A Brief History of Chaos 804

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APPENDIX A1. A BRIEF HISTORY OF CHAOS 804

Kepler’s Rudolphine tables had been a great improvement over previous tables, and Kepler was justly proud of his achievements. He wrote in the introduc- tion to the announcement of Kepler’s third law, Harmonice Mundi (Linz, 1619) in a style that would not ﬂy with the contemporary Physical Review Letters editors:

What I prophesied two-and-twenty years ago, as soon as I discovered the five solids among the heavenly orbits–what I firmly believed long before Appendix A1 I had seen Ptolemy’s Harmonics–what I had promised my friends in the title of this book, which I named before I was sure of my discovery–what sixteen years ago, I urged as the thing to be sought–that for which I joined Tycho Brahé, for which I settled in Prague, for which I have devoted the best part of my life to astronomical contemplations, at length I have brought to light, A brief history of chaos and recognized its truth beyond my most sanguine expectations. It is not eighteen months since I got the first glimpse of light, three months since the dawn, very few days since the unveiled sun, most admirable to gaze upon, burst upon me. Nothing holds me; I will indulge my sacred fury; I will triumph over mankind by the honest confession that I have stolen the Laws of attribution golden vases of the Egyptians to build up a tabernacle for my God far away from the confines of Egypt. If you forgive me, I rejoice; if you are angry, I 1. Arnol’d’s Law: everything that is discovered is can bear it; the die is cast, the book is written, to be read either now or in named after someone else (including Arnol’d’s posterity, I care not which; it may well wait a century for a reader, as God law) has waited six thousand years for an observer. 2. Berry’s Law: sometimes, the sequence of an- tecedents seems endless. So, nothing is discovered Then came Newton. Classical mechanics has not stood still since Newton. for the first time. The formalism that we use today was developed by Euler and Lagrange. By the 3. Whiteheads’s Law: Everything of importance has end of the 1800’s the three problems that would lead to the notion of chaotic been said before by someone who did not discover dynamics were already known: the three-body problem, the ergodic hypothesis, it. and nonlinear oscillators. —Sir Michael V. Berry

Writing a history of anything is a reckless undertaking, especially a history of A1.1.1 Three-body problem something that has preoccupied at one time or other any serious thinker from ancient Sumer to today’s Hong Kong. A mathematician, to take an example, might Bernoulli used Newton’s work on mechanics to derive the elliptic orbits of Kepler see it this way: “History of dynamical systems.” Nevertheless, here comes yet and set an example of how equations of motion could be solved by integrating. another very imperfect attempt. But the motion of the Moon is not well approximated by an ellipse with the Earth at a focus; at least the eﬀects of the Sun have to be taken into account if one wants to reproduce the data the classical Greeks already possessed. To do that one has A1.1 Chaos is born to consider the motion of three bodies: the Moon, the Earth, and the Sun. When the planets are replaced by point particles of arbitrary masses, the problem to be solved is known as the three-body problem. The three-body problem was also I’ll maybe discuss more about its history when I learn a model to another concern in astronomy. In the Newtonian model of the solar more about it. system it is possible for one of the planets to go from an elliptic orbit around the —Maciej Zworski Sun to an orbit that escaped its dominion or that plunged right into it. Knowing if any of the planets would do so became the problem of the stability of the solar (R. Mainieri and P. Cvitanovi´c) system. A planet would not meet this terrible end if solar system consisted of two celestial bodies, but whether such fate could befall in the three-body case remained unclear. rying to predict the motion of the Moon has preoccupied astronomers since antiquity. Accurate understanding of its motion was important for deter- After many failed attempts to solve the three-body problem, natural philoso- T mining the longitude of ships while traversing open seas. phers started to suspect that it was impossible to integrate. The usual technique for

803 appendHist - 20Mar2013 ChaosBook.org version15.9, Jun 24 2017 chapter 19 806 published a ergodic, then ff uld show that the mechanical aotic dynamics to make a great s and wrong for nterest in under- n proving the er- nd more realistic scillator does, so kho t of the theory of ng it the status of llators. The prob- ann had combined rstanding of phys- echanics. To eval- llator. But real in- nonlinear problem to create statistical ng on his method, t the computed av- édid. nce. Though Sund- cult. By the end of often. This hypoth- as an important step e up to Weirstrass’s say that a trajectory t approximation one ormulate it using the anics and was called ffi e proof of the hypoth- . This does not make ChaosBook.org version15.9, Jun 24 2017 ystem would visit every y extend our understand- erent set of mathematical ff

A1. A BRIEF HISTORY OF CHAOS Proving the ergodic hypothesis turned out to be very di comes as close as desired to any phase space point. mechanics, deriving thermodynamics from theuate equations the of heat m capacitysimplifying of assumption even of ergodicity: a that simple thepart dynamical system, s of Boltzmann the had phase spaceesis allowed was by conservation extended laws to equally otherthe averages ergodic used hypothesis. in statistical It mech was reformulated by Poincaréto was the ergodic hypothesis of Boltzmann.the Maxwell mechanics and Boltzm of Newton with notions of probability in order wanted to compute, are neededman’s to achieve work numerical deserves converge betterexpectations, credit and than the series it solution gets, diding not it of “considerabl the did solar not system.’ liv The work that followed from Poincar A1.1.2 Ergodic hypothesis The second problem that played a key role in development of ch that showed that it was possible to take a finite-dimensional recently developed methods of Hilbert spaces [A1.3]. This w and reformulate it asthe a infinite-dimensional problem linear easier, problem but it does allow one to use a di twentieth century it hasquite only a been few others. shown Early true on,esis for as was a a broken mathematical few down necessity, th system intosystem two was parts. ergodic (it Firstit would one would go would go show near near that any eacherages point) point made and equally mathematical then often sense. and onegodic regularly Koopman wo hypothesis took so when tha the he first realized step that i it was possible to ref tools on the problem.von Neumann Shortly proved a after versiona Koopman of theorem started the [A1.4]. lecturi ergodic hypothesis,the He givi computed averages proved would that make if sense. the Soon afterwards mechanical Bir system was much stronger version of the theorem. A1.1.3 Nonlinear oscillators The third problem thatchaotic was dynamical very systems influential was in the the work developmen on the nonlinear osci lem is to construct mechanical models that would aid our unde can construct a model of a musical instrument as a linear osci APPENDIX ical systems. Lord Rayleighstanding came how musical to instruments the generate problem sound. through In his the i firs struments do not produceLord a Rayleigh simple modified tone this forever simple as model the by linear adding o appendHist friction - 20Mar2013 a . t 3 √ 805 erent er got ff ffl ectively in ff under- tories of any a strip. This eries in a ablish a mathe- o show that he ions at di erved quantities e-body problem Swedish mathe- rical methods or sult did not pre- ates valid for all er according r collide, try se values the onserved quantity nates as a function n any one has ever not solve the prob- ved quantities that king them go away ther, but rather that ns, the homoclinic he best submission. , Mittag-Le three-body problem uantities that do not n the King Oscar II’s ries are e om prize- Poincaré’s e importance of state ifold. The trick is not ties. This problem was ChaosBook.org version15.9, Jun 24 2017 gh the king would have merical results. The con- erent ways [A1.1, A1.2]. , but rather as a function of t ff Acta Mathematica . Quite rapidly this gets exponentially close to one, the ra- t 3 √ − e −

