Chaos: A Program Collection for the PC PDF
Preview Chaos: A Program Collection for the PC
Chaos H.J. Korsch H.-J. Jodl T. Hartmann Chaos A Program Collection for the PC Thirdrevisedand enlarged edition With250 Illustrations,ManyNumerical Experiments and aCD-ROM ABC ProfessorHansJürgenKorsch ProfessorHans-JörgJodl TUKaiserslautern FBPhysik Erwin-Schrödinger-Str.46 67663Kaiserslautern Germany TimoHartmann UniversitätRegensburg InstitutfürTheoretischePhysik 93040Regensburg Germany LibraryofCongressControlNumber:2007940051 ISBN978-3-540-74866-3 SpringerBerlinHeidelbergNewYork ISBN978-3-540-63893-32nded.SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright. Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember 9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations areliableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:2)c Springer-VerlagBerlinHeidelberg2008 Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:bytheauthorandEDV-BeratungFrankHerwegusingaSpringerLATEXmacropackage Coverdesign:eStudioCalamar,Spain Printedonacid-freepaper SPIN:11936657 543210 Preface to the Third Edition Ithasbeenthirteenyearssincetheappearanceofthefirsteditionofthisbook, and nine years after the second. Meanwhile, chaotic (or nonlinear) dynamics is established as an essential part of courses in physics and it still fascinates students,scientistsandevennonacademicpeople,inparticularbecauseofthe beauty of computer generated images appearing frequently in this field. Quitegenerally,computersareanidealtoolforexploringanddemonstrat- ing the intricate features of chaotic dynamics. The programs in the previous editions of this book have been designed to support such studies even for the non-experienced users of personal computers. However, caused by the rapid development of the computational world, these programs written in Turbo Pascal appeared in an old-fashioned design compared to the up-to-date stan- dard.Evenmoreimportant,thoseprogramswouldnotproperlyoperateunder recent versions of the Windows operating system. In addition, there is an in- creasing use of Linux operating systems. Therefore, for the present edition, all the programs have been entirely rewritten in C++ and, of course, revised and polished. Two version of the program codes are supplied working under Windows or Linux operating systems. We have again corrected a few passage in the text of the book and added somemorerecentdevelopmentsinthefieldofchaoticdynamics.Finallyanew program treating the important class of two-dimensional discrete (‘kicked’) systems has been added and described in Chap.13. Kaiserslautern, H. J. Korsch, H.-J. Jodl, and T. Hartmann August 2007 Preface to the First Edition The problem, expressed in its general form, is an old one and appears under manyguises.Whyarethecloudsthewaytheyare?Isthesolarsystemstable? Whatdeterminesthestructureofturbulenceinliquids,thenoiseinelectronic circuits,thestabilityofaplasma?Whatisnew,thatistosaywithrespectto Newton’sPrincipiapublishedthreehundredyearsago,andwhichhasemerged over the last few decades, is the heuristic use of computers to enhance our understanding of the mathematics of nonlinear dynamical processes and to explore the complex behavior that even simple systems often exhibit. It is the purpose of this book to teach chaos through a simultaneous reading of the text and interaction with our selected computer programs. The use of computers is not only essential for studying nonlinear phe- nomena, but also enables the intuitive geometric and heuristic approach to be developed, taught to students, and integrated into the scientist’s skills, techniques, and methods. • Computers allow us to penetrate unexplored regions of mathematics and discover unforeseen links between ideas. • Numerical solutions of complex nonlinear problems – displayed by graphs or videos – as opposed to analytical solutions, which are often limited due to the approximations made, become possible. • The visualization of mathematics will also be a focus of this book: one goodgraphorsimulationvideothathighlightstheevolutionofacoherent complex pattern can be worth more than a hundred equations. • Nonlinear problems are almost always difficult, often having unexpected solutions. • Inattemptingtounderstandthedetailsofthecomputersolutiononemay uncover a new kind of problem, or a new aspect that leads to deeper understanding. • Appropriate graphical displays, especially those that are constructed and composed on the screen as one interacts with the computer, will improve the ability to choose from among promising paths. This procedure natu- VIII Preface to the First Edition rally complements the usual approaches of experiment, theoretical formu- lation and asymptotic approximation. • The benefits of the computational approach