Lorenzparadigm English

Chaos, 154c 2013 Springer Basel AG Poincare Seminar 2010

The Lorenz Attractor, a Paradigm for Chaos

Etienne Ghys

Abstract. It is very unusual for a mathematical or physical idea to disseminateinto the society at large. An interesting example is chaos theory, popularizedby Lorenzs buttery eect: does the ap of a butterys wings in Brazilset o a tornado in Texas? A tiny cause can generate big consequences!Mathematicians (and non mathematicians) have known this fact for a longtime! Can one adequately summarize chaos theory is such a simple mindedway? In this review paper, I would like rst of all to sketch some of themain steps in the historical development of the concept of chaos in dynamicalsystems, from the mathematical point of view. Then, I would like to presentthe present status of the Lorenz attractor in the panorama of the theory, aswe see it Today.

Translation by Stephane Nonnenmacher from the original French text1

1. Introduction

The Lorenz attractor is the paradigm for chaos, like the French verb aimeris the paradigm for the verbs of the 1st type. Learning how to conjugate aimeris not sucient to speak French, but it is doubtlessly a necessary step. Similarly,the close observation of the Lorenz attractor does not suce to understand all themechanisms of deterministic chaos, but it is an unavoidable task for this aim. Thistask is also quite pleasant, since this object is beautiful, both from the mathemat-ical and aesthetic points of view. It is not surprising that the buttery eect isone of the few mathematical concepts widely known among non-scientists.

In epistemology, a paradigm is a dominant theoretical concept, at a certaintime, in a given scientic community, on which a certain scientic domain bases thequestions to be asked, and the explanations to be given.2 The Lorenz attractorhas indeed played this role in the modern theory of dynamical systems, as I will try

1http://www.bourbaphy.fr/ghys.pdf2Tresor de la Langue Francaise.

2 E. Ghys

Figure 1. The Lorenz attractor

to explain. The Lorenz dynamics features an ensemble of qualitative phenomenawhich are thought, today, to be present in generic dynamics.

According to the spirit of this seminar, this text is not written exclusively formathematicians. The article [81] is another accessible reference for a description ofthe Lorenz attractor. The nice book Dynamics beyond uniform hyperbolicity. Aglobal geometric and probabilistic perspective by Bonatti, Daz and Viana givesan account of the state of the art on the subject, but is aimed at experts [16].

In a rst step, I wish to rapidly present two past paradigms which have beensuperseded by the Lorenz attractor: the (quasi-)periodic dynamics and the hy-perbolic dynamics. Lorenzs article dates back to 1963, but it was really noticedby mathematicians only a decade later, and it took another decade to realize theimportance of this example. One could regret this lack of communication betweenmathematicians and physicists, but this time was also needed for the hyperbolicparadigm to consolidate, before yielding to its nonhyperbolic successors3. In a sec-ond step I will present the Lorenz buttery as it is understood today, focussing onthe topological and statistical aspects. Then, I will try to sketch the general pic-ture of dynamical systems, in the light of an ensemble of (optimistic) conjecturesdue to Palis.

Chaos theory is often described from a negative viewpoint: the high sensi-tivity to initial conditions makes it impossible to practically determine the futureevolution of a system, because these initial conditions are never known with total

3For a historical presentation of chaos theory, see for instance [8].

The Lorenz Attractor, a Paradigm for Chaos 3

precision. Yet, the theory would be rather poor if it was limited to this absenceof determinism and did not encompass any deductive aspect. On the contrary, Iwant to insist on the fact that, by asking the good questions, the theory is able toprovide rich and nontrivial information, and leads to a real understanding of thedynamics.

There remains a lot of work to do, halfway between mathematics and physics,in order to understand whether this little ordinary dierential equation can ac-count for meteorological phenomena, which initially motivated Lorenz. Long is theway between these dierential equations and the true NavierStokes partial dif-ferential equations at the heart of the physical problem. Due to my incompetence,I will not dwell on this important question.

I prefer to consider the buttery as a nice gift from physicists to mathemati-cians!

I thank Aurelien Alvarez, Maxime Bourrigan, Pierre Dehornoy, Jos Leys andMichele Triestino for their help when preparing those notes. I also thank StephaneNonnenmacher for his excellent translation of the French version of this paper.

2. Starting with a few quotations

I would like to start by a few quotations, which illustrate the evolution of theopinions on dynamics across the last two centuries.

Let us start with Laplaces famous denition of determinism, in his 1814Essai philosophique sur les probabilites [41]:

We ought then to consider the present state of the universe as the eectof its previous state and as the cause of that which is to follow. Anintelligence that, at a given instant, could comprehend all the forcesby which nature is animated and the respective situation of the beingsthat make it up, if moreover it were vast enough to submit these datato analysis, would encompass in the same formula the movements ofthe greatest bodies of the universe and those of the lightest atoms. Forsuch an intelligence nothing would be uncertain, and the future, like thepast, would be open to its eyes.

The fact that this quotation comes from a (fundamental) book on probabilitytheory shows that Laplaces view on determinism was far from naive [38]. Welack the vast intelligence he mentions, so we are forced to use probabilities tounderstand dynamical systems. Isnt that a modern idea, the rst reference toergodic theory?

In his little book Matter and Motion published in 1876, Maxwell insistson the sensitivity to initial conditions in physical phenomena: the intelligencementioned by Laplace must indeed be innitely vast [48]! One should notice that,according to Maxwell, this sensitivity is not the common rule, but rather andexception. This debate is still not really closed today.

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There is a maxim which is often quoted, that The same causes willalways produce the same eects.To make this maxim intelligible we must dene what we mean by thesame causes and the same eects, since it is manifest that no event everhappens more that once, so that the causes and eects cannot be thesame in all respects.[. . . ]There is another maxim which must not be confounded with that quotedat the beginning of this article, which asserts That like causes producelike eects.This is only true when small variations in the initial circumstances pro-duce only small variations in the nal state of the system. In a greatmany physical phenomena this condition is satised; but there are othercases in which a small initial variation may produce a great change inthe nal state of the system, as when the displacement of the pointscauses a railway train to run into another instead of keeping its propercourse.

With his sense of eloquence, Poincare expresses in 1908 the dependence toinitial conditions in a way almost as fashionable as Lorenzs buttery that we willdescribe below, including the devastating cyclone [62]:

Why have meteorologists such diculty in predicting the weather withany certainty? Why is it that showers and even storms seem to come bychance, so that many people think it quite natural to pray for rain or neweather, though they would consider it ridiculous to ask for an eclipse byprayer?We see that great disturbances are generally produced in regionswhere the atmosphere is in unstable equilibrium. The meteorologists seevery well that the equilibrium is unstable, that a cyclone will be formedsomewhere, but exactly where they are not in a position to say; a tenth ofa degree more or less at any given point, and the cyclone will burst hereand not there, and extend its ravages over districts it would otherwisehave spared. If they had been aware of this tenth of a degree they couldhave known it beforehand, but the observations were neither sucientlycomprehensive nor suciently precise, and that is the reason why it allseems due to the intervention of chance.

Poincares second quotation shows that he does not consider chaos as anobstacle to a global understanding of the dynamics [61]. However, the contextshows that he is discussing gas kinetics, which depends on a huge number ofdegrees of freedom (positions and speeds of all atoms). Even though Poincare hasrealized the possibility of chaos in celestial mechanics (which depends on muchfewer degrees of freedom), he has apparently not proposed to use probabilisticmethods to study it4.

4Except for the recurrence theorem?


You are asking me to predict future phenomena. If, quite unluckily, Ihappened to know the laws of these phenomena, I could achieve this goalonly at the price of inextricable computations, and should renounce toanswer you; but since I am lucky enough to ignore these laws, I willanswer you straight away. And the most astonishing is that my answerwill be correct.

Could this type of ideas apply to celestial mechanics as well? In his 1898 article onthe geodesics of surfaces of negative curvature, after noticing that a tiny changeof direction of a geodesic [. . . ] is sucient to cause any variation of the nal shapeof the curve, Hadamard concludes in a cautious way [34].

Will the circumstances we have just described occur in other problemsof mechanics? In particular, will they appear in the motion of celestialbodies? We are unable to make such an assertion. However, it is likelythat the results obtained for these dicult cases will be analogous tothe preceding ones, at least in their degree of complexity.[. . . ]Certainly, if a system moves under the action of given forces and itsinitial conditions have given values in the mathematical sense, its futuremotion and behavior are exactly known. But, in astronomical problems,the situation is quite dierent: the constants dening the motion areonly physically known, that is with some errors; their sizes get reducedalong the progresses of our observing devices, but these errors can nevercompletely vanish.

Some people have interpreted these diculties as a sign of disconnectionbetween mathematics and physics. On the opposite, as Duhem already noticed in1907, they can be seen as a new challenge for mathematics, namely the challengeto develop what he called the mathematics of approximation [23].

One cannot go through the numerous and dicult deductions of celestialmechanics and mathematical physics without suspecting that many ofthese deductions are condemned to eternal sterility.