A1. A BRIEF HISTORY OF CHAOS This problem, whose solution would considerably extend our Given a system of arbitrary mass points that attract each oth to Newton’s laws, under theto assumption that find no a two points representationvariable eve that of is some the known function coordinatesseries of converges of time uniformly. and each for all point of as who a s standing of the solar system, ... The Sundman’s series are not used today to compute the trajec In an attempt to promote the journal The interesting thing about work Poincaré’s was that it did the variable 1 three-body system. That isthrough more series simply that, although accomplished divergent, by produce betterformal nume nu map and the collision regularization mean that the se To solve the problemallowed for Sundman all to times obtain hetimes, a solving used the series a problem expansion that conformal wascompetition. for map proposed by the into Weirstrass coordin i the permission of the Kingmatical Oscar competition. II Several of questions Sweden were and posedpreferred Norway (althou only to one), and est the prizeOne of of 2500 the kroner would questions go was to formulated t by Weierstrass: dius of convergence of the series. Many terms, more terms tha appendHist - 20Mar2013 integrating problems was tochange find with the time conserved and quantities, allow q one to relate the momenta and posit ona´’ submissionPoincaré’s won the prize.were He analytic showed in that the conser introduced methods momenta that and were positions veryspace geometrical could flow, in not the spirit: exist. rolepoints. th of T periodic orbits and their cross sectio lem posed. He did not findof a time function for that all would give times. theit He coordi could did not not show be that doneand it with was trying the impossible Bernoulli to ei technique integrate. ofwinning finding Integration memoir, a would but c seem itmatician unlikely was fr Sundman. accomplished by Sundman theone showed had Finnish-born to that confront to the two-body integratethrough collisions. a the He trick did thre known that as by regularizationto ma of expand the the collision coordinates man as a function of time APPENDIX times. The first signcame from on a the result impossibility ofthat of Bruns were integrating that the polynomial showed that include there the the were possibility momenta no of and cons more complicated positions.settled conserved by quanti Poincaréand Bruns’ Sundman re in two very di chapter 15 808 t Arnol’d, topological initiated the (R. Mainieri) nosov proved al systems for t of time with Andrew; these rmation theory, ohlin connected nd Sinai applied e confusing. The arried on with the entropy and use it 7 review article on an area preserving Kolmogorov went t takes you from a rstanding garnered systems Axiom-A. e hyperbolic struc- eemed to have spo- namical system, on ng about these prob- her with Pontryagin study dynamical sys- he great advantage of shoe and enumeration he next step was pub- ChaosBook.org version15.9, Jun 24 2017 that many of the details iant sets of the map into uggestion was taken by s. Coarse systems were udy dynamical systems. on or spring force func- d a chain of developments ect the overall picture of the state space: ff erentiable dynamical systems” [A1.7]. There are ff

A1. A BRIEF HISTORY OF CHAOS erential equation (in the form of vector fields) to the global ff In Russia, Lyapunov paralleled the methods of Poincaréand The path traversed from ergodicity to entropy is a little mor Smale collected his results and their development in the 196 In the fall of 1961 Steven Smale was invited to Kiev where he me strong Russian dynamical systems school [A1.6]. Andronov c study of nonlinear oscillators and in 1937 introduced toget the notion of coarse systems. They were formalizing theon unde how these oscillators work do not a from the study of nonlinear oscillators, the understanding general character of entropy was understoodken by to Weiner, who Shannon. s where In he 1948 Shannon discusses published thefar his beyond entropy and results of suggested on the a info definition shift of transformation. the metric entropy of transformation in order to classify Bernoulli shifts. The s description in terms of horseshoes. his student Sinai andthese the results results to published measure-theoreticallished in notions in 1959. of 1965 entropy. by Inpapers T Adler 1960 showed and R that Palis, one and could also define the Adler, notion Konheim, of Mc topological as an invariant to classify continuous maps. In 1967 Anosov a appendHist - 20Mar2013 dynamical systems, entitled “Di lems. Smale’s result catalyzed their thoughts andthat initiate persisted into the 1970’s. Anosov, Sinai, and Novikov. He lectured there, and spent a lo ture, and it was inIn honor Kiev of Smale this found result a receptive that audience Smale that named had been these thinki tion by a little bit.eliminating And exceptional changing cases the in system the athe mathematical little concept analysi that bit caught has Smale’s t attentiontems. and enticed him to A1.3 Chaos with us Anosov. He suggested a series ofwhich conjectures, all points most (as of opposed to which a A non–wandering set) admit th there will still be limit cycles if one changes the dissipati within a year. It was Anosov who showed that there are dynamic The emphasis of the paper is on the global properties of the dy and ordering of all its orbits; the use of zeta functions tohow st to understand the topologylocal of di the orbits. Smale’s accoun disjoint stable and unstable parts; the existence of a horse APPENDIX many great ideas in this paper: the global foliation of invar chapter 15 807 Lord bits of ng, and ffi (R. Mainieri) artwright and t. It prompted ible codes grew han a pure tone was taken up by regardless of the dic system could created two basic ntroduction of the n a surface of con- r Pol, Du n 1898 Hadamard ynamical invariant s by introducing a te general, such as lassify these trans- ed the basis for the illator played a role tion for Mary Lucy t with the curvature two directions. One t a physical billiard al systems as trans- stems known as the ncreteness of experi- e first to show that a re is everywhere un- godic hypothesis was s result escaped most hat many of the struc- ns by Artin, Hedlund, ChaosBook.org version15.9, Jun 24 2017 ’ where he showed that physicists did indeed fol- The Theory of Sound on the ergodic hypothesis were erentiable and it would be nice ff ff had found a ‘remarkable curve’ in a ff

A1. A BRIEF HISTORY OF CHAOS . Morse in 1923 showed that it was possible to enumerate the or ff The theorems of von Neumann and Birkho The work of Lord Rayleigh also received vigorous developmen Rayleigh introduced a series ofthe methods notion that of would a proveinitial limit qui conditions. cycle, a periodic motion a system goes to A1.2 Chaos grows up two dimensional map; it appeared to be non-di Hayashi. They found other systems inand which classified the the nonlinear possible osc motions ofments, these and systems. the This possibility co ofCartwright analysis and was J.E. Littlewood too [A1.5], much who of settures out tempta conjectured to by prove the t experimentalists andlow theoretical from the equations of motion. Birkho a ball on asymbolic surface code of to constant each negative orbitexponentially curvature. and with the showed He that length did of the thi and the number H. code. of Hopf it poss With was contributio eventuallystant proven that negative the curvature motion was of ergodic. aphysicists, The ball o one importance of exception thi beingwas Krylov, a who dynamical system understood on tha aconcentrated surface along of the negative curvature, lines bu ofphysical billiard collision. can Sinai, be ergodic, who knew was Krylov’s th work well. many experiments and some theoretical development by van de models for the spring.models By for a the clever musical useand instruments. of decay negative with These friction time models he when have not more stroked. t In his book published in 1912 and 1913. Thisdirection line of took enquiry an developed in abstractformations approach of and measurable considered spaces dynamic intoformations themselves. in a Could meaningful we way? c Thisconcept lead of Kolmogorov entropy to for the dynamical i it systems. With became entropy possible as to aBernoulli d classify systems. a The set other line ofin that abstract developed trying dynamical from to the sy er findnot mechanical have systems stable that are orbits,published ergodic. as a these An paper would ergo withthe break a motion ergodicity. playful of title balls So of onstable. i ‘... surfaces This of billiards constant dynamical ..., negative systemBirkho curvatu was to prove very useful and it APPENDIX to see if a smoothLittlewood flow lead could to generate the such work ahorseshoe of curve. construction Levinson, of which The S. in work Smale. turn of provid C appendHist - 20Mar2013 810 zeta , uantum culty with initiated by ffi nics. The set (P. Cvitanović) its of a dynam- and even today p get a sense of e di rate subjects: 1. by the action of aplacian operator uelle had already pens to be exact.” tary to each other [A1.10], so that it n classical system s of Poincaréwho his led to Selberg’s A1.12, A1.13], and echanics, but could . Starting out from trace formulas hich to evaluate the e with a set of peri- ime soon. stepping stone in the sing the shorter orbits ChaosBook.org version15.9, Jun 24 2017 n terms of rapidly con- eciality. In Gutzwiller’s ent advances are equally orm. Such a relation had sional vector spaces. The quantum theory erent lines of reasoning and driven ff . — Joe Keller, after being asked to define applied Pure mathematics is a branch of applied mathematics. mathematics , initiated by Poincaréand put on its modern footing by initiated by Riemann and formulated as a spectral prob- spectral determinants

A1. A BRIEF HISTORY OF CHAOS , one can say that zeta functions arise in both classical and q number theory and erent motivations, the three separate roads all arrive at ff The fact that these fields are all related is far from obvious, This account is far from complete, but we hope that it will hel Once one develops an understanding of the topology of the orb understanding of quantum mechanics, in particular when the the practitioners tend to cite paperswords only [A1.13], from their “The sub-sp classical periodic orbits are ais crucial chaotic. Thisemphasized situation the is importance very ofnot periodic satisfying have orbits had when any in one idea classicalof think of energy m what levels they and couldsince mean the they for set are quantum essentially of mecha related periodic throughbeen orbits a found Fourier are earlier transf complemen byon the mathematicians Riemannian in surfaces the withtrace study constant formula of negative in the curvature. 1956 L which T A has exactly posteriori the same form, but hap 3. analytic lem by Selberg [A37.2, A1.15]. Following di Bohr, with the modern ‘chaotic’ formulation by Gutzwiller [ functions mechanics because the dynamicallinear evolution evolution (or can transfer) operators be on infinite-dimen described appendHist - 20Mar2013 classical chaotic dynamics Smale [A1.7], Ruelle [A39.14], and many others, 2. The history of periodic orbit theory isinspired rich and by curious; rec more than a century of developments in three sepa by di Ruelle’s zeta function isSmale’s that observation it that does a not chaotic dynamical converge system very is well dens odic orbits, Cvitanovićused these orbits as a skeleton on w averages of observables, and organized suchverging cycle calculations expansions. i This convergence isused attained as by a u basis for shadowing the longer orbits. perspective on the field. It is not a fad and it will not die anyt A1.4 Periodic orbit theory could be used to compute the average value of observables. Th APPENDIX ical system, one needsgeneralized to the be zeta able function to introduced compute by its Artin properties. and R Mazur 809 d is still far as, although tein inspired es needed to statistical metropolis, Stein, efore these pa- s in the context hurston was giv- ns of the statisti- rked out in 1976 The topology of e maps the codes of 1-dimensional tion group which rtitions divide the e work of Ruelle. obtained from the t Sinai introduced opology of the or- ry of the universal- e extended version applied in physics, The energies could opy of a dynamical ions with statistical amical systems, the l systems and statis- theory to dynamical cept of shadowing of he invariant measure tanović. ribe the dynamics of ecified by a symbolic to studying particular , and Stein, or Milnor at map into each other. measure notions to the ChaosBook.org version15.9, Jun 24 2017 haos. Feigenbaum intro- eomorphisms of surfaces.” ff

A1. A BRIEF HISTORY OF CHAOS A lecture of Smale and the results of Metropolis, Stein, and S The work in more than one dimension progressed very slowly an Markov partitions allow one to relate dynamical systems and After Smale, Sinai, Bowen, and Ruelle had laid the foundatio cal mechanics approach to chaotic systems,cases. research turned The simplestthe case orbits to for consider parabola-likeand is maps Stein was 1-dimensional [A1.8]. worked maps. The outby more in Milnor general 1973 and 1-dimensional by case Thurston Me was ineventually wo got a published widely in circulated 1988 preprint, [A1.9]. whos Feigenbaum to study simple maps. Thisity lead him in to quadratic the maps discove andsystems. the application Feigenbaum’s of work ideas wassystems; from the a field- culmination complete analysis in of the a nontrivialduced study transition many to new c ideas intoled the him field: to the introduce use functionalscaling of equations function the in which renormaliza completed the the study linktical of between mechanics, dyn dynamica and thescaling presentation functions. functions which desc from completed. The firstbits in result two in dimensions trying (theand to equivalent Thurston’s understand of work) was Metropolis, the obtained Stein t bying Thurston. lectures Around 1975 “On T the geometry and dynamics of di the notion of entropyof to studying the the study entropy ofMarkov associated partitions dynamical in to systems. 1968. a dynamical It system wa tha chanics; this has been atopological very framework fruitful laid relationship. down in Itstate Smale’s adds space paper. of the Markov dynamical pa systemEach into box nice is little labeled boxes by th aaround, code and inducing the a dynamics on symboliccover the all state dynamics. the spac space, From Sinaisystem. was the able number to In define of 1970 thethere box Bowen notion was came of presumably entr up some independently flowpers got with of published. information the Bowen back also same introduced andchaotic ide the forth orbits. important b con We do notmechanics know were whether clear at this to pointRuelle everyone. understood the They that relat the became topology explicit ofcode, in the and orbits th that could be one sp couldbe associate formally an combined ‘energy’ in to aof each the ‘partition orbit. system. function’ to generate t APPENDIX Thurston’s techniques exposed inbut that much lecture of have the not classificationnotion been that of a Thurston ‘pruning developed front’ can formulated be independently by Cvi appendHist - 20Mar2013 chapter 24 section 27.4 chapter 28 chapter 23 chapter 18 812 s by . The usion (A1.1) ff [A1.31] , the as- Recycling of et al. y person was (for level sum 1989), G. Gu- oning. One was lic dynamics of ort to prove that p der the watchful ost cited periodic ts are involved in ff ations into papers he cycle expansion from studies of scal- ults, has barely been ered by Jensen, Bak II. Applications 1.27, A1.28, A1.29]. wo long f the period-doubling ete by the exact cycle oth ODEs and PDEs). ChaosBook.org version15.9, Jun 24 2017 s for both discrete-time minants. This observa- ) is coded by transition mulas, such as the cycle Stosszahlansatz ation of cycles is not the related. In periodic orbit and (submitted March 1987). Here , ) n ( and Tang [A1.32] in 1984.) Even as it j ns th order estimate of the leading Perron- Λ ff − n e = I. Cycle expansions j Invariant measurement of strange sets in terms t cients [A1.35, A1.36, A1.37], such as di ffi , using chaotically across a spatially-periodic array. ) ff n , , and applied to the attractor Hénon (Eq. (4) of the above j ) x n ( ( probabilistic assumptions. The classical Boltzmann equa- A s β e weight of an unstable prime periodic orbit j any t n f exact Fix [A1.30, A1.27] are a better read. A1. A BRIEF HISTORY OF CHAOS X papers [A1.30, A1.27]: ∈ j x (submitted March 1988) [A1.33]. However, the two long paper = et al. 1 The first attempt to make cycle expansions accessible to ever Several applications of the new methodology are worth menti Cvitanovićderived ‘cycle expansions’ in 1986-87, in an e The first published paper on these developments was Auerbach sumption that velocities of colliding particles are not cor appendHist - 20Mar2013 theory all correlations are included in cycle averaging for expansion for a particle di tion for the evolution of 1-particle density is based on Frobenius eigenvalue paper). (The to the trace formula is presented as an was written, the heuristics ofexpansions, this and paper was yet, rendered mysteriously, obsol orbits this papers. might be one of the m condensed into Phys. Rev. Letter, of cycles constants without the accurate calculation of theoperator leading [A1.27, dozen A1.28, eigenvalues A1.34]. o of Another deterministic breakthrough was transport t coe Exploring chaotic motion through periodiconly orbits a ‘level sum’ approximation (23.40), Artuso there is no ‘entropy barrier’,problem. and the exponential prolifer main lesson was thatnumbered one I should and never II; part splitglanced II, theory at which and by covers many anyone. applic interesting res (21.6)) had been conjectured by Kadano strange sets tion led to the study of convergence(iterated of maps) spectral determinant and continuous-time deterministicCycle flows expansions thus (b arose not from temporalings dynamics, in but period-doubling and cycle-mapThis renormalizations work [A was donenaratne, in and collaboration E. with Aurell R. (PhD Artuso 1984-1989), (PhD and 1987- it was written un eye of parrot Gaspar in Faca, Fundaçaõde Porto Seguro, as t the mode-locking dimension forand critical Bohr circle [A1.25] maps is discov the universal; Hénon map, the but same now kindthe in attractor of Hénon renormalization (the periodic ‘time’. pruning orbi front The conjecture symbo [A1.26] APPENDIX graphs, topological entropy is given by roots of their deter exercise 4.1 section 22.1 remark 22.2 chapter 38 chapter 18 chapter 22 chapter A45 = th n ) 811 L compute elated to approach the ctitioners, un- s fair and bal- ics is built upon e could ces, log det ( ion of classically ] never attempted igenvalues (global oblems; they relate la’ gives the semi- no more ‘clever re- e traced back to the olute convergence’, k on hyperbolic sys- dic orbits embedded oretic zeta functions tician’s perspective), where S. Smale pro- plement summations ). We are grateful for er orbits, and the Ising model configu- analogous to the one uantum systems. The function for counting orbits [A1.18, A1.19, his is to observe that erent accounts cover- iant measures are ex- s [A1.24]). uring than what prac- ate determinants. One n of a determinant is a urns out that the more ler Zeta functions were ff ChaosBook.org version15.9, Jun 24 2017 rect, but prescient) zeta te long time averages of hyperbolic deterministic . Under these circumstances, -cycles sum. For unstable, hyper- n cycle expansions erence between the shorter cycles estimate of exponentially or super-exponentially [A1.60]. ff ff erence falls o ff