clearly depend on the avail- abilityofvariousgraphicaldisplays:smalleffectsthatmaysignalnewphe- nomena(zoom);propermappingofdatainplaceofasearchforstructures involuminousprintoutsofcolumnsofnumbers;pictureswhichclearlypro- duce an insight into the physics; color or real-time videos which enhance perception, enabling one to correlate old and new results and recognize unexpected phenomena. • Interactive software in general, or educational software in physics, must provide the ability to display in one, two or three dimensions, to show spatial and temporal correlations, to rotate, displace or stretch objects, to acquire diagnostic variables and essential summaries or to optimize comparisons. Are we providing this kind of training in our universities, so that our students may learn this method of working at the ‘nonlinear’ frontier? We have to find new methods to teach students to experiment with computers in the way that we now teach them to experiment with lasers, or deepen their knowledge of theory. Therefore, the philosophy of our approach in this book is to practice the use of a computer in computational physics directed at a convincing topic, i.e., nonlinear physics and chaos. Our programs are written in such a style that physical problems can quickly be tackled, and time is not wasted to program details such as the use of algorithms to solve an equation, ortheinputandoutputofdata.Ofcourse,thestudentwilleventuallyneedto mastertheelementsofprogramminghimselfashecomesclosertoindependent research. Therefore, this book is aimed at those who have completed a course of study in physics and are on the threshold of research. Another advantage is that ‘mini-research’ can be carried out, thanks to the nature of the topic, chaos, and to the tool, the computer. These allow one to discuss physical problems which are only mentioned in textbooks nowadays as a potential topic of study, e.g., the double pendulum. From the point of view of the complexity of the mathematics and physics, this book is designed mainly for students in the third or fourth year in a scienceorengineeringfaculty.Inalimitedway,itmightalsobeusefultothose working at the frontiers of nonlinear physics, since this topic is relatively new and far from having well-established solutions or wide applications. This book is organized in the following way: in Chap.1 ‘Overview and Basic Concepts’,weattempttointroducetypicalfeaturesofchaoticbehavior andtopointoutthebroadapplicabilityofchaosinscienceaswellastomake the reader familiar with the terminology and theoretical concepts. In Chap.2 ‘NonlinearDynamicsandDeterministicChaos’,wewilldevelopthenecessary basis, which will be deepened and applied in subsequent chapters. Chapter 3 ‘Billiard Systems’ and Chap.4 ‘Gravitational Billiards’ will treat two of the ‘classical examples’ of simple conservative mechanical systems. In Chap.5, Preface to the First Edition IX the class of different pendula, such as kicked, inverted, coupled, oscillatory or rotating, is representatively discussed through the double pendulum. Phe- nomena appearing in chaotic scattering systems are represented by the three disk scattering in Chap.6. The subsequent chapters treat systems explicitly dependent on time: namely, in Chap.7 a periodically kicked particle in a box ‘Fermi Acceleration’, and in Chap.8 a driven anharmonic oscillator ‘Duffing Oscillator’. The celebrated one-dimensional iterated maps are the topic of Chap.9, and the observed period-doubling scenario can be studied via the physical example of nonlinear electronic circuits in Chap.10. Numerical ex- perimentswithtwo-dimensionalmapsareconsideredinChap.11‘Mandelbrot and Julia Sets’, while Chap.12 ‘Ordinary Differential equations’ provides a quite general platform from which to investigate systems governed by cou- pled first-order differential equations. Finally, further technical questions of hardwarerequirements,programinstallation,andtheuseoftheprogramsare addressed in the appendices. Most chapters follow the same substructure: – Theoretical Background – Numerical Techniques – Interaction with the Program – Computer Experiments – Real Experiments and Empirical Evidence – References Many books and articles have been written on chaotic experiments, and some of the ‘classical’ experiments are mentioned in Chap.1. Therefore, the last subsection in each chapter is intended to give the reader confidence to progress from his studies on the computer to real experiments and empirical evidence; e.g., comparing the trajectory of a double pendulum in reality and on the screen. Of course, some aspects of the system are better studied in the computer experiment, others in the real one; in addition, they complement each other, e.g., looking for bifurcations in a nonlinear electronic circuit on theoscilloscopeandonthescreen(Chap.10).Oneisatfirstimpressedbythe apparently chaotic motion of a real double pendulum, but deeper insight into thestructureofthischaoticbehaviorisgainedbylookingatPoincar´emapsin phasespace.Theexperimentschosenhereandbrieflyreported(fordetailssee the cited literature) are mainly for educational purposes, to be reconstructed andusedinstudentlaboratoriesorinlectures.Therefore,theyarenotmeant to represent those experiments investigated in current nonlinear research. Themosteffectivewayofusingthisbookmaybetoreadachapterwhile working simultaneously on the computer using the appropriate program. As already mentioned, the reader should use the programs — rather than pro- gram major parts himself — in much the same way as he would use standard service software in combination with commercial research apparatus. On the other hand, the use of our programs should not be a simple push button pro- cedure, but involve serious interaction with the software. For example, some parameters, initial values, boundary conditions are already preset to execute X Preface to the First Edition numerical experiments discussed in the book, while other numerical experi- ments described in detail require changes in the preset parameters. Further studies are suggested and the reader may proceed independently, guided by some hints and the cited literature. The programs are flexible and organized insuchawaythathecansetuphisowncomputerexperiments,e.g.,definehis ownboundaryinabilliardproblemorexplorehisfavoritesystemofnonlinear differential equations. Reading this book and working with the programs requires a knowledge of classical mechanics and a basic understanding of chaotic phenomena. The short overview on chaotic dynamics and chaos theory in Chaps.1 and 2 can not serve as a substitute for a textbook. Within the last decade a number of such books have been published reflecting the rapid growth of the field. Somefocusonexperiments,someconcentrateontheory,somedealwithbasic concepts, while others are simply a selection of original articles. The reader should consult some of these texts while exploring the nonlinear world by means of the computer programs in this book. Wehopethattheselectedexamplesofchaoticsystemswillhelpthereader togainabasicunderstandingofnonlineardynamics,andwillalsodemonstrate the usefulness of computers for teaching physics on the PC. Mostoftheprograms,atleastintheirpreliminaryversion,weredeveloped bystudentsduringaseminar‘Computer Assisted Physics’ (1985–1990).With the aid of two grants (PPP 1987–1989, PPPP 1991–1994), we were able to improve, test, standardize and update those programs. Chapter 1 contains a list of all coauthors for every program. We wish to acknowledge funding from the Bundesministerium fu¨r Bildung und Wissenschaft (BMBW), from theUniversityofKaiserslauternviatheKultusministeriumofRheinland-Pfalz and the Deutsche Forschungsgemeinschaft (DFG) for the hardware. We are indebted to the Volkswagenstiftung (VW) for supporting one of us (H.-J. J.) to finish this book during a sabbatical. Finally,wewishtorecognizethecontributionofundergraduateandgradu- atestudentsandresearchassociateswhoworkedsoenthusiasticallywithuson problemsassociatedwithchaosandontheuseofcomputersinphysicsteach- ing. We would particularly like to thank the graduate students Bjo¨rn Baser andAndreasSchuchfortheirconsiderableassistanceindevelopinganddebug- ging the computer codes, the instructions for interacting with the programs as well as the large number of computer experiments. Finally, we would like tothankFrankBenschandBrunoMirbach,whoreadpartsofthemanuscript and made many useful suggestions. Kaiserslautern, H. J. Korsch and H.-J. Jodl May 1994 Contents 1 Overview and Basic Concepts.............................. 1 1.1 Introduction ............................................ 1 1.2 The Programs........................................... 5 1.3 Literature on Chaotic Dynamics........................... 8 2 Nonlinear Dynamics and Deterministic Chaos ............. 11 2.1 Deterministic Chaos ..................................... 12 2.2 Hamiltonian Systems .................................... 13 2.2.1 Integrable and Ergodic Systems ..................... 13 2.2.2 Poincar´e Sections.................................. 16 2.2.3 The KAM Theorem ............................... 18 2.2.4 Homoclinic Points ................................. 20 2.3 Dissipative Dynamical Systems............................ 22 2.3.1 Attractors........................................ 24 2.3.2 Routes to Chaos .................................. 26 2.4 Special Topics........................................... 27 2.4.1 The Poincar´e-Birkhoff Theorem ..................... 28 2.4.2 Continued Fractions ............................... