Math. Proc. Camb. Phil. Soc. (1978), 83, 451
With 2 text figures
Printed in Great Britain
451
The topological classification of Lorenz attractors
BY DAVID RAND
University of Warwick
(Received 8 August 1977)
(Revised version 4 October 1977)
1. Introduction
The Lorenz attractor is a strange attractor which has been proposed as an explicit
model for turbulence ((4), compare (5)). First studied by E. N. Lorenz as a truncation
of the Navier-Stokes equations (2), it has since attracted the attention of mathematicians because of its particularly interesting dynamical properties.
Attractors of this type, henceforth called Lorenz attractors, are particularly delicate
vis a vis perturbations of the vector field. An excellent description of Lorenz attractors
is given by J. Guckenheimer in (l) and their properties have been clarified still further
by R. F. Williams in (6).
The delicacy of Lorenz attractors manifests itself as follows: every neighbourhood of
a vector field containing a Lorenz attractor contains a two-parameter family of vector
fields with Lorenz attractors such that no two of these attractors are homeomorphic.
This family is detected using artifacts called kneading series which describe the way
in which the trajectories near the attractor perform their characteristic dance of wheel
and counterwheel. Kneading series were first introduced by J. Milnor and W. Thurston
to study the dynamic properties of certain endomorphisms of an interval of the real
line. R. F. Williams (6) adapted the idea for use in the context of Lorenz attractors and
used them to show that Thorn's wS conjecture is false.
In this note we attempt to answer some questions raised by (6); in particular, to show
that the kneading series determines the homeomorphism type of the attractor. To do
this we introduce a kneading invariant which is essentially the same as Williams's
sequences but which follows Milnor's development (3) more closely. We construct a
piecewise linear model for the Poincare" map when the attractor has a symmetry as in
Lorenz's original example, and describe some properties of the periodic orbits.
Roughly speaking, Williams constructs Lorenz attractors as follows. Let L be the
branched surface shown in Fig. 1 (cf. (6), fig. 1; (2), p. 138). Let <j>t, t > 0, be a semiflow
defined on L so that 0 is a saddle point with stable and unstable manifolds as shown
and the Poincar6 map/: I -> I (f(x) is the point of/ where the trajectory through x next
meets /) has a graph like that shown in Fig. 2 and satisfies the hypotheses (l)-(3) of
Section 2. Then the Lorenz attractor associated with (L,<j>t) is the inverse limit
(L, $t) oiL with respect to $t. This is a singular space L together with a,flow$t. Generically each point & 4= 0 of L has a neighbourhood which is either the Cartesian product of
a Cantor set with a 2-disk, or the product of a 1-disk with the cone over a Cantor set
(the latter if $ is contained in the unstable manifold of 0 = limO, but £ =t= 0).
452
DAVID RAND
Fig. 2
Of course virtually all of the information about this attractor is contained in the
Poincare" m a p / a n d it is this map which we study below. The step of relating/to the
Lorenz attractor is very adequately covered by Williams's exposition (6).
2. The Lorenz map
In the following sections we study the Poincare" m a p / : / - » • / where / = [ — l . + l ]
and/satisfies the following conditions:
(1) / i s differentiable at x provided x # 0,
(2) lim/(z) = + 1, lim/(x) = - l,/(0) = 0,
(3) there exist a > 1 such that 8f/8x(x) > a for all x # 0.
Of course, a > 1 implies that if J is any subinterval of/ there exists a positive integer
k such that/*(«/) contains 0.
3. The kneading invariant off.
Ifze/let
, . .j.
I
ko(x) = |
1-1 if
and let kt(x) = k^x))
n
1 if x > 0,
0 if i = 0,
x<0,
for i = 1,2,3
00
Definition 1. The formal power series k(x) — 2 ^i(^) ** is called the kneading series ofx.
i=0
The map XHk(x) is strictly monotone increasing (i.e. x < y implies k(x) < k(y)), if
we endow Z[[t]] with the lexicographical ordering:
i=0
< 2
f=0
Topological classification of Lorenz attractors
453
if the first non-zero coefficient of
»=o
is positive. Thus/"(a;) = x if and only if k(x) = -Pn_i(<) (1 — t")"1, where i^,.^) is a polynomial in t of degree n — 1.
Also, if Z[[t]] has the topology induced by the metric,
\t = 0
the following limits exist:
k(x+) = lim k(y),
k(x~) = ]imk(y).
Definition 2. The pair (k+, k_) = (&(0+), k(0~)) is called the kneading invariant of/.
If there is any uncertainty as to which map we are referring to, we denote these by
k+(f) and k_(f).
Of course, this definition is motivated by the kneading sequences of R. F. Williams
(6). However, it is important to note the distinction between his kneading sequences and
the kneading invariants defined here for the homoclinic case where / m (l) = 0 or
/ n ( — 1) = 0. In this case the two definitions do not agree, though there is an obvious
correspondence. The reason for choosing the definition used here will become obvious
from the results which follow, and particularly those in Sections 6,7 and 8. This
approach follows Milnor(3).
The map x v-y k(x) is continuous at points x such that fn{x) =# 0 for allTO^ 0. If
n
f (x) = 0 with n minimal then
k(x+) =
provided x 4= 1, and
provided x 4= — 1.
^
It is also fairly easy to characterize the image of k. If xe (— 1,1) and k(x) = ^ 6iti
i=0
then for all n > 0,
<=n
But k+ = 1 + tk{ - 1+) and k_ = - 1 + tk(l-), so that for all n > 0,
k_+ 1.
(1)
i=»
Also note that,
S IV* = 0 , k_ > £ ^< i "" or
£ ^t*-* > A+
(2)
for each n ^ 0. Thus:
Definition 3. A formal power series
i=0
with coefficients in {— 1,0, + 1} is said to be (k+, k_)-admissible if and only if it satisfies
(1) and (2) for all n > 0.
454
DAVID R A N D
LEMMA 1. For every xe( — 1,1), k(x) is (k+, k_)-admissible. Conversely, every (k+, k_)admissible series is equal to k{x) for some x e (— 1,1).
Proof. We only have to prove the second statement. Let
0 = £ 6tft
i=0
be a {k+,&_)-admissible series and let x = su-p{yel\k(y) < 6). Then, by (1), x is well
defined. Clearly
k{xr) < 6 < k(x+)
so if k is continuous at x we have 6 = k(x). Otherwise fn(x) = 0 for some n. Taking n
minimal, we have that
k{x) + tnk_ < 6 s; k(x) + tnk+.
I t follows that 0£ = kt(x) for 0 < i < n, and
This latter implies that di = 0 for alii ^ n. because ^ is admissible. Hence di = k^x)
for alii > 0 and therefore 6 = k(x).\
Thus the periodic admissible sequences correspond bijectively with the periodic
points of/, and k+ and k_ thus determine the periodic points of/. The converse is also
true if a > J2 and as a straightforward corollary we can deduce that the kneading
invariant is an invariant of the homeomorphism type of the attractor (6).
LEMMA
2. / / a > ^/2 then the periodic points off are dense in I.
Proof. We show that any open interval J contains a periodic point. Assume without
loss of generality that J <= (— l, 0).
Let 70 = J and Ii+1 be the longest component of/(7i)\{0} or if these components have
the same length let it be the one on the same side of 0 as J.
For each i Js l , / - 1 / ^ = / t is a contraction sending It into 7 ^ . Now a > ^/2 implies
that for n large enough 7n = (— 1,0) (see, for example, (6)). Thus we have a contraction
so 70 contains a point of period ^ n.
PROPOSITION 1. (i) k+ and k_ determine the periodic orbits, (ii) The periodic orbits
determine k+ and
k_ifa>*j2.
Proof. We only need to prove (ii). This follows from the following lemma.
LEMMA 3. Suppose that a > ^/2. For every integer n there exist periodic points x and y
in I such that k+ = k(x)modtn and k_ = k(y) mod tn.
Proof. Fix n. Let «/e (0,1] be the smallest number such that/ n (y) = 0. Then for all
ze(0,y), k(z) = k+modtn. Now by the previous lemma there exists a periodic point
ze(0,y). Thenfc(z) = k+modtn.
Topological classification of Lorenz attractors
455
Proof of Proposition 1. Let P be the set of all periodic points in (0,1). Then zeP
implies k(x) > k+. Let k = m£{k(x) \xeP}. Then k ^ k+. If k > k+ there exists n ^ 0
such that kmodtn > k+ modt n . By Lemma 8 there exists xeP such that
k{x)modtn = k+mod.tn.
Then & > k(x) contradicting the definition of k. Thus k = k+. Similarly for k_.
