .ISU j
t h i s can be w r i t t e n i n t h e form
Thus we can s o l v e t h e equation by s e t t i n g each i n d i v i d u a l term
Setting
t h e equation i s s e p a r a t e d i n t o
4A (u;) J
which i s solved by
s.1
2 ?dmdz.
j=l
Introducing t h e polynomial R ( z ) = -A(z)Q(z) become
-A(z)
t h e d i f f e r e n t i a l equations (15)
l
n
l-I
- j=l 1 &ET
Z=U+
dt
p,n-1
f o r p = O,l,...,n-1.
J
are i n t e g r a l s i n involution and it i s i n t e r e s t i n g
Clearly the 'l1,'l2,...,qn t o r e l a t e them t o the
i du = 6
r a t i o n a l i n t e g r a l s derived before.
Far t h i s purpose
we write
I t is c l e a r t h a t the coefficients are functions of the
as p a r t i a l fractions. tll,...,tln
and hence i n t e g r a l s i n involution.
We claim
t h a t the Mv = Fv a r e
the previous integrals. We sketch tlie proof:
Since both Q (z)/A(z) and
are integrals o f the m t i o n orbit.
it s u f f i c e s t o identify them f o r some point of an
For t h i s we t a k e the zeroes, say z = q . of Q(z). 3
has u = 0 from (18) and hence x = 0
+d therefore the l a s t expression becomes
which vanishes precisely f o r z = ul,u 2i...,u (18) it i s c l e a r t h a t f o r u ao,al,
'
...,an,ql, '
...,%
Hence (19)
j
= 0
of R ( z ) .
A t these points one
n'
From the d i f f e r e n t i a l equations
one has u. i s equal to one of the roots 3
Because of the
ordering (9') it follows
has the same roots and poles as (20) when u
both expressions a r e integrals
j
= 0. Since
they agree up t o a f a c t o r , which i s 1 by
comparing t h e asymptotic behavior.
Hence we have the i d e n t i t y
This formula shows t h a t the FV a r e functions of 'll,q2,...,nn
, hence
integrals
i n involution. References [l]
C. Neumann, De problemate quodam mechanico, quod ad primam integralium
ultraellipticorum
classem revocatur, Journ. reine Angew. Math.
56, 1859,
pp. 46-63. [21
k K. Uhlenbeck, Minimal 2-spheres and t o r i i n S (informal p r e p r i n t , received 1975)
131
R. Devaney,
.
Transversal homoclinic o r b i t 3 i n an integrable system
preprint On t h e separation of Hamilton-Jacobi equations: [4]
E. Rosochatius,
b e r Bewegungen eines Punktes Dissertation a t Univ.
Gbttingen, Druck
von Gebr. Unger,
Berlin, 1877
(available a t Library of the Math. I n s t i t u t , a t t i n g e n )
151
P. S t l c k e l , h e r d i e Integration
gleichung Halle
mittels t
1891
der Hamilton-Jacobischen Differential-
Separation der Variablen
.
Habilitationsschrift ,
(available a t Library of the Math. I n s t i t u t , M t t i n g e n ) . 8.
H i l l ' s Equation
(a) I n recent years remarkable progress has been achieved i n the description of the spectrum of the H i l l ' s equation, including the description and construct i o n of those periodic p o t e n t i a l s which belong t o a given spectrum, the soc a l l e d inverse s p e c t r a l problem. intriguing
We can e n t e r i n t o t h i s complicated and
s u b j e c t only t o a very l i m i t e d extent.
W e want t o show t h a t the
integrable system of d i f f e r e n t i a l equations of Section connected with the H i l l ' s equatipn i n the case of a f i n i t e
i s intimately gap p o t e n t i a l .
W e begin with a description of the spectrum H i l l ' s equation and the r e l a t e d inverse problem.
The H i l l ' s equation
i n h i s lunar theory (1)
--
--
s o called because H i l l encountered i t
i s of t h e form
(-
D
where D = d/dx and q ( x ) = q ( x
2
+
+ q ( x ) ) 4(x) =
$(XI
,
1 X E R
1) i s c a l l e d the p o t e n t i a l of the operator,
which i s assumed t o be continuous ( a t l e a s t ) .
With t h i s operator one can
associate a number of spectra dependins on the domains of d e f i n i t i o n and ti:e
boundary condition. Perhaps the most c o m n problem is to consider i n a dense linear manifold of 2
'
L (R-) i n which the operator is essentially s e l f adjoint.
In this case the
spectnnn is continuous and consists i n general of i n f i n i t e l y many intervals extending to
+
("band spectrumw). In exceptional cases one has oslly
f i n i t e l y many such intervals, one being i n f i n i t e l y
long.
The end points of the above continuous spectrum are given by the operator
- D2 + q
2 considered i n a dense l i n e a r manifold of L (0,23 of periodic per
functions of period 2 (not 1) with the boundary condition I#(= + 2) = Qtx). This operator has a discrete spec-
X1<'X2<
A,<
(2)
X3<_.X4<
...
where every second inequality i s i n the s t r i c t sense. coalesce they become double eigenvalues, like f o r q The spectrum for the operator
I f two eigenvalues 0.
on the whole line is given by the intervals
which may touch each other i n exceptional cases. The t h i r d case of a boundary condition is given by considering
- D2 + q
in
2
a dense l i n e a r manifold of L (0,l) with the boundary condition
.$(O) = 4(1) = 0
(3)
This problem gives r i s e t o a discrete spectrum, with s-le
... which
eigenvalues lil,l.$,,
l i e i n the follawing intervals
i n particular
i f A2j-l
= X2
&en a l l 3 members X2 j-l ., X2
li agree. j The determination of these spectra is a standard and well knmn application
of spectral theory. y, (x,M
, y2 (x,X)
r
In order t o describe its solution one introduces a basis
of (1) normalized by
The Floquet multipliers are the eigenvalues of the matrix
Since its detenainant.is 1 (by the constancy
t h e Floquet
8f the Wronskian)
multipliers have a product 1 and as sum the trace .of the above matrix, which
i s called the discriminant.
A(A) = yl(l,A) + y;(l,A)
.
I n order t o have a periodic solution of period 1 o r 2 one needs Floquet multipliers (+1,+1) o r (-1,-11, eigenvalues
Obviously
...
,A il,X2 ,
i.e.
A ( A ) = +2 o r = -2.
This shows t h a t the
are given a s the mots of the equation
the eigenvalues for the boundary
condition
(3) a r e given as roots
of the equation
are e n t i r e functions of order 1/2. This implies t h a t
A l l functions y1,y2,A
Hadamard's factorization theorem applies and we have, f o r example,
since 51
j
* j2r 2
r j
This constant c i s equal t o 1 a s one v e r i f i e s from
jW.
the asymptotic behavior
Thus
y2 ( 1 , A )
i s uniquely determined by t h e spectrum vl,p2,..
..
(b) The inverse s p e c t r a l problem can be formulated as follows. (i) Which sequences AO ,A1,
(ii) Given such admissible
q(x)
... occur as spectra AO, A1 ,A2, ... find and construct a l l potentials A2,
giving r i s e t o t h i s spectrum
(iii)Which additional data allows us t o f i x q(x) uniquely.
The answer t o these questions can be found i n the quoted references. The answer t o (i)i s contained i n ~ar'cenkoand Ostrovskii, t h a t t o (ii)i s the
subject of mst of the references.
A s t o (iii)we point out t h a t the prescrip-
tion of t h e additional spectrum pl ,U2, q uniquely.
... f o r the boundary
condition ( 3 ) fixes
Moreover, one has the e x p l i c i t formula m
which suggest For a point
the following a l t e r n a t e procedure:
5 determine the eigenvalues
1~ (5) ,p2 1
(c) ,... f o r the boundary
condition
2 i n L (6,5+1).
Then the above formula gives m
q(6) = 1, +
(6')
Thus giving an e x p l i c i t We w i l l t r e a t here t h a t h2j-l
1
.(h2j-l + A*j -. 21Jj(c)) j=1 formula f o r q from these various spectra.
only the case of a f i n i t e gap p o t e n t i a l , i.e. we assume
= h2j holds f o r a l l b u t a. f i n i t e number of subscripts j. For simpli-
c i t y we w i l l assume h2j-l<
hZj
for
j =1,2
h2j-1 = h2j
for
j > N
,...,N
,
although the general case can be t r e a t e d i n the same way. It i s not even obvious t h a t t h i s case can occur, and l e s s how t o find the
corresponding potentials. We w i l l show (A)
...<
The simple eigenvalues ho< 'h < 1
ha
determine the double eigen-
values, and hence A (h) uniquely. (B) The s e t of potentials q ( x ) belonging t o a s e t of admissible ho <
... <
A2N
form a torus TN of dimension N; the q ' s are given as
h y p e r e l l i p t i c functions. (C)
On t h i s torus T~
the flow q(x)
flow of a p a r t i c l e on a sphere
-+
q(x
sN
+
t) i s the same as the integrable
under the influence of a quadratic
p o t e n t i a l (discussed i n t h e previous section) when
r e s t r i c t e d t o a torus
obtained by f i x i n g N i n t e g r a l s F Thus, i n a sense, the flow q(x)
+
= c
j
q(x
j
,j
= 0,1,
...,N.
+ t) f o r the general case can be
i d e n t i f i e d with the r e s t r i c t i o n t o a torus of the corresponding mechanical pxoblem on an i n f i n i t e dimensional sphere! (c) Proof of (A) : Since OD
1
A ( A ) is a function of the order we have 2 2N OD 2
j=O
0
and
where
h : a r e the zeroes of Ae(h) which a r e r e a l and located i n 3
and the corresponding terms cancel i n
Hence f o r j > N a l l three numbers agree
fi
the quotient
A ' o
J =1
= c"
(1
-
h A-
Note t h a t the left-hand s i d e i s =
-
0 T3 ) i f we s e t
5 $'
A(A) = and 8(X) i s obtained a s a
2 cos $(A)
hyperelliptic integral
$(A)
= .L
f fi j=l
2N R(s) =
j=O This formula is due t o Hochstaedt. By adding a constant t o q so t h a t $(O)
(s
(s
- A:)
ds ,ma
- A ~ ).
we can assume t h a t A0 = 0 and w i l l normalize $
2 = 0, and s e t $ ( z ) = 6 ( z )
i.e.
Im z > 0
W e can i n t e r p r e t w = 8 ( z ) as a mapping o f the upper h a l f plane the s l i t domain obtained by deleting v e r t i c a l slits a t
= 8(
)
W e ha-to
(Schwarz-Chriatoffel 's fornula) choose the parameters h
j'
Re w = 8
(
G
into
)
.
A ' so that j
B(h2j-1) =
8
(
q = jn, so
t h a t A ( A .) = 2 cos j w = 2 ( - 1 ) j . Thus we can parametrize the spectrum most 23 e f f e c t i v e l y by choosing N p o s i t i v e numbers hl,h2,. ,h and define 8 (z) a s
..
the u ~ i q u es c h l i c h t conformal mapping of
Im z > 0
N
onto the domain obtained
from I m w > 0 by deleting t h e slits Rew=+ja, and such t h a t , with z+
2
=
2
5,
8 ( z .) = f o r j = 1,2,.
9
..,N;
O<
z h =
+ aj ,
and 8(0) = 0.
+
Given the s p e c t r m , however, a l s o the
A.
= 0,
A1,.
..A
ih
j Moreover, f o r large values of z we have
depend only on the N parameters hl,h2,.
j
+aj +
8(z'.) =
9
This shows t h a t the 2N numbers X1,...,)IZN
eigenvalues A
ImwLthj.,
cannot be chosen a r b i t r a r i l y , b u t
..,$.
...,% as well a s the o t h e r
A; ,A;,
(j
> 2N) are uniquely determined.
Consequently, the spectrum
2N
determines h: and a l l the double eigenvalues , hence the 3
.
discriminant A (1)= 2 cos 8 (A)
This takes care of (A)
.
(dl Description of the p o t e n t i a l i n tenns of an a u x i l i a r y spectrum. We ask f o r the s e t of a l l p o t e n t i a l s belonging t o a given spectrum of the type (7). To f i x a p o t e n t i a l one may use another spectrum, e.g.
the eigenvalues belongN
ing t o the boundary conditions (3) ( a c t u a l l y t h i s gives r i s e t o 2 p o t e n t i a l s i n general.) I n t h i s part we w i l l be sketchy
and r e f e r t o the paper (e.g. Trubowitz [ G I ) .
W e mentioned t h a t f o r any p o t e n t i a l q(x) = q ( x
+
1) t h e eigenvalues p
j
f o r (3)
l i e i n the interval
5 pj 5 X 2 j
'
Conversely we wish t o construct a p o t e n t i a l q f o r a r b i t r a r y eigenvalues p the above i n t e r v a l , where t h e A . a r e given according t o ( 7 ) . 3 q(O) from (6). I n order t o f i n d q ( x ) we subject .q q(x)
-+
q ( x + t) so t h a t t h e eigenvalues p
. are
3
j
in
W e can compute
t o the t r a n s l a t i o n
taken i n t o p. (t) i n the above 3
interval.
We w i l l derive a d i f f e r e n t i a l equation f o r these functi.ons p. (t) 3
and. integrate them; thus we can recover q ( t ) ffom the formula (6) i n a unique way. Below we w i l l derive these d i f f e r e n t i a l equations which show t h a t f o r increasing t the eigenvllues o s c i l l a t e i n the i n t e r v a l
5 pj 5 A2j
A2j-1
back and forth, the double ~ l u e d n e s s coming from the unspecified sign of
&.
We can make the choice unique i f we assign the sign of
each point, effectively making each i n t e r v a l i n t o a c i r c l e . cation of pj i n A2j-l p o t e n t i a l t h i s way.
5 pj 5 A Z j
2
and a sign of {A
((Y-4
7 A (PI-4 t o .
For any'specifi-
one finds a unique
A complete proof would require the verification t h a t q ( t )
so determined i s of period 1. We forego t h i s proof and turn t o the determinat i o n of the d i f f e r e n t i a l equation. (e) The d i f f e r e n t i a l equations f o r q(x) + q ( x q (x + t) we replace the parameters p1,p2,. ask f o r the d i f f e r e n t i a l equations
+
..,%
t)
.
I f we replace q(x) by
by p l ( t )
,...,% (t)and we We w i l l write t h i s
describing t h i s flow.
d i f f e r e n t i a l equation i n two d i f f e r e n t forms: I n e x p l i c i t form it has the form ap a m N (8) -i= dt A'(p.1 where A(z) = (pj z) o r i n the i m p l i c i t form
3
-
j=l
o
(9)
for p =
1 for
o,i,
...,N-2
p = N-1
The l a t t e r follows from the former by the observation t h a t the left-hand side
i s the residue
sum f o r the i n t e g r a l
taken over a large c i r c l e . under ( c ) .
These are the
d i f f e r e n t i a l equations asked f o r
The l a t t e r form exhibits the familiar Abelian sums of the d i f f e r -
en ti a l s
& 2VGii.F of the f i r s t kind on the .Riemann surface defined by w2 =
- It(=).
W e derive t h e above d i f f e r e n t i a l equations (see E. Trubauitz): eigenvalues p
j
W e view the
as functionals o f q and compute the d i f f e r e n t i a l
6lJ.
L= $2
where
4.
3
69 j i s t h e normalized eigenfunction of
1:
L@j=-~;+q$j=pj@j;
lp.dx=l..
This i s e a s i l y v e r i f i e d by taking the differen&
of t h i s r e l a t i o n
with
respect t o q ,
Taking the i n n e r product 1
Noting
with
4.
3
we get
t h a t the f i r s t term vanishes as L i s s e l f a d j o i n t , we have the s t a t e d
relation. Therefore
we have f o r the t r a n s l a t i o n
Using the d i f f e r e n t i a l equation dlJ =
the d i f f e r e n t i a l equation
we replace
-4
=
4. q 3
- 4;
0
0
4.
Clearly, x = 0.
3
3
(1,
l.~4 j j
and ndtai'n
.