Indeed, a mathematical deduction is of no use to the physicist solong as it is limited to asserting that a given rigorously true propositionhas for its consequence the rigorous accuracy of some such other propo-sition. To be useful to the physicist, it must still be proved that thesecond proposition remains approximately exact when the rst is onlyapproximately true. And even that does not suce. The range of thesetwo approximations must be delimited; it is necessary to x the limitsof error which can be made in the result when the degree of precisionof the methods of measuring the data is known; it is necessary to denethe probable error that can be granted the data when we wish to knowthe result within a denite degree of approximation.

Such are the rigorous conditions that we are bound to impose onmathematical deduction if we wish this absolutely precise language to be

6 E. Ghys

able to translate without betraying the physicists idiom, for the termsof this latter idiom are and always will be vague and inexact like theperceptions which they are to express. On these conditions, but onlyon these conditions, shall we have a mathematical representation of theapproximate.

But let us not be deceived about it; this mathematics of approx-imation is not a simpler and cruder form of mathematics. On the con-trary, it is a more thorough and more rened form of mathematics,requiring the solution of problems at times enormously dicult, some-times even transcending the methods at the disposal of algebra today.

Later I will describe Lorenzs fundamental article, dating back to 1963, which bearsthe technical title Deterministic non periodic ow, and was largely unnoticedby mathematicians during about 10 years [44]. In 1972 Lorenz gave a conferenceentitled Predictability: does the ap of a butterys wings in Brazil set o a tor-nado in Texas?, which made famous the buttery eect [45]. The three followingsentences, extracted from this conference, seem to me quite remarkable.

If a single ap of a butterys wing can be instrumental in generating atornado, so all the previous and subsequent aps of its wings, as can theaps of the wings of the millions of other butteries, not to mention theactivities of innumerable more powerful creatures, including our ownspecies.If a ap of a butterys wing can be instrumental in generating a tor-nado, it can equally well be instrumental in preventing a tornado.More generally, I am proposing that over the years minuscule distur-bances neither increase nor decrease the frequency of occurrence of var-ious weather events such as tornados; the most they may do is to modifythe sequence in which these events occur.

The third sentence in particular is scientically quite deep, since it proposes thatthe statistical description of a dynamical system could be unsensitive to the ini-tial conditions; this idea could be seen as a precursor of the SinaiRuelleBowenmeasures which, as I will later describe, provide a quantitative description of thistype of chaotic system.

3. The old paradigm of periodic orbits

3.1. Some jargon

One should rst set up the jargon of the theory of dynamical systems. The spaceson which the motion takes place will almost always be the numerical spaces (and most often 3). Sometimes more general spaces, like dierentiable manifolds (e.g., a sphere or a torus) will represent the phase space of the system. Thetopology of and the dynamics can be strongly correlated, this interaction beingthe main motivation for Poincare to study topology. However, we will ignore thisaspect here . . .


The dynamics is generated by an evolution dierential equation, or equiva-lently a vector eld (assumed dierentiable) on . Each point on is thestarting point of a trajectory (or an orbit) on . Without any further assumption,this trajectory may not be dened for all time : it could escape to innityin nite time. If all trajectories are dened for for all , the eld is said tobe complete: this is always the case if is compact. In this case, for and , we may denote by () the position at time of the trajectory startingat for = 0. For each the transformation : " () is a dieren-tiable bijection on (with dierentiable inverse ): it is a dieomorphism of . Obviously 1+2 is the composition of 1 and 2 , and () is called theow generated by the vector eld . In other terms, represents the vast in-telligence Laplace was dreaming of, which would potentially solve all dierentialequations! When a mathematician writes Let be the ow generated by , heclaims he has made this dream real. . . Of course, in most cases we make as if weknew , but in reality knowing only asymptotic behavior when goes to innitywould be sucient for us.

One could object with some ground that the evolution of a physical systemhas no reason to be autonomous, namely the vector eld could itself depend ontime. This objection is, in some way, at the heart of Maxwells argument, whenhe notices that the same cause can never occur at two dierent times, becauseindeed the times are dierent. I will nevertheless restrict myself to autonomousdierential equations, because they cover a suciently vast range of applications,and also because it would be impossible to develop such a rich theory without anyassumption on the time dependence of the vector eld.

The vector elds on are not always complete, but they may be transverseto a sphere, such that the orbits of points on the sphere enter the ball and cannever exit. The trajectories of points in the ball are then well dened for all 0.In this case, is in fact only a semiow of the ball, in the sense that it is denedonly for 0; this is not a problem if one is only interested in the future ofthe system. The reader not familiar with dierential topology can, as a rst step,restrict himself to this particular case.

It is customary to study as well discrete time dynamics. One then chooses adieomorphism on a manifold and studies its successive iterations = ( times) where the time is now an integer.

One can often switch between these two points of view. From a dieomor-phism on , one can glue together the two boundaries of [0, 1] using ,such as to construct a manifold with one dimension more than , which car-ries a natural vector eld , namely , where is the coordinate on [0, 1] (seeFigure 2). Notice that the vector eld is nonsingular. The original manifold can be considered as a hypersurface in , transverse to , for instance given byxing = 0. is often called a global section of in . The orbits () of travel inside and meet the global section when is integral, on the orbits ofthe dieomorphism . Clearly, the continuous time ow on and the discrete

8 E. Ghys

x

(x)

V [0, 1] V

Figure 2. A suspension

time dynamics on are so closely related to each other, that understandingthe latter is equivalent with understanding the former. The ow , or the eld , is called the suspension of the dieomorphism .

Conversely, starting from a vector eld on a manifold , it is often possibleto nd a hypersurface which meets each orbit innitely often. To each point one can then associate the point (), which is the rst return point on along the future orbit of . This dieomorphism on is the rst return map.The eld is then the suspension of the dieomorphism .5

Yet all elds are not suspensions. In particular, vector elds with singular-ities cannot be suspensions, and we will see that this is the case of the Lorenzequation. . . Nevertheless, this idea (due to Poincare) to transform a continuousdynamics into a discrete one is extremely useful, and can be adapted, as we willsee later.

3.2. A belief

For a long time, one has focussed on two types of orbits:

the singularities of are xed points of the ow : these are equilibriumpositions of the system;

the periodic orbits are the orbits of points such that, for some > 0 onehas () = .

The old paradigm which entitles this section is the belief that, in general,after a transient period, the motion evolves into a permanent regime which iseither an equilibrium, or a periodic orbit.

Let us be more precise. For a point in , one denes the -limit set (resp.the -limit set) of , as the set () (resp. ()) made of the accumulationpoints of the orbit () when + (resp. ). The above belief consistsin the statement that, for a generic vector eld on , and for each ,the set () reduces either to a single point, or to a periodic orbit. It is notnecessary to insist on the ubiquity of periodic phenomena in science (for instance in

5In general the return time is not necessarily constant, and the ow needs to be reparametrizedto obtain a true suspension.


astronomy). The realization that other types of regimes could be expected occurredamazingly late in the history of science.

In fact, the rst fundamental results in the theory of dynamical systems,around the end of the nineteenth century, had apparently conrmed this belief.Let us cite for instance the PoincareBendixson theorem: for any vector eld onthe two-dimensional disk, entering the disk on the boundary and possessing (tosimplify) nitely many xed points, the -limit sets can only be of three types: axed point, a periodic orbit or a singular cycle (that is, a nite number of singularpoints connected by nitely many regular orbits). See Figure 3.

Figure 3. A xed point, a limiting periodic orbit, and a cycle.

It is easy to show that the case of a singular cycle is unstable, that is, itdoes not occur for a generic vector eld; this remark conrms the above belief,in the case of ows on the disk. Interestingly, Poincare did not explicitly notice the(rather simple) fact that, among the three asymptotic limits, one is exceptional,while the two others are generic. Robadeys thesis [63] discusses the concepts ofgenericity in Poincares work. The explicit investigation of the behavior of genericvector elds started much later, probably with Smale and Thom at the end of the1950s.

The case of vector elds on general surfaces needed more work, and was onlycompleted in 1962 by Peixoto [55]. There is no PoincareBendixson type theorem,and the -limit sets can be much more complicated than for the disk. Yet, thesecomplex examples happen to be rare, and generally the limit sets are indeedequilibrium points or periodic orbits.

We shall now introduce an important concept, which we will later widelygeneralize. A singularity 0 of a vector eld (on a manifold ) is said to behyperbolic if the linearized vector eld at this point (namely, a matrix) has noeigenvalue on the imaginary axis. The set of points the orbits of which convergeto 0 is the stable manifold of 0: it is a submanifold (0) immersed in , whichcontains 0 and has dimension equal to the number of eigenvalues with negativereal parts. Switching to , one similarly denes the unstable manifold (0)of 0, of dimension complementary to that of (0) (since the singularity ishyperbolic).

If is a periodic orbit containing a point 0, one can choose a ball ofdimension 1 containing 0 and transverse to the ow. The ow then allows todene Poincares rst return map, which is a local dieomorphism in dened

10 E. Ghys

near 0 and xing that point. If the linearization of this dieomorphism at 0 doesnot have any eigenvalue of modulus 1, then the orbit is said to be hyperbolic. Theset of points such that () coincides with the periodic orbit is the stablemanifold of , denoted by (). As above, it is an immersed submanifold. Theunstable manifold () is dened similarly.

