PUBLICATIONS MATHÉMATIQUES DE L’I.H.É.S.
ROBERT F. WILLIAMS
The structure of Lorenz attractors
Publications mathématiques de l’I.H.É.S., tome 50 (1979), p. 73-99
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THE
STRUCTURE
OF
LORENZ
ATTRACTORS
by
R.
F.
WILLIAMS
(
1
)
Dedicated
to
the
memory
of
Rufus
Bowen
Introduction.
The
system
of
equations
x
=
—
i
ox
+1
oy
y=^x—y—xz
8
———3^
ofE.
N.
Lorenz
[7]
has
attracted
much
attention
([3],
[lo],
[12])
lately,
in
part
because
of
its
relation
to
turbulence.
Lorenz
obtained
this
system
by
"
truncating
"
the
Navier-
Stokes
equation;
it
offers
a
striking
example
of
a
strange
attractor,
vis-a-vis
Ruelle-
Takens
[n].
We
present
the
Ruelle-Takens
idea
briefly.
In
order
that
any
type
of
motion
be
observable,
the
set
of
initial
conditions
leading
to
this
motion
must
be
of
positive
measure.
This
essentially
says
that
the
motion
must
be
bound
to
an
attractor.
Until
recently,
mathematicians
knew
of
only
two
types—steady
state
attractors
(or
sinks)
and
periodic
attractors.
Thus
when
a
persistent
motion
was
seen
to
be
neither
steady
state
nor
periodic,
it
was
termed
((
random
9?
or
"
chaotic
53
,
and
stochastic
mathematics
was
invoked.
It
is
just
this
non
sequitur
that
Lorenz
was
attacking;
his
article
is
entitled
"Deterministic
aperiodic
motion
53
(1963).
Though
many
scientists,
especially
experimentalists,
knew
this
article,
it
is
not
too
surprising
that
most
mathematicians
did
not,
considering
for
example
where
it
was
published.
Thus,
when
Ruelle-Takens
proposed
(1971)
specifically
that
turbulence
was
likely
an
instance
of
a
"
strange
attractor
5
'
1
,
they
did
so
without
specific
solutions
of
the
Navier-Stokes
equations,
or
truncated
ones,
in
mind.
This
proposal,
controversial
at
first,
has
gained
much
favor.
In
particular,
the
paper
of
Guckenheimer
(see
below)
gives
a
geometric
description
(
1
)
This
research
partially
supported
by
NSF
Grant
No.
MPS
74-06731
AOi.
321
10
74
R.F.WILLIAMS
of
what
seems
to
be
going
on
in
the
system
(L)
and
it
is
indeed
a
strange
attractor.
(To
prove
this,
one
would
have
to
make
certain
estimates;
meanwhile
computer
printouts
surely
indicate
this
is
about
right.)
This
aids
the
advocate
of
strange
attractors
in
two
ways:
it
adds
a
fairly
simple
example
to
our
knowledge,
and
at
the
same
time,
one
that
comes
up
naturally.
Meanwhile,
the
estimates
needed
to
tie
the
system
(L)
to
the
geometric
work
of
Guckenheimer
or
the
present
paper
have
not
been
made.
Though
the
current
work
is
of
independent
interest,
it
would
certainly
be
enhanced
by
such
a
direct
connection.
We
begin
by
summarizing
a
theorem
of
[3].
Theorem
(Guckenheimer).
—
There
is
an
open
set
oSf
in
^,
the
space
of
all
vector
fields
on
R
3
,
such
that:
1)
if
XeJif,
then
X
has
a
two
dimensional
attractor
(herein
called
Lorenz
attractor)
which
contains
a
singular
point,
2)
there
are
two
dense
subsets
^,
^CJSf
such
that
the
attractors
for
X
in
SS
are
topologically
distinct
from
those
for
YeJ^.
Here
we
improve
upon
Guckenheimer's
result
by
showing
that
there
are
uncountably
many
topologically
mutually
distinct
Lorenz
attractors.
Therefore
this
answers
in
the
negative
a
question
asked
by
R.
Thorn
[13].
In
particular,
we
show
that
the
obvious
cc
kneading
sequences
"
are
invariant
under
homeomorphisms
near
the
identity.
Briefly,
these
sequences
tell
to
which
side
of
the
singular
point
its
own
unstable
manifold
passes,
in
its
various
<(
trips
"
around
the
attractor.
In
the
process
of
proving
this,
we
develop
a
cell-structure
of
Lorenz
attractors,
and
a
singular
fibration
into
a
figure
eight
space,
Bo.
We
proceed
to
show
that
the
kneading
sequences
can
be
thought
of
as
infinite
words
in
the
monoid
of
positive
words
ofTCi(Bo).
The
second
main
tool
is
a
kind
ofpre-zeta
function,
Y],
whose
arguments
x,y
are
the
generators
of
this
monoid.
The
function
T]
can
be
computed
in
the
following
sense.
First
there
is
a
(possibly,
in
fact,
usually
infinite)
matrix
B(A:,^).
That
is,
B
is
a
pairing
on
certain
symbols
S,
with
values
either
x,
j,
or
o.
Then
^trB
1
7]==2j————
i
Z
where
one
must
take
care,
as
7^(Bo)
is
not
abelian.
Finally
we
show
that
T]
is
a
topological
invariant
and
that
the
correspondence
between
the
kneading
sequences
and
T]
is
one-to-one.
This
proves
our
basic
proposition,
that
the
kneading
sequences
are
topological
invariants.
More
precisely:
Theorem.
—
There
is
a
positive
number
A
such
that,
if
the
attractors
Ax
and
Ay,
for
X5
YeoSf,
are
homeomorphic
via
a
homeomorphism
within
A
of
the
identity
(G°-sense),
then
X
and
Y
have
the
same
kneading
sequences.
The
number
A
is
the
(<
diameter
59
of
the
hole
(see
Figure
i)
or
about
30
for
the
equations
of
Lorenz.
322
THE
STRUCTURE
OF
LORENZ
ATTRACTORS
75
From
here
it
is
just
set-theoretic
topology
to
show
that
the
(oS-conjecture
of
Rene
Thorn
(
1
)
—
long
thought
to
be
false
—
is
indeed
false.
That
is,
the
Lorenz
attractors
are
not
generically
countable
up
to
topological
type.
This
differs
significantly
from
Guckenheimer's
result
inasmuch
as
one
of
his
two
dense
subsets
is
not
a
Baire
set,
and
hence
has
no
existence,
generically.
Another
basic
geometric
fact
about
Lorenz
attractors
is
brought
out,
and
used
as
a
strong
tool.
This
is
the
fact
that
these
attractors
are
real
objects,
in
ordinary
euclidian
3-space,
and
that
they
consist
of
many-many
two-dimensional
layers,
stretching
from
front
to
back
in
our
line
of
sight.
It
follows
that
these
layers
are
linearly
ordered,
by
this
front
to
back-ness.
For
example,
see
the
stereoscopic
computer
printouts
of
Rossler
[10].
We
conclude
the
introduction
with
two
types
of
comments.
First,
we
use
branch
manifolds
([17],
[tS])
in
our
proofs,
and
would
like
to
call
the
reader's
attention
to
the
sketches
in
Lorenz's
original
(1963)
paper
[7].
Also
his
comments,
particularly
about
his
Figure
3,
correspond
quite
well
to
the
author's
theorem
G
[18].
Secondly,
we
emphasize
below
certain
nice
aspects
of
Lorenz
attractors.
They
have
a
relative
2-manifold
structure,
are
orientable,
have
a
smooth
line
as
boundary,
form
a
singular
fiber
bundle,
and
have
a
rich
cell-complex
structure;
in
a
sense,
all
of
this
depends
continuously
on
the
original
equation.