3
-1/2
where t h e dot i n d i c a t e s A-derivative. 2
cj(o)
-
2 -
Thus -1
- $ .3( I ) = y 2 ( l r p3. l
I f we use t h e r e l a t i o n y 1y 2 ' 2
4: + 2
i s a multiple of t h e s o l u t i o n y 2 ( x r p . ) whi& a l s o vanishes a t
One computes
we have y
by
-
Y;Y2=
.
I
( ~ ~ ( l , u ) - ~ - y ; ( l 3, p . ) )
1 and the f a c t t h a t f o r h = 11.
= 0 we find Y1 Y;
Therefore we have the i d e n t i t y
= 1
-
j
, and
x = 1
and
Since y2 (1.1) has as zeroes the ho,X1,..
.
+
(k=1, 2,.
..I
2
4-A (A) as zeroes the
and
we obtain the d i f f e r e n t i a l equation up to a constant.
We forego
determination of the constant, which i s found from the asymptotic
the
behavior f o r large A. I n f a c t , Trubowitz uses these d i f f e r e n t i a l equations i n the i n f i n i t e dimensional case i n t h e form
which i s valid i n t h e general case and uses them spectral problem i n the periodic case. admits a uniform liipschitz estimate
t o solve the inverse
The point is t h a t the right-hand side
t o allow global solution
of the d i f f e r -
e n t i a l equation. However, we r e s t r i c t ourselves t o the f i n i t e gap case.
For example, i f
N = 1 the d i f f e r e n t i a l equation for F( = p1 becones
which gives the e l l i p t i c p-function plus a constant. Formula ( 6 ' ) shows t h a t the corresponding potential i s also an e l l i p t i c function with r e a l period 1. Thi.s i s the same equation. ( f ) A mechanical description of the d i f f e r e n t i a l equations (8) o r ( 9 ) . McKean and Trubowitz derived
belonging t o the eigenvalues AZj.
42
m
1
(10) The
an i n t e r e s t i n g i d e n t i t y
E
j=o
. depend on
3
E. @
3
They satisfy.the r e l a t i o n
2 (x) = 1 where 21
.
the AO ,A1, A2,.
f o r the eigenfunctions
E
j
=
A(A2
.)
y;(l,AZj)
.. only and a r e positive
.
i f A2 j- A, j-l > 0, and
are equal t o zero i f A 2 j gaps
-
h2j-1 = 0.
Thus i n our case (7) of f i n i t e l y many
t h i s i d e n t i t y reduces t o
Setting
we see t h a t the x = (x xl,...I$) a r e r e s t r i c t e d t o t h e u n i t sphere. t o derive t h e d i f f e r e n t i a l equations on t h e u n i t the t r a n s l a t i o n q(x)
-+
q(x
+
sphere
We wish
which correspond to
t ) . For t h i s purpose we use the d i f f e r e n t i a l
equations f o r $ which give 2j
-
(11)
The f a c t t h a t lxI2 = 1 i s i n v a r i a n t and
<x,%>
+
1
= 0.
under the
flow
implies t h a t < x , h = 0
Thus , taking the inner production with x we
from the d i f f e r e n t i a l equation:
- 1;12
= qlx!
2
-I
2 Azjxj
d
W e notice t h a t t h i s system. (ll),(12) is precisely t h e integrable system o f the p a r t i c l e moving on the sphere under the influence of a quadratic p o t e n t i a l discussed i n a previous section. Here q plays t h e r o l e of the normal force N and h x2 of the quadratic p o t e n t i a l . We can make use of our information jL.o 2) j about the w c h a n i c a l problem t o gain information about the s p e c t r a l problem.
1
For example, (12) y i e l d s t h e i d e n t i t y
f o r the p o t e n t i a l . for
4
I n McKean-Trubcrwitz (p. 223)
one finds -another formula
of the form
Taking the difference of the two expressions we f i n d
NOW, the left-hand side is
the mechanical prlblem,
];I2
+
1 A2 jx2j , i.e.
twice the t o t a l energy f o r
which is c l e a r l y a constant.
Its value i s given by
the right-hand side. N (gj Identification of the torus T
t i o n of translation q (XI
q as i n (12).
+
q(x
.
Thus w j see t h a t t h e d i f f e r e n t i a l equa-
+ t) agree with the mechanical problem
But clearly not a l l solutions give
r i s e to periodic potentials.
Which solutions correspond t o the N-gap potentials described by (711
They
form an N-dimensional torus which we wish t o identify. For t h i s purpose we use the previously discussed i n t e g r a l s '"vy,, yvx,,) 2 2 Fv = XV +
C PfV
-
I2v- i2p
-- @2,,@;v)2 I211
(@2v@;p
and the corresponding function
Since the integrals-are i n involution it i s c l e a r t h a t the manifolds Fv = c
v
are t o r i , provided they a r e regular, compact and connected.
Therefore it i s
t o be expected t h a t the desired manifold corresponding t o the N-gap potential i s described i n the algebraic form Fv = cv (O,l,...,N).
case.
But instead of giving the values of t h e constants
zeroes of
Q instead. Z
This i s indeed the c
v we describe the
Since the asymptotically
we have
where poles.
u are the zeroes of j
QZ.
Thus Qz i s characterized by t h e zeroes and
THEOREM.
The f i n i t e gap p o t e n t i a l s q(x) defined by (7) are characterized by
This follows from a scrutiny o f Proposition 1, p. 171 of McICean and
COWJUAFtY 1.
well defined, i.e.
F
N
1 v=o
E
2 on t h i s torus are ~= 1 $. A~l l solutions ~
of period 1 defining t h e potentials
COWJLLARY 2.
Z
have specified values, defining an N-dimensional torus
k on the tangent bundle of periodic
the function Q i s
For a given simple spectrum X0.&i,...,h2N
Since the X2j-l
i n t e r l a c e the A
q (x
+
.
t)
it follows t h a t on this
2j
torus a l l the F have positive values. k A l l solutions $2V a r e h y p e r e l l i p t i c functions of x and s o are the corres-
ponding p o t e n t i a l s according t o (131, both periodic of period 2 or 1. One can determine a p a r t i c u l a r p o t e n t i a l with given simple spectrum h,IXII--.IX,N
by s e t t i n g
vj
= X2j-1.
as the "origin" on the torus 'FN.)
(In McKean-Trubowitz
t h i s i s choser
Then q(x) i s an even function and the
a r e the roots of y; ( 1 , X ) 2j Therefore f o r t h i c choice of t h e p o t e n t i a l q(x) one has A2j-1
"j
a r e roots
of
y2 ( 1 , X )
while the h
Since the roots of y2(l.z). yi(1,z) i n h2j-l
(h <
-
2j
coalesce i f j > N
.
the
function.
obove expression is i n f a c t a r a t i o n a l
It is suggestive t o investigate the potentials belonging t o other t o r i , i.e.
t o d i f f e r e n t values of the constants
c
j
= F
e l l i p t i c , but now i n general quasi-periodic.
j'
A l l these solutions are hyper-
Since very l i t t l e is known about
the s p e c t r a l theory of quasi-periodic potentials it would be worthwhile investigating even these s p e c i a l examples
which present themsexves through
t h i s surprising connection between H i l l ' s equation and the mechanical problem. (This connection was found by E. Trubowitz and the author.)
However, t h i s
approach has not y e t been c a r r i e d out. References 111 A. A. Ihbrovin,
V. B. Matveev and
S. P.Novikov,
Nonlinear equations
of Korteweg de Vries type, f i n i t e zone l i n e a r operators varieties,
Russ. Ma&.
Surveys 31 (1976)
and Abelian
59-146.
121 V. A. MarEenko and I. V. ~ s t m v s k i i , A characterization of t h e Bpectrum of H i l l ' s operator, [3]
H. Hochstadt,
Mat. Sbornik 97 (139) 1975, pp. 493-554.
On the determination of H i l l ' s e m a t i o n from i t s spectrum,
a c h . Rat. Mech. Anal., [4]
Vol. 19 (1965) 353-362.
H.P. M~Keanand P. van Moerbeke, Inventiones Math.
30 (1975)
The spectrum o f H i l l ' s equation,
217-274.
151 H. P. McKean and E. Trubowitz, H i l l ' s operator and h y p e r e l l i p t i c function theory
i n the presence of i n f i n i t e l y many branch points, Comm. Pure
Appl. Math. 161 E. Trubowitz,
29 (1976) 14 -226. The inverse problem f o r periodic p o t e n t i a l s , Corn. Pure
Appl. Math. 30 (1977) 171 N. Levinson, (1949) 25-30.
321-337.
The inverse Sturm-Liouville problem, Mat. Tidsskr. B.
CEN TRO INTERNAZION ALE MATEMATICO ESTIVO
(c.I.M.E.)
HOPF BIFURCATION FOR INVARIANT TORI
A.
CHENCINER
HOPF B I F U R C A ~ OFOR N INVARIANT TORI by A.Chenciner- I.M.S.P.
(*I
Mathematics Dept. P a r c V a l r u s e
- 06034 NICE Cedex - France
I want t o d e s c r i b e some j o i n t work w i t h Gerard IOOSS o f Nice U n i v e r s i t y connected w i t h t h e R u e l l e and Takens d e t e r m i n i s t i c approach t o t u r b u l e n c e . The main r e f e r e n c e s are [R.T.], Navier-Stokes
[C.I. I.], [C.I.2.].
e q u a t i o n s can be t h o u g h t o f as a flow i n some i n f i n i t e dimensio-
n a l Banach s p a c e o f divergence-free v e c t o r - f i e l d s i n R~ w i t h some Sobolev norm ( f o r r e g u l a r i t y p r o p e r t i e s a n a l o g o u s t o t h o s e found i n t h e f i n i t e dimensional c a s e , even though w e - d e a l w i t h unbounded o p e r a t o r s , s e e
[XI.
As a
c e r t a i n parameter v a r i e s ( f o r ex. t h e T a y l o r number i n Couette-Taylor e x p e r i ment) t h e f l o w i s a t first supposed t o undergo a sequence o f b i f u r c a t i o n s o f t h e b p i ? t y p e , e v e n t u a l l y l e a d i n g t o a s t a b l e ( t h i s means h e r e a s y m p t o t i c a l l y s t a b l e ) invariant r-torus,
r n o t t o o b i g s a y r = 4).
The t r a n s i t i o n s f r o m a s t a b l e e q u i l i b r i u m t o a s t a b l e c l o s e d o r b i t and f r o m b s t a b l e c l o s e d o r b i t t o a s t a b l e invariant 2-torus a r e w e l l understood "generic" phenomena ( s e e [R.T.];
h e r e " g e n e r i c " i s i n t h e v e r y s t r o n g senSe
t h a t a f i n i t e number o f q u a n t i t i e s b e non zero); on t h e o t h e r hand, t h e n e x t b i f u r c a t i o n s , s t a r t i n g with t h e t r a n s i t i o n
f r o m a s t a b l e i n v a r i a n t 2-torus
t o a s t a b l e i n v a r i a n t 3-torus a p p e a r t o b e h i g h l y e x c e p t i o n n a l ( i n any r e a s o n n a b l e s e n s e , measure o r c a t e g o r y ) : it i s t h e purpose o f what f o l l o w s t o d e s c r i b e , and d i s c u s s on an' example, s u f f i c i e n t c o n d i t i o n s f o r t h i s last b i f u r c a t i o n t o o c c u r ( f o r t h e g e n e r a l case, see [C.I.I.$. F i r s t , as i n [R.T] we r e p l a c e t h e f l o v s b y maps, t h u s d e c r e a s i n g by one t h e dimension o f i n v a r i a n t s e t s ; more p r e c i s e l y , suppose t h a t we c o n s i d e r
(*I
-
-
- -
-
p p
S l i g h t l y expanded v e r s i o n of a c o n f e r e n c e g i v e n a t t h e Bressanone CINE s e s s i o n on dynamical systems, J u n e 1978.
an i n v a r i a n t 2-torus f o r a flow and let us make the assumption t h a t the r e s t r i c t i o n o f the flow t o t h i s torus admits a cross section (i.e.
a trans-
verse closed curve on which t h e Poincarg r e t u r n map can. be definedli such a cross- section e x i s t s f o r example i f the flow on the 2-torus i s close t o a quasi-periodic one, o r i f the 2-torus r e c e n t l y b i f u r c a t e d f r o m a closed orbit. W e extend t h i s Poincarg-map to g e t a cross-section o f the flow i n a neighbor-
if the r e s t r i c t i o n t o the transverse closed curve of
hood of the 2-torusi
the normal bundle of the 2-torus i n ambiant space
i s trivial
( an
assump-
t i o n which can be e a s i l y achieved by eventually adding variables) the flow i n the neighborhood of the i n v a r i a n t torus i s described by a map
1 1 F : T x V , T x 0
+T
1
x E , T
1
xO
where%' i s a neighborhood o f 0 i n the Banach space E
, and T1 i s
This leads t o the following problem (compare w i t h [R.T.]
the c i r c l e .
where b i f u r c a t i o n
from a closed o r b i t i n t o a 2-torus i s reduced to Hopf b i f u r c a t i o n for maps ) :
1 k Let F : T x 'lr -+ Tx I E be a C mapping. c1 Suppose t h a t f o r p negative small enough, F admits an i n v a r i a n t c i r c l e 1 P goes t o 0, and g e t t i n g close t o T x 0 ( i n v a r i a n t by F,) when exponentially stable.
> 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 <
1
vertical strips, O<j<m
f
Then
n
n
j
-0<
x
C2
0 for
Q n f(Q) n f(Q
x .c f
n f(Q))
We assume
-1 A2
= Q n fQ
n f 2Q c o n s i s t s of 2
f J Q c o n s i s t s of 2n v e r t i c a l s t r i p s , and O<jIn j =~C1 x I where C1 is a Cantor s e t and I i s an i n t e r v a l .
Similarly,
C1
f(Q)n Q by A1 and A2.
fJQ = I x C 2
is a c a n t o r s e t .
with
The s p l i t t i n g
C2
TxA
fJQ = -<j <m i s given by the horizontal
a Cantor set, and
-1 and v e r t i c a l subspaces, and we may take 4, = a , C = 1. The dynamics of C
= 2
f l ~ can be conveniently described a s follows.
1's and
be the s e t of b i - i n f i n i t e sequences of
product (compact-open) topology, and w r i t e elements of x(i) -
= 1 or
U(Z) ( i ) = ll(i
2
and
Define t h e s h i f t map
i c 2.
+ 1); that
is
a
z2
2's by
a : C2 + C2
s h i f t s a sequence 5
Let
with t h e
5 where by
one Step t o the l e f t -
2
Now i f h(x)
E
i
A, then
. f (x) c A1
Then one can prove t h a t hfh-' in
h : A += C 2
is a homeomorphism and
hood of
x
a r e dense
2 k.
+(i
+ k)
i s p e r i o d i c i f and only i f t h e r e i s a
E2
E
= ~ ( i ) for a l l
Z, choose a
k > 0 such t h a t
The sequence g
If
i.
y
and
5 c E2
is periodic and l i e s i n
f
=(i) = z ( i )
for
lil
s k
~ ( i =)
for
lil
> k
i s any neighbor-
y ( i ) = ~ ( i )f o r
where
f :X + X
and
f
-k s
S
o
equivalence
h
Y
g
g :Y + Y
satisfying
hf = gh.
mappings.
= (xl,yl) -
-1 2 (-axl
= (b
a diffeomorphism of
from
R
to
R
-
- y1 IR
2
.
+
C) ,xl)
.
g
if
The topological f.
For a given
f,
f.
Fix numbers
0 < b < 1, and consider the one-parameter family of mappings 2
2'
a r e homeomorphisms, one says
t o use a s a model f o r
Remark. Horseshoes a c t u a l l y occur i n simple
(y,-ay2-bee)
C
IA.
preserves a l l t h e dynamical s t r u c t u r e of
one would l i k e t o f i n d a simple
and
a r e dense i n
is topologically conjugate ( o r topologically equivalent) t o +
k
+ 1)
1 mod(2k
Thus t h e p e r i o d i c o r b i t s of
U.
t h e r e is a homeomorphism h : X
gc(xl,yl)
k > 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
(3)
7
&
i s a p o s i t i v e constant.
0
0
fl
-nk n,6 kEZf (AL- )
= (iklkEZ E Z 2,
n f-nk(~nb). k€Z 4c
A"' = ik+l
i s any f i n i t e
1's and 2's, then
sequence of
h(&) =
...,iO,.. .,im)
(i,m,i-m+l,
So
fn%(&) r An'& ;or ik
f o r a l l k, so
(&)
is
h : H2 + A
(3) shows t h a t
h
1
-
i s a s i n g l e point. all
.
f%(&) = hu(i.)
k.
fnk(f%(&)) r
I f we l e t A =
1, onto, and conjugates
i s continuous.
Hdnce
o
with
Set
n
nk n,6 f (A1 uAiPl,
k€ 2
fnlA.
Statement
Thus, h i s a homeomorohism s i n c e
Z2
is
compact. The proof of (1) and (2) a s w e l l a s t h e hyperbolicity of same estimate a s the following b a s i c r e s u l t known a s the proposition (2.5) ( A 1 - l e m a ) [ 39 1. with a hyperboZic periodic p o i n t Let
A
be
u-disk meeting
~ P ( A contains ) n>O
wS(p)
Let
f be a
p, q d Zet
D~ be a
transuerseZy
u-disks a ~ b i t r a rZir
c1
close
A-lemma. diffeomorphism u-disk i n wU(p).
a t some p o i n t
m
involve the
A
D ~ .
x.
Then
proof.
Let
(u,v) . be coordinates on
lRS x I R ~ . We assume
The general case is then obtained by replacing
f.
some
frh.
Since wU(p)
and
wS(p)
c1
are
f
by some
t$ :
U
c1
lRs x I R ~ is a
+
Thus,
and
by
A
d i f feomorphism so t h a t
centered a t
(~,t$)
i s a neighborhood of
U
fn
manifolds, we can use t h e im-
p l i c i t function theorem t o produce a coordinate c h a r t so t h a t they become f l a t .
i s f i x e d by
p
p
p
and
t$(p) = (0,0),
n t$-l((v;.o)> c ws(p), and t$'l((u=o)) c ~ ~ ( Replacing ~ 1 . x by f '(x) and n 1 -1 A by f (A) n U f o r some nl > 0, we may assume x E 4 ( ( ~ 0 ) ) and A c U.
i=
Write
t$feel
so that for
(u,v)
near
(0,0),
7
i s given by
< 1 and
vl=Bv+f2(u,v)
ordinates
may be chosen t h i s way by l i n e a r algebra).
t$
tial deri~atives f Choose in
IA
flu(u,v)I
K > 0
then
<
SO
Pick
z = (u,v)
K.
that
Suppose
(5'
< Aand
n
be such t h a t i f
I
6, > 0
6 > 0 1
i c e
> 0
2
B
f2v(u,v))nl
so that
n
n
w'(~)
is invariant,
and
lu1 <
is a vector i n lul <
if 0
implies R
x
and
x
W
-
lRu with /v1
IfLv(
<$.
g
+ 0.
Set
Ifn(31
< 61.
tangent t o
= +(XI
Let
-f n 2.-A ,
IK
I f 2 U ( u , ~ ) > 1.
# 0 and
Letting
- XK-1+ Iflvl ' C ' I ~ -I I f Z u l ~ l n l 1 - I f Z U 1 ~ I'll
q + O
f (u,O) 1 0, s o we may choose 2
imply IRs
a r e a l l zero.
forall
1
,nl), we have
~ ~ l '+l l
We assume t h e par-
I (u,v) 1 < 9,we have
2
i s a vector i n
i s a p o i n t with
X
illnl
>
(6,n)
lv1 < 62
(6,n)
and
s o t h a t whenever
+
I(B
< 1 ( t h e co-
f2u(0,0), and f2v(0,0)
(A( < X
so that
A < 1
n = ((A).
and
(0,0), flv(O,O),
Then choose
IRU.
+
lu
with
JB-'~
Au+fl(u,v),
u = 1
I
< K, and 191 TT(S,O) =
-
where
K
1
If write
=
sup
1 (UYV)1
fi(z) T
IflV(u,v)I. ~ 6 ~
{(u,v) : lul < 611 lvl < 62)
Tm(E,n)
for
0 L i
6
m
-
1, and we
= ((m,qm), then
and
Now i t e r a t e s of p o i n t s i n
-n f
h
near
(v = 0)
f i r s t s t a y near
(v = O),
(0,0), and f i n a l l y they s t a y near (u = 0 ) . In the f i r s t n -2case, vectors tangent t o i t e r a t e s of f A stay i n the sector S ( I R ~ , I R ~ ) . 2 I n the second c a s e , a l l the . p a r t i a l d e r i v a t i v e s flu, flv' f 2 u , f2v a r e then they s t a y near
s m a l l , s o the tangent vectors converge t o case, If
lv
I
R U exponentially. I n t h e l a s t
i s small, s o an estimate s i m i l a r t o t h a t of t h e f i r s t case shows
t h e tangent vectors remain close t o -n N o w choose n, > 0 s o t h a t f 3 ~ U= I$-'({o)
xu).
x
2
shows t h a t hence t o
U f n ( b ) contains u-disks a r b i t r a r i l y n>0 DU i t s e l f .
c1
The above estimate -n c l o s e t o f 3 ~ u ,and
Returning t o t h e proof of t h e homoclinic theorem, t h e reader may use estimates l i k e those of Proposition (2.5)
t o show t h a t i f
i s l a r g e , then some s e c t o r S (T(6DU),TD')
K
6 > 0
is small, and n
over A;'~UA;'~ i s i n v a r i a n t and ex-
panded by Tfn. Also, t h e complement of S (T(6DU) ,TDS) i s i n v a r i a n t and expanded
K
~ f - ~Having . done t h i s , statements (1) and (2) a r e proved by induction on Also, i t is immediate from theorem (2.2) t h a t hyperbolic f o r
fn.
For more d e t a i l s s e e [55] o r
[25].
A =
k
fnk(AYs6 u
is
m.
3.
Homoclinic classes, shadowing lemma, and hyperbolic basic sets.
In this lecture we will derive some simple consequences of the homoclinic theorem and the A-lemma, and we will study the structure of hyperbolic sets. Let H(f)
# 0.
H(f)
u
be the set of hyperbolic periodic points of f and ass-
For p
wU(x)
E
,
d(f)
03(p)
is the orbit of p, and we let wU(o(~))
be the unstable manifold of the orbit of p.
=
Similarly, we let
X€O(P)
I$(o(p))
=
U
wS(x)
~ €(p0 )
Define a relation
be the stable &ifold
-
on H(f)
of the orbit of p.
by saying p
non-empty transverse intersection with wS(o(q)) empty transverse intersection with wS(o(p)).
-q
if wU(o(p))
and wU(o(q))
has a has a non-
This relation is clearly re-
flexiveand symmetric. It follows from the A-lemma that it is transitive. For if pl of wU(o(pl))
- p2
and p2
- p3, let
n #(o(p2)),
z
be a point of transverse intersection
and let z'
.
of wU(o(p2)) and wS(o(p3)). j Z' wU(f 3p2) n us(£ j4p3). T
From the A-lemma for f
be a point of transverse intersection
jl p ) n #(f 1 be so that fT(pi)
Say z Let
T
, we have
6
wu(f
that wU(f
jl pl)
j2 p2) =
pi
and for i = 1,2,3.
1 contains disks which C
1 j2 j -j j accumulate on wU(f pZ), so f ZyU(f Ipl) contains dish which C u j3 j -j +j accumulate on W (f p2). Hence, wU(f lpl) has non-empty transverse intersections with
wS(fj4P3) .
verse intersections with wS(o(p or h-stated to q if p
Similarly, wU(o(p ))
1
.
) ) has non-empty trans3 We say p is homocZinicaZly related
- q, and we call the equivalence class of
homoclinic class or h-class.
Denote the h-class of p by Hp(f).
p
its
Note that
the homoclinic theorem (2.3) gives that every transverse homoclinic point of
a p
i s a l i m i t of a . sequence
H(f)
E
ql,q2,.
..
in
H(f)
where each
qi
is
H ( f ) s o(p) and H ( f ) i s f - i n v a r i a n t . It i s P P easy t o show, a s we wS11 s h o r t l y , t h a t H ( f ) 2 O(P) i f and only i f p has P a t r a n s v e r s e homoclinic p o i n t . I n t h e l a t t e r . r a s e , t h e c l o s u r e of H ( f ) P e q u a l s t h e c l o s u r e of t h e o r b i t s of t h e t r a n s v e r s e homoclinic p o i n t s of p.
h-related t o
p.
Clearly
Let
Proposition (3.1) (Birkhoff [ 3.1).
f :X
+
be a homeomorphism of the
X
complete metric space .X. Assume the topology for and for every open s e t x
E
U
X
i s dense i n X.
c X,Ufnu
n2O
has a countable base, Then there i s a point
whose forward and backward o r b i t s are dense i n X.
X
Proof. Since
Let
uf"Ya
uf"Yan:Ou nIO dense i n
n20 X.
{ValacA b e a c o u n t a b l e open base f o r t h e topology of
i s dense and open, s o is
X.
uf"Va, and, consequently, s o i s n0' By t h e B a i r e Category Theorem, fnva n fnva1 5 acA nLO n2 0 But any x E 8 h a s b o t h i t s forward and backward o r b i t s dense
u
Note t h a t t h e preceding proof a c t u a l l y g i v e s t h a t t h e s e t of p o i n t s whose forward and backward o r b i t s a r e dense is r e s i d u a l .
A homeomorphism
f :X
+
X
which h a s dense o r b i t i s c a l l e d t o p o l o g i c a l l y t r a n s i t i v e .
Proposition (3.2).
a cZosed,
1
fjr
> r2
1
and
> $r2
Let
{rl)
U1
r1 > r 2 r2
=
7
r
3
U2
f T where
a Hp(f)
is
i s topologically t m n s i t i o e .
f
T
rl
and
r i , l e t us w r i t e
i s a common p e r i o d of
implies that
wU(rl)
imply t h a t
rl > r3.
f o r any i n t e g e r and
Closure H ( f ) P
h a s a non-empty t r a n s v e r s e i n t e r s e c t i o n w i t h
A-lemma a p p l i e d t o
ve see that
r
the s e t
H(f),
For h y p e r b o l i c p e r i o d i c p o i n t s
-
wU(rl)
By t h e
E
f-invariant s e t on which
Proof. if
For any p
c1
accumulates on
Also, i f
r
1
> r
2'-
wS(r2)
rl
and
r1 > 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
<(XI (W:(x))
t h e s t a b l e (unstable) s e t of
X.
Theorem (3.3) fh'ipsch and Pugh feomorphism, k 2 1, Zet TxM = :E
E
be t h e d i s t a n c e function induced
n
= w:(x,£)
size
= {y
x
d
c M : d(fnx,f Y) Z E f o r n > 01, and l e t WE(x) =
w:(x) (
Let
e,:E
G
1).
Let
-
f :M
be a hyperboZic s e t for
A
x x A, and endow
small, and x
[ 17
TM with an.a&ted
M
be a
ck d i f -
with s p l i t t i n g
f
Then, for
metric.
E
> 0
A,
k and wU(x,f) are C disks through x varying E continuously with x i n the ck topology.
11) w:(x,~)
(2)
i s tangent a t x
w:(x,f)
t o E'
It follows from t h e d e f i n i t i o n s of
d ( x ) c c(fx)
and
(
X
w:(x)
and w:(x,f)
i s tangent
and
that
w:(x)
) c W ( f X ) Also, theorem (3.3) implies t h a t
x
f o r x e A. Therefore, and wU(x) = u n ~ : ( f - n x ) Ws(x) = u f ' V ( f n x ) ~ 0 ' 1-120 a s i n the case of periodic points, wS (x) and wU(x) a r e ck i n j e c t i v e l y immersed copies of Euclidean spaces. A homeomorphism
f : X +- X
expansive i f t h e r e is an x = y.
Any such
Proposition ( 3 . 4 ) . sive.
E
E
> 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
o(x) = {fix : i r 2 )
(E
Thm,
t w:(x).
t h e r e i s a neighborhood
E ,
compactness allows one t o choose
orbit for
5 0, then y
0, then
> 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 <
oO > 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
<
Y , % ~ )5
for a l l
..
d(fNi+j
j ~
%i) f
'
+
?, and
for
NOW l e t i.
is a
y
j
3c6 < 1-rN
f o r each
0 5j
W:
0
(x) n
{ ~ i } - - < i < ~be a
for
0
for
and
i
5j
I N .
r -shadowed. 1 f , then
0 5j
5 N, one
such t h a t
A
2 N , d(f
Ni+j
Let
small enough t h a t
6
WE0(y)
'
~*%i+j)
x,y
> 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) <
implies
such t h a t each
d(f j xi,xi++)
~ ' - ~ 2< c1. Observe t h a t i t is enough
For i f we can do t h i s and
d(x,y) <
6 > 0
d(z,y) 5 cd(x,y).
6-pseudo-orbit
for
2.
?
in
A , and l e t
b > 1 be
an .integer. We w i l l produce a point Once t h i s is done, we l e t d(f j Nw,x.) I
for a l l
( E
J
Now, we.have that for
j
w
wb
d(f ( x ~ - ~,xb) ) < 6.
d ( ~ ~ ~ <x 6~ - ~ )
(aj
d(f
I f we s e t
zj+l
satisfies
(1)j+l
z
j
Set
and
N
z
for
i=O
fmNrw:
E
for
1 j 1 5b.
{ab), and then
=
-
xb. Assume, inductively,
such t h a t
k-l - N i zj~%-j+k ) < 3 c 6 ( 1 h )
-
d(f j N w ,x ) 5 b j
a s required.
j
N
(lIj
Nk
such t h a t
be an accumulation point of
2 1, we have a point
and
A
E
(£ % ( j + l ) ) n 0 (2)j+l.
wS
€0
I ( k 5 j .
( Z ) 1, then we claim t h a t j
zj+l
F i r s t observe t h a t by (1) j , d(%-fyZj)
< 26, so
zj+1
< -
~ - ~ 2 c 6f o r For
~r t
-
k
I d(p"b-(j+l)
Czca
< 2c6. -
N
c , d(f zj+lyi?xb(j+l))
9"b-j)
+
Thus,
which is (1) j+l.
< 6
S(z),wehave f zjc1 e w EO
Since
,zj)
can be defined.
r l o o , by the choice of d(Zj+l"L(j+l) ) <
N
d(f %-(j+l)
d(f
2 0.
l l k ( j + l ,
1, t h i s gives
+ 1, it
2 5k zj
d(Pzj+l.rb(j+l)+k
)
5 2c6
+6
< 3c6
.
~f
gives
which is (2) j+l. Now, s e t t i n g ub = 2ibzZb, 3c6 -N 1-X
for
< E
IjI(b
~ o r o t l m y(3.6).
FOP
and (c)
E
10
smaZZ, set w ~ ( A=) xuA~E(x) and
6
of
A
such that
nf%
ne Z
structure. f-l.
u
S
Wc(A) =
c w~(A)
ns
Proof.
by
d(fjNwb,xj) 5
asrequired.
fiere is a ne-ighborhmd UW;(X). xe A
a
and (2IZb give us
=
Statement (c) follows from (a) and (b) and t h e l o c a l product We prove only (a) s i n c e t h e proof of (b) follows replacing
Let
r > 0 be small and choose
6
e (
0
f
s o t h a t every
E
pseudo-orbit i n A can be --shadowed by an o r b i t i n 2 6 6 E (o.+) SO t h a t d(x,y) 5 6 implies d(fx,fy) 5
.
A.
Let
Choose
U = B6(A) =
Iy
M : d i s t (y,A)
E
xi
+
Let
d(f y,f x) 5 d(f y,xi) A neighborhood
neighborhood f o r
of
A
-shadowi'hg i t .
2
i
E
x) 5 2
E
+2 =
Then, f o r
so. x
E,
2 0,
w:(~).
E
A. x
whose forward o r b i t s t a y s
is a c t u a l l y forward asymptotic t o a point of
A
i
a s i n Corollary (3.6) i s c a l l e d a fundamental
A
Note t h a t (3.6a) implies t h a t any p o i n t near
.
i s a 6 pseudo-orbit 1-
+ d(xi,f
U
fl\fnu, then f o r each i 2 0 t h e r e is an n
SO
be an o r b i t i n ,i
i
I
54,
6
E
5 6.
d ( f x,xi)
61
i {f y}
x
i
AsucC that
d(f i + l ~ , ~ i + l )
If
( 61.
A.
The next r e s u l t a s s e r t s t h a t t h e o r b i t s t r u c t u r e s of hyperbolic s e t s p e r s i s t under perturbation.
Theorem (3.7) ( S t a b i l i t y ) . Let I\ be a hyperbolic s e t for product structure.
There are neighborhoods- U of
i n ~ i f f b so that i f
g
N, then A(g)
E
g with a local product structure.
b
g
: A -+ ~ c g ) such that
Proof.
Let
U
3
gh
g
l o c a l product s t r u c t u r e .
c1
near
Gl pseudo-orbit f o r
(0,-5-)
i
f
in
Since
g.
n
P ( u ) i s a hyperne Z ~ ( g )c i n t U, i t has a
~ ( g )=
Also, s i n c e
.
f 1 ~ ( f ) Let A(£) can be d(x,y) 1 6
g be such t h a t
n g n ( ~ ) ,then f o r each n
d(xi,g X) 2 6.
varies continuously with
To produce t h e homeomorphism
be such t h a t
U = B6(A(f)), and l e t
x E A(g) =
g
f , then
be an expansive constant f o r
E
h
is
by theorem (2.2).
6
n
g
g
Then, l e t
ngnui s a hyperbolic s e t for
be a small fundamental neighborhood f o r A a s i n
bolic set for
every
N of f
A
If
Ler
and
g
.
> 0
and
M
Moreover, there i s a horneomrphisrn
= h f,
Corollary (3.6)
E
=
in
A
with a local
f
i
61 e (0 , E ) be such t h a t E
T; -shadawed by an o r b i t .
t h e r e i s an
6 1-pseudo-orbit.
d(fx,fy) 2
implies
d(gx,fx)
d ( f ~ ~ , x ~ 5 +d(fxi,fgix) ~ )
i+l d(g X , X ~ + ~ ) , {xi} ( ~ ~ ,i s a
h, we f i r s t l e t
i>
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
<-invariant s e t and
alCA (or sometimes
The set
A.
ZA i s
i t s e l f ) is c a l l e d a
CA
s a s h i f t of f i n i t e type. meorem (3.8). f
.
Let
A
be a hyperbolic basic s e t for a C'
Then there are a matrix A of
tinuous surjection
s : CA
-+
0's
and
and a finite-to-one con-
1's
A so that the following diagram conrmutes
This important r e s u l t was proved by S i n a i [53] by Bowen [ 4 ] f o r general basic s e t s . of
when
A = M, and l a t e r
The proof involves s p e c i a l coverings This i s t r e a t e d
by l o c a l product s e t s c a l l e d Markov p a r t i t i o n s .
A
diffeomorphism
n i c e l y i n [ 7 1. The mapping of
s-l(x)
matrix
A
n
is
i s bounded by [ 91.
1
-1
f o r each
N2
The space
on many p o i n t s i n
x where
and t h e c a r d i n a l i t y
CA
i s t h e order of t h e
N
codes t h e a c t i o n of
CA
f
on
i n a very
comprehensible way, and can be used t o prove many f a c t s about instance the minimal sets i n
A
a r e zero-dimensional
can compute the number of periodic p o i n t s of period each
f
IA.
For
[5], [22], and one n
of
f 1681
for
n 2 1.
Remark:
1. Theorem (3,8) holds f o r hyperbolic s e t s with l o c a l product struc-
1
t u r e s ( i .e. witheut .assuming f A -has asdense .orbit)
.
However, i f
A
is a
hyperbolic s e t with a l o c a l product s t r u c t u r e , one can prove t h a t ~ ( [ A f )=
Alu
...uAn
is a finite union of subsets Ai such that f (Ai has a dense
orbit. Here ~ ( ]A) f
is the non-wandering set of
f restricted to A.
decomposition is similar to the one we will give in Proposition (4.2).
This Since
one is usually interested in studying recurrence phenomena of f (A,it is. no harm to assume at the outset that f 1 A
has a dense orbit.
4.
Hyperbolic Limit S e t s
Let u s now d e s c r i b e . s o m e diffeomorphisms a l l of whose r e c u r r e n c e is hyperbolic. A point
x
M
E
O
..
i s a n w - l i m i t p o i n t of
f
and a sequence of i n t e g e r s Similarly, O
x
..
such that
fni(yl
as i
+x
The set of a - l i m i t p o i n t s of l i m i t p o i n t s of Let
La(f)
y
when
L(f)
+
i f there a r e a
and a sequence
g
-+ w.
i s denoted
a(y)
be t h e set of a - l i m i t p o i n t s of
t h e l i m i t s e t of
o r b i t s approach
y
f
and t h e s e t of a-
i s denoted w(y).
t h e s e t of w l i m i t p o i n t s of L(f)
i f there a r e a point y E M -n x a s i + a). srich t h a t f i(y)
i s an a - l i m i t p o i n t of
L(f)
f.
f.
Define
f,
and l e t
La(£)
be
~ ( f ) = c R ( ~ ~ ( f ) u ~ ~W ( ef )c)a.l l
It i s c l e a r l y c l o s e d and i n v a r i a n t , and a l l
i n t h e f u t u r e and p a s t .
We s h a l l s t u d y t h e s i t u a t i o n
is hyperbolic.
F i r s t we r e c a l l t h e t o p o l o g i c a l n o t i o n of t h e index of a mapping. See [ 1 0
3
f o r a g e n e r a l treatment of t h i s theory.
u n i t b a l l i n Eln
and
n-1 aA=S
be t h e
continuous map w i t h no f i x e d p o i n t s on on
t o be t h e degree of t h e map
A
denoted
Ind ( f , A).
D=$(A),
and
f
If
$:Eln + R"
(n-1)-sphere. aA,
A
Let If
be t h e c l o s e d f :A
-+
R~
one d e f i n e s t h e index of
x + - e f l
for
x E
sn-l.
This i s
D
to
R~
without f i x e d p o i n t s
on
aD, set
D.
It does n o t depend on t h e o r i e n t a t i o n p r e s e r v i n g homeomorphism It i s a s t a n d a r d f a c t t h a t i f
in
D.
f
is a n o r i e n t a t i o n p r e s e r v i n g homeomorphism,
i s a continuous map from
~ n d ( ,f~ ) = ~ n d ( $ - ' f $ , A).
is a
We c a l l
1nd(f ,D)
Ind(f,D)#O,.
then
t h e index of
f
f
on
$.
has a fixed point
1 For example, c o n s i d e r t h e mappings f (x,y)= ( ~ , 2 y , ) f 2 (x,~)=($,+
1 f3(x,y)=(7 x,-2y)
on IR
1nd ( f l , , ~ ) = -1 and
2
.
If
A
,
y)
i s t h e u n i t d i s k i n R2 , one may compute
Ind(f2, A) = Ind(f ,A) = 1. More g e n e r a l l y , i f 3
L : LRn + R ~i s a l i n e a r hyperbolic automorphism, then we may compute i t s index on t h e u n i t d i s k
(-llU
Ind(f A) =
if
A
a s follows.
u = dim wU(o ,L) , then
If
preserves t h e o r i e n t a t i o n of
L
w U ( o , ~ ) , and i t
9
is
(-11~'~ if
L r e v e r s e s t h e o r i e n t a t i o n of
% IfI . D is any
IndCf, A)
n-disk containing
wU(o,I,). 0
I n any case,
i n i t s i n t e r i o r , then
IndcL,D) = Ind(L,A)!.
Proposition 14.11.
If L ( f l
<s hyperbo tic, then the periodic orbits are
dense in L l f l . Let
Proof.
E'@?=T
M LCf 1
b e t h e continuous s p l i t t i n g of
by the d e f i n i t i o n of hyperbolicity.
Then
TLCf)M given
.uLn
LC£) = Liu..
where
Li n L = 4 and each L i s a closed £-invariant s e t on which dim EU and j i dim Es a r e constant. Let x E Li f o r some lSiSn, and l e t y E M be such n t h a t x E ~ b ) . Since u b ] LCf) = Lj and t h e c o l l e c t i o n {L. } 3 i=1 c o n s i s t s of c l o s e d d i s j o i n t s e t s , one s e e s t h a t ~ ( y )c Li. Let exp be
.
U
-
t h e exponential map of a Riemann m e t r i c on For
z
z 1' 2
near x,
let
TM adapted t o
:T M zi p a r a l l e l t r a n s l a t i o n along t h e geodesic from
+
6>0,
and
and l e t
U =
(u,v)
r E:
a small product neighborhood of Since
~ ( y )c Li,
Li f o r a l l nsN, n f Zy a r e near x Let fjy
C
T
Z1'Z2
:E
: lu Is6
x.
Write
there i s an integer
and t h e r e a r e i n t e g e r s in
Li
T M be t h e map induced by z2 z t o z2. Choose a small 1
1
Iv s6)
.
Then
expxU
is
U1 = exp U. X
N>O
so that
N
SO
fny
that
i s near n f ly and
U1.
be t h e connected component of
remains near
A.
for
n1sjsn2,
fnly
in
U1
n -n n f 1 2u1.
Since
i t keeps behaving hyperbolically, and
we s e e t h a t
C
and
f n2-n1~ a r e a s i n f i g u r e 4.1.
I;
(
f
n2 v
Figure 4.1
f P2'9w i t h Consider t h e hyperbolic l i n e a r map @ = .rZ o ~ Z n 2 1 1 zl = f n l y and r 2 = f 2y. We have t h a t @ maps Tz M t o Tz M and i t s 1 1 index on a neighborhood of 0 i n T M i s +I. But from f i g u r e (4.1) z 1 fn2-nl has no f i x e d p o i n t s on t h e boundary of C and i t s index on C i s t h e same a s t h a t of
@
on a small neighborhood of
f n z m n ~ has a f i x e d p o i n t i n in
Lo(£).
The proof f o r
C.
La(£)
TZ M. So 1 This proves t h e p e r i o d i c p o i n t s a r e dense
is similar.
Proposition 14.21 LSpectraZ Decomposition). L ( f l = AIu...uAn
0 in
If L (f) is hyperbolic, then
where {Ail is a disjoint coZZection of closed invariant
sets with periodic points dense. Moreover, for each i, topoZogically transitive, and Ai Proof. (4.1)
Let that
h-classes.
Per(f)
implies
.
pl,p 2 y . . .
which converge t o a p o i n t manifold theory.
has a Zocal product structure.
denote the s e t of p e r i o d i c p o i n t s of
L(£) = C&er ( f ) For i f
f /hi is
Observe t h a t
f
W e know by
has only f i n i t e l y many
i s an i n f i n i t e sequence of p o i n t s i n
y,
then f o r l a r g e
The same argument shows t h a t
H Cf). = H (f). P q
f.
Now, j u s t t a k e
H
P1
i,j, pi-
pj
by s t a b l e
CIHp (f) n C%(f)
...H Pn f
( ),
)
Per(£)
# 4
t o be t h e
distinct h-classes, and let only need to prove that
If x, y
E
E
Ai
u8b) E
is small, then any point z
>O
Ca
on $(y),
so
n w:(y)
in ) x ( : w
with
accumulates on W : ( X )
and
wS(o (Pi))
and
accumulates
is a limit of transverse homoclinic points of
2
We
is a point of transverse intersection of W;(X)
(£1 Pi But wU(o(pi)) =
Ai
= C&
(f) be the h-closure of pi. pi has a local product structure. I$
obi), i.e.,
E
transverse intersection of wU(o(pi))
and W~(O(~ )). In the proof of i proposition (3.2), we saw that such points were limits of the orbits of Hence, by theorem (2.3),
transverse homoclinic points of pi. of points in H (£1. pi
CoroZZmy 14.3).
M =
u
If =
$(X)
xaL (f )
Thus, z
for wu(x) Remark:
CRH (fl. pi
Llf)
C)
i s hyperbolic, then
W~CX). That i s , each stable and unstable s e t i n
x
E
L (f) = Alu
(4.2),
product structure. If x (3.64,
is a limit
xeL (£1
Proof. From
By
E.
z
w:(~)
E
...uAn where each
hi has a local
M, then the w-limit set of x is in some
for some y
E
Ai.
Then
wS (x)
=
W~O,).
The proof
is similar. One can strengthen the condition that L(f)
requiring that all of the non-wandering set Q(f)
is hyperbolic by be hyperbolic.
If in
addition one assumes the periodic points of f are dense in Q(f),
then one
gets what is called an Axiom A diffeomorphism. Proposition (4.2) was first proved by Smale in [56] for Axiom A diffeomorphisms, and Corollary (4.3) was first proved in [16] for Axiom A diffeomorphisms. Proposition (4.1)
(with
a different proof) and the present treatment of (4.2) were given in [28].
Let us lo'ok at some examples.
1. This f i r s t e x a m p 1 e . i ~an e x t e n s i o n of t h e horseshoe map t o an Axiom A diffeomorphism
f
conjugate of t h e 2 - s h l f t , p2. and
W e take
p2
s2
on h(f)
,
whose l i m i t s e t c o n s i s t s of t h e previous
and two hyperbolic f i x e d p o i n t s
t o be a s i n k ( a t t r a c t i n g f i x e d . p o i n t ) a s i n f i g u r e (4.2),
t o be a sourke ( r e p e l l i n g f i x e d p o i n t ) "at
pl
p and 1
m."
Figure 4.2 2.
Anosov d i f feomorphisms These a r e diffeomorphisms
f
f o r which t h e whole manifold
i s hyperbolic.
The simplest examples a r e t h e following.
M = T~ = IRn/zn
be t h e n-torus,
Thus,
A can be represented a s an
determinant a s a map of lRn = ES
@
fl. n
R
Assume A to
tRn
and t h e s p l i t t i n g
map
XI Es
X
E'
(B
.S
en
be an automorphism.
m a t r i x w i t h i n t e g e r e n t r i e s and
so that
A(E')
1. F u r t h e r ,
EU p r o j e c t s t o an
c o n t r a c t i n g and
-t
-
A
There is a d i r e c t sum decomposition
n EU and a norm on R
1 ~ 1 <~ 1,~ 1and I A I E I
nxn
A : E"
Let
has no eigenvalues of norm 1. W e t h i n k of
also.
-1 u
with
and l e t
M
-u A/E
= ES,
ACE^)
A induces a m a p Ti-invariant
expanding.
i s c a l l e d a l i n e a r t o r a l automorphism.
Hence,
= EU,
: T"-+ T"
splitting .-
ES
A i s Anosov.
$
EU
The
A simple example i s given
I n a l l known cases, an Anosov diffeomorphism
f : M+M
has
L(f) = M.
But, t h i s property has only been proved under a d d i t i o n a l r e s t r i c t i o n s on t h e t o p ~ l o g yof
It holds i f
M.
M
i s homeomorphic t o a t o r u s [ 23
1,
on
c e r t a i n manifolds c a l l e d i n f r a n i l manifolds [23], o r i f dim ES o r EU = 1 [27].
However, i t can be proved t h a t an Anosov diffeomorphism s a t i s f i e s
Axiom
A ([ 7
1
o r [ 281 1.
Three o t h e r a d d i t i o n a l problems on Anosov
diffeomorphisms are: a.
If
f : M +M
i s Anosov, is t h e u n i v e r s a l covering space of
d i f f eomorphic t o
R*?
b.
Does every Anosov diffeomorphism have a f i x e d point?
c.
I s every Anosov diffeomorphism topologically conjugate t o an
in£r a n i 1 manifold automorphism b e e [I11
3.
M
for definition)?
Gradients Let
,+ : M
+R
degenerate c r i t i c a l points.
be a Given a
c2
real-valued function with non-
C
d e f i n e s t h e gradient vector f i e l d grad 4 g
Riemann m e t r i c
g
on
E
T M.