In 1959 Smale dened the vector elds which are now called of MorseSmaletype [72]. These elds are characterized by the following properties:

has at most nitely many singularities and periodic orbits, which are allhyperbolic;

The and -limit sets of all points are singularities or periodic orbits; the stable and unstable manifolds of the singularities or the periodic orbitsintersect each other transversally.

x0

W s(x0)

W u(x0)

W s()

B

Figure 4. Hyperbolic xed point and periodic orbit

In this article Smale formulated a triple conjecture. Before stating it, I rstneed to explain the fundamental concept of structural stability, introduced in 1937by Andronov and Pontrjagin [5]. A vector eld is structurally stable if thereexists a neighborhood of (in the 1 topology on vector elds) such that all elds in this neighborhood are topologically conjugate to . This conjugacy meansthat there exists a (generally non-dierentiable) homeomorphism in which mapsthe orbits of to the orbits of , keeping the time orientations. For such elds ,the topological dynamics is qualitatively insensitive to small perturbations. Thisconcept of structural stability thus belongs to the mathematics of approximationcalled for by Duhem. Andronov and Pontrjagin had shown that certain very simpleelds on the two-dimensional disk are structurally stable. The most naive exampleis given by the radial eld = / /, for which all points convergetowards the (singular) origin. If one perturbs , the new eld will still have anattracting singularity near the origin: is thus structurally stable.


I can now state Smales conjectures:

1. Given a compact manifold , the MorseSmale elds form an open dense setin the space of all vector elds on .

2. All MorseSmale elds are structurally stable.3. All structurally stable elds are of MorseSmale type.

The rst and third conjecture are false, as Smale will soon himself discover.But the second one is true. . . Smales motivation for these conjectures was clear:he wanted to prove the (then dominant) belief in a permanent regime equilibriumpoint or periodic orbit for a generic system.

In 1962 Peixoto proved the 3 conjectures if is a compact orientable sur-face [55].

The rst of Smales conjectures is surprising, since Poincare or Birkho al-ready knew it was false: a ow can have innitely many periodic orbits in a stableway. About this period, Smale wrote in 1998 [77]:

It is astounding how important scientic ideas can get lost, even whenthey are aired by leading scientic mathematicians of the precedingdecades.

It was explicitly realized around 1960 that the dynamics of a generic vector eldis likely to be much more complicated than that of a MorseSmale eld. Theparadigm of the periodic orbits yielded to the next one that of hyperbolic systems which I will describe in the next section. But periodic orbits will continue to playa fundamental role in dynamics, as Poincare had explained in 1892 [59, Chap. 3,Sec. 36]:

In addition, these periodic solutions are so valuable for us because theyare, so to say, the only breach by which we may attempt to enter anarea heretofore deemed inaccessible.

One more remark before going on towards the Lorenz attractor . . . The dynamicswe are considering here are not assumed to be conservative. One could discuss forinstance the case of Hamiltonian dynamical systems, which enjoy very dierentqualitative behaviors. For example, the preservation of phase space volume impliesthat almost all points are recurrent, as follows from Poincares recurrence theorem.A conservative eld is thus never of MorseSmale type.

In the domain of Hamiltonian vector elds, the old paradigm is quasi-periodicmotion. One believes that in the ambient manifold, many orbits are situated oninvariant tori supporting a linear dynamics, generated by a certain number ofuncoupled harmonic oscillators with dierent periods. The typical example is theKeplerian motion of the planets, assuming they do not interact with each other:each planet follows a periodic trajectory, and the whole system moves on a torusof dimension given by the number of planets. The KAM theory allows to showthat many of these invariant tori persist if one perturbs a completely integrableHamiltonian system. On the other hand, there is no reason to nd such tori for ageneric Hamiltonian system.

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One should thus keep in mind that we are discussing here a priori dissipativedynamics. It is amazing that eminent physicists like Landau and Lifschitz have fora while presented turbulence as an almost periodic phenomenon, with invariant toriof dimensions depending on Reynolds number. Only in the 1971 second edition oftheir famous treatise on uid mechanics have they realized that almost periodicfunctions are too nice to describe turbulence.

4. Second attempt: hyperbolic dynamics

4.1. Hadamard and the geodesics on a bulls forehead

In 1898 Hadamard publishes a remarkable article on the dynamical behavior of thegeodesics on surfaces of negative curvature [34]. This article can be considered asthe starting point of the theory of hyperbolic dynamical systems and of symbolicdynamics. It probably appeared too early since, more than 60 years later, Smalehad to follow again the same path followed by Hadamard before continuing muchfurther, as we will see.

Figure 5. Hadamards pants

Hadamard starts by giving some concrete examples of negatively curved sur-faces in the ambient space. The left part of Figure 5 is extracted from his article.This surface is dieomorphic to a plane minus two disks, which allows me to callit (a pair of) pants, and to draw it like on the right part of Figure 5.

The problem is to understand the dynamical behavior of the geodesics onthis surface. That is, a point is constrained to move on , only constrained by thereaction force. At each moment the acceleration is orthogonal to the surface: thetrajectories are geodesics of , at constant speed.

An initial condition consists in a point on and a tangent vector to onthis point, say of length 1. The set of these initial conditions forms the manifold = 1 of dimension 3, called the unitary tangent bundle of . The geodesic ow acts on : one considers the geodesic starting from a point in a certain direction,and follow it during the time to get another point and another direction.

hell02From [34] with kind permission from (c) Elsevier (???).

... depends on whether they have the back archive


The main property of the negative curvature used by Hadamard is the fol-lowing: every continuous path drawn on the surface between two points can bedeformed (keeping the boundary points xed) into a unique geodesic arc joiningthe two boundary points.

One can nd three closed geodesics 1, 2, 3 cutting into 4 parts. Threeof them correspond to the three ends of , and the fourth one is the convexcore, which is a compact surface with boundary given by the three geodesics (seeFigure 6).

g1

g2g3

Figure 6. The core of the pants

Let us now consider a geodesic " () . If the curve intersects1, 2 or 3 at time 0, to exit the core and enter one end, the property I have justmentioned shows that for any > 0 the curve () remains in this end and cannotcome back in the core. In fact, one can check that () goes to innity in that end.Conversely, if a geodesic enters the core at time 0, it remains in one of the endswhen . One can also show that no geodesic can stay away from the corefor ever. There are thus several types of geodesics:

() is in the core for all ; () comes from one end, enters the core and exits into an end; () comes from one end, enters the core and stays there for all large ; () is in the core for suciently negative, and exits into an end.The most interesting ones are of the rst type: they are called nowadays

nonwandering orbits. Hadamard analyzed them as follows. Let us join 1 and 2by an arc 3 of minimal length: it is a geodesic arc inside the core, orthogonal to1 and 2. Similarly, let 1 connect 2 with 3, and 2 connect 1 with 3. The arcs are the seams of the pants. If one cuts the core along the seams, one obtainstwo hexagons 1 and 2 (see Figure 7).

Let us now consider a geodesic of the rst type, that is entirely containedin the core. The point (0) belongs to one of the hexagons, maybe to both if it lies

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c3

c1

c2

Figure 7. Three cuts in the pants leading to two hexagons

on a seam, but let us ignore this particular case. When one follows the geodesic from (0) by increasing the time , one successively intersects the seams 1, 2, 3innitely many times6. One can then read an innite word written in the 3-letteralphabet {1, 2, 3}. If intersects the seam 1, the next seam cannot be 1: it willbe either 2 or 3. The word associated to a geodesic will thus never contain twosuccessive identical letters.

Similarly, one can follow in the negative time direction and obtain a word.These two words make up a single bi-innite word () associated with the geo-desic .

Actually, Hadamard only treats the closed geodesics, which are associatedwith the periodic words. In 1923 Morse will complete the theory by coding thenonperiodic geodesics by bi-innite words [50]. In the sequel I will mix the twoarticles, and call their union HadamardMorse.

The main result of HadamardMorse is the following:For any bi-innite word in the alphabet {1, 2, 3} without repetition, there

exists a geodesic realizing that word. The geodesic is unique if one species thehexagon containing (0).

Of course, the uniqueness of the geodesic should be understood as follows:one can move the origin of into () without changing the word, as long as does not meet any seam between = 0 and = . Uniqueness means that if twogeodesics of type 1 are associated with the same word and start from the samehexagon, then they can only dier by such a (short) time shift.

The proof of this result is quite easy (with modern techniques). Startingfrom a bi-innite word, one takes the word of length 2 obtained by keepingonly the rst letters on the right and on the left. One then considers a path : [, ] " which follows the word and starts from 1 or 2. Moreprecisely, one chooses a point 1 in 1 and 2 in 2, and the path is formed by2 geodesic arcs alternatively joining 1 and 2 and crossing the seams as indicated

6This fact is due to the negative curvature.


in the word . This arc can be deformed, keeping the boundaries xed, intoa geodesic arc : [, ] . One then needs to show that this path convergesto a geodesic : when , which is easy for a modern mathematician(using Ascolis theorem etc.). Uniqueness is not very dicult to check either.