It
is
a
pleasure
to
thank
Dennis
Pixton
for
his
helpful
conversations.
Also,
J.
Milnor
for
his
conversations
about
work
on
kneading
sequences
he
and
W.
Thurston
have
done
recently,
in
another,
basically
more
difficult
connection.
Michael
Kervaire
for
his
hospitality
and
encouragement
at
the
third
cycle
in
Geneva.
Finally,
and
most
important,
the
long
conversation
with
W.
Parry,
in
part
about
his
early
papers
on
maps
like
our
Poincare
map
f\
in
particular
he
seems
to
have
singled
out
the
property
we
call
I.e.o,
locally
eventually
onto
(Prop.
i,
§
2).
i*
Use
of
branched
manifolds.
Our
point
of
departure
is
to
describe
a
type
of
semi-flow,
q^,
^eR^",
on
a
certain
smooth
branched
manifold
L
of
dimension
2.
Then
{L,
<p^,
^eR"^}
forms
an
inverse
system,
and
its
inverse
limit
L==lim{L,
<p^,
teR^}
inherits
a
flow
^,
^eR.
These
L,
9^
are
the
Lorenz
attractors.
There
are
several
additional
steps,
required
to
show
that
these
L,
^
are
indeed
attached
to
the
differential
equations
of
Lorenz.
First,
there
are
analytic
estimates
to
be
made
on
the
stable
and
unstable
manifolds
of
the
singular
point.
This
task
has
(
1
)
First
proposed
by
Thorn
in
about
1967
and
restated
in
the
volume
on
Hilbert's
problems
[20],
p.
59.
323
76
R.
F.
WILLIAMS
been
carried
out
by
various
researchers,
on
computers
([7],
[lo],
[12]),
the
only
way
now
known.
The
missing
step
involves
a
novel
and
fascinating
problem,
which
I
state
here
as
a
conjecture.
Conjecture,
—
There
is
a
vector
field
X^p,
transversal
to
the
flow
0^
of
the
Lorenz
equation,
such
that
for
each
t
and
each
x
near
the
attractor,
^.X^==^X^,
c=c{x,
t)e{o,
m)
where
o<X<i
and
m>o
are
independent
of
x
and
t.
Thus
X
determines
a
strong
stable
(oriented)
line
bundle.
Next,
one
needs
to
prove
a
strong
stable
manifold
theorem
for
the
Lorenz
attractors,
along
the
lines
of
the
Hirsch-Pugh
[5]
version
of
the
Smale
formulation
[13]
for
hyperbolic
systems,
and
related
to
the
Hirsch-Pugh-Shub
paper
[6].
However,
one
familiar
with
these
tech-
niques
will
have
little
trouble
making
this
step;
admittedly,
this
should
be
done
in
print,
but
should
probably
await
a
more
general
description
of
Lorenz
structures.
Finally,
one
needs:
a)
to
proceed
from
the
actual
attractors
to
the
artifact,
L,
9^,
^o;
b)
to
proceed
from
L,
9^,
^eR
to
a
vector
field
(^differential
equation)
in
some
neigh-
borhood
of
R
3
.
These
two
steps
were
treated
in
great
detail
in
the
author's
papers
([17],
[i8])
for
the
case
of
diffeomorphisms.
Admittedly,
this
too
should
be
done
in
print;
mean-
while,
those
familiar
with
this
earlier
work
will
have
no
trouble
in
these
last
two
steps.
As
a
final
remark,
note
that
we
do
not
use
the
assumption
that
the
equations
(and
hence
the
attractors)
of
Lorenz
are
symmetric
(see,
e.g.
[12]).
This
generality
seems
natural
to
us.
On
the
other
hand,
all
our
work
is
(or
can
be)
done
symmetrically,
so
that
the
theorems
apply
in
the
symmetric
case
as
well.
2.
The
branched
manifold
L.
Let
L
be
the
branched
manifold
of
Figure
i
FIG.
i
2U