A1. A BRIEF HISTORY OF CHAOS -cycles’ contribution and the exact n ). In this way the spectrum of an evolution operator becomes r L For a long time the convergence of such sums bedeviled the pra Distant history is easily sanitized and mythologized. As we M. Gutzwiller was the first to demonstrate that chaotic dynam periodic orbits [A1.10], and 3.for the Riemann 1956 surfaces Selberg number-the of constant curvature [A37.2]. That on using these infinite setsexplicit was computations, not and Gutzwiller clearover only at anisotropic attempted all. Kepler to periodic im orbits Ruellerations by [A39.17] [A39.14 treating (In them retrospect, as periodic Gutzwiller orbits was one lucky; includes, the it worse t convergence one get spectra of these operators are given by the zeros of appropri the trace formulas perform thethe same role spectrum in of all lengths ofeigenstates), (local the dynamics) and above pr to for the nonlinearthat spectrum geometries Fourier of they transform e play plays for a the role circle. present, our vision is inevitably more myopic; for very di way to evaluate determinants is to expand them in terms of tra summations’ than the Plemelj-Smithies cumulant‘resummation’, evaluatio and their theory istitioners considerably of more quantum reass chaos fear: there is noappendHist - 20Mar2013 ‘abscissa of abs its traces, i.e. periodic orbits. A deeper way of restating t following sources: 1. theposed foundational as 1967 “a review wild [A1.7], idea infunction this over direction” periodic a orbits, (technically 2. incor the 1965 Artin-Mazur zeta classical quantum spectrum as a sumA1.20, over A1.12]. classical Equally periodic important was D.tems, Ruelle’s 1970’s where wor ergodic averages associated with natural invar Contrary to what some literature says, cycle expansions are ing the same recent history, see V.and Baladi [A1.16] M. (a V. Berry mathema [A1.17] (a quantumany chaologist’s comments perspective from theanced. reader that would help make what follow unstable periodic orbitschaotic in quantum his systems, 1960’s where work the on ‘Gutzwiller trace the formu quantizat the period tr (log til the mathematically rigorous spectralflows, determinants and for the closely related semiclassicalyrecast exact in Gutzwil terms of highly convergent a relatively few short periodic orbitsquantities lead measured to in highly chaotic accura dynamics andidea, of in spectra for a q nutshell, is that long orbits are shadowed by short pressed as weighted sums on thein infinite the set underlying of chaotic unstable set perio [A1.21, A1.22]. This idea can b bolic flows, this di APPENDIX term in a cycle expansion is the di remark 5.1 section 15.4 remark 23.1 0 of 814 = 0 et al. ρ β bits in (A1.2) Unstable (submitted Chaos and e. After the t is one to cite? er the unstable ) by taking the eenside [A1.57] zantsev [A1.59] l measure tic attractors, as he kibutz Kiryat n the vast major- odel, and for his d in ref. [A1.79], le, the description he first time. Ever ctually computing , a special case of sums weighted by ng on “ c orbit’s expanding e dynamical (or Ru- .” This meeting was attractor recent contributions, Do not faint because e authors, but not for ]. There are no cycle 127 of them) for Ku- per [A1.56], [A1.31], in which the assical quantization as ChaosBook.org version15.9, Jun 24 2017 d millennium, and hav- ence has its virtues. n papers and some of the 0 (no escape), and d them, as well as a useful et al. = s . S and Tang 1984 and Auerbach ff ∈ M j x , embedded in a chaotic attractor. Each periodic , meeting, the Maryland group wrote a series of n j 1 Λ n f Fix X ∈ j , Eq. (14) in their paper: x j Λ →∞ of long period lim n j [A1.58] for being the first to determine relative periodic or x = 1). The first paper does cite Auerbach )

A1. A BRIEF HISTORY OF CHAOS = S et al. M ( 0 ρ Depending on the context, one should also cite 1) Zoldi and Gr So you have now written a paper that uses periodic orbits. Wha While derivations of (A1.1) by Kadano point is weighted byFloquet the multipliers inverse of the product of its periodi a mixing hyperbolic attractorperiodic points is given by the limit of a sum ov This is an approximate(A1.1), level sum with formula leading for Perron-Frobenius natural eigenvalue measure periodic orbits and the dimensions of multifractal chaotic computation of associated dynamical averages, at the meeti ing a continuously updated, hyperlinked and reliable refer this webbook is available on (gasp!) the internet - it’s thir a Zeta function over unstableexpansions periodic in orbits these [A1.12, papers A1.13 orin in Scholarpedia.org). Bowen’s work (see, If for youperiodic-orbit examp have weights, cite computed the something first using paperup-to-date that reference, introduce which in this case is ChaosBook.org. anything using periodic orbitsyou and can are safely reluctant credit to Ruelleelle) [A1.22, refer zeta A39.14] to function, for and deriving Gutzwiller th for formulating semicl same approximate level sum seems tosince have then, been various published cyclist for teams t cite exclusively their ow Work by Sinai-Bowen-Ruelle is smarterity and of more ‘chaos’ profound publications tha from the 1980s on. If you are not a mathematicians of the 1970’s. (observable Related Nonlinear Phenomena: Where do we go from here? organized by Moshe Shapiro and Itamar Procaccia and held in t Anavim. A great“Where meeting, do and we Celso go Grebogipapers from was on here?” unstable in periodic the orbits, or audienc ‘UPOs’. In the first pa September 1987), the focuswas was the on fashion fractal in dimensions the of late chao 1980’s. They prove that the natura for being the second to determine unstable periodic orbits ( ramoto-Sivashinsky, on a domain2) larger López than what wasa studie spatio-temporal PDE (complexfor Landau-Ginzburg), being and the 3) first Ka to determine periodic orbits in a weather m variational method for finding periodic orbits. We love thes their ‘escape-time weighting’. APPENDIX 1987 were heuristic, Grebogi, Ott and YorkeappendHist 1987 - 20Mar2013 prove (A1.2 chapter 29 chapter 21 section 23.7 r- 813 such ected } ff k p , about cycle ··· 2 riptych of arti- section suspen- bits by starting p er not available, sions is a , y not just a game. ization of chaotic 3] tend to cite this 1 r continuous time eory had failed to lic repeller. More m (classical, semi- r Wintgen obtained p le expansion by in- f spectral determin- ating maps, so Cvi- cal interest are con- og-log plot of level d the periodic-orbit { determinant [A1.40] at Gaspard and Rice ncations such as the n 1992 P. E. Rosen- om a small set of its 0 bounces determine rporated into dynam- take cycle expansions so in 1991 Cvitanović e [A1.50]. For people ce quantum resonances ts on the periodic-orbit trace formula [A1.51]. ChaosBook.org version15.9, Jun 24 2017 nuous time flows along e accuracy [A1.39] (sub- ... . equal to or smaller than the most max ffΛ 45. . 0 ≃ , with the cuto γ max Λ | ≤ k of Dahlqvist and Russberg [A1.46, A1.47] and Dettmann and Mo p ff