Here are a few qualitative consequences: If two bi-innite words and coincide from a certain index on, the corre-

sponding geodesics , will approach each other when +: they are asymp-totes. There exists such that the distance between () and ( + ) tends to0 when +. Nowadays we would say that and belong to the same sta-ble manifold. Of course, a similar remark applies when and coincide forall suciently negative indices: the geodesics then belong to the same unstablemanifold. Starting from two innite words + and without repetitions, one can of

course construct a bi-innite word coinciding with + for suciently positiveindices, and with for suciently negative indices. Hence, given two nonwan-dering geodesics, one can always nd a third one which is an asymptote of the rstone in the future, and an asymptote of the second one in the past. One can evenarbitrarily x any nite number of indices of . Obviously, this result implies asensitive dependence to the initial conditions: an arbitrarily small perturbation ofa given geodesic can make it asymptote to two arbitrary geodesics, respectively inthe past and in the future. Another interesting property: any nonwandering geodesic can be approached

arbitrarily close by periodic geodesics. Starting from a bi-innite word, it suces toconsider a long segment of this word and to repeat it innitely often to construct anearby periodic geodesic. Therefore there exist countably many periodic geodesics,and their union is dense in the set of nonwandering points. The dynamical behaviorof these geodesics is thus much more complicated than for a MorseSmale ow.

To end this brief account of these two articles, I shall mention some conse-quences HadamardSmale could have easily obtained, but did not notice. The description of the dynamics of the geodesics does not depend on the

choice of metric with negative curvature. One could almost deduce that for twometrics of negative curvature on , the geodesic ows are topologically conjugate.We are close to the structural stability, proved by Anosov in 1962 [3]: the geodesicow of a Riemannian metric with negative curvature is structurally stable. More-over, Gromov showed that, on a given compact manifold, the geodesic ows of twoarbitrary metrics of negative curvature are topologically conjugate [30].7

Another aspect, implicit in these articles, is the introduction of a dynamicson a space of symbols. The space of all sequences () {1, 2, 3} is compact,and the shift of the indices () " (+1) is a homeomorphism which codesthe dynamics of the geodesic ow. More precisely, one should take the subshift of

7Actually, the two metrics could be dened on two dierent manifolds with isomorphic funda-mental groups.

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R112

R221

R223

Figure 8. Dynamics on the rectangles

nite type formed by the sequences without repetitions, and (even more precisely)one should take two copies of this space, corresponding to the two hexagons.

Finally, by extrapolating a little, one could see in these articles a subliminalconstruction of Smales horseshoe, an object I will soon describe. Indeed, considerall unit vectors tangent to on a point of a seam, heading towards another seam.The geodesic generated by such an initial condition starts from , crosses 1 or2, and lands on . These unit vectors form twelve rectangles: one has to choose, , then 1 or 2, which gives twelve possibilities, and for each one we shouldindicate the starting and arrival points on the seams. These twelve rectangles denoted by 1 ,

2 are embedded in

1 , the unitary tangent bundle of ,transversely to the geodesic ow.

Strictly speaking, the union of these rectangles is not a global section forthe geodesic ow, since they do not meet all the geodesics: the seams themselvesdene geodesics not meeting . Nevertheless, one can dene a rst return map, which is not dened in the whole of , and which is not surjective. Startingfrom a unit vector in , the corresponding geodesic crosses a hexagon and landson another cut. The tangent vector at the exit point is not necessarily in , sincethe geodesic could then exit the second hexagon along 1, 2 or 3 instead ofanother cut. If this tangent vector is still in , we denote it by (). The domainof denition of is the union of 24 vertical rectangular zones, two in eachrectangle, and its image is formed by 24 horizontal rectangular zones. Each ofthe 24 vertical rectangular zones is contracted by along the vertical direction, andexpanded along the horizontal direction, and its image is one of the 24 horizontal


rectangular zones. Figure 8 displays the two vertical zones in 112 and their imagesin 221 and

223.

This simultaneous appearance of expansion and contraction is clearly at theheart of the phenomenon, but it was not explicitly noticed by HadamardMorse.An easy observation: if one slightly perturbs the geodesic ow (not necessarilyinto another geodesic ow), will still be transverse to the ow, and the returnon will have the same shape, with slightly deformed rectangular zones; thenonwandering orbits will still be coded by sequences of symbols, accounting forthe sequences of rectangles crossed along the evolution. The structural stability(at least of the nonwandering set) can be easily deduced from this observation.

It is surprising that this article of Hadamard, containing so many originalideas, could stay unnoticed so long. Yet, Duhem described this article in such acolorful way as to attract the attention even of non-mathematicians [23].

Imagine the forehead of a bull, with the protuberances from which thehorns and ears start, and with the collars hollowed out between theseprotuberances; but elongate these horns and ears without limit so thatthey extend to innity; then you will have one of the surfaces we wishto study. On such a surface geodesics may show many dierent aspects.There are, rst of all, geodesics which close on themselves. There aresome also which are never innitely distant from their starting pointeven though they never exactly pass through it again; some turn con-tinually around the right horn, others around the left horn, or rightear, or left ear; others, more complicated, alternate, in accordance withcertain rules, the turns they describe around one horn with the turnsthey describe around the other horn, or around one of the ears. Finally,on the forehead of our bull with his unlimited horns and ears therewill be geodesics going to innity, some mounting the right horn, oth-ers mounting the left horn, and still others following the right or leftear. [. . . ] If, therefore, a material point is thrown on the surface studiedstarting from a geometrically given position with a geometrically givenvelocity, mathematical deduction can determine the trajectory of thispoint and tell whether this path goes to innity or not. But, for thephysicist, this deduction is forever useless. When, indeed, the data areno longer known geometrically, but are determined by physical proce-dures as precise as we may suppose, the question put remains and willalways remain unanswered.

4.2. Smale and his horseshoe

Smale has given several accounts of his discovery of hyperbolic systems around19608 (see for instance [77]). This discovery is independent of the previous contri-butions of Poincare and Birkho, which however played a role in the subsequentdevelopments of the theory. It seems that Hadamards article played absolutely

8On the beach in Copacabana, more precisely in Leme!

18 E. Ghys

no role at this time. Nevertheless, Smale constructs a counterexample to his ownconjecture, according to which MorseSmale ows form an open dense set in thespace of dynamical systems: the famous horseshoe. It is a dieomorphism onthe two-dimensional sphere 2, thought of as the plane 2 with an extra pointat innity. is assumed to map a rectangle in the plane as shown on Fig-ure 9, by expanding the vertical directions and contracting the horizontal ones.The intersection () is the union of two rectangles 1, 2.

R R1 R2

(R)

Figure 9. Smales horseshoe

One can assume that the point at innity is a repulsive xed point. Thepoints of which always stay in , that is (), form a Cantor set, homeo-morphic to {1, 2}. Each of its points is coded by the sequence of rectangles 1, 2successively visited by its orbit. In particular, the periodic points form an innitecountable set, dense in this Cantor set. Smale then establishes that the horseshoeis structurally stable, a rather easy fact (nowadays). If one perturbs the dieomor-phism into a dieomorphism , the intersection () is still made of tworectangular zones crossing all along, and one can still associate a single orbitto each sequence in {1, 2}, allowing to construct a topological conjugacy between and , at least on these invariant Cantor sets. There just remains to extend theconjugacy in the exterior, using the fact that all exterior orbits come from innity.

Smale publishes this result in the proceedings of a workshop organized inthe Soviet Union in 1961 [73]. Anosov tells us about this hyperbolic revolutionin [4].

The world turned upside down for me, and a new life began, having readSmales announcement of a structurally stable homeomorphism with aninnite number of periodic points, while standing in line to register for


a conference in Kiev in 1961. The article is written in a lively, witty, andoften jocular style and is full of captivating observations. [. . . ] [Smale]felt like a god who is to create a universe in which certain phenomenawould occur.

Afterwards the theory progresses at a fast pace. The horseshoe is quicklygeneralized by Smale (see for instance [74]). As I already mentioned, Anosov provesin 1962 that the geodesic ow on a manifold of negative curvature is structurallystable9. For this aim, he conceives the concept of what is known today as anAnosov ow.

A nonsingular ow on a compact manifold , generated by a vector eld , is an Anosov ow if at each point one can decompose the tangent space into three subspaces .() . The rst one .() is the linegenerated by the vector eld, and the others are called respectively unstable andstable subspaces. This decomposition should be invariant through the dierential of the ow, and there should exist constants > 0, > 0 such that for all and , :

() 1 exp() ; () exp()(here is an auxiliary Riemannian metric).

Starting from the known examples of structurally stable systems (MorseSmale, horseshoe, geodesic ow on a negatively curved manifold and a few others),Smale cooked up in 1965 the fundamental concept of dynamical systems satisfyingthe Axiom A (by contraction, Axiom A systems).