TM, one
by
gx (gradg 4 (x); Y) = Txb Cg), f o r
x
EM,
Y
It is e a s i l y seen t h a t
grad 9 i s a C' vector f i e l d . g a r e t h e orthogonal t r a j e c t o r i e s of t h e l e v e l s e t s of :$
X
Its s o l u t i o n curves
Let
f
be i t s
time-one map. Then
LC£)
c o n s i s t s of hyperbolic f i x e d p o i n t s and coincides with t h e
c r i t i c a l p o i n t s of 4. Simple examples on t h e 2-sphere below.
s2
We have drawn i n v a r i a n t curves.
and t h e 2-torus
T~
a r e pictured
F i g u r e 4.3
4.
I n t h e f i r s t t h r e e examples, t h e l i m i t s e t was a c t u a l l y equal t o t h e
non-wandering set, s o each
sL h a s L ( f )
f
s a t i s f i e d Axiom A.
T h i s example of
f i n i t e l y many h y p e r b o l i c f i x e d p o i n t s w i t h
Q(f)
f
on
infinite
and n o t h y p e r b o l i c .
F i g u r e 4.4 The c i r c l e s r e p r e s e n t s o u r c e s and s i n k s and t h e r e a r e two s a d d l e p o i n t s p1
and
p2.
W e have t h a t
u W (pl) nWs(PZ)
t r a n s v e r s e i n t e r s e c t i o n s , and
u
S
c o n s i s t s of two o r b i t s of
W (p2) n W (pl)
of non-transverse i n t e r s e c t i o n s . General diffeomorphisms w i t h
Also, L(f)
c o n s i s t s of a n o r b i t
o(x)
Q ( f ) = o(x) u L ( f ) . h y p e r b o l i c may b e viewed a s looking
somewhat like these exampleswithcomplicated hyperbolic sets replacing the fixed points. We
now briefly consider the concepts of structural stability and 3-
stability. A diffeomorphism f : M
+
M is called
if
b&b&
bh&M&$
there is a neighborhood N of f in ~ i f f h such that for each g in N,
.
there is a homeomorphism h : M + M such that h f = This means that 8 g ghg the entire orbit structure of f persists under 'C small perturbation. If f satisfies Axiom A, then one says f satisfies the strong transversality condition if wU(x) Recall that wU(x)
is transverse to wS(x)
for each x
M.
E
are manifolds in the Axiom A case.
and 'W (x)
Theorem ( 4 . 4 ) (Robbin, Robinson [50], [51J).
If
f
s a t i s f i e s &om
the strong trm2sversaZity condition, then f
i s structumZZy stable.
A and
The extended horsehoe diffeomorphism in example 1 satisfies Axiom A and strong transversality, so it is structurally stable. It is amusing and non-trivial to try to prove directly that this diffeomorphism is structurally stable. Theorem (4.4) had been proved by Palis and Smale in the case when Q (f]
is finite 691 , and by Anosov for Anosov diffeomorphisms. We remark
that if L(f]
is hyperbolic and f satisfies the strong transversality
condition, then f satisfies Axiom A 1281.
Hence, theorem (4.4) holds
It has been conjectured that Axiom A and
with these weaker assumptions.
strong transversality are necessary and sufficient for structural stability.
A weaker concept of stability than structural stability is that of Q-stability. One says f in Q-stable if there is a neighborhood N in ~ i f f h such that for each g h
g
: Il(f]
+
If f
Q(g)
E
of f
N there is a homeomorphism
such that h f = ghg. g
satisfies Axiom A and 3(f)
= bcf] = A1u
...uAn
is the spectral
decomposition, then a
i~ and, for each Osj
%orem
( 4 . 5 ) (Smale [ 573 )
.
ws(ni
If f
)
..A ir
,.
is a sequence A
CV&
such that A = Ai i~ r
# 9.
jl-1
*
satisfies Axiom A and has no cycles, then
Again, it is sufficient to assume that L(f)
is hyperbolic and there
are no cycles 1281. Also, in this case there is the conjecture that'Axiom A and no cycles is equivalent to O-stability.
In the direction of the con-
verses to theorems (4.4) and (4.5) see Pliss [ 421, Mang [ 211, and theorem (1.2) in 1301. While there has been relatively little progress on the precise converses to theorems (4.4) and (4.5),
it has been shown that somewhat stronger notions
of stabilities do characterize Axiom A and strong transversality and Axiom A and no cycles. Say that f is ab~o&t@
Q-stable if there are a neighbor-
hood N of f in ~ i f f band a constant k>O so that for g < N
there is
a homeomorphism h : Q(f) .+ Q(g) such that gh = h f and the CO distance g g g from h to the inclusion i : Q(f) -+ M is less than k times the CO g distance from f to g. Guckenheimer 1671, extending earlier work of Franks
[64J,proved that an absolutely Q-stable f must satisfy Axiom A and must have no cycles.
The analogous result holds for absolute structural stability
1651. Another characterization of Axiom A and strong transversality is Franks' time-dependent stability [66]. The main importance of the Axiom A and no cycle diffeomorphisms is that, at present, they give the largest open set of diffeomorphisms whose orbit structures are well understood.
5.
Attractors
Let
f :M
- topology c1
be a
M
-+
diffeomorphism.
A closed
is an a t t r a c t o r i f t h e r e i s a compact neighborhood
A
U
£-invariant s e t of
A
such
that
f(U) c i n t U , n f n ( u ) 4 A , and £ [ A has a dense o r b i t . Thus, f o r ria every x E U , W(X) c A where w (x) is t h e w - l i m i t s e t of x. The open = ( y c M :. ~ ( y )c A ? = U f - % is c a l l e d t h e b a s i n of n20 a g i ~ e ndiffeomorphism i t is important t o describe i t s a t t r a c t o r s . set
w'(A)
A.
For
They g i v e
t h e time e v o l u t i o n of c e r t a i n open s e t s i n M. I f an a t t r a c t o r
i s hyperbolic then one has considerable information
A
about its s t r u c t u r e .
Let us begin with a few examples.
The simplest example i s , of course t h e o r b i t of a p e r i o d i c sink.
i s the o r b i t
o(p)
i s a union of
n
of a p e r i o d i c p o i n t
p
of a period n
This
T fn P has a l l its eigenvalues of norm l e s s than 1. In t h i s c a s e t h e r e is a small n-1 neighborhood U of p such t h a t fn(u) c i n t 8, and w s ( o ( ~ ) ) = [ ) , f - m ( u f ' ~ ) 0 j=o open c e l l s which a r e permuted. by
such t h a t
f.
At t h e o t h e r end of t h e spectrum, we have t h e t o p o l o g i c a l l y t r a n s i t i v e Anosov diffeomorphisms which were described i n t h e l a s t s e c t i o n . A t h i r d and i n t r i g u i n g example is known a s t h e solenoid.
Let
s1
{z
and l e t
D2
(z
E
$
I
: z 1 = 1) be t h e u n i t c i r c l e i n t h e complex plane,
: lzl
Consider the mapping Z
2
W
+)
I f we think
f ( s l x D2) c i n t ( s l f i g u r e 5.1.
x
D2)
s 1)
be t h e u n i t 2-disk.
f :S1
s1 and
x
D2
f
x
D2
-c
s1
x
D~
defined by
a s the s o l i d t o r u s i n wraps
1 S x D2
R
f (z, w) =
3
,
then
around i t s e l f twice a s i n
Figure 5.1 2 2 2 f ( I z ) x D ) c Cz 1 x D , and
Also,
t h e two d i s k s
f(I.1
2 D )Uf((-z)
x
2 2 f ( s l x D ) n ( z 1 x D~ 2
x D ) =
I(z2,
i ,
-
z +w u ( ( z2, - 5
n
A =
that
< 11. Using theorem
: IwI
( z , V) E A , W;(Z, w))
-
c
]w/
1)
L
it i s easy t o verify
(2.2)
is a hyperbolic set.
fn(S1 x D') n2 0
.
c o n s i s t of
Also, f o r each
D~ c wS((z, w)).
X
2
It i s e a s y t o s e e t h a t
For t h i s , i t s u f f i c e s which meet end,
t o show t h a t i f
t h e n f o r some
A,
g ( z ) = z2
let
If I
(a)
for
n
1
>
n 2 N
Let
1 r : S
x
for
i > 0,
expands (:w
D
0 and a p o i n t
a point Since that
z
( z ' , w') rw:((z',
E
2
zl
s1
-+
E
.
w'))
1
S
implies
,
g n ( ~ )= S
( 2 , ~ ) )f o r s m a l l
.
s1
x
D~
+. Toward t h i s
there i s an i n t e g e r
> 0
E
S
1
,
E
N > 0
.
1
be t h e p r o j e c t i o n .
and a s m a l l
1
open sets i n
Note t h a t
i s an i n t e r v a l i n = S
are
V
fn(U n A) n V n A +
g - n l ( ~ ) such t h a t
U n A
rfn2wu((z' ,w')) E
E
S1
and
U
n 2 0,
i s an i n t e r v a l i n
such t h a t
-i
f l ~h a s a dense o r b i t u s i n g p r o p o s i t i o n (3.1)
Let
,
(z,w)
E
V n A.
Taking
s o we may choose an i n t e g e r
2 f n l ( f z l ] x D )c V.
Now pick
such t h a t wU((z',w')) c U n A . we can f i n d a n
Then, t h e r e i s a p o i n t
p
in
n
2
> 0
such
n f 2 ~ ( ( z , w ) ) m ( z 1 I x D 2. n +n f
(U n A ) n V n A s $
Hence,
f
-n
n f 1p c V n A ,
2 p ~ U n A and
as required.
The construction of t h e solenoid (as a hyperbolic a t t r a c t o r ) t o Smale
.[ 56
so
i s due
1.
It l e a d s t o a general c o n s t r u c t i o n of one dimensional hyperbolic
a t t r a c t o r s due t o Williams
To describe t h i s construction we need
[ 611.
some d e f i n i t i o n s . Let
01
Let
S be t h e s e t of f u n c t i o n s
: IR +
and
IR
$2 : IR
+
R
be defined by
B
8
$2
where
B varies
through t h e r e a l numbers. The graphs of elements of
S have i n f i n i t e c o n t a c t with
A compact branched 1-manifold
K
is compact Hausdorff topological
space s a t i s f y i n g t h e following property. such t h a t
each point
x
in
B
There is a f i n i t e subset
B
c
K
has a neighborhood which i s homeomorphic t o
a f i n i t e union of graphs of elements of
S, and each
x
E
K
-B
neighborhood which i s homeomorphic t o a r e a l open i n t e r v a l . Typical p i c t u r e s of branched 1-manifolds a r e in Figure 5-2.
Figure 5.2
has a
The s 6 t
in
B
i s c a l l e d the branch s e t
K
-
There i s a f i l i i t e open covering t h e r e i s a homeomorphism Iba Y:
= ((u, (;(u))
4;
=
R2 :
: Ua
a
S
(
of
such t h a t f o r each
K
where
Ya
K.
Ya
P
... u Y:
and
. ?he open i n t e r v a l s correspond
, = 1 , .n
( (Ua,(a)]
and f o r each of
a,
Iba(Ua)
f to
0
IR
-1 (a
.
cr
defines a
a s usual by saying t h a t a function
d i f f e r e n t i a b l e s t r u c t u r e on
f :K
extends t o a
cr
cr
is
IR
-P
to
tangent bundle.
function- from neighborhood
Since t h e graphs of two elements of
One defines
Riemann metrics and
branched 1-manifold can be
C
embedded i n
We now always assume t h a t with a fixed Riemann metric.
Let
1 '1
1 so that
For example, i f
(p),
CI
+
K
n g(z) = z , n > 0 K
and l e t
on
r
TK.
s o t h a t t h e r e a r e constants
1 K = S
be t h e map
,
g
cr
0.
2
XEK,
then
n10, g
described
and V E Tx K.
i s expanding.
t o be a wedge of two c i r c l e s , say g
cr
branched 1-manifold
I T ~ ( ~ " ) ( Vr ) c~ l n l v [ f o r a11
Another example is t o take S n S2 1
for
be t h e induced norm on
An expanding map of K i s a CLmap g : K c > 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
.
- l),
p r 0,
one has Vander Pol's equation which comes up Y n facuum tube circuits (see e.g. [ 591).
Under certain conditions on. f
the system x = v
in [ 15]),
stable periodic solution.
on IR
2n
.
If n
-
(as in theorem 10.2
i"
f(u) du has a single asymptotically
If one has n
such systems, one gets the system
This system has a unique invariant attracting n-torus. 2
3, there are small perturbations of both systems (1) and
(2)
which possess non-trivial hyperbolic attractors. As J. Ford pointed out to us, a recent paper in Science [12] gives related experimental results. In particular, the broad band noise spectrum in figure 2B of
[12 ] may be
due to a non-trivial hyperbolic attractor. 2.
Williams has extended theorems (5 .l) and (5.2) to higher dimensional
"expanding" attractors
[63], and has given general conditions for
topological 'equivalenceof one dimensional attractors 162 1.
6. Attractors
- ergodic theory
We begin with some notions from ergodic theory. Let f : X + X be a homeomorphism of the compact metric space X. invariant Borel probability measures p
on X.
in X.
for every Borel set B.
That is, p
be the set of M(f) is a
E
such that p(x) = 1, and
regular Borel non-negative measure on X p(f-l~) = p(B)
Let M(f)
Let 8 be the o-field of Borel sets
is called ergodic if whenever B a 8 and
The measure p
f (B), = B,
we have p(B) = 0 or 1. That is, any invariant p-measurable set has measure zero or one. An equivalent condition can be given in terms of real valued functions ( : X
+
This means that
4
R.
Such a function 4 : X
+
R is invariant If
QOf = $.
is constant on orbits of f. Then p
is ergodic if and 1 only if any invariant function 9 in L 01) is constant almost everywhere. 1 (of (, then ((a) I(dp for p-almost ( a L ()I) satisfies That is, if
-
all x.
-
From the Riesz representation theorem, one may think of M(f)
subset of the dual space CQ)*
where C(X)
is the space of continuous
real-valued functions on X, and we set )I(() = gives a topology on M(f) sequence p i for each space. M(f)
E
E
M(f)
i
$dp for $
E
C(X).
)I
E
M(f)
if and only if pi($)
With this topology, M(f)
It is also a convex subset of
C(X)
*,
6
Y
is the point mass at y,
6£kX) k=O
~($1
and the extreme points of is non-
then any weak limit of a
subsequence of
($
+
becomes a compact metrizable
are the ergodic invariant measures of f. Note that M(f)
empty because if
This
(called the weak or vague topology) so that a
converges to
C(X).
as a
is in M(f). n ~ l
The most b a s i c r e s u l t of ergodic theory i s the following theorem.
Theorem (6.1) (Sirkhoff ergodic theorem). and Zet p
the. compact metric space X,
E
Let f be a homeomorphism of M(f). For any ( E L1 (u),
there is a set A c X: of- p-measure 1 such that for . n-1
lim n-
1. $ (f kX) n k=O
for
x
.exists. Moreover, if we set
i
then
€,A,
x
A ,
E
A
E
~'(p),
i, and
jof =
( (x) =
14
n-
n-1
- 1
$(fkx)
k*0
l i d p = I(dp. *
I.f t h e measure
11
i n theorem (6.1) i s ergodic, then t h e f u n c t i ~ n(
must b e constant p-almost everywhere,. s o where.
Thus, f o r p-almost a l l
t h e o r b i t of If
M
x
m(E) = JM
If
M
J
t h e time-averages
approach t h e space average
n-1
;; 1
almost every((fkx)
along
k=O
I
J(dp.
i s an o r i e n t a b l e compact manifold, t h e r e is a n a t u r a l Borel
measure which can be defined. and define
x,
;&) = [idp = [(dp
m(() = XEaY
j
f o r any (
(ow
M where
xE
r*m(E) = m(?;'~).
m on
R
w
be anowhere vanishing n-form, n=dimM,
E C (MI.