4. The dependence of the kneading invariant onf.
Let T denote the set of Poincare maps as defined at the beginning of Section 2. We
put the C-metric p0 on T, where
\
xel
If/, ger write/ > g if/(a;) > g(x) for all se/\{0}.
2.f>g implies k+(f) > k+(g) and k_(f) > k_(g).
Proof. It is clear that / > g implies that kf(x) > kg(x) for all xel. Consequently
&+(/) ^ k+(g) and k_(f) ^ k_(g). We can now rule out the possibility of equality because
if/, ger are such that f(x) ^ g(x) for all xel, then k+{f) = &+(gr) implies that
/ " ( - I ) = gr"(-l) for all TC^O
and &_(/) = MflO implies/ n (l) = gn(l) for allTC> 0.
To prove this suppose that k+(f) = k+(g). If/ n ( - 1) 4= 0 and gn( - 1) + 0 for all n 2 0
it follows immediately that/ n ( - 1) = gn( - 1) for all n Js 0 because if/ m ( - 1) 4=fifm(- 1 )
with m ^ 0 minimal and, say, gm(-l) < / m ( - 1), t h e n / * y ( - 1) s$ 0 </">+*>(_ 1) for
0. Thus k+(f) = k+(g) implies that gm+*>( - 1) > /"sfm( - 1). But for 0 ^ q < p,
— 1) and/9<7m( — 1) lie to the same side of 0 so if p is chosen minimal
PROPOSITION
If/"( - 1) = 0 the same fact implies that if gn{ - 1 ) = 0 then/ r ( - 1) = gr{ -1) for all
r > 0. If gn( — 1) 4= 0 we have that k+(g) and hence kg( — 1) has periodic coefficients. It
follows that — 1 is periodic contradicting the fact that g e T. Consequently
/«(_1) = ^ ( _ 1 )
for all n > 0. A similar proof works when gr{ — 1) = 0 and for the results regarding k_.
This completes the proof of the Proposition.
1. In any C-open subset of T there exists an uncountable subset whose
elements have mutually distinct kneading invariants.
COROLLARY 2. In any O°-open subset ofr there are uncountably many conjugacy classes.
COROLLARY
Remark. Two maps/, ger are conjugate if there is a homeomorphism h: I~>I such
that/oA = hog.
Remark. We only work with the Cf0-metric here because it is possible that
lim/'(x) = +oo or lim/'(x) = — oo.
arfO
xj.0
This is the case for the Lorenz attractor as described in Guckenheimer(i). Consequently the C^-metric is not well defined. However, it is clear from Corollary 1 that
456
DAVID RAND
Corollary 3 will still hold for any reasonable metric, for example one based on the vector
field inducing the semiflow on the branched manifold L, even though it may take
account of differentiability. Thus from Proposition 1 we deduce, with a little work, that
every <72-neighbourhood of a vector field with a Lorenz attractor contains uncountably
many vector fields with non-homeomorphic attractors, as was proved in (6).
5. Kneading is a complete invariant
THEOREM 1. Iff, ger have the same kneading invariant then they are conjugate via an
orientation preserving homeomorphism.
Proof. Let
A n (/) = U /-"(0) and
A(/) = U A n (/).
s=0
n=0
Equality of kneading invariants of/ and g implies that there is an order preserving
bijection xn'- A n (/)->• An(gr) such that xjA~n-i(f) = Xn-i- This is because, by Lemma
1, these points correspond to the polynomials of degree < n — 1 which are admissible.
Moreover, if x e An then Xnf(x) a n d 9Xn(x) belong to An(g) and these have the same
gr-kneading series whence Xnf(x) = 9Xn(x)- Thus Xn°f=9°Xn o n A n = A «(/)Let hn: Is*I be the piecewise linear homeomorphism such that
hn(±l) = ±l,
hJKn = Xn
and hn is linear outside Am.
By the strict monotonicity of x i-» k(x), A(/) and A(g) are dense in I. Using this fact
we see that {An} is a Cauchy sequence in the C-topology on the space of continuous
maps of / into / . Thus h = lim hn exists and is continuous.
n—»-oo
The mapping h agrees with xn o n A « whence, by the density of A, h is strict monotone
and hence a homeomorphism. Also hof=goh
on An for each n ^ 0 so hof and
goh agree on a dense subset of / . By continuity, we have that hof = goh. Thus h
is the required conjugacy and the theorem is proved.