Consider a compact set invariant through a dieomorphism . issaid to be a hyperbolic set if the tangent space to restricted to points of admits a continuous decomposition into a direct sum: = , invariantthrough the dierential , and such that the vectors in are expanded, whilethose in are contracted. Precisely, there exists > 0, > 0 such that, for all , , , one has:

() 1 exp() ; () exp().A point in is called wandering if it has an open neighborhood disjoint fromall its iterates: () = for all = 0. The set of nonwandering points is aninvariant closed set, traditionally denoted by ()10.

By denition, is Axiom A if () is hyperbolic and if the set of periodicpoints is dense in ().

Under this assumption, if is a nonwandering point, the set () (resp.()) of the points such that the distance between () and () goes to 0when tends to + (resp. ) is a submanifold immersed in : it is the stable(resp. unstable) manifold of the point . An Axiom A dieomorphism satisesthe strong transversality assumption if the stable manifolds are transverse to theunstable ones.

9Surprisingly, he does not seem to know Hadamards work.10Although the notation looks like one of an open set.

20 E. Ghys

An analogous denition can be given for vector elds : one then needsto continuously decompose into a sum . . In particular, thisdenition implies that the singular points of are isolated in the nonwanderingset, otherwise the dimensions of the decomposition would be discontinuous.

Smale then states three conjectures, parallel to the ones he had formulatednine years earlier:

1. Given a compact manifold , the Axiom A dieomorphisms satisfying thestrong transversality condition form an open dense set in the set of dieo-morphisms of .

2. The Axiom A dieomoprhisms satisfying the strong transversality conditionare structurally stable

3. The structurally stable dieomorphisms are Axiom A and satisfy the strongtransversality condition.

The second conjecture is correct, as shown by Robbin in 1971 and by Robinsonin 1972 (assuming dieomorphisms of class 1). One had to wait until 1988 andManes proof to check the third conjecture.

However, the rst conjecture is wrong, as Smale himself will show in 1966 [75].As we will see, a counterexample was actually available in Lorenzs article fouryears earlier, but it took time to the mathematics community to notice this result.Smales counterexample is of dierent nature: he shows that a defect of transver-sality between the stable and unstable manifolds can sometimes not be repairedthrough a small perturbation. However, Smale introduces a weaker notion thanstructural stability, called -stability: it requires that, after a perturbation of thedieomorphism into , the restrictions of to () and of to () beconjugate through a homeomorphism. He conjectures that property to be generic.

Smales 1967 article Dierential dynamical systems represents an importantstep for the theory of dynamical systems [76], a masterpiece of mathematicalliterature according to Ruelle [69].

But, already in 1968, Abraham and Smale found a counterexample to thisnew conjecture, also showing that Axiom A is not generic [1]. In 1972, Shub andSmale experiment another concept of stability [70], which will lead Meyer to thefollowing comment:

In the never-ending quest for a solution of the yin-yang problem moreand more general concepts of stability are proered.

Bowens 1978 review article is interesting on several points [18]. The theory ofAxiom A systems has become solid and, although dicult open questions remain,one has a rather good understanding of their dynamics, both from the topologicaland ergodic points of view. Even if certain dark swans have appeared in thelandscape, destroying the belief in the genericity of Axiom A, these are still studied,at that time, as if they were hyperbolic. Here are the rst sentences of Bowensarticle, illustrating this point of view.

These notes attempt to survey the results about Axiom A dieomor-phisms since Smales well-known paper of 1967. In that paper, Smale


dened these dieomorphisms and set up a program for dynamical sys-tems centered around them. These examples are charming in that theydisplay complicated behavior but are still intelligible. This means thatthere are many theorems and yet some open problems. [. . . ] The lastsections deal with certain non-Axiom A systems that have received agood deal of attention. These systems display a certain amount of Ax-iom A behavior. One hopes that further study of these examples willlead to the denition of a new and larger class of dieomorphisms, withthe Axiom A class of prototype.

It is not possible here to seriously present the theory of hyperbolic systems, thereader may try [36]. I will still describe a bit later some of the most importanttheorems, which give a more precise account of the ergodic behavior of thesesystems.

Still, I have to cite one of the major results due to Bowen which allows tounderstand the dynamics a` la Hadamard. Assume one covers the nonwandering set() by nitely many boxes ( = 1, 2, . . . , ), assumed compact with disjointinteriors. Let us construct a nite graph, with vertices indexed by , and such thateach oriented link connects to if the interior of () is nonempty (as asubset of ()). Let us denote by the closure of the set of sequences {1, 2, . . . , }corresponding to the innite paths on the graph, namely the sequences of indices() such that two consecutive indices are connected in the graph. One saysthat is a subshift of nite type. The collection of boxes is a Markov partition ifthis subshift faithfully codes the dynamics of . One requires that for each sequence , there exists a unique point = () in (), the orbit of which preciselyfollows the itinerary : for each , one has () (). But one also requiresthat the coding : () be as injective as possible: each ber 1() is niteand, for a generic point (in Baires sense in ()), it contains a single itinerary.Bowen establishes that the nonwandering sets of hyperbolic sets can be coveredby this type of Markov partition. The usefulness of this result is clear, since ittransfers a dynamical question into a combinatorial one.

Nowadays, Axiom A systems seem to occupy a much smaller place as wasbelieved at the end of the 1970s. The hyperbolic paradigm has abandoned itsdominant position . . . Anosovs following quotation could probably be expandedbeyond the mathematical world [4].

Thus the grandiose hopes of the 1960s were not conrmed, just as theearlier naive conjectures were not conrmed.

For a more detailed description of the hyperbolic history one can also read theintroduction of [36] or of [54]. See also What is . . . a horseshoe by one of theactors of the subject [69].

4.3. How about Poincares innitely thin tangle?

I barely dare to set a doubt on Poincares legacy in this seminar bearing his name.Many authors claim that Poincare is at the origin of chaos theory. His role is

22 E. Ghys

doubtlessly very important, but maybe not as much as is often claimed. Firstly,the idea that physical phenomena can be very sensitive to initial conditions wasnot totally new, as we have seen with Maxwell. Second, Poincares contributionson these questions seem to have been largely forgotten in the subsequent years,and have only played a minor role in the further development of the theory. Mostof Poincares results were rediscovered. More essentially, the central idea that Iwish to present here is that chaos theory cannot be restricted to the powerlessstatement that the dynamics is complicated: the theory must encompass methodsallowing to explain the internal mechanisms of this dynamics. The following famousquotation of Poincare illustrates this powerlessness facing the complexity of thedynamics [59, Chap. 33, Sec. 397]:

When we try to represent the gure formed by these two curves and theirinnitely many intersections, each corresponding to a doubly asymptoticsolution, these intersections form a type of trellis, tissue, or grid withinnitely ne mesh. Neither of the two curves must ever cut across itselfagain, but it must bend back upon itself in a very complex manner in or-der to cut across all of the meshes in the grid an innite number of times.

The complexity of this gure is striking, and I shall not even tryto draw it. Nothing is more suitable for providing us with an idea of thecomplex nature of the three-body problem, and of all the problems ofdynamics in general, where there is no uniform integral and where theBohlin series are divergent.

Poincares major contribution11 was to realize the crucial importance of ho-moclinic orbits. Lets consider a dieomorphism on the plane, with a hyperbolicxed point at the origin, with stable and unstable subspaces of dimension 1. Apoint (dierent from the origin) is homoclinic if it belongs to the intersection of the stable and unstable manifolds of the origin. The orbit of con-verges to the origin both in the future and in the past.

It is not necessary to recall here in detail how this concept appeared in 1890.On this topic, I recommend the reading of [12], which relates the beautiful storyof Poincares mistake in the rst version of his manuscript when applying for KingOscars prize [12]. The book [11] also discusses this prize, insisting on the historicaland sociological aspects rather than the mathematical ones.

Poincares setup was inspired by celestial mechanics, so preserves the area,and he had rst thought it was impossible for and to cross each otherwithout being identical. He deduced from there a stability theorem in the threebody problem. When he realized his error, he understood that these homoclinicpoints where meets transversely are not only possible, but actually therule. He also understood that the existence of such a point of intersection impliesthat the geometry of the curves and must indeed constitute an inn-itely thin tangle. Poincare did not try to draw the tangle, but to illustrate thecomplexity of the situation he shows that the existence of a transverse homoclinic

11in this domain!


Figure 10. A homoclinic orbit

intersection induces that of innitely many other homoclinic orbits: the arc of

situated between and () crosses the unstable manifold innitely often.In [68] Shilnikov wonders why Poincare does not attach more importance to

Hadamards article, where one may obviously detect the presence of homoclinicorbits: an innite sequence of 1, 2, 3 without repetitions and taking the value 1 onlynitely many times denes a geodesic on the pants, which converges towards theperiodic orbit 1 both in the future and in the past. But Poincare thinks that [60]:

The three-body problem should not be compared to the geodesics onsurfaces of opposite curvatures; on the opposite, it should be comparedto the geodesics on convex surfaces.

The opposition hyperbolic/elliptic, negative/positive curvature is not new.The next step in the analysis of homoclinic orbits is due to George Birkho

in 1935: through a cute geometrical argument, he establishes that one can ndperiodic points arbitrarily close to a transverse homoclinic point [13].

Then comes Smales rediscovery 25 years later. If one considers a rectangle near the stable manifold, containing both the origin and the homoclinic point,like in Figure 11, then a sucient iterate on acts like a horseshoe. One thus

Figure 11. A homoclinic point and its horseshoe

24 E. Ghys

obtains a description of the internal mechanism of this thin tangle, for instanceusing sequences of symbols 0, 1 as we have seen above. The presence of innitelymany periodic orbits becomes obvious. One can read in [68] an account on theevolution of the ideas around these homoclinic orbits.

5. The Lorenz attractor

How could one explain the lack of communication in the 1960s and 1970s betweena theoretical physicist like Lorenz and a mathematician like Smale? The rst onewas working on the East Coast and the second one on the West Coast of theUnited States . . . According to Williams [82], one reason would be the journalwhere Lorenz published his article12:

Though many scientists, especially experimentalists, knew this article, itis not too surprising that most mathematicians did not, considering forexample where it was published. Thus, when RuelleTakens proposed(1971) specically that turbulence was likely an instance of a strangeattractor, they did so without specic solutions of the NavierStokesequations, or truncated ones, in mind. This proposal, controversial atrst, has gained much favor.

It seems that Smale had very few physical motivations when cooking up his the-ory of hyperbolic systems, while physics itself does not seem to encompass manyhyperbolic systems. This is at least Anosovs point of view [4]:

One gets the impression that the Lord God would prefer to weaken hy-perbolicity a bit rather than deal with restrictions on the topology of anattractor that arise when it really is 1960s-model hyperbolic.

Even nowadays, it is not easy to nd physical phenomena with strictly hyperbolicdynamics (see however [35, 39]). In my view, one of the main challenges of thispart of mathematics is to restore contact with physics.

5.1. Lorenz and his buttery

Lorenzs 1963 article [44] is magnicent. Lorenz had been studying for a few yearssimplied models describing the motion of the atmosphere, in terms of ordinarydierential equations depending on few variables. For instance, in 1960 he describesa system he can explicitly solve using elliptic functions: the solutions are stillquasiperiodic in time [42]. His 1962 article analyzes a dierential equation in aspace of dimension 12, in which he numerically detects a sensitive dependence toinitial conditions [43]. But it is the 1963 paper which for good reasons leadhim to fame. The aim of the paper is clear:

In this study we shall work with systems of deterministic equationswhich are idealizations of hydrodynamical systems.

12One may also wonder whether the prestigious journal where Williams published his paper [82]is accessible to physicists.


After all, the atmosphere is made of nitely many particles, so one indeed needsto solve an ordinary dierential equation in a space of huge dimension. But ofcourse, such equations are highly intractable, and one must treat them throughpartial dierential equations; in turn, the latter must be discretized on a nitegrid, leading to new ordinary dierential equations depending on fewer variables,and probably more useful than the original ones. Lorenz discusses the type ofdierential equations he wants to study.

In seeking the ultimate behavior of a system, the use of conservativeequations is unsatisfactory [. . . ]. This diculty may be obviated byincluding the dissipative processes, thereby making the equations nonconservative, and also including external mechanical or thermal forcing,thus preventing the system from ultimately reaching a state at rest.

A typical dierential equation presenting both viscosity and forcing has the fol-lowing form:

=,

+

where

vanishes identically13 and

is positive denite. Thequadratic terms represent the advection, the linear terms

corre-

spond to the friction and the constant terms to the forcing. Lorenz observes thatunder these conditions, the vector eld is transverse to spheres of large radii (thatis, of high energy), so that the trajectories entering a large ball will stay thereforever. He can then discuss diverse notions of stability, familiar to contemporarymathematicians; periodicity, quasiperiodicity, stability in the sense of Poisson, etc.The bibliographic references in Lorenzs article include one article of Poincare, butit is the famous one from 1881 [57]. In this article, founding the theory of dynam-ical systems, Poincare introduces the limit cycles, and shows particular cases ofthe BendixsonPoincare theorem, introduces the rst return maps etc., but thereis no mention of chaotic behavior yet; chaos will be studied starting from the 1890memoir we have discussed above, which Lorenz seems to have overlooked. An-other bibliographic reference is a dynamical systems book by Birkho publishedin 1927. Again, this reference precedes Birkhos works in which he almost ob-tains a horseshoe . . .

Then, Lorenz considers as example the phenomenon of convection. A thinlayer of a viscous uid is placed between two horizontal planes, set at two dierenttemperatures, and one wants to describe the resulting motion. The high parts ofthe uid are colder, therefore denser; they have thus a tendency to go down dueto gravity, and are then heated when they reach the lower regions. The resultingcirculation of the uid is complex. Physicists know well the Benard and Rayleighexperiments. Assuming the solutions are periodic in space, expanding in Fourier

13This condition expresses the fact that the energy

2 is invariant through the quadraticpart of the eld.

26 E. Ghys

series and truncating these series to keep only few terms, Salzman had just ob-tained an ordinary dierential equation describing the evolution. Simplifying againthis equation, Lorenz obtained his equation:

/ = + / = + / = .

Here represents the intensity of the convection, represents the temperaturedierence between the ascending and descending currents, and is proportionalto the distortion of the vertical temperature prole from linearity, a positive valueindicating that the strongest gradients occur near the boundaries. Obviously, oneshould not seek in this equation a faithful representation of the physical phenome-non . . . The constant is the Prandtl number. Guided by physical considerations,Lorenz is lead to choose the numerical values = 28, = 10, = 8/3; it was agood choice, and these values remain traditional. He could then numerically solvethese equations, and observe a few orbits. The electronic computer Royal McBeeLGP-30 was rather primitive: according to Lorenz, it computed (only!) 1000 timesfaster than by hand . . .

But Lorenzs observations are nevertheless remarkably ne. He rst observesthe famous sensitivity to initial conditions14. More importantly, he notices thatthese sensitive orbits still seem to accumulate on a complicated compact set, whichis itself insensitive to initial conditions. He observes that this invariant compactset approximately resembles a surface presenting a double line along which twoleaves meet each other.

Thus within the limits of accuracy of the printed values, the trajectoryis conned to a pair of surfaces which appear to merge in the lowerportion. [. . . ] It would seem, then, that the two surfaces merely appearto merge, and remain distinct surfaces. [. . . ] Continuing this processfor another circuit, we see that there are really eight surfaces, etc., andwe nally conclude that there is an innite complex of surfaces, eachextremely close to one or the other of the two merging surfaces.

Figure 12 is reprinted from Lorenzs article. Starting from an initial condition, theorbit rapidly approaches this two-dimensional object and then travels on thissurface. The orbit then turns around the two holes, left or right, in a seeminglyrandom way. Notice the analogy with Hadamards geodesics turning around thebulls horns.

14The anecdote is quite well known I started the computer again and went out for a cup ofcoee. . . It was told in the conference Lorenz gave on the occasion of the 1991 Kyoto prize,A scientist by choice, which contains many other interesting things. In particular, he dis-cusses there his relations with mathematics. In 1938 Lorenz is a graduate student in Harvardand works under the guidance of G. Birkho on a problem in mathematical physics. He doesnot mention any inuence of Birkho on his conception of chaos. A missed encounter? On theother hand, Lorenz mentions that Birkho was noted for having formulated a theory of aes-thetics. Almost all Lorenzs works, including a few unpublished ones, can be downloaded onhttp://eapsweb.mit.edu/research/Lorenz/publications.htm.


Figure 12. Lorenzs diagram

Besides, Lorenz studies the way the orbits come back to the branching linebetween the two leaves, which can be parametrized by an interval [0, 1]. Obviously,this interval is not very well dened, since the two leaves do not really comein contact, although they coincide within the limits of accuracy of the printedvalues. Starting from a point on this interval, one can follow the future trajectoryand observe its return onto the interval. For this rst return map [0, 1] [0, 1],each point has one image but two preimages. This corresponds to the fact that,to go back in time and describe the past trajectory of a point in [0, 1], one shouldbe able to see two copies of the interval; these copies are undistinguishable onthe gure, so that two dierent past orbits emanate from the same point of theinterval. But of course, if there are two past orbits starting from one point, thereare four, then eight, etc., which is what Lorenz expresses in the above quotation.Numerically, the rst return map is featured on the left part of Figure 13. Workingby analogy, Lorenz compares this application to the (much simpler) following one: () = 2 if 0 1/2 and () = 2 2 if 1/2 1 (right part ofFigure 13). Nowadays the chaotic behavior of this tent map is well known, butthis was much less classical in 1963 . . . In particular, the periodic points of areexactly the rational numbers with odd denominators, which are dense in [0, 1].Lorenz does not hesitate to claim that the same property applies to the iterationsof the true return map. The periodic orbits of the Lorenz attractor are thusdense. What an intuition!

There remains the question as to whether our results really apply tothe atmosphere. One does not usually regard the atmosphere as eitherdeterministic or nite, and the lack of periodicity is not a mathematicalcertainty, since the atmosphere has not been observed forever.