That is, define
i s t h e c h a r a c t e r i s t i c function of a Borel s e t E:
i s not o r i e n t a b l e , l e t
Take t h e measure
Let
r: % + M be an o r i e n t a b l e 2-to-1
and l e t
Any measure
n*m
m
on
w i l l be c a l l e d Lebesgue measure on M.
be t h e measure on
M
M
covering.
defined by
induced by an n-form on
Dividing by
or
M
mm), we w i l l assume
m(M) = 1. Ruelle has proved t h e following theorem. p E M(f)
containing
is t h e s e t of p o i n t s x, p(U)>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)
with and
,
be an
V
pA(v)
g2 be continuous functions such t h a t
and
xctu
and
C&J
with
U
$(A)
n>O a n d x E
fl@)-A,
xF
s
s
xu
we have
k xU(f X ) 2 Jgldp 5 pA(F) 2 pA(u)-e.
n
k=O
xcLU(fkx)
i s a r b i t r a r y , and
pA(C&U)
5
-
I
42duA s pA(v)
pA(CIJJ)+E=
pA(U), we get
Before proceeding t o t h e proof of theorem (6.21,
l e t us note that it
implies a celebrated theorem of Anosov.
Theorem 16.3).
Let
f
be a C'
diffeomorphism and suppose f
topoZogicaZZy transitive Anosov
preserves a measure v which is absoZuteZy
continuous with respect to Lebesgue measure m. Proof.
Since
v
also.
Thus,
Let
6
E C(M).
Let
A c M
Then v
be such t h a t
i s absolutely continuous with respect t o
Q1 (x) A
= lim
n*
n-1
- 1
g (fkx)
k = ~
my
is ergodic.
m(A) = 0 and
we have
v(A) = 0
is constant v-ame. Hence lidv
-
I+dpAmv(M). By the bounded convergence
theorem and the fact that v is f-invariant.
'
0 ,
4dv =
14 duA. Hence,
-. Remarks. being C2
v
vA
and
v is ergodic.
1. Theorem (6.2) and (6.3) hold under weaker assumptions than f
.
The proof given here works just as well (with straightforward
changes) if f is
.O
is H3lder continuous of order e(.
c1
and its derivative
Anosov'points out in.[ 21 that theorem
(6.3) ,holds if the modulus of continuity ~ ( r )of Tf satisifes b
I
dr <-
for some b>O
0
.
I have not checked the details, but I expect that the proof here gives (6.2)
(and hence (6.3) as well) under this assumption. Theorem (6.2) does
not hold for all
C'
f. An example where it fails can be obtained by
embedding the Bowen's example of a horseshoe with positive measure 181 in a hyperbolic attractor. However, it is not known whether theorem (4.3) is false for C1 f. 2. The Bernoulli shift B (pl,. o :
EN + EN
PI, p
on
..,pN)
be the full N-shift with
IN N= { 1,...,N} E .
- ,.PN be positive numbers such that a,. .,,N
Bore1 sets of
by p({i})
IN.
Then
=
pi.
Let
is defined as follows. Let
1
Let
pi = 1. Define the measure
i=l
5
be the product measure on the
is invariant under the shift
0,
and the pair
(u,C)
A measure preserving
is called the Bernoulli shift B(pl, ...,pN).
transformation T : X + X with probability measure m is called 8ehnou&% if (T, m)
is measure-theoretically conjugate to some B(pl,
...,pN).
This
U'
means there is a .measurabletransf.ormation S : X A c X, B
c
EN
with m(A) = 1, ;(B)
onto B, and uS(x1
=
-t
EN, and subsets
= 1 such that S maps A
bijectively
for x r A.
ST(x1
A , theorem of Otnstein [38] says that
theoretically conjugate to B(ql, ...,qr)
B(pl,.
..,pN)
is measure
if and only if
If 4 A is topolagicelly rnixlng, then HA is Bernoulli, !Che proof involves representing flh as a finite-to-one quotient of a subshift of finite type
lA
lA
via theorem (3.8),
and obtaining
pA
from a measure on
which can be shown to be Bernoulli. For details, see [ 7 1.
3. Bowen and Ruelle have proved [ 7 1, [70] that if f satisfies Axiom 2 A and is C , then m-almost all points x in M are forward asymptotic to attractors. This also holds if ~ ( f ) is hyperbolic. Putting this together with theorem (6;2), one sees that if L(f)
is hyperbolic, there are
finitely many ergodic invariant measures which describe the forward asymptotic behavior of m-almost all points in M. We now proceed towards the proof of Ruelle's theorem (6.2). follow [ 7J except that our actual. construction of u,
We shall
comes from [ 61
which in turn was motivated by [47.]. Let f : M + M
be a
attractor for f. Let For x
E A,
C"
diffeomorphism and let A be a hyperbolic
EY)
be small, and let n2.I be a positive integer.
{y
14 :d(f 1y, fjX) s E
let ~:(x,n)
=
E
for osj cn).
x
W e might c a l l
x
of s i r e
(n).
is a s i n t h e d e f i n i t i o n of hyperbolicity, then ws(x,n)
h>l
<(I)
and
Y;-~(X).
Let
If
is nearly t h e
E
product of set
t h e s t a b l e s e t of
be Lebesgue measure on the open
m
E
ws (A).
Propoeition 16.4). There i s an ergodic f-invariant measure with the fottaring property. such that for any x
E
For
E.>O
pA E
m a t t , there i s a constant
M(f)
cb > 0
A and n>0.
W e defer t h e proof of proposition (6.4).
Proof of theorem (6.2). Let E>O
so t h a t
and l e t
+
d($(x), 4(y))<6 whenever
E(4.6) = {x
M :x
E
Let
C(M.
E
d(x,y)<s.
6>0 be fixed,and choose For
-0,
set
Cn (@,6) f o r i n f i n i t e l y many
n]
We f i r s t claim
Assume (1) is proved f o r t h e moment. f
preserves s e t s of m-measure zero, we have
f o r each n2O. Letting But i f
36 =
A(+)
But
-m
wS(A) =
f a r m21,
-U mrl
u
n20
fy(A),
r e get that
(@(A) n E ( 4 ,
:I).
E(4, 36)
i s f-invariant, and
m(f-"(~;(A))
n E(4, 36)) = 0
As
so m(#(A)
n E(+, 36)) = 0.
m(U(wS(A) n E(4, m2l then
x
(
1
1 3)) = 0.
-($1
iqlies
be a countable dense subset of
Now, l e t Then,
is1
= 0,
xu&)
we have
and i f
x
E
wSh)-
n-1
1
1 . n*
k=O
c@),
.
= 14dpA
We now prove (1).
F i x &O
a s follows. For
A n Cn(4,
such t h a t
If
26)
(a)
<(x,n)
n W;(y,
Cb)
$(x,n)
n $~y,n) =
y E W;(A) n ~ ~ ( 36) 4 , for
z E Cn@, 26)
f(z,n)
U U
k=N x
E
%
Id i f x
E
E
Rn, y Rn, y
n a , and y E.
%
E
E
Rn
$(z)
and and x
with
By t h e maximality of E
\.
Nsk
# y. z
E
A,
then
Rn,
Then
c $~z,k> c ~ : ~ ~ r , k ) .
one has
E,
x
be a maximal subset of
Rn
4 f o r some NSBn and x
by Proposition (6.4). of
let
for
0
by t h e choice of
p ( z , n ~n ~ ; b , k } $
Y
k) =
e N ,
s+l.. . 0.f
%,
and 'successively d e f i n e s u b s e t s
A n C (4, 26) n
A and $ E C@),
.
~ ( f
and l e t
W;(X,U
Now,
$(x,k)
C
%c Ck($,6).
Ck&,
26),
so i f
x
E
I $ , by t h e choice
Also, by t h e choice of t h e
isadisjointunion.
SO
%'s
But, by t h e ergodic theorem (6.1), sime
N"
pA i s ergodic;
t h i s l a s t number approaches zero a s
Thus, p u t t i n g C2) and (3) together, we get t h a t
and t h i s implies (1). We a r e now ready t o produce t h e measure
C6.4).
Our construction of
p,,
t h a t i t avoids the use of Mar-
needed t o prove proposition
p,
is based on [ 6 3. This has t h e advantage p a r t i t i o n s , and, hence, l e a d s t o a s h o r t e r
proof of theorem (6.2) than one f i n d s i n [ 7 ] or [70].
However, one pays t h e
p r i c e t h a t i t is not r e a d i l y apparent from t h e construction here t h a t the measure
pA is Bernoulli.
Nevertheless, i t is of some value t o give proofs This is because t h e r e a r e c e r t a i n
of theorem (6.2) without Markov p a r t i t i o n s . non-hyperbolic a t t r a c t o r s (e.g.
those in 1183) f o r which Markov p a r t i t i o n s
do not e x i s t , but f o r which one still has t h e p o s s i b i l i t y of having theorem (6.2). To construct fixed point.
A
we f i r s t obseme t h a t we may assume t h a t -f has a
pA,
I f not, we choose
n>O s o t h a t
fn
has a fixed point
~f
in
i n A a r e dense n-1 A , = C ~ W ~ (n ~A , ~then ~ ) f n ~ l = ~ l ,and A = f hl. k-0
(Proposition (4.1) gives t h a t t h e periodic points of
i n A).
p
It may be t h a t
U
A1
i s a proper subset of
single periodic non-fixed sink). Proposition (6.4),
then s e t
f o r any Bore1 set
E.
f
p,
A
(for instance, when A
p e A,
is a
If we f i n d )lh f o r f n a s in 1 n-1 k k f*pAl. Here f,pA (El = pA (f-% n Al) = k*O 1 1
One may check t h a t t h i s
Now, w e assume t h a t that this forces
f
and
f (p) = p.
pA works f o r
f
on h .
W e w i l l s e e i n lemma (7.1)
t o have periodic points of a l l high periods.
By passing t o two 2-to-1
coverings of
M,
and replacing
f
by
f
4
,
i f necessary, we may assume t h a t
ie
ES and. EU
o r i e n t a b l e , and
M
i s o r i e n t a b l e , each of t h e bundles
Tf lES and
Tf
IE'
preserve o r i e n t a t i o n .
For amusement t h e reader should examine t h i s covering f o r t h e
Plykin
example. Let
g be a smooth Riemann metric on M.
form
w
is,
/$dm =
/&
wU(x) i s a
c2
on M
Then
g
which we use t o d e f i n e Lebesgue measure f o r every continuous function submanifold of
metric ,on wU(x)
and t h i s induces a
1
C -u-form
g
on
x
E
A,
p o s i t i v e continuous function.
f w
= $ (x) wx
f (x)
That
M.
For each x E A ,
wU(x)
where
.
C
1
Thus, f o r each EU* is Y wU(x), and
where in
is a
$ (x)
The p o s i t i v i t y comes from t h e f a c t t h a t the
~f 1 EU preserves orientation. * (fn) wfncX) = $ (fn-$1 . (f "-2) ...$ (x) .w , and X
EU i s oriented, and
bundle
Qne can check t h a t (f-'}*
*
we have
on
restricts to a
y E wU(x) , and x E A , we have a u-form w E A'(E u* ) Y Y t h e dual space of EU The forms w vary c1 with y Y' Y continuously with y i n A. For each
m
$ r C(M).
s o t h e metric
M,
induces a volume
wx
-
$(x)-'
1
$ (x) = Jac (Txf E):
wfx.
F o l l o ~ i n gBowen and Ruelle
t h e Jacobian of
1
Txf .E:
Set
1 7 J , we c a l l $U(x) = -log $ (x) =
l o g $(x)-?
L e m (6.5). Proof.
$U is X5lder continuous. The bundle
x
I-+
E~
X
i s ~ b ' l d e rcontinuous since
theorem ( 6 . 4 ) i n [17], and the metric x
W+
w
X'
and
x
i s &."lder continuous.
1
g
is
cm
on M.
T f EU ' a r e Hiilder continuous. X
X
f
is
c2
by
Hence, the maps
T h i s implies t h a t
$U
Note t h a t ,
Thus,
e
snoucx1
measures how much
- volurne.
n
f x
pA'
Now, we can construct of
f
in
Zn '
1
of period
A
e
%oU(p)
n.
,
.
Let
Let
6x
and s e t
(n) It is easy t o chnck t h a t
measure on A .
p n
P,
works.
1 and
-
p = n I 1 n ' pe~er(n)
p n
4p.
i s an £-invariant probability
Per(n)
with c e r t a i n weights.
which converges, and set
{p
n
p
"
= lim i-
pn
i
.
1. Thus, we vlll i n c i d e n t a l l y show t h a t i f
i
then f o r any
the sequence
%
Define
x.
s,P(P)
i
{pm 1 a r e subsequences of
i
, 'hi
be t h e point pass a t
M(f); i . e . ,
E
be t h e s e t of periodic points
Observe t h a t t h e proof we s h a l l give a p p l i e s t o every
convergent subsequence of {pn
Per(n)
It counts t h e elements of
Choose a subsequence This
T n f a l ~ i n x contracts the f r
4
E
CL),
{pn}
such t h a t
f ? d p = J4dv.
pmi
That is,
p
p
-
v.
and Hence,
{p n 1 a c t u a l l y converges t o pA.
We need t o qhow (6.6) x r h
and
For any small n21,
E>O
t h e r e is a constant
then
m(<E
b,n))
cEp,,(g(x,n))
and (6.7)
pA i s ergodic.
These f a c t s w i l l be proved i n the next section.
CE>O
such t h a t i f
7.
The measure PA.
I n . t h i s s e c t i o n we s h a l l prove (6.6) and (6.7) t o complete the proofs of W e w i l l need s e v e r a l t e c h n i c a l lemmas.
Proposition (6.4) and theorem (6.2).
Our notation w i l l be i n the context pf s e c t i o n 6.
Our f i r s t lemma is a
strengthening of t h e shadowing lemma c a l l e d t h e s p e c i f i c a t i o n lemma. E
> 0 , a positive integer
P' > 0 , a s e t of points
f i n i t e s e t of p o s i t i v e i n t e g e r s €-specifies
(xl,nl),
(2)
for
(3)
for 1 < i
(x n ) 2: 2
011
-nl,
r
,...,(xr,nr)
d(f'xl,f'q) arid
0
xl,
i t e r a t e s of
after
q
41
sE
x2.
n
q
P, the next
€-shadow n2
the f i r s t
i t e r a t e s of
Again a f t e r a delay of n3
i t e r a t e s of
iterates
r
P if
have
i t e r a t e s of
€-shadow the f i r s t
q
€-shadowing t h e f i r s t
t h e o r b i t of
we
i t e r a t e s of
q
and
j ( ni,
nl
r- i n A, a& a
we say t h a t a p o i n t
with delay
LE,
then a f t e r a delay of
€-shadow the f i r s t n2 n3
2
...,nr'
..+ni-l+(i-l)P+j
The idea i s t h a t the f i r s t i t e r a t e s of
nl,nq,
xl,...,x
Given
of
x
r
q
P, the next
x3, e t c .
Finally,
and a delay of
Let
con-
P(E) > 0
such
,...,(xr ,nr) may be €-specified with delay
For
n A.
x
E
A
and
E
> 0, write
we f i r s t prove
> 0, there i s a
f
that any sequence
(xl ,nl)
!l%en for any
be a hyperbolic s e t for
A
f.
= w:(x)
again,
P
taining a fixed point of
$(x)
1
closes up.
Lemma (7.1). fSpecification Zemna).
Proof.
n
E
^u
W (x) = wU(x) E
E
n A and
P (€1
.
(a)
d > 0 , t h e r e is an i n t e g e r
f o r any n ,N(E)
and
x,y r A
Once (a) i s proved and pseudo-orbit
in
Then, given
(xl,n1), -
for
1i i < r
...f P 1 y r .
y
xl,fxl,.
f
T
=
y
Let
be an o r b i t i n
52 ~ .
T
f q = q , and
ix.)1
.,.
P(E) = P = N(6 ( E ) )
.
n f-P(E);68(Xi+1) y
be the f i n i t e n r-1 ,f x ~ , Y ~ , ~ Yr ~ . , .
..
P
repeating
d(fT+jq,f:)
Let
6(~)-
ni + rP be the length of y . Since d(f y r , l ) 2 6 , i=l t o t h e l e f t and r i g h t gives an i n f i n i t e 6-pseudo-orbit 7.
Let
o(q)
y i r ;:(f
n
n
..
