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INTRODUCTION TO LECTURE SERIES 191 ONNON LINEAR DYNAMICS AND CHAOSDr Marc PELEGRINHonorary Scientific AdviserONERA / CERTB P 402531055 TOULOUSE CEDEXFRANCEI do not intend to give a summary of what Prof P. Coulet. Ph.Guicheteau, C. Houpis and JJ. Slotine will talk about during thisLecture Series - it's merely a matter for the Conclusions.encourage us to examine the importance of the part played by thecorrelation.I will try to mention some aspects of non-linear dynamics and/orchaos which p,.Laps a,, margin-al with regard to the core of thesubject which will be developed during these 3 days; eventually theycould be commented on or discussed during the Round Table.The next concept to be introduced is order and disorder. Completedisorder is independence. All other definitions are merely negativesuch as this one: a sufficient (but not necessary) condition to say thata system is not disorganized is that correlations between thefluctuating parameters - if any - are not null.Most of us are involved in engineering studies or designs: althoughaeronautics and space are not mentioned in the title of the LS, it ispertinent that these two fields will be dominant during the lectures,However, it is always fruitful to look around and compare thedifferent approaches to the subject. This is why Prof Houpis andGuicheteau will speak mainly about aeronautics and space-relatedproblems, Prof Slotine will comment about robotics and Prof Couletwill cover the general subject both from a theoretical viewpoint andon application in various domains (fluid flows, optics. chemical andbiological systems).1. SOME BASIC DEFINITIONS AND COMMENTSYA system is an ensemble of components well delimited in space andtime; outside the system is the external world. If there is no masstransfer across the delimiting surface the system is said to be"closed"; if there is no heat transfer - or mote generally no radiationexchanged with the external world, the system is said to be "isolated".The complexity of the system is difficult to precise; it implies thespecification of observability scales; a trivial example is thedifference in the concept used for the engineer who designs a turbineand the physicist who studies molecular transformations; both of themuse the entropy concept. For the engineer, entropy S is defined asdS-dQ/T Q heat flux exchanged T absolute temperature; for thephysicist entropy is the logarithm of the probability that an event ofa given complexion (in terms of molecule arrangement) happens. Thisnotion of scale is of prime importance as we will see later,The correlation concept is very useful in the study of systems (see,for example, the proceedingF of the International Symposium on "Thecorrelation" ref2l),Correlation apparently hears different meanings for mathematicians.physicists and engineers. The correlation is commonly used as adevice to quantify an uncertainty in physics phenomena. Theuncertainty can be a fundamental one, like in the quantum mechanics,or can be produced by the large number of parameters which areconsidered, or it can simply be an appearance like in phenomenalinked to the determinist chaos. To understand such situations, it isnecessary to enlarge the concept of correlation and even to go beyondits limits (concept of linkage, resemblance, distance). That's why itseems worthwhile considering the question.Finally the perception and comprehension mechanism of the humanbrain, as well as the "neuronal machines" which are being developed,iL/For further details see ref I I the paper presented by A. Favre"Correlations Spatio-Temporelles, determinisme et chaos".-.Hazard is generally associated to probability; however, completedisorder is considered to be a manifestation of hazard, but. from astochastic point of view, complete disorder escapes from laws ofprobability: there is a contradiction.Then we arrive at chaos, which is normally associated with disorder.The deterministic chaos has been introduced by Poincart around 1892in his famous books (3 volumes) on "Les Mdthodes Nouvelles de lam6canique c6leste" 1892-93-99 reprinted by Dover Publications Inc.1957.It seems that nobody has really considered the problem stated byPoincard until the '60s though the Bdnard's curls have been deeplystudied both on an experimental and a theoretical basis. Nobodyquestioned - and still questions - the fact that organized large motionsof molecules (water, for example) appear gently from completelyunorganized motions when external parameters (the heat flux in th.case) varies continuously in one direction; it is worthwhile noting thatthe container in which the Benard's curls