To summarize, this article contains the rst example of a dissipative and physicallyrelevant dynamical system presenting all the characteristics of chaos. The orbitsare unstable but their asymptotic behavior seems relatively insensitive to initial

hell02From [44] p. 138 with kind permission from the American Meteorological Society 1963.

28 E. Ghys

Figure 13. Lorenzs graphs of rst return maps

conditions. None of the above assertions is justied, at least in the mathematicalsense. How frustrating!

Very surprisingly, an important question is not addressed in Lorenzs article.The observed behavior happens to be robust : if one slightly perturbs the dier-ential equation, for instance by modifying the values of the parameters, or byadding small terms, then the new dierential equation will feature the same typeof attractor with the general aspect of a surface. This property will be rigorouslyestablished later, as we will see.

Everybody has heard of the buttery eect. The terminology seems to haveappeared in the best-selling book of Gleick [29], and be inspired by the title of aconference by Lorenz in 1972 [45].

5.2. Guckenheimer, Williams and their template

The Lorenz equation pops up in mathematics in the middle of the 1970s. Accordingto Guckenheimer [32], Yorke mentioned to Smale and his students the existence ofthis equation, which was not encompassed by their studies. The well-known 1971article by Ruelle and Takens on turbulence [67] still proposes hyperbolic attractorsas models, but in 1975 Ruelle observes that Lorenzs work was unfortunatelyoverlooked [65]. Guckenheimer and Lanford are among the rst people to showsome interest in this equation (from a mathematical point of view) [31, 40]. Thenthe object will be fast appropriated by mathematicians, and it is impossible to givean exhaustive account of all their works. As soon as 1982 a whole book is devotedto the Lorenz equation, although it mostly consists in a list of open problems formathematicians [79].

I will only present here the fundamental works of Guckhenheimer andWilliams, who constructed the geometric Lorenz models [33, 82] (independently

hell02From [44] p. 139 with kind permission from the American Meteorological Society 1963.


from Afraimovich, Bykov and Shilnikov [2]). The initial problem consists in justi-fying the phenomena observed by Lorenz on his equation. We have seen that thisequation is itself a rough approximation of the physical phenomenon. Proving thatthe precise Lorenz equation satises the observed properties is thus not the mostinteresting issue. Guckenheimer and Williams have another aim: they consider thebehaviors observed by Lorenz as an inspiration, in order to construct vector elds,called geometric Lorenz models, satisfying the following properties:

for each or these elds, the set of nonwandering points is not hyperbolic, sinceit contains both nonsingular points and a singular one

the elds are not structurally stable; the elds form an open set in the space of vector elds.

Let us rst consider a linear vector eld:

= ,

= ,

=

with 0 < < < . Let be the square [1/2, 1/2] [1/2, 1/2]{1} 3. Theorbit starting from a point (0, 0, 1) in this square is (0 exp(), 0 exp(),exp()). Call the triangular zones where these orbits intersect the planesgiven by { = 1}. They are dened by the equations = 1, 12 / and > 0, so they are triangles with their lower corner being a cusp. One thenconsiders the zone (box) swept by the orbits starting from until they reach, to which one adds the future orbits of the points in { = 0;1/2 1/2; = 1} (which never intersect ), as well as the wedge {1 1; =0; = 0} (see Figure 14).

The Lorenz vector eld we will construct coincides with this linear vectoreld inside the box . Outside one proceeds such that the orbits exiting from

Figure 14. The box

30 E. Ghys

Figure 15. A Lorenz geometric model

come back inside the square . One then obtains a vector eld dened only insidea certain domain sketched in Figure 15.The main objective is to understand the dynamics inside , but one can alsoextend the vector eld outside this domain to get a globally dened eld. Such avector eld is a geometric Lorenz model.

f

Figure 16. Return maps

An important remark is that not all the points of the triangles originate fromthe square: the tips do not, since they come from the singular point at the origin.There are several ways to organize the return from onto ; one can make surethat the Poincare return map : has the following form:

(, ) = ((), (, )).

Technically, one requires that (, ) > 1/4 for > 0 and (, ) < 1/4for < 0. Furthermore, that the map : [1/2, 1/2] [1/2, 1/2] satises thefollowing conditions:

1. (0) = 1/2, (0+) = 1/2;2. () >

2 for all in [1/2, 1/2].


The second conditions implies that for all interval contained in [1/2, 1/2], thereexists an integer > 0 such that () = [1/2, 1/2]15

To describe the structure of the orbits inside the box, Williams introducesthe concept of template. Figure 17 is reprinted from [15]16: We are dealing with a

Figure 17. The template

branched surface 3 embedded in space, on which one can dene a semiow ( 0). A semiow means that : is dened only for 0 andthat 1+2 = 1 2 for all 1, 2. The trajectories of the semiow are sketchedon the gure: a point in has a future but has no past, precisely because of thetwo leaves which meet along an interval. The rst return map on this interval ischosen to be the map dened above. The dynamics of the semiow is easy tounderstand: the orbits turn on the surface, either on the left or on the right wing,according to the signs of the iterates ()

We shall now construct a ow starting from the semiow using a well-knownmethod, the projective limit. One considers the abstract space of the curves : which are trajectories of in the following sense: for all and +, one has (()) = (+ ). Given a point on , to choose such a curve with (0) = amounts to selecting a past for : on goes backwards in timealong the semiow, and at each crossing of the interval, one chooses one of thetwo possible preimages. The map " (0) thus has totally discontinuousbers, which are Cantor sets. The space is an abstract compact set equippedwith a ow dened by ()() = (+ ), which now makes sense for all .

15The fact that the graph of does not resemble Figure 13 is due to a dierent choice of notations.16Incidentally, this gure shows that the quality of an article does not depend on that of itsillustrations . . .

hell02From [15] p. 52 with kind permission from the Elsevier 1983.

32 E. Ghys

Let us x a Lorenz model generating a ow and associated with a rstreturn map . Williams shows in [82] that: There exists a compact contained inthe box , such that:

the -limit set of each point in is contained in ; is invariant through , and the restriction of on is topologicallyconjugate with acting on ;

is topologically transitive: it contains an orbit which is dense in ; the union of all periodic orbits is dense in .

We have now justied Lorenzs intuition, according to which the attractor be-haves as a surface within the limits of accuracy of the printed values.

We also see that the topological dynamics of the vector eld is completelydetermined by that of the map . To understand it, one uses the notion of kneadingsequence introduced around the same time in a more general context. Take thetwo sequences , = given by the signs of (0) et (0+) for 0 (thesign of 0 is dened to be +). Obviously, two applications of the above typewhich are conjugated by a homeomorphism (preserving the orientation) dene thesame sequences , , and the converse is true. Back to the vector eld, these twosequences can be obtained by considering the two unstable separatrices startingfrom the origin, which will intersect the square innitely many times, either inthe part > 0 or in the part < 0. The two sequences precisely describe theforward evolution of these two separatrices. It should be clear that these elds arestructurally unstable, since a slight perturbation can change the sequences , .

Guckenheimer and Williams prove that the two sequences contain all thetopological information on the vector eld. Precisely, the establish the followingtheorem:

There exists an open in the space of vector elds in 3 and a continuous map from into a two-dimensional disk, such that two elds , on are conjugatethrough a homeomorphism close to the identity if and only if () = ( ).

We wont dwell on the proofs, but will insist on an important point: theLorenz ow, although nonhyperbolic, is still singular hyperbolic. Namely, ateach point of the attractor , one can decompose the tangent space into a directsum of a line and a plane , such that the following properties hold:

and depend continuously of the point , and are invariants throughthe dierential of the ow ;

the vectors in are contracted by the ow: there exists > 0, > 0 suchthat for all > 0 and all , one has () exp();

the vectors in are not necessarily expanded by the ow, but they cannot becontracted as much as the vectors in . Precisely, if and areunitary, one has () 1 exp()() for all > 0;

the ow uniformly expands the two-dimensional volume along : det(

) exp()for some > 0 and all > 0.


For each one may consider the set () of the points 3 such that thedistance () () goes to zero faster than exp(). It is a smooth curvewith tangent in given by . The collection of these curves denes a foliation ofan open neighborhood of , justifying the terminology attractor: all the pointsin this neighborhood are attracted by and their trajectories are asymptotes oftrajectories in . The branched surface is constructed from the local leaves ofthis foliation. Of course, one needs to show that all these structures exist, and thatthey persist upon a perturbation of the vector eld.

Hence, the open set in the space of vector elds detected by Guckenheimerand Williams trespasses the hyperbolic systems, but the dynamics of this type ofelds can still be understood (at list qualitatively). The specicity of these eldsis that they are, in some sense, suspensions of maps on the interval, but alsocontain a singular point in their nonwandering set. In some sense, their dynamicscan be translated into a discrete time dynamics, but the presence of the singularpoint shows that the rst return time on is unbounded. It would be too naiveto believe that this type of phenomenon, together with hyperbolic systems, suceto understand generic dynamics. We will see later some other types of phenom-ena (dierent from hyperbolicity) happening in a stable manner. But the Lorenzphenomenon I have just described is doubtlessly one of the few phenomena rep-resenting a generic situation. We will come back later to this question.