# 0.
b e such t h a t any
-P(E) W6(xl). S Let 'xr) n f nl-1 P-1 ,f xl,yl,fyl,. ,f yl,x2,.
r w;(f
-
6 = 6(c)
choose
such t h a t whenever
n $.(y)
f%:(x)
E-shadowed by an o r b i t .
...,(xr,nr),
-
and
6-pseudo-orbit
> 0, l e t
E
can be
A
we have
N(E) > 0
q
E
Per
A
E-shadowing
~f
2~
7.
Then, f o r any i n t e g e r
j,
i s small, expansiveness gives
T.
We now prove ( a ) . Let
p
be t h e fixed point of
6 = 6 ( ~ ) be such t h a t whenever -u Since W (p)
point.
~ ( 6 ( € ) )> 0 < 6
and
eh-L(E) <
SO
and
that for
in
d(x,y)
6 0 \,?,ere 3
zn,zm r A
h > 1
For
E
u
5 6,
> 0
small, l e t
is a imique
n fI2(y)
-
is
L(r) > N ( ~ ( E ) ) be such t h a t
as i n the d e f i n i t i o n of hyperbolicity.
nl 2 N(E), w r i t e
such t h a t
A.
a r e . dense in.' A, t h e r e i s an i n t e g e r u n > N ( ~ ( E ) ) and . x E A,, we have dist(x,f%613c~))
d i s t ( ~ , f - ~ ~ ~< ( p6.) ) Let
N(E) = ~ L ( E ) . Then, i f Pick
-s W (p)
f
me
%/3(P),
nl = n
+m
zm E ;:13(P)
u
with
n
and
, d(x,fazn) (f-"zn),
Set
m 2~(€1. < 6
and
and
# 0 by t h e choice of 6. I f we l e t wn r WEI2(x) n WEj2 s -m 2 Z. wm r WEl2(y) n w : ~ ~ ( ~ ~ then Q , d(fnwn,zn) 5 h - n E2, and d(fqwm,zm) 5 X Hence,
s So, ~ r / ~ ( f ~ nw ~ )~ ~ ~ ( #f 0. - ~But w ~then, ) f%r(x)
and f-%:(y)
3
f-%:/2(wm)
3
W8E /2 (f-mwm),
3
f%r,z(wn)
3
w:/*(fnwn)
and we get that
This proves (a) and lemma (7.1).
L e m (7.2).
Let
E >
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
<(x,n)
and n
Lema (7.4). (VoZme Zema)
for aZt
x
E
Lema ( 7 . 5 ) . a
A
.
2
E >
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 >
(-&,IS),
~ ( v ~ ) .
E
(-6,6)
0
n D ( V ~ =) Al
< n ~ i i s 9 for (3)
mag i n t e r s e c t
u A2
n gt
and t h e index of
on
i=1,2.
gn has no fixed points on aD(vo) and t h e index of v n gv on D(vo) i s 0 f o r v E (-&,&I.
See f i g u r e 8.5.
Figure 8.5
Now f o r some a between
~~~z
point det
z
in
D(vo)
< 1 for
s
with eigenvalue w 6 D(vo)
a r e near the d i s s i p a t i v e o r b i t be
1 and
p
where
Now t h e r e i s a
t o the eigenspace of
and
1111
cr
t , i t follows t h a t
1. I f
D(vo)
is very near
s i n c e most of the i t e r a t e s o(p).
has a fixed
g :
x, then
j g ~ ( v ~ 0) 2, j i n ,
Thus, the eigenvalues of
T~~~
must
< 1.
n g -invariant curve 5 a
through
~~~t
1. This uses t h e so-called
corresponding t o
center manifold theorem (see 1181)
.
n Then, gal?
r e a l i n t e r v a l with a fixed point of d e r i v a t i v e g n / < t o obtain a fixed point we r e a l i z e t h i s perturbation a s
zl
in gnlc
<
z ( r < -) tangent
is a diffeomorphism of a 1, s o we c l e a r l y can perturb
with d e r i v a t i v e l e s s than
where
g =
@of,
then
zl
1. If
i s a periodic
sink for
g.
Lema (8.31.
p
Let
cr diffeonnorphism
and q be
with q 4 o(p)
f
Then f
tangent a t some point.
w'(o(~), g)
Let
Let E
g such that
are
o(p)
W ~ ( O ,g) ( ~ ) and
g, and
small, s e t
U
and
yc = C(I,W~(X) n BE(x))
ws(o(~)) are
s yE
i s an i n t e r v a l about x
and
fiys
i n a set
z
i:
= ~ ( x , W ~ ( x )BE(x)) n
Here, a s usual, d is t h e d i s t a n c e f h c t i o n
(€1.
is small enough, then
E
w U ( o ( ~ ) ) and
denote t h e connected component of a point
C(z,F)
> 0
If
E
cr perturbed to
m y be
x be the p o i n t a t which
where BE (x) = Iy E M : d(x,y)
on M.
W ~ ( O ( ~and ) ) ?(o(~))
Suppose
have a tangency arbitpru*iZy near p.
Proof.
F. For
.
are hyperbozic periodic orbits of
and o(q)
tanent.
h-related hyperbolic periodic points of a
in
i s an i n t e r v a l about
yU
E
, and,
#(XI
for
i
in
x
wU(x),
i u f yE n B~~ (3 = 0
.i 0,
n B ~ € ( x )= 0.
We w i l l produce sequences of i n t e r v a l s
11,12,... and
J1,J2,
...
such
that (a)
Ii c wU(o(q))
(b)
Ii
(c)
for
+
1 : . and
and Ji
+
Ji c WS(o(q))
y
s
i n the
E
n 2 1 and l a r g e
topologies
C'
Suppose these sequences have been found. t e e s we .may f i n d a diffeomorphism
is tangent t o
Ji
near
x
and
+ cr
o r b i t under
g
.
Since the tangency of
w i l l pass near
produce t h e sequences
(Ii)
p.
and
the analogous construction of t h e Let r be such t h a t
be a common period of w ~ ( and ~ ~ ?(P2) )
0 and f
Then f o r l a r g e
n
n BE (x) = 0.
Ji
i, (b) guaran-
near the i d e n t i t y such t h a t
$(n) = q
g = +of, then (a) and (c) guarantee t h a t Ji c w'(o(~) ,g)
-
fanIi n BE(x)
i
for +(Ii)
4 BE(x).
q c
$(Ii)
I f we then s e t
W ~ ( O ,g) ( ~ ) and
( ( I ~ ) and
i s near
Ji
x, i t s
Thus, t o prove t h e lenrma we only need t o (Ji). J 's i p
and
We w i l l produce the
Iils
and leave
t o the reader. q.
Let
a r e Cangent a t
pl
x.
E
o(p)
Let
D
and
p2
E
o(p)
be en i n t e r v a l
in
w ~ ( ~ , . such ) that I
{p, ,XI c D
- aD'
such t h a t
pj
A
an i n t e r v a l i n Since p
and
q
o(q) n ocp) E
> 0
for
n
-
b,
s
so that
W (p ') 2
0
u
and
of transverse
z
- aD' .
D'
z
be
D'
Because
W ~ ( O ( ~n) W ) ~ ( O ( ~=) 0, ) s o we may choose a small
we have
and pi small tubular neighborhood
r
f-'D, and l e t 1rjs.r aD' and x 4 D'.
we may choose a p o i n t
a r e h-related,
i n t e r s e c t i o n of . W ~ ( O ( ~ and ))
-
D'
t
u
x d
and
f - ' ~ n B,(x) 123ST neighborhood N ' of D' such t h a t gives a t y p i c a l s i t u a t i o n when
T
-
0.
D such t h a t
of
N
faz
n N =
Also, we may take a small tubular
u
fJN' n BE(x)
osj ST
= 1.
Observe t h a t
-
6.
Figure (8.6)
D may meet
D'
in
s e v e r a l places
Figure 8.6
It
is a small i n t e r v a l about
I
z
in
wU(o(q))
and
E
is small,
then (dl
f-"I
(e)
if
n BE(x) =
0
for
n
20
and
for
i s t h e smallest non-negative i n t e g e r such t h a t
k > 0
k f 1n B 0
S
)
+ 0,
i < k.
the
fil
c
0
u
f-'N I<jl~
u
u
OSj
f'N'
Now, f o r l a r g e interval
Lemma ( 8 . 4 ) .
f ' " ~ ~n Be (x) =
cr perturbed t o
may be
and W'(O(~),g) Rwof.
0
for
n
Let p be. a perio&c point i n A.
X.
so that
g
p
i s periodic for
g
Then f
and W ~ ( O ,g), (~)
have a tangency arbitrarity near x.
Using corollary (3.6), l e t
> 0
E
be small enough s o t h a t i f U nB2,(x)
7 0. Let
z1,z2 c A
be such t h a t
WS(z2) a t
x.
so that
2
n 6
no
N
such that wU(h) and
f
U = Iy r M : d(y,A) 2 ~ 1 then , /)f%=
choose
u y,.
near
C'
2 1, a s required.
Let A be a hyperbolic basic set for
are tangent a t a point
w'(A)
i s an i n t e r v a l
D, so c ( f T i l , ~ ) n B,(x)
guarantee t h a t
N'
c ( f T i l , ~ ) is afi
(2.5) insures t h a t
Ii = C(f T i1,N) n BE(x), then (d) , (e) , and t h e construction of
I f we l e t and
near
C'
A-lemma
i, the
A and n WU(zl) is tangent t o
implies
f-"x
( 0 , ~ ) so that n a small i n t e r v a l i n f OW(:f E
E
~ r ( f - ~ zc ~U ) and
Pick
fnx r W:(fnz2).
fj~&(xn ) B6(x) = 0 f o r -n 0 zl) o B6 (x) containing
1j 1 x.
n o
no > 0 c
Let
Then, fYy
U. u
U
Then be
n B6 (x)
d f o r n 2 1. Similarly, t h e r e i s an i n t e r v a l , y s c WS(z2) about x such t h a t fnys n B6(x) = 0 f o r n 2 1. Since the o r b i t s of t h e homoclinic p o i n t s =
of
p
a r e dense i n
A, t h e r e a r e sequences ( r i ) i k l ' ( S i ) i r l of homoclinic -n n 0 0 points of o(p) such t h a t ri -+ f zl and si -t f z as i + For 2 -n u 0 zl), SO t h e r e a r e i n t e r v a l s l a r g e i, WZE(ri) i s near w:~ (f
-.
-
I
c W , r
) c W ~ ( O ( ~ such ) ) that
topology.
i, one has
For l a r g e
there are intervals
Ji
i + - ,and, f o r l a r g e
c
W;,(si)
iand all
proof of lemma (8.3), i f we l e t with
$(w) =
W,
for
f
"0
Ii
+
yu
i n the
cr)
0
be an appropriate function
$
6
-n f
O J ~
cr
near i d
g = $og, then we have, f o r
some l a r g e (a)
(actually
f o r n 2 1. Similarly, -n 0 c W ' ( O ( ~ ) ) such t h a t f Ji + ys a s n-n n ~ 1 ,f OJ n B (x) = 0. A s i n the i 6 f-nmO~i n B6(x) =
w / B (x), and we s e t
i, n $(f O I ~ ) is tangent t o
c1
near
x
(b)
p
(c)
f
is a hyperbolic periodic p o i n t f o r g. n -n I c W ~ ( O g( ~ ) and f O J ~c w S ( o ( ~ ),g)
This proves lemma (8.4)
.
.
'b
Now we can prove theorem (8.1). Let
be a wild hyperbolic s e t f o r
A
p e r i o d i c point
p. L e t ,
be a neighborhood of
N
w ~ ( A ( ~ )is ) tangent somewhere t o
then
n 5 1, l e t
For
b o l i c periodic o r b i t s of in
sink
s(q)
s e t of in
of
n, and l e t
g
such t h a t
To s e e t h i s l e t
g
E
s e e s t h a t we may choose t h i s
(8.3),
introduce a tangency of
gl
gl
be t h e s e t of diffeomorphisms
p
Per(gl)
6
i s dense
ff n Let
and
p = p(g).
w U ( o ( ~,gl) )
From t h e proof of lemma (8.4), one
so that g2
g
3
q
E
Per(gl)
leaving p
and
and
q
p
-
W (o(q) ,g2) somewhere.
p
6
a t some
WS (o (p) ,g3)
g4
Perk3)
near
q.
q.
Now,
unaffected and
s
so that
Now, l e m a (8.2) gives us a 1 ; of
g
be t h e s e t of a l l hyper-
Per(n,g) n H P (g)
E
so that
W ~ ( O (g~3) , has a tangency with
within
Hn
q
W ~ ( O (g2) ~ ) , and
(8.3) again enables us t o perturb
s(q)
Per(g)
and (8.4), we can prove
N, and l e t
lemma (8.3) says we may perturb t o
1 ?;;.
is d i s s i p a t i v e .
p(g)
E
i s tangent somewhere t o W ' ( O ( ~ ,gl). )
d(x,q) <
is i n N,
g
t h e r e i s a periodic ~ e r ( n , g )n H p(g) ' 1 d(s(q) ,q) < 2 Clearly, ff n is an open sub-
q
By lemma (8.4) we may perturb t o
and
so that i f
W ~ ( A ( ~ ) and ),
Finally, l e t
Using lemmas (8.2),
N.
N.
g.
such t h a t f o r each
N
f
~ e r ( n , g ) be the s e t of hyperbolic p e r i o d i c p o i n t s of
of period l e s s than o r equal t o
g
f containing t h e d i s s i p a t i v e
,q
E
Applying Per(g3)
,
x with
g which has a s i n k
n~
is r e s i d u a l i n N. rill To complete the proof of theorem (8.1), we need only remark t h a t i f g 6 8, Thus,
and with
q
n
E
ff
n
H (g),
P
+
-.
is dense and open i n
then
q
is a
N.
So
8 =
l i m i t of a sequence
q
E
per(ni,g)
nH
P (g) As we have already noted i n s e c t i o n 3, t h i s follows from the
homoclinic theorem (2.3)
.
i
A s an i n d i c a t i o n of t h e f a c t t h a t wild hyperbolic s e t s occur frequently, we have
kt p be a dissipative hyperbolic saddle point of a cr
!i'heorem (4.5).
on M 2, r r 2.
diffeomorphisrn f some point
Suppose w ~ ( ~and ) w " ( ~ ) are tangent a t
cr
m e n arbitrarity
x.
near
there i s a d i f f e m r p h i s m g
f
having a wild hyperbbZic set near the orbit of
x.
For a proof of t h i s theorem, s e e 1321. The condition t h a t
u ~ ( and ~ )
W' (p) b e tangent somewhere occurs very
For instance, i t
n a t u r a l l y i n one-parameter f a m i l i e s of diffeomorphisms.
frequently.occurs i n a r c s between two s t r u c t u r a l l y s t a b l e diffeomorphisms
'
The study of such a r c s , o r more g e n e r a l l y , parametrized
of d i f f e r e n t types.
This
systems of diffeomorphisms and flows, i s c a l l e d b i f u r c a t i o n theory.
We r e f e r t h e
i s a subject of wide scope with many i n t e r e s t i n g a p p l i c a t i o n s . reader t o [24], (141, [13], [34], [35] f o r more information.
We now d e s c r i b e some s p e c i f i c diffeomorphisms of t h e two-disk Thinking of
wild hyperbolic s e t s .
M
2
,
D'
with
D~ as a subset of any two manifold
t h e s e diffeomorphisms provide examples on
M
2
.
F i r s t , we need some
preliminaries on Cantor s e t s . A Cantor set
each point of
F
is a compact s u b s e t of t h e r e a l l i n e
is a l i m i t p o i n t of
F
Given such a s e t we may w r i t e i n t e r v a l and of for
F.
Let
U-2 Fo
and
Ui,
- F = 1-2
a r e unbounded.
EFi?
%-
V Uj.
Ui
where
Fl=
a defining sequence f o r
F.
F
0
F.
i z 0 , l i e s i n a s i n g l e component
...
and
F
Uils
i s empty.
i s an open
Ui
each
We c a l l t h e
=
Then
0sJ
an ordering of t h e bounded gaps of Each
and t h e i n t e r i o r of
be t h e smallest c l o s e d i n t e r v a l containing
1 1 1, Fi =
We c a l l
U-l
R
F
such t h a t
B
t h e gaps
F, and define,
F =
n
izo
Fi.