A1. A BRIEF HISTORY OF CHAOS in the data set. 4103384077693464893384613078192 Λ . p 0 ··· Λ = 1 billiards are only toys, but quantization of helium is surel p γ D Λ | A 3-disk billiard is exceptionally nice, uniformly hyperbo 2 Physicists tend to obsess about matters weightier than iter It is pedagogically easier to motivate sums over periodic or The Copenhagen group gave many conference and seminar talks often than not, goodor symbolic dynamics its for grammar a is given not flow finite, is or eith the convergence of cycle expan Try to extract thissums from (A1.1)! a direct Prior to numerical cycleachieved simulation, by expansions, or applying the Markov a best approximations l accuracy towas th the 1 spectral significant digit, riss [A1.48] might be helpful. The idea is to truncate the cyc that paper as the first paper on cycle expansions. tanovićand Eckhardt showed that cycle expansions reprodu of Eckhardt’s 3-disk scatterer [A1.38] tomitted rather impressiv February 1989).cles (submitted Gaspard September and 1988) Rice aboutclassical published the and same a quantum 3-disk scattering) lovely syste qvist [A1.40, t [A1.43, A1.41, A1.44], in A1.42]. his PhDants I thesis, with combined their the symmetry magic factorizations o to [A39.17, A1.45] ridiculous to accuracy; forthe example, classical periodic escape orbits rate up for a to 3-disk 1 pinball to be by non-hyperbolic regions of state space. In those cases tru cluding only the shadowing combinations of pseudo-cycles unstable with discrete time dynamical systems, buttinuous most in flows of time. physi flows The were weighted introduced averages by Bowen,sions of weighted who periodic by treated orbits the them ‘time fo asical ceiling’ zeta function, Poincaré functions and by were inco Parrysteeped and in Pollicott quantum [A1.49] mechanics and it Ruell alland looked very Eckhardt unfamiliar, reformulated spectral determinantsthe for conti lines of Gutzwiller’sAs derivation a of consequence, quantum the mechanicians semi-classical [A1.17, A1.52, A1.5 By implementing cycle expansions in 1991,a the surprisingly group of accurate Diete helium spectrumshortest [A39.11, cycles. A1.55] This fr happeneddo 50 so years and after 20systems old years [A1.12]. quantum after th Gutzwiller first introduced his quant expansions. In December 1986, Cvitanovićpresented resul description of the topology of Lozi and Hénon attractorsappendHist - an 20Mar2013 APPENDIX stability cuto example 2.2 chapter 30 816 ich dynam- unction space mputation and imilar problems llenge in a way uential Lorenz’s studied spatially eater importance ristic features of y within the past radigmatic model them seems hope- t half century. In urbulence brought full Navier-Stokes to find and to treat ce. Laminar flows insight into the na- ded in the infinite- ion of dynamics led the 1-spatial dimen- 73, A1.74]. Further ions associated with , known today as the ators [A1.69]. Hopf etric state space anal- s), turbulent states set te space. The Proper ciated with the phase mensional state space, ciently large viscosity. spatio-temporal PDEs. e flow is, however, not olutions of the Navier- nal forces, with instan- ChaosBook.org version15.9, Jun 24 2017 ural’ or SRB measures rameter has been rigor- ffi tems with chaotic behav- s observation that viscos- ic orbits, heteroclinic con- culties of these important problems ffi unstable manifolds. /

A1. A BRIEF HISTORY OF CHAOS In a seminal 1948 paper [2.4], Ebehardt Hopf visualized the f Hopf noted “[t]he great mathematical di Today, as we hope to have convinced the reader, with modern co of allowable Navier-Stokes velocity fieldsparameterized as by an viscosity, boundary infinite-di conditions andtaneous exter state of a flowcorrespond represented to by equilibrium a points, point globally in stable this for state su spa Poincarésections, key roles played by equilibria, period nections, and their stable As the viscosity decreases (as thein, Reynolds represented number by increase chaotic state spaceity trajectories. causes Hopf’ a contraction ofto state his space key volumes conjecture: under that the long-term, act typically observed s dimensional state space of allowed‘inertial states. manifold,’ is Hopf’s well-studied manifold inIts the finite mathematics dimensionality of forously non-vanishing ‘viscosity’ established pa in certainpresciently settings noted that by “the Foias geometricalthe picture and most of collabor important the phas problemis of the the theory determination offlow”. of turbulence. the Of Hopf’s probability gr call distributions for asso understanding probability distribut ior [A1.7, A1.70, A39.1, A39.14]. Stokes equations lie on finite-dimensional manifolds embed the phase flow has indeed proven to be a key challenge,has played one a in critical role wh in understanding dissipative sys ical systems theory hasparticular, made the Sinai-Ruelle-Bowen the ergodic greatest theory progress of in ‘nat the las lessly barred. However,the there hydrodynamical is phase flow no occur doubt in a that much many larger characte class of s are well known and at present the way to a successful attack on governed by non-linear space-time systems.ture of In hydrodynamical order phase to flows gain we are, at present, forced simplified examples within that class.” Hopf’sysis call of for geom simplified modelstruncation first [A1.72] came of to the fulfillment Rayleigh-Bénard convection with sta the infl Orthogonal Decomposition (POD) models ofthis boundary-layer type t of analysis closer to physical hydrodynamics [A1. experimental insights, the wayproblem to is a no successful longer attack “hopelessly on barred.” the We address the cha significant progress has proved possible for systems such as sion Kuramoto-Sivashinsky flow [A1.75, A1.76],of which is turbulent a dynamics, pa asextended well dynamical systems. as one of the most extensively APPENDIX Hopf could not divine,two employing decades, methodology explained in developed depth onl in this book. appendHist - 20Mar2013 section 27.4 chapter 28 not 815 →∞ n without ort than ff ts to take such re e this limit? The thods that lie at atistical mechan- sition was meant inant, convergent eason why linear e, Gutzwiller and one prefer a limit ry fond of baker’s sets. This was set end to cite Rugh’s iting earlier papers e message: Heuris- xpansion curvature l theories of turbu- le; and the natural te averages s, throw in “a sense ansions were intro- las (that are derived uilt into the spectral of state space flows, chanics and stochas- Ω ull sum of all states. act cycle expansions blem [A1.67], he in- le expansions are expansions in the un- ant is not a long-time ly more and more un- ; faster convergence is las for dynamical aver- ChaosBook.org version15.9, Jun 24 2017 to 61, A1.62, A1.63]. The The correlation spectrum α exact [A1.60], which proves that the zeros of spectral det- . 0 ρ A1. A BRIEF HISTORY OF CHAOS is everywhere singular, with support on a fractal set, with i 0 ρ limit. In actual computations it would be madness to attempt Taking a limit to obtain a proof is good mathematics, but in st If you intend to determine and use periodic orbits, here is th →∞ exact periodic orbits sums, and exactages periodic of orbits observables? formu It ismaps not [A1.66], even which, wrong. being Perhaps piecewiseterms, if linear, one one have does is no not ve cycle appreciatedeterminants e the and shadowing their cancelations b cyclethinkers expansions. stop at the That level sum might (A1.2). be the r A1.5 Dynamicist’s vision of turbulence The key theoretical concepts thatlence form are the rooted basis in ofSpiegel. the dynamica work In 1889 Poincaré’s of analysis Poincaré,troduced Hopf, of the Smale, the geometric Ruell three-body approach pro the to core dynamical of systems the theory and developed me here: qualitative topology appendHist - 20Mar2013 stable periodic orbits [A1.33, A1.30,tic A1.27]; in mechanics quantum they me areof semi-classically a exact. heuristic So sum why such would as (A1.2) to the exact spectral determ the heuristic approximations), notobtained smart by for computations utilizing theof shadowing dynamical that zeta is functions built and into spectral the determinants. ex Cyc heuristic, in classical deterministic dynamics they are limit even more impossible towhole compute. point And of why cycle would expansions one is take that it is smarter to compu ics a partition functionLikewise, is its not ergodic a limit theorylimit; of cousin, it anything; the is it spectral the is determin duced exact the in sum f a over non-rigorous allnot manner, periodic to on orbits. frighten purpose a [A1.33]: Cycle novice, the exp innocent expo of Borel measurable constructing limit, as longer andstable, longer periodic exponentially orbits growing are inmeasure exponential number, and non-computab proof is well withinpaper mathematicians’ as comfort the paper zone, on soof ‘Fredholm they Grothendieck” determinants’, for and, t good as measure alway on [A1.16, cycle A1.64], expansions. without c erminants are indeed the Ruelle-Pollicott resonances [A1. right in the elegant PhDfor thesis hyperbolic of analytic H. maps H Rugh’s in 1992, tic ‘level sums’ are approximationshere, to in the ChaosBook, exact and trace Gaspard formu monograph [A1.65] with no mo APPENDIX n ) Die 818 statistical ish-speaking ied group the- Slater, inventor 90] started . That is when n only describe ered the picture its elaborations, highly influential igner and Johann ... it was obvious n with the group- 30, after a visit to y heralded as hav- ut Kuramoto-Siva- pest’ wrote papers ’ is gradually being were remarks made r as the rotation and utrage” he and other tokes calculations of o other piece of work J. C. Slater, published ally, the correct trans- ] that the fluid dynam- groups as unnecessary ional regimes have be- ChaosBook.org version15.9, Jun 24 2017 e, but everyone felt that s at the hands of Wigner, (as opposed to e, forced to read chapter 10. ff dynamical — Fabian Wale Groups and Physics - Dogmatic Opinions of a Se- How many Tylenols should I takegroup with theory, this?... still (never took needuse to to be this convinced beyond mind-numbing that formalizations.) there is any “The mathematical elegance and profundity of Weyl’s book that plagues us to this very day.