appear is not an isolatedsystem.Major works have been provided by Lorentz on the Rayleigh-Benardconvection and by Ruelle and Takens-' on the turbulence in the '60s.At that time high subsonic civilian planes and supersonic ones in themilitary domain came into being. The power of the jet engine andtheir high consumption (double flux jet engines did not yet exist)imposed a careful study of the aerodynamic drag of the plane. It wasrecognized that the big problem was in the boundary layer and, inparticular, the transition laminar/turbulent in the extrados of thewings. Aerodynamists shall try to arrive at a full laminar boundarylayer along the entire wing chord.The transition is not yet fully understood but progress is made everyday. perhaps thanks to scientists working on chaos, as this transitionbelongs to a more general phenomena, contrary to the Bernard'scurls, the passage from organized flow structure to unorganized flowstructure.Extreme chaos is the molecular chaos which is an undeterministicchaos governing thermodynamics. We will come back to it a littlefurther on.A pure determinist chaos is represented in certain types of "fractals".Discovered - or rediscovered - in 1975 by Henoit Mendelbrot, fractalscan have regular but incredibly complex structures (such ones issuefrom a triangle) or fully irregular structures though issued from amathematical iteration.2/ For bibliographic references see Footnote under Paragraph 6.S

IE-IInitially fractals did not seem to have any application except, perhaps,in the beauty of the presentation. Later on, it wac -'ecognized that,firstly fractals may represent a good approximation of real featuressuch as maritime coast, secondly, that the "auto-similarity" which canbe considered as the intrinsic property of fractals, is certainly ageneral property of nature. For examples, the structure of a tree isabout the same if we look at it from very far or very near; types oframifications are similar, the cosmos structur- seems to comply withthe rule of auto-similarity (star with planets: galaxy with stars, clusterof galaxies.).A fractal dimension has been defined (the fractal dimension of thecoast of Brittany is 1.26 and the "dust of Caritov" has a fractaldimension of 0.6309).Anotlher apparent contradiction lies in the dimension of a structureand in 1890 G. Peano described a geometrically defined curve whichcan cover a square completely: any point of the square can be reachedby the curve. Then, what is the dimension of a square, 2 as normallyassumed or I because it is anything else than a curve?On the other hand, a phenomenon that can be called anti-turbulencehas been discovered and studied by John Russel in 1834. This is thes )liton wave which is an isolated wave whose amplitude can be large(a few decimeters on a canal 5m wide). This wave can travelthousands of meters without any modification in shape. The twoproblems rising from this fact are that first, how is the wave producedand second, how can it travel without a quick damping? The answersare not yetwell established. It seems that the soliton starts from thecomplex and apparently unorginized turbulence which is created infront of an old-fashioned boat. This is why we have said above thatsuch a phenomenon is sometimes associated with anti-turbulence. Asto the absence of degradation of this isolated wave, we can only saythat some non-linear interaction should be produced between thebottom of the canal and the wave.Indeed it is rather difficult to produce a soliton in the laboratory;some success has been obtained with large canals: it has always beennoted that the energy involved in tis process is quite a criticalparameter: too little energy causes a quick dumping of the waveproduced, too much energy creates just turbulent motion. Some nonlinear partial derivative equations give an acceptable representation ofthe phenomenon (KdV equation of DJ. Kortweg and C. de Vries).It is time now to comment about the contradiction between theclassical mechanics (which implies reversibility) and thermodynamics(which implies irreversibility, the entropy of an isolated system canonly increase),As a first remark, is the assumption that molecules, at leastmonoatomic ones, behave like interacting bodies in macroscopicmechanics absolutely true? If we answer yes, then thermodynamicsshould be reversible and the entropy should no longer be amonotonous quantity. If we answer no, then thermodynamics is aseparate branch of physics and everything has to be reconsideredfrom the molecule behaviour.Today and for the next 2 days, I propose to you that the answer isyes. How to justify this worldwide accepted assumption? The basicpoint is