It i s obtained by specifying
Pi
of
Pi,
and d l v i d e s
Pi
into
two components. and let
Let
cil
- Ui
be t h e component of
Fi - Ui
cir be t h e component of
t o t h e l e f t of
t o t h e r i g h t of
Ui
Ui,
as i n
f i g u r e 8.7.
Figure 8.7
Let
J
0 be t h e length of an i n t e r v a l
Finally, s e t We c a l l
r(F) = sup{r({Fi))
T(F)
t h e thickness of
and s e t
i s a defining sequence f o r
: (Fi)
F.
It measures t h e s i z e of
F).
F i n a certain
sense. Let us consider some examples.
0 < B < 1. The middle s e t
Let
F(B)
i s defined a s follows. Let BR(Fo)
Let
0
centered a t t h e midpoint of
ponents Fll
F be a closed i n t e r v a l .
Fll
and
F12
and
F
SO
that
Form
F2
be t h e open i n t e r v a l of length
Uo
Fob Then
Continue i n t h i s manner defining
F,
Fli
a c e n t r a l l y placed open i n t e r v a l of length
then
Fi
F(6) = F:B),
0 ~The~Fi 's.
zi
because we have taken out s e v e r a l gaps of
i i Fi = 2 a
.
L(Uli)
Ul19U12
B$(c).
a
i
.
from
= Bl(Fli). c
1-B T(F(B)) = a = 28
.
F
SO T(F(B))
Finally, s e t
F(8)
It i s
a t each s t a g e . -+
as
8
-+
0.
i s 0 s i n c e t h e measure
The reason f o r thickness is t h e following.
of
1-' 2 = a,
Thus, i f
i n t h e example a r e not a defining sequence f o r
Observe t h a t t h e Lebesgue measure of each of
and
components of length
i*
easy t o check t h a t
has two com-
by removing from each component
Fi-l
i s a union of
- Uo
by removing open i n t e r v a l s
i s centered i n
Uli
F1 = Fo
If F and G are Cantor s e t s with
L e m 8.6.
T(F)-T(G) > 1 and
neither i s contained i n a gap of the other, then F n G # 0. Wof.
Let
{Pi)
and
(G 1 b e defining sequences of
respectively such t h a t
T({F~)).T({G~))> 1. Since n e i t h e r
i n a gap of t h e o t h e r ,
Fo n Go #
If
(*)
c
G
.
Pi
which meets
which meets
Suppose
c
ponents of
c
- Ui
Pi
F-gap
and
cr
c
contains
G,
s o F meets
c n G $
0. . I f , c
F Ui c c.
Let
c
,,
cr
is
Fi+l be t h e cow
a s i n f i g u r e 8.8.
Figure 8.8
cr
Assume by way of c o n t r a d i c i t i o n t h a t ce
Fimeets
such t h a t
by removing t h e
ct
both
G, then
t h e r e is nothing t o prove, so.assume t h a t
a l s o a component of Fi
G 'is
.
is a component of
is obtained from
nor
G.
h a s been proved, we have t h a t each
Let us prove (*)
F
G
We prove t h e following statement.
0.
is a component of
a component of Once (*)
F and
i
a r e i n G-gaps.
If
c,
both
and c,
c
do not meet
r
and
c
r
G.
Thus,
were i n unbounded
G g a p s , then we would have one of t h e Falowing s i t u a t i o n s . (a)
Go
l i e s t o t h e l e f t of
ce
(b)
Go
l i e s t o t h e r i g h t of
c
Go c
cr
3 .
Now (a) and (b) c o n t r a d i c t t h e assumption t h a t .c n G f G
i n the
F-gap
0 , and
( c ) would put
.:u
Hence, a t l e a s t one of
ct
and
c
r
is i n a bounded Ggap.
Suppose
G U..
c4, is i n t h e bounded G g a p
J
The argument i s similar i f
cr
is i n a
uG j
as i n
bounded Drgap. Let
c;
and
c
1
r
be t h e aomponents of *Gj+l
adjacent t o
f i g u r e 8.9.
Figure 8.9
Now
cr
cannot be i n
uG j
f o r t h i s would give
c c IJG contrary t o hypothesis. j
Then we have,
In case 1, we have f i g u r e 8.10.
which gives
l C e c e - ecc;) ~($1
In case 2, we have
c'. r
1, a contradiction.
L(U~C) c
r So we have f i g u r e 8.11.
is contained i n another G-gap
G
Uk
which i s i n
Figure 8.11
If
c"
is t h e component of
F
c i r c Ui,
adjacent t o
Gk+l
-< 1, a
so
Now we describe our examples. with
G Uk and t o its l e f t , then
contradiction.
This proves (*)
. cr
A l l of our diffeomorphisms w i l l be
r 2 2 fixed. F i r s t r e t u r n t o the horseshoe diffeomorphism a s defined i n s e c t i o n 2.
-
This is a diffeorarphinm
f
fQ n Q = A1
U
Q
such t h a t
that
~ f 1 f - = l ~ ~
This time we a d j u s t t h e r i g h t s i d e s of
of
A2
and
IR2
to
R2
with
A1
and
f o r which t h e r e is a square
r:
A2
Tflf-l~~=
s o t h a t t h e - l e f t s i d e s of
Q
Q and
A2
want the l e f t lower corner of such t h a t t h e boundary of
I
Q
d i s j o i n t rectangles such with Q
and
coincide, and we take
Q
a
A1 E
Figure 8.12
.
2
coincide and
1.1 (-i,T).
t o be a hyperbolic fixed point
i s i n w ~ ( u~ w) ' ( ~ )
O < o < -1
We a l s o p
for
See f i g u r e (8.12)
.
f
'
The shaded a r e a i s of
Q , and l e t y
i n half.
A
1
Let
uA
x be t h e midpoint of t h e base
be a v e r t i c a l l i n e segment through
Recall t h a t
n
dividing
x
f-lA1
f * ~h a s t h e f o p ( i n t e r v a l ) x (Cantor s e t ) . I f n 1. 3 2'
-
-.
-
From t h e way we have defined low Q such t h a t
I
f , t h e r e is an i n t e r v a l
31 c A2, f(a1) c A1,
and
f (I) n i n t Q = 0
in
wU(p) be-
a s i n figure
8.13a.
Figure 8.13a
We can modify fl(r)
through a curve of diffeomorphisms t o
o f f a neighborhood of
= f (2)
tangent t o
f
Figure 8.13b
$(p,fl)
at
x
Al u A2,
fl
so that
I c ~ ~ ( p ~ and f ~ )f l,( I )
is
a s i n f i g u r e 8.13b.
This i s most e a s i l y accomplished a s follows. Let Let g
<
be a
borhood of wS(p). Let
be a curve joining a p o i n t
cr
<
f(1)
diffeomorphism such t h a t and
Then take
$ maps a p i e c e of
-Q
and
g(z) = z
f(1)
x
as i n f i g u r e 8.13a.
o u t s i d e a small neigh-
t o a curve tangent a t
x
to
fl = $of.
F~ be t h e v e r t i c a l coordinate curves near t h e r i g h t boundary of
Q, and l e t
be the horizontal coordinate curves near the bottom of
'F
Then
is a f o l i a t i o n of n neighborhood of
f1fU
S
a f o l i a t i o n of a neighborhood of so that
f l ~ U and
WE (x)
f o r small
, and f
f 1(1) E
.
Q.
is
Let .us choose
$
a r e tangent along a v e r t i c a l curve which contains
y,
and a l l of these tangencies a r e only f i r s t order contacts of the corresponding curves.
This gives us Figure 8.14.
Figure 8.14
U i s a small neighborhood of
Thus, i f
the projection onto
y , then
x i n M, and
r e s t r i c t e d t o any curve i n
n
unique non-degenerate maximum which l i e s on y. neighborhood of
let
fl(I)
-
be a small rectangular
up s l i g h t l y t o give a new
f 2 , then we claim
nf;(3 = n f n ( q ) i s a a d hyperbolic set f o r n n The proof of t h i s f a c t is as follows. For
:B
g
near
-
(nfg))
T(n(f2))
f 2 and
o5n
n
g2( gja n 05j Sn
u.
5 a, l e t q ( i ( g ) ) =
-
~ ' ( f ~ )i P ( h ( f 2 ) ) n y since
f
= f
and
and 3(Jl(f2))
FU(f2) = P(Jl(f2))
n
n25
-a50
and
n u; and
n
and ?(Jl(f2)).
and
wu(h(f2))
, re-
a r e Cantor Sets.
Y
off a small neighborhood of
Figure 8.15 shows 5 ( h ( f 2 ) )
f2.
~t is evident t h a t ijso(n(f2))
a r e two families of curves i n Q(d(f2))
spectively, and 'ijs,(n(f2)) n y
F'(~)
flFU has a
Q.
I f we push ncc2)
Let
is
r :U + y
Y.
Note t h a t
f-'~.
set
FS(f2) =
Figure 8.15
We assume
f2
is chosen s o t h a t
s a T(F (f2)) = - = T ( F ~ ( ~ * )>) 1. From 0
t h e construction, i t is c l e a r t h a t t h e s m a l l e s t closed i n t e r v a l s i n taining
FS(f ) 2
and
FU(f2) i n t e r s e c t .
By lemma (8.61,
y
FS(f2) and
conpU(f2)
themselves i n t e r s e c t . For
g
cr
near
contain the s e t s are
C'
near
f2, there are f o l i a t i o n s
-s
Wm(A(g))
~ ' ( f*)
W:
and
and
See lemma (9) i n [32]
that
, respectively.
C'
curve
These f o l i a t i o n s
c1
tangent l i n e
Now,
FS(g)
c1
near
c ( ~ ( g ) )n y(g),
a r e Cantor s e t s , and n e i t h e r is i n a gap of the
I f we show t h e i r thicknesses a r e near FS(g) n ~ ~ ( fg 0) .
gFU(g) which
which i n t u r n is
y(g)
f o r a proof of t h i s .
FU(g) = z ( h ( g ) ) n y(g)
other.
and
f FU(f ) , r e s p e c t i v e l y , and have 2 2
and
f i e l d s which a r e tangent along a y.
(A(g))
fS(g)
-a0 '
then lemma (8.6) w i l l show
Any point i n t h i s l a s t s e t w i l l be a tangency of ~ ( f ~is) a wild hyperbolic
wU(A(g)) n W ~ ( A ( ~ )and ) , we w i l l have shown t h a t set. Let us prove t h a t
r (FS(g))
i s near
-a0 '
he proof f o r
~ ~ ( gis)
similar. For
n
large, l e t
be t h e component of
I
be a component of
[ ~ ~ -(A(g)) 1
It i s s u f f i c i e n t t o prove t h a t
- Z(A(g))]
-
is near
Z ( A ( g ) ) n y(g), and l e t
n y(g) a -
0
'
Let
J
which i s adjacent t o ~ ; ( ~ ( g ) )be the
I.
component of
p(g)
in w
~(g))( n~
5, and
let
r : y (g)
mapping induced by p r o j e c t i n g along the curves of near
C'
c1
is
~ ' ( f ~ )y(g) ,
near
.
I1 = rI
Thus, i f
w : ( ~ ( ~ ) ) be t h e
F'(~).
1
T
F ~ ( ~i s)
Since
cr
w : ( ~ ( ~ ) ) is
y ( f 2 ) , and
~ : ( ~ ( f ~ ) one ) , has t h a t the norm of t h e d e r i v a t i v e y (g)
+
is near 11.
1 on
J1 = rJ, i t s u f f i c e s t o prove
and
near
is near
a -
8 '
unit vector i n For
g
I
i Tz g 11, and i
-
i s a u n i t vector i n
v;
T ~ , ~ ~ J ~ . i
f 2 , t h e r e is a homeomorphism h : ~ ( g -+ ) ~ ( f ~near ) the
near
inclusion i : A(g) A(g) and h(aJ1) = a (Jl(f2))
n2
such t h a t
where
I (f ) 1 2
f2h and
-
hg.
Also, h(aIl) = a(Il(f2))
Jl(f2)
are intervals i n
wU(p(f2)), and l ( f Z - 1 1 ~(f 2))
*
~(f~--'J1(f2))
i s near
Since h
i A ( g ) , i t follows t h a t n-1 ai
s u f f i c e s t o show t h a t
Let
/
>
a 7
for
all
j-
is near
Is,
- ail
is1 i
1
it s u f f i c e s t o show t h a t
18i
1I i
since so that
f
2
S
n
-1 g
+ cl)
Thus i t
agn-15, ) In-1 ai n-1 Bi a 1. Since = n (1 is1 'i i=l Bi
-
For g
f 2, one has
near
, T I B ~-
- -,
ai
I
is small.
be t h e maximum of the c u r v a t u r e of
K 2 (g)
- 1, and l e t K 3 (g) be t h e maximum norm of t h e on Q n g-lc. Observe t h a t K2(f2) = K ( f ) = 0
is piecewise l i n e a r on
4 ( 2 ~K (f ) 11 2
-a8 '
is near
i, s o we only need t o show t h a t
I
second d e r i v a t i v e s of
.
1cgn-l~l)
is small.
l
l*i
-1 Kl (g) = sup TZg I , l e t zeq
i i g I1 u g I2 f o r
6
diam
-
-1Q n f 2 Q.
5<
E ,
3 2
Given
E
and then suppose
> 0, choose
g
El
> 0
i s close enough
to
f2
< -
3 i d.iam.5 f o r
and, so,
that
SO
Kl(g) ( 2Kl(f2),.maxiK2(g) ,K3(g)) < el,
Ilfii - ail
Notes:
16 i
5
n -'I.
and
i
i
L(g I1 u g J1)
Then,
5 4(2K1 (f 2) E1 + el)diam
6<
E.
The modification Plykin's example a t t h e beginning of this sec-
t i o n i s a v a r i a t i o n of examples f i r s t s t u d i e d by Simon [52]. p a r t s of t h i s s e c t i o n a r e v a r i a t i o n s of r e s u l t s i n
The remaining
[26], [29], and [32].
The f i r s t examples of open s e t s of diffeomorphisms with non-hyperbolic hclosures were given by Abraham and Smale [ I ] , and v a r i a t i o n s of these were described, by Shub [54], and studied i n Hirsch, Pugh, and Shub [18].
Concluding remarks. We have t r i e d i n these l e c t u r e s t o present some of t h e methods and r e s u l t s which have been developed i n recent years t o describe non-trivial recurrence i n ordinary d i f f e r e n t i a l equations.
We understand hyperbolic
systems q u i t e well, and they provide a wide range of examples.
On t h e
other hand, i t is c l e a r t h a t t h e r e a r e many examples of non-hyperbolic recurrence which have t o be faced. For f u t u r e development, we wish t o s i n g l e out s e v e r a l problems.
1. Find examples of hyperbolic s e t s i n models of s p e c i f i c physical systems.
2.
I n p a r t i c u l a r , f i n d non-trivial hyperbolic a t t r a c t o r s .
Develop a s t r u c t u r e theory f o r non-hyperbolic recurrence.
iff^(^ 2 )
instance, i s it t r u e generically i n
that
For
L(f)
has
Lebesgue measure zero?
3.
If
Nc
iff^(^2) i s
i s t h e r e an
4.
f c N
an open s e t of non- R -stable diffeomorphisms
which has a wild hyperbolic s e t ?
(a, CA) is a s u b s h i f t of f i n i t e type, and
If with
o(F) = F,
symbols.
we c a l l
(o, F)
5.
E
B,
Is t h e r e a r e s i d u a l s e t
B c
i s closed
if f r ~such t h a t f o r
each
(Thom) Is t h e r e a r e s i d u a l s e t
c CA
a s u b s h i f t on f i n i t e l y many
C 1 H ( f ) i s a finite-to-one P on, f i n i t e l y many symbols ? f
F
B c
quotient of a s u b s h i f t
iff^^
t h e union of the b a s i n s of the a t t r a c t o r s of
such t h a t f o r f
f
E
B
is dense i n M?
,
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