Remark. The conjugacy constructed in the proof of Theorem 1 is orientation preserving. If k±(f) = —kzf(g) we can construct an orientation reversing conjugacy in a
similar way.
COROLLARY 1. The set {k+,k_, —k+, —k_} is a complete invariant for the homeomorphism, type of a Lorenz attractor if a > ^2.
Proof. That the set {k+, k_, — k+, — k_} is an invariant of the homeomorphism type
follows from Williams(6) (cf. Proposition 1). The converse follows immediately from
Theorem 1 subject to the obvious amendments for the orientation reversing case, and
some detailed considerations of the constructions of (I) and (6).
Of course, the theorem implies a stronger result.
COROLLARY
2. Two Lorenz attractors with a. > ^2 are conjugate if and only if the two sets
| K + , AJ_,
of kneading series are identical.
«;_(.,
k_j
Topological classification of Lorenz attractors
457
6. The cutting invariant
The cutting numbers measure the numbers of times the interval / is cut up by the
discontinuity after a number of iterations.
Definition 4. The nth cutting number yn of/is the cardinality of the set
The cutting invariant of/ is the formal power series
n=0
THEOREM
2. In Z[[t]],
k
n+1
Proof. The function x->k(x)modt
is a step function with discontinuities at the
r
points x e [ — 1,1] such that/ (x) = 0 for 0 < r ^ n. If r is minimal the size of the discontinuity is k(x+) — k(x~) mod tn+1 = tr(k+ — k_) mod tn+1 (see section 3). There are
y r such discontinuities. Adding up all these jumps we see that
= 2yrF(ifc+-Jfe_)mocUB+1.
k(l~)-k{-l+)
r=0
Letting n->oo we have
,.
,
,.
< +» _ /».
r. »
from which the result follows directly.
The rate of growth of the numbers yn as w->oo is measured by s = lim sup %V7»f
Since 1 < y n ^ 2 n it follows that 1 < s ^ 2. We can sharpen this estimate.
LEMMA 4. a ^ s ^ 2.
Proo/. Since/-1(a;)\{0} is non-empty for all a;e (— 1, l)\{0} it follows that y n + 1 ^ yn
for all w > 0.
Now a ^ inf/'(a;). Thus after n+ 1 iterations there must have been no less than
xel
an+i cu ttings.
Consequently,
n
2 y{ > a n+1 .
But y i + 1 > y t for alii > 0 whence
Therefore
y n + 1 > a.n+1/{n +1)
from which it follows immediately that s > a.
7. TAe 2>ieceM«se linear model for the symmetric case
Lorenz's original example (2) possessed a natural symmetry arising from the physics
which we can interpret as/( — x) = —f(x) or k+ = — k_. Throughout this section and the
next we assume that / possesses this symmetry. In this section we construct a ' conjugacy' between / a n d a piecewise linear function.
458
DAVID R A N D
The first thing to note is that r = s'1 is the radius of convergence of y considered as a
power series in the complex variable t. But r < 1 by Lemma 4. Thus the radius of convergence of y is smaller than that of k+ — k_ considered as a complex power series. It
follows from Theorem 2 that k+ — k_ must have a zero z on \t\ = r. But for all N ^ 0,
N
N
N
n=0
n—0
0
so we have (k+ — k_) (r) = 0, implying that k+(r) = k_(r) = 0 by symmetry.
Let K: /-*- U be defined by
•"•W — 2J
i=0
K J: r
i\ )
•
Then by our choice of r, i£ is continuous. Moreover, if x < y, by an argument similar to
that used in the proof of Theorem 2, we have that
Hy-)-k(x+) = (k+-k_) _S yAx,y)t\
where yn(x, y) is the cardinality of the intersection of/-n(0)\/-(™-1)(0) with (x, y). Using
the fact that (k+ — k_)(t) > 0 for 0 < t < r, we deduce that K(y) ^ K(x). Using the
continuity of K we deduce that K is monotone.
It is now easy to calculate the image of K since K( — 1) = — s and i£(l) = s.
Let .F: [ — s, s] -»[ — s, s] be the map defined as follows:
I
THEOREM
3. The map Kqf[—l,
sy-s
if y > 0,
0
sy + s
if 2/ = 0,
if «/ < 0.
1] owto [ — s, s] is continuous, monotone and satisfies
Kof=FoK.