A1. A BRIEF HISTORY OF CHAOS A. John Coleman writes in “It was at this point that Wigner, Hund, Heitler, and Weyl ent According to John Baez [A1.91], the American physicist John If you are not fan of chapter 10 “Flips, slides and turns,” and A1.82, A1.83]. Considerably more unstable, higher-dimens shinsky. It is only a model; it was not until the full Navier-S come accessible [A1.84]. Of course, nobody really caresEckhardt, abo Kerswell and collaborators [A1.85, A1.86,ics A1.87 community started to appreciate that the which were incomprehensible toory... those The practical like consequences me appeared who toto be be had negligibl in not the stud mainstreamas one a had feeling to of learn outragethat about at it. a the great I turn many had whichtheoretical other what the I approach physicists subject ca to were had the disgusted taken such problem. as as I ‘Slater As had has I bee slain heardI the have later, ‘Gruppenpest”. done there was I so believe universally that popular.” n lation is ‘the group plague’] ... The authors of the ‘Gruppen Gruppenpest of the ‘Slater determinant,’ is famousto for physics. having He dismissed wrote: with their ‘Gruppenpest:’ the pest of the group theory [actu physics community. In the preface of the second edition in 19 nior Citizen [A1.92]: [Theory of Groups and QM] was somewhat traumaticthe USA, for Weyl wrote, the “It has Engl been rumored that the ‘group pest cut out of quantum physics.Lorentz This groups is are certainly concerned; not ....” true Inin in 1975, the the so autobiography famous of fa MIT physicistphysicists described felt the at “feeling the of incursion o ofWeyl group et theory al. into physic In 1935,treatise when Condon on and the Shortley “Theory published of their Atomic Spectra”, SlaterappendHist was - 20Mar2013 widel analysis of wall-bounded flows is now feasible [A1.88]. A1.6 Gruppenpest APPENDIX you are not alone.the Or, articles at by least, twovon you young Neumann were mathematical [A1.89], not physicists, and alone Eugene Wigner’s in W 1931 the Gruppentheorie 1930s [A1. te- ac- 817 quali- c orbit as ee Ginelli utation re- Spiegel was ay. There are that turbulent rd ‘turbulence’ is turbulence?” rbulent flows is ow dimensional odic orbits were st, dynamics on a moto-Sivashinsky pf’s incorrect the- for the given PDE at Caltech in 1964 r convection equa- is often finite and years later, in 1996 very useful to me.” y one that could be another one; that’s e the long-time dy- onset of spatiotem- essentially reduced ther, and continued vided both a e-periodic solutions on.” What Hopf ues. Spiegel met Kraich- erek, “This is turbu- quencies as building ional Poincaréreturn near recurrences. For turbulence moved Ed. , “Flow follows a reg- ay was the demonstra- ChaosBook.org version15.9, Jun 24 2017 s were introduced [A1.81, ations) dynamics of this of contracting directions heal ∞ -dimensional PDE state spaces? For dis- quantitative predictions ∞ projects) that the periodic orbit theory be applied to infini / [A1.79] proposed (in what is now the gold standard for exem- and highly accurate et al. ), by choosing the unstable manifold of the shortest periodi

A1. A BRIEF HISTORY OF CHAOS ectively finite dimensional. For a more precise statement, s s ( ff f → said and where he said it remains deeply obscure to this very d [A1.80]. s The 1996 project went as far as one could with methods and comp How is it possible that the theory originally developed for l Hopf, however, to the best of our knowledge, never suggested The first paper to advocate a periodic orbit description of tu sources available, until 2002, when new variational method appendHist - 20Mar2013 et al. even low-dimensional; perturbations along the sipative flows the number of unstable, expanding directions themselves, and play only a minornamics role is in e cycle weights - henc dynamical systems can work in the with the given boundary conditions and system parameter val tually dimensional flows, such asmodel the as a Navier-Stokes, laboratory using for theporal exploring chaos. the Kura The dynamics main close conceptual to advancetion the that in the this high-dimensional initial (16-64 for mode trunc Galërkin dissipative flow can be reduced to anmap approximately 1-dimens the intrinsic curvilinear coordinate fromthe which first to time measure fordetermined any numerically. nonlinear What PDE, was somestrange novel attractor about 1,000 this embedded unstable work? in peri to a Fir high-dimensional 1-dimensional space dynamics. was Second, the solutions found pro tative description flow should be analyzed inof terms of the ‘recurrent defining flows’, PDEs. i.e.Robert tim Kraichnan’s The research associate. story so Kraichnanular told far solution him goes for like a this:turbulence.” while, It then in was not another 1960 too one, Ed In clear, then 1962 but Spiegel Kraichnan’s switches and vision to Derek of Mooretions investigated a which set seemed of to 3rd follow orde going one from periodic periodic solution, solution then tolence!” ano periodic and solution. Derek said Ed “This toldand is D came wonderful!” back He gave very a angry.they lecture kept They asking pilloried and him hefrom there. could their not 1966 “Why paper answer, is [A1.77] so th onnan he periodic and expunged solutions. told the In him, wo 1970 “This visionKraichnan of said: turbulence “That of yours wasn’t has my been vision, that was Hopf’s visi plary ChaosBook.org APPENDIX papers that lump him togetherory with of Landau, turbulence,’ as a the proposal ‘Landau-Ho blocks to of deploy turbulence. incommensurate This fre was Landau’simplemented guess at and the was time. the onl thus the 1966 Spiegel and MooreChristiansen paper [A1.77, A1.78]. Thirty 820 erg, were not the fault ential for he- um consists of uantum theory ectrum and the planetary orbits he problem, and tained a “partial tum theory lusion principle, He wrote enthu- ctrum . . . I think r neutral helium. drogen ratio was fied logic of clas- y consistent. undations seemed hanik” from 1925 st months to make f the helium spec- rturbative scheme. antum theory ey missed in 1913- se cases where d that the two kinds Sommerfeld theory, tion with Max Born n the treatment ect. The concluding eagues he wrote that l mechanics into tary orbits around the of orbits (a suggestion ations. The next level ff ck of understanding of g classical trajectories ChaosBook.org version15.9, Jun 24 2017 ered a bout of hay fever, and the old quantum theory ff