Poincare's statement according to which a closed and isolatedsystem will necessarily pass in the "vicinity" of any initialconfiguration from which it started :Y. I am unable to comment about"vicinity", however, it does not cancel what is said below.In the famous statement called Maxwell's devil, two vessels initiallyat different pressures (or temperatures) communicate through a smallhole. Everybody knows that after a while the pressure will be thesame in the two vessels. Poincard's statement says - just wait and insome moments pressure (or temperature) will be different. Nobodyhas yet observed such a phenomenon.The de- il was looking at the speed of molecules; he let pass throughthe hole only those which had a "great velocity". If pressure in thevessels was equal in the beginning, they will not be "ter a while.An explanation ma. come from the fact that there are 6.023.10"molecules per mol . Can the "classical mechanics" be extended tosuch numbers of "components"? Previously we said yes. but if wenow compute the time it can take to arrive at an "abnormal" moleculerepartition, we find (from C. Marchal):- Initial pressures in the vessels 1.4 10' Pa 1and 0.6 l0s Pa.- The mode of choice of molecules which passes through the holedoes not notably influence the results.- If we measure the pressure with 10- Pa accuracy, we can noticefluctuations with regard to the average solution at a rate of aboutonce per two years.- If we look for variations of 5m Pa th. prohability of finding sucha division is 10 ' before 1.4 10321 years.- The a priori probability that the pressure in one vessel wouldbe 1.4 l0W Pa or higher is 10 M with M 3.5 1016: it is not zero!.It is hard to say that this is a final answer to the dilemma. However.we could not deny that the return time to initial conditions(Poincard's return time) exists; unfortunately we could not pass frommacroscopic systems to microscopic systems on a continuous basis.State vector of dimension 100, even 1,000 are now considered(flexible structures, for example) in the macroscopic domain; in fluiddynamics state vectors are on the order of 6,023 x 10" per mole!The last concept we want to mention is the stability concept. At firstglance stability and chaos seem opposite - like stability andinstability: this is not always true.Is stability a measurable quantity - like mass, or an identifiablequantity - like temperature? There are many definitions of stability,sometimes contradictory: in fact, it is a subjective quality whichshould be defined in the context of the field considered. The referencesystem in which the system evolves should he defined: stability m ywell exist in a given reference system, but no longer exist in anotherreference system. Stability seems to be a dominant factor in aircraftor missile control - or for that matter, of any type of vehicle.However, stability and manoeuvrability are two opposing factorswhich intervene in aircraft control: for civilian aircraft, stability is thedominant factor; for military aircraft or missiles, manoeuvrability isthe dominant factor. The above are some of the reasons which led tothe organization of a Workshop on "Stability" for the AGARDcommunity.121Basically, stability is related to irreversibility, which means no energydissipation for linear systems. While linear systems are very rare, theyoften represent a suitable approximation of non-linear systems.Stability is also a matter of accuracy. Take, for example, the rotationof the earth: is it stable or unstable? This question has no meaningunless the range of accuracy we are looking for, and in fact. thewhole context can be specified. Thanks to the accuracy of existingatomic clocks, it can be demonstrated that dail) variations of theorder of Ims yearly or pluri-annual variations of the order of tens ofms, occur in a pseudo periodic manner. However, angular velocity isnecessarily decreasing long-term; this is mainly due to the water/earthfriction in tides. In the pre-Cambrian period (400 M years ago) theday was only 15 hours long (that is Ims lost every 10 years)! Whathas been said about the angular velocity of the earth could also besaid about the direction of the earth's momentum. At the pole. (thetrace of the rotational vector moves continuously in a circle of about2m in diameter. However, for all human activities the eart

-Les syst mes liniaires (y compris les equations ý coefficient variable dans le temps) compares aux systimes non-liniaires. Les diff rents types de non-lindarit6: les caract ristiques courbes, les sauts, ]a bifurcation.-La dynamique non-lin aire; sensibilit6 aux conditions initiaies et/ou incertitudes concernant les paramitres du systime.