Proof. We only have to prove the last statement. If x e / ,
•n-\J\-c>) ~
2u
K
i\]\x))-r
= sK(x) — sko(x).
8. Periodic points of the symmetric map
Again we assume that k+ = — k_ as in Lorenz's original example.
If 6 e Z[[t]] then by |0| we denote max {6, - &}. Then it follows directly from Definition 3 that if
m
i =0
and if for all n > 0,
or
= 0
Topological classification of Lorenz attractors
459
then 2 0i ' i - 1 is (k+, — &+)-admissible. Actually we can drop the condition k_ = — k+
if we replace the condition
S Bttf-
> k+
> max{&+, — &_}.
by
i=n
It follows immediately that if 0 = 2 <M* has periodic coefficients 5ie{— 1,0,1}
r=0
then 0 is (k+, — &+)-admissible if, and only if, for all n ^ 0
> k+
or
(3)
2 0*** = 0.
<=n
Thus we deduce the following result:
3. Ifk+ = — k_ the map x-> k(x) induces a bijection between the periodic
points off and the periodic formal power series
PROPOSITION
00
ft — V ft ti
with coefficients in { — 1,0,1} which satisfy condition (3) for all n ^ 0.
Definition 6. A formal power series
0 = 2 0,**
is said to be self-admissible if, for all n ^ 0,
CO
> , (/; I
i=n
^ U.
We now give some examples of self admissible series. These next few remarks are
essentially due to Leo Jonker.
Examples. (1) Let a(l) = 1 and a(2n) = a(2n-1)-tin-1oc(2n~1).
Then
is periodic of period 2 n and is self-admissible.
(2) If k * 1 is odd, let
i(fc-5)
a(2™+1. k) = a(2") - t2na(2n) - t*n+1a(2n) +13-2"
2
Then a'(2n.k) = a(2n.k)j(i -t*1*)-1 is periodic of period 2n. k and is self-admissible.
THEOREM 4. (i) For any positive integer r, oc'(r) is the maximum element in the set of all
periodic self-admissible series of minimal period r.
(ii) Suppose that a.'{k) < oc'(l). Then iff has a periodic point of period k it has one of
period I.
(iii) The series a'(r) exhibits the following order:
a'(3) < a'(5) < a'(7) < ... < a'(2.3) < a'(2.5)...
< ... < a'(2 2 .3) < a'(2 2 .5) < ... < a'(2«) < a'(2"-1) < ... < a'(2).
460
DAVID R A N D
Proof. We only prove (iii) since the rest is proved in a forthcoming paper of L.
Jonker(7).
Suppose that x is of period k. Then 6 = min{|&(y)|: y lies in the orbit of a;} is self
admissible and 6 > k+. By (i) <x(k) ^ d so oc(l) > a(k) > k^. Thus a(l) satisfies condition
(3) for all n ^ 0 and is therefore (k+, — &+)-admissible. By Lemma 1 this implies that
6 = k(z) for some z e / and we are done because in this case z is periodic with primitive
period I.
I would like to acknowledge my indebtedness to the work of Guckenheimer, Williams
and Milnor and thank David Fowler, Leo Jonker and Christopher Zeeman for several
helpful conversations regarding Lorenz attractors.
NOTE. Since completing this paper the author has received a preprint of a paper by
J . Guckenheimer which proves Theorem 1 in a different manner.
REFERENCES
(1) GUOKENHEIMER, J. A strange, strange attractor, in J. Marsden and M. McCracken. The Hop/
bifurcation (Springer Lecture Notes in Applied Mathematics, 1976).
(2) LORENZ, E. N. Deterministic nonperiodicflow.J. of Atmospheric Sciences 20 (1963), 130-141.
(3) MILNOB, J.' The theory of kneading' and 'A piecewise linear model for kneading', Unpublished
notes, 1976.
(4) RUELLE, D. The Lorenz attractor and the problem of turbulence, to appear in The Proceedings of the Conference on Quantum Models and Mathematics, Bielefeld, 1975.
(5) RUELLE, D. and TAKENS, F. On the nature of turbulence. Comm. Math. Physics 20 (1971),
167-192; 23 (1971), 343-344.
(6) WILLIAMS, R. F. The structure of Lorenz attractors (preprint, 1976).
(7) JONKER, L. Periodic orbits and kneading invariants (Warwick University Preprint, 1977).