A1. A BRIEF HISTORY OF CHAOS ect were its death knell. orts of the young generation, including Kramers and Heisenb This is not surprising, for the principles used are not reall ff ff (...) the systematic application of the principles of the qu (...) gives resultsthe in motion agreement of with a experimentof single only the in electron motion tho is of considered; the two it electrons fails in even the i helium atom. (...) A complete systematic transformation of thestrives. classica a discontinuous mechanics is the goal towards which the quan For a while Heisenberg thought that he had the ionization pot Since the late 1890’s it had been known that the helium spectr That year Heisenberg su Why did Pauli and Heisenberg fail with the helium atom? It was to compute the spectrum ofTo helium a using first Born’s approximation, systematic they pe of reproduced corrections the turned earlier calcul out to be larger than the computed e lium, which he hadsiastic obtained letters by to a Sommerfeld simple and perturbative was scheme. drawn into a collabora of no avail. paragraph of Max Born’s classicsums “Vorlesungen it Atommec über up in a somber tone [A1.94]: he wrote to Rutherford, “Ia have used serious all my attempt spare to timereally in that solve the at the la last problem I“the have of theory a was ordinary clue worked to helium out theagreement spe in problem.” with the To fall other the of coll measurements.” 1916”while Nevertheless, and by of the having Bohr- and ob largeHeroic successful e for hydrogen, was a disaster fo the orthohelium and parahelium lines.of In helium 1915 lines might Bohr be suggeste associated withthat two turned distinct out shapes to be wrong). In 1916 he got Kramers to work on t Zeeman e nucleus; their wave aspectsobscure, were yet but to Bohr’s be answercorrect discovered. for to The the five fo once-ionized significantmarched helium figures on, to and until hy hard by 1924 to it ignore. reached an The impasse: old the q helium sp sical theory. The electrons were supposed to describe plane set of rules bringing some order to a set of phenomena which de was dead. In 1926trum. he gave He the used first wavenone quantitative mechanics, of explanation electron which o spin belonged and toof the electrons the Pauli were old cast exc quantum away theory. for nearly As half a a result, century. of the old quantum mechanics, but rather it reflected their la the subtleties of classical mechanics. Today we know what th APPENDIX 24, the role of conjugate points (topological indices)appendHist alon - 20Mar2013 819 et it into the quantum the- were working rtley’s treatise .. o use of group t along without r and mountain ed to have seen rs of the Lie al- a seminar, at the ry at all; it was a just committed a ysics journal that Tanner and Klaus at what they were ce at the Niels Bohr Gruppenpest’ must ns of group theory. ust penned are s time?” WIGNER: ectrum. mitted to a Springer sly, argues that what aragraphs to the role ffi er [A1.93].” t previous knowledge ff ChaosBook.org version15.9, Jun 24 2017 e relations between math- y kind of guy, always wearing san- ff Inward Bound: of Matter and Forces in — A. Pais, the Physical World In 1913 Otto Stern and Maxup Theodor for Felix a von Laue walk went upand the talked Uetliberg. about On physics.the the In new top particular they atom sat they model down the talked of ‘Uetli about Bohr. Schwur:’ If There that andto crazy then be model of right, they Bohr then made turned theydidn’t. out would leave physics. It did and they

A1. A BRIEF HISTORY OF CHAOS The failure of the Copenhagen School to obtain a reasonable . From AIP Wigner interview: AIP: “In that circle of people you Never mind that the ‘Copenhagen School’ refers not to the old One afternoon in May 1991, Dieter Wintgen is sitting in his o A1.7 Death of the Old Quantum Theory Institute beaming with themajor unparalleled mischief. glee The of starting a words of boy the who manuscript has he has j ing “slain the Gruppenpest”.are Pages fascinating 10 reading and in 11 thisof of context. group Condon and theory They Sho in devoteit.” their three book. p This is First followed theyend by say, of a “We which lovely manage Weyl anecdote. protested totheory that ge In but Dirac 1928 had that Dirac said in gave heDirac fact would replied, most make “I of n said his thatof arguments I group were would theory!” applicatio obtain the Mackey,Slater results in and withou the Condon article referred andgebra to Shortley of previou did SO(3) was as to “angulardoing rename momenta” was the and physics generato create and not the esoteric feeling mathematics.” th with in Berlin, was there“No. much interest On in the group opposite.be theory abolished.” at “It Schrödinger coined thi is the interesting, and expression,ematics representative and ‘ of physics, th thatphysics Wigner’s paper journal. was originally It sub wasmight rejected, take and it when Wigner von wasAnnals Neumann seeking of told Mathematics. a him Wigner ph was not happy to to worry, accept he his would o g dals and holed outclimber. jeans, He the worked German around flavor theRichter of clock to with a complete his 90’s students the leftdone Gregor work winge back that in 1916: Bohr a himself ‘planetary’ would calculation have of the lov helium sp ory, but to something else. The old quantum theory wasappendHist - no 20Mar2013 theo APPENDIX Wintgen was 34 years old at the time, a scru 822 erentials. For the ff he problem has no aining his notion of etry of billiards in a re robust: look at the in di he Andronov-Pontryagin amical systems with sin- ChaosBook.org version15.9, Jun 24 2017 ozi and Byelikh attractors v-Pontryagin school from he problems in dynamical ty, with singularities, and . Katok and J.-M. Strelcyn o Brazilian mathematician . The next year Smale pub- le was already lecturing on l with his topological back- ries that approach the set by iven that the stability grows ral stability, a notion that got r as we can trace the idea, it one that; but Smale was the y Hadamard. Smale attended ems: ained by excising the regions dering set. Smale was not the s hits the singularities. Even strong as Anosov’s), but with set. In order to establish some h an infinity of periodic points pment. As an example, take the . A. Katok and J.-M. Strelcyn proved n For each paper cited in dynamical systems In the mid 1970’s A. Katok and Ya.B. Pesin tried erentials are nonhyperbolic. The points do not separate ff neighborhood around it. Given that the Lebesgue measure is , Ya.B. Pesin proved that the Lyapunov exponent is non-zero, ǫ n Notion of global foliations. Levels of ergodicity.

A1. A BRIEF HISTORY OF CHAOS , or faster. n ǫ A1.1 A1.2 In mid 1980’s Ya.B. Pesin studied the dissipative case. Now t In the systems that are uniformly hyperbolic, all trouble is Ya.G. Sinai, L. Bunimovich and N.I. Chernov studied the geom 1. Anosov flow and the stability grows not faster than (distance) α ńnattractor,Hénon already the di appendHist - 20Mar2013 that do not separate. Hence there are 3 levels of ergodic syst uniformly, but the analogue of the singularity set can be obt that the Lyapunov exponent is non-zero. invariant Lebesgue measure.tying Assuming together uniform Lebesgue measure hyperbolici not and faster discontinuities, than and (distance) g satisfy these conditions. and that SRB measure exists. He also proved that the Lorenz, L very detailed way. A.discontinuity Katok set, and take Ya.B. an Pesin’s idea was much mo to use geometry tocarried establish positive out Lyapunov the exponents. programgularities. A and developed They a studied theory uniformlysets of hyperbolic of general systems dyn singularities. (as more Under important iterations are the a points densecontrol that set over never hit how of they the point singularity approach the set, one looks at trajecto some given ǫ Commentary Remark him enthusiastic about dynamical systems,ground. as It it was blended from wel discussionssystems with that Peixoto lead him that to Smale his gotlished 1960 his t paper result on on Morse the inequalities hyperbolicfirst structure to of consider the non–wan a hyperbolic point, Poincaréhad already d the horseshoe as a structurally stable dynamical system wit Lefschetz) was the closest Smaleschool. had It ever was from come Peixoto until that then Smale to learned about t structu first to introduce a global hyperbolic structure. By 1960 Sma andpromotinghisglobalviewpoint.Remark (R.Mainieri) transversality. One ofPeixoto. Thom’s disciples Peixoto (who introduced had Smale learned t the results of the Androno a seminar of Thom in 1958 or 1959. In that seminar Thom was expl APPENDIX literature, there are many results thatnotion went of into global its foliations develo thatgoes we back to attribute RenéThom; local to foliations Smale. were already used As b fa 821 k - in Bethe today if all this e old quantum hat it took–and f periodic orbits Would quantum voice over your- k in Seattle about order to approach the topological in- figured out how to , A1.55] understood oved it so much that ?” ; while the rest of us o Hans’ secret place: ChaosBook.org version15.9, Jun 24 2017 et al. erent if in 1917 Bohr and Kramers ff

A1. A BRIEF HISTORY OF CHAOS “It would not matter at all!” Bethe was very annoyed. He responded with an exasperated loo Since then the calculation for helium using the methods of th One is also free to ponder what quantum theory would look like use the helium classical 3-body dynamics to quantize helium Deutschinglish (if you have everself): talked to him, you can do the appendHist - 20Mar2013 APPENDIX was not accounted for, andin they nonintegrable had systems. no idea of the importance o mechanics has been fixed. Leopold anddices Percival in [A1.95] 1980, added and in 1991the Wintgen and role collaborators of [A39.11 periodic orbits.were preparing Dieter had to good sharpen reasonsthe our to dreaded pencils gloat 3-body and problem, supercomputersmuch they in else–is just described went in ahead this book. and did it. W was worked out in 1917.helium In and 1994 cycle Predrag expansions Cvitanovićgave to–inter a alia–Hansafter tal Bethe, the who talk l he pulledthe Predrag best aside lunch and they onmechanics trotted campus over look (Business t di School). Predrag asked: “ , 73 824 Int. J. (World (Wesley, , 82 (1965). (Amer. Math. ries, J. Indian 81 f motion. erential equations erval,” in A. Dold mit sets for trans- compact Riemann ff ts, Commun. Swiss oups in weakly sym- ntization conditions, momentum space and ChaosBook.org version15.9, Jun 24 2017 , 25–44 (1973). , 315 (1931). 15 17 , 180 (1945). , 1979 (1967). 8 20 , 225 (1972). 465 (Springer, Berlin 1988). 25 , 587 (1932). J. Comb. Theo. 33 1342, Celestial Encounters, The Origins of Chaos and Chaos in Classical and Quantum Mechanics Dynamical Systems, Proceedings, U. of Maryland Poincaréand the Three Body Problem , 47 (1956). Ann. Math. Proc. Nat. Acad. Sci. USA 20 , 343 (1971). erentiable dynamical systems, Bull. Amer. Math. Soc. ff Statistical Mechanics, Thermodynamic Formalism Positive transfer operators and decay of correlations 12 , 4839 (2012). 37 Lec. Notes in Math. (Princeton Univ. Press, Princeton NJ 1996). , , 55:531–534, 1992. metric Riemannian spaces withMath. applications Soc. (N.S.) to Dirichlet se Control 747 (1967). Scientific, Singapore, 2000). of the second order, J. London Math. Soc. the bound states of an atom, J. Math. Phys. and B. Eckmann, eds., 1986-87 Stability (Springer, New York, 1990). surface, Comm. Pure Appl. Math. Soc., Providence R.I., 1997). formations on the unit interval,” Phys. Soc. J. Math. Phys. Reading, MA, 1978). refsAppHist - 11feb2013 [A1.6] A. M. Lyapunov. The general problem of the stability o [A1.7] S. Smale, Di [A1.15] H. P. McKean, Selberg’s trace formula as applied to a [A1.8] N. Metropolis, M.L. Stein and P.R. Stein, “On finite li [A1.17] M. V. Berry, Martin Gutzwiller and his periodic orbi [A1.5] M. L. Cartwright and J. E. Littlewood, On nonlinear di [A1.4] J. von Neumann, References [A1.1] J. Barrow-Green, [A1.16] V. Baladi, [A1.3] B. O. Koopman, [A1.14] A. Selberg, Harmonic analysis and discontinuous gr [A1.2] F. Diacu and P. Holmes, [A1.10] M. Artin and B. Mazur, On periodic points, Ann. Math. [A1.18] M. C. Gutzwiller, Phase-integral approximation in [A1.13] M. C. Gutzwiller, [A1.9] J. Milnor and W. Thurston, “On iterated maps of the int [A1.11] D. Ruelle, References [A1.12] M. C. Gutzwiller, Periodic orbits and classical qua 823 se is cient to give him Carleson [A1.96, ffi and L.-S. Young [A1.99]. e, with no regard for e by Ya.G. Sinai and is house that studied n. ion is a one sentence their numerical results ay an important paper neither of extreme ide- to for the first time by s discussed above, pro- t [A1.105]. In August h symmetry for generic erman Physical Society, oclaw, Poland), pointed m Pais; neither had any hanique was Céleste not lps. ChaosBook.org version15.9, Jun 24 2017 nearthed a reference that 101]. We asked about the ng [A1.102]. The idea is zet function on the other. iche, who had previously ional equation ansatz. The es described above [A1.30], t find any evidence in their ional equation conjecture is y taking the real part of the le expansions; the cycles are quantization was anticipated deemed su oit the self-duality of the Rie- ded the topological phase, and counts of the demise of the old at the appropriate cycle length. search inspiration. ff (based on Ya.B. Pesin’s comments) A very appealing proposal in the context The first hint that chaos is afoot in quantum mechanics The tale of appendix A1.7, aside from a few personal erent; the claim is that the optimal estimate for low eigen- ff singularity set: For the Hamiltonian systems the general ca + Sources. Berry-Keating conjecture. Einstein did it? A1.5 A1.4 A1.3 by a single external criterion, such as the maximal cycle tim studied by A. Katok andL. J.-M. Bunimovich. Strelcyn, The and dissipative case the is billiards studied cas by Ya.B. Pesi A1.97, A1.98]. A morereadableproofis givenin M. Benedicks ff The real life challenge are generic dynamical flows, which fit 3. case: Hénon The first proof was given by M. Benedicks and L. 2. Anosov flow the topology and thenot curvature in corrections. its final While form the yet, it funct is very intriguing and fruitful re alized settings, Smale horseshoe on one end, and the Riemann Remark recollections, is in large part liftedquantum from theory Abraham Pais’ [A1.103, ac A1.104],1994 as Dieter well Wintgen as died in Jammer’s a accoun climbing accident in the Swiss A refsAppHist - 11feb2013 M. Sieber, G. Tanner andsupport D. this Wintgen, claim; and F. P.numerical Dahlqvist Christiansen results. find and The that usual P. Riemann-Siegel Cvitanovićdomann formulas no expl and other zetaHamiltonian functions, flows. but Also thereposal from is in the no its point current evidence form of ofcut belongs hyperbolic to o suc the dynamic category of crude cyc values of classically chaoticcycle quantum expansion systems of is the semiclassical obtained zeta b function, cut o of semiclassical quantization isto due improve cycle to expansions M. bycycle Berry expansions imposing and that unitarity they J. as use Keati a arebut the the funct same philosophy as is the quite original di on the credit for being thepaper pioneer two of people quantum chaos from [A1.13,recollection that A1. era, of Sir the Rudolf 1917shows Peierls article. that and in Abraha However, early Theo 20sPoincaré’s Méchanique Geisel Céleste [A1.67]. Born has did u have afollowed In study Einstein 1954 group as meeting Fritz professor inout Re of h to physics J.B. in Keller Breslauby that (now the Keller’s Wr long geometrical forgotten semiclassical written paper by by the A. physicist Einstein whoand at [A1.100]. the the most time In famous was thisKeller scientist the [36.3], w of president 41 of years his later. G time,Wintgen But came and before students to Ian recycled Percival be inclu the referred enough Helium to atom, complete Bohr’s knowing original Méc program. Remark Remark remark. Einstein being Einstein, this one sentence has been REFERENCES was given in a note by A. Einstein [A1.100]. The total discuss , 20 826 , edited ambridge, , eds. B. Eck- mber theorem cs cattering from a zation of bound l systems, J. Stat. zeta functions, in tion of chaotic dy- tization of chaotic lly chaotic repellor, nsions for classical on of the scattering of cycle expansions, or bound chaotic sys- nqvist, and P. 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