As we have mentioned, the geometric models for the Lorenz attractor havebeen inspired by the original Lorenz equation, but it wasnt clear whether theLorenz equation indeed behaves like a geometric model. This question was not re-ally crucial, since Lorenz could clearly have made other choices to cook up his equa-tion, which resulted from somewhat arbitrary truncations of Fourier series. Lorenzhimself never claimed that his equation had any physical sense. Nevertheless, thequestion of the connection between the Lorenz equation and the Guckenheimer-Williams dynamics was natural, and Smale chose it as one of the mathematicalproblems for the next century in 1998 [78]. The problem was positively solvedby Tucker [80] (before the next century!). The goal was to construct a square adapted to the original Lorenz equation, the rst return map on , and to checkthat they have the properties required by the geometric model. The proof uses acomputer, and one needs to bound from above the errors. The major diculty which makes the problem quite delicate is due to the presence of the singularpoint, and the fact that the return time may become very large . . . For a briefdescription of the method used by Tucker, see for instance [81].

6. The topology of the Lorenz attractor

6.1. Birman, Williams and their can of worms

We have seen that the periodic orbits of a geometric Lorenz model are dense inthe attractor. To better understand the topology of the attractor, Birman andWilliams had the idea to consider these periodic orbits as knots. A knot is a

34 E. Ghys

closed oriented curve embedded in space without double points. A topologist willconsider that two knots are identical (the technical term is isotopic) if it is possibleto continuously deform the former into the latter without any double point. Nowthe questions are: which knots are represented by at least one periodic orbit of theLorenz model? Can a single knot be represented by innitely many periodic orbits?Beyond knots, one can also consider the links, which are unions of nitely manydisconnected knots. Each collection of nitely many periodic orbits denes a link.Since the periodic orbits are dense in the attractor, one can hope to approximatethe latter by a link containing a huge number of periodic orbits. The article [15]nicely mixes topology and dynamics.

Figure 18. A few periodic orbits

A priori, this study of knots and links should be performed for each geometricLorenz model, that is for each return map , or more precisely for each choiceof a pair of kneading sequences , . However, we may restrict ourselves to theparticular case of the multiplication by two case, by setting 0() = 2 + 1for [1/2, 0[ and 0() = 2 1 for [0, 1/2]. A point in [1/2, 1/2] isfully determined by the sequence of signs of its iterates by 0, and each sequenceof signs corresponds to a point in [1/2, 1/2]. This is nothing but the dyadicdecomposition of numbers in [1/2, 1/2]. If is the return map on [1/2, 1/2] fora given geometric Lorenz model, one can associate to each point [1/2, 1/2] theunique point () [1/2, 1/2] such that for each 0 the numbers () and0 (()) have the same sign. This denes an injection : [1/2, 1/2] [1/2, 1/2]such that = 0 , and one can thus think that 0 contains all the one-dimensional dynamics we are interested in. Of course, the map is not always abijection, depending on the specic geometric Lorenz model. We will study themaximal case of 0, since it contains all the others. Strictly speaking, 0 cannot be


a rst return map of a geometric Lorenz model, since its derivative does not blowup near = 0. However, starting from 0 one may construct, like in the previoussection, a topological semiow on the branched surface, and then a ow through aprojective limit. This topological ow is perfectly adapted to our problem, whichis the nature of knots and links formed by the periodic orbits of the geometricLorenz models.

Each periodic orbit of the Lorenz ow can be projected onto the template,so it is associated with a periodic orbit of . One could fear that this projectioncould modify the topology of the knot through the appearance of double points.But this does not happen because the projection on the template occurs alongthe (one-dimensional) stable manifolds, and clearly a stable manifold can meet aperiodic orbit on at most one point (two dierent points cannot simultaneously beperiodic and asymptotes of each other).

The periodic points of 0 are easy to determine: they are the rational numberswith denominators of the form 2(2+1), with . If is such a periodic point ofperiod , one constructs a braid as follows: inside the square [1/2, 1/2] [0, 1], for = 0, . . . , 1 one connects the points ( (), 0) and ( +1(), 1) by a segment,such that the segments climbing to the right are below those climbing to theleft, like on Figure 19. Then, one closes the braid as usual in topology, to obtain

Figure 19. A Lorenz link

hell02From [15] p. 48 with kind permission from the Elsevier 1983.

36 E. Ghys

a knot (). Starting from a nite number of periodic points 1, 2, . . . , , oneobtains a link. Those are called the Lorenz knots and links.

Before describing some of the results obtained in [15, 83], I should remindsome denitions related with knots. An oriented knot is always the boundaryof a oriented surface embedded in the three-dimensional space (called its Seifertsurface). The minimal genus of such a surface is the genus of the knot. If a knotcannot be obtained as the connected sum of two nontrivial knots, it is said to beprime. A knot is said to be chiral if it cannot be continuously deformed into itsmirror image. Given a knot embedded in the three-dimensional sphere 3 (unionof 3 and a point at innity), it is said to be bered if the complement 3 breson the circle; this means that there exists a family of surfaces with boundaries,parametrized by an angle 1, which all share the same boundary but do notintersect each other outside their boundary, and which altogether cover the wholesphere. Near the knot, the surfaces looks like the pages of a book near the binding.Finally, given two disjoint knots 1 and 2, one can dene as follows their linkingnumber : choose an oriented surface with boundary 1, and count the algebraicintersection number of 2 with this surface. This integer enl(1, 2) is independentof the chosen surface, and enl(1, 2) = enl(2, 1).

Here are a few properties of the Lorenz links:

The genus of a Lorenz knot can be arbitrary large. The linking number of two Lorenz knots is always positive. The Lorenz knots are prime. The Lorenz links are bered. The nontrivial Lorenz knots are chiral.

The Lorenz knots are very particular. For instance, using a computer and the tablescontaining the 1 701 936 prime knots representable by plane diagrams with lessthan 16 crossings, one can show that only 21 of them are Lorenz knots [28]. Moreinformation on the Lorenz knots and the recent developments on the subject can befound in [22, 14]. Surprisingly, Ghrist has shown that if one embeds the branchedsurface in 3-space by twisting one of the wings by a half-turn, the resulting owis universal : all the links are represented by (nite collections of) periodic orbitsof that ow [25].

6.2. The right-handed attractor. . .

Try to see the attractor as a topological limit of its periodic orbits, and take intoaccount the fact that each nite union of periodic orbits denes a bered link; howdo these circle brations behave when the number of components of the link tendsto innity? In a way, one would like to see the complement of the attractor itselfas a bered object . . . In [27] I propose a global description of these brations,based on the concept of right-handed vector eld.

Consider a nonsingular vector eld on the sphere 3, generating a ow. Let us call the convex compact set formed by the probability measuresinvariant by the ow. This space contains, for instance, the probability measures


equidistributed on the periodic orbits (if any), as well as their convex combinations.A general invariant probability measure can be seen as a generalized periodic orbit(a foliated cycle in Sullivans terminology). We have seen that two knots in thesphere always dene a linking number; I show that one can also dene a linkingquadratic form enl : . The ow is then called right-handed if enl onlytakes positive values. For the Lorenz ow, enl is only nonnegative, but it is positivewhen restricted to invariant measures which do not charge the singular point. Thisfact can be directly read on Williamss template, since two arcs of trajectoryonly have positive intersections when projected onto the plane. The Lorenz owis thus not strictly right-handed.

One of the main results in [27] is that a right-handed ow is always bered,in the following sense. One can nd a positive Gauss linking form on the ow.Precisely, there exists a (1, 1)-dierential form on 3 3 minus the diagonal,such that:

if 1, 2 : 1 3 are two disjoint closed curves, their linking number isgiven by the Gauss integral

1(1),2(2)(

11

, 22 )12; if 1, 2 are two distinct points, then 1,2((1), (2)) > 0.Interestingly, such a Gauss form directly provides a bration on the comple-

ment of each periodic orbit. Indeed, for any invariant probability measure, theintegral

() =

,(,()) ()

denes a closed and nonsingular 1-form on the complement of the support of .If is supported on a periodic orbit, the form admits a multivalued primitive,which denes a bration on the circle of the complement of the orbit.

Even though the Lorenz ow does not strictly belong to this frame, I showin [27] how this construction allows to better understand the results of Birmanand Williams I have described above.

6.3. From Lorenz back to Hadamard. . .

Hadamard studies the geodesics on negatively curved pants, and shows that theycan be described by the sequences of symbols enumerating the crossed seams.Williams analyzes the trajectories of the Lorenz attractor by a semiow on theLorenz template, itself described by two sequences of symbols, following thewings (left or right) successively crossed by two limiting trajectories. It is thus notsurprising to nd a connection between these two dynamics. Such a connectionwas indeed exhibited in [26].

Following a nonwandering geodesic on the pants , after each crossing witha seam one may consider to turn right or turn left to reach the next seam. It isthus possible to associate to each nonwandering geodesic a bi-innite sequence ofleft/right symbols. Yet, this new coding is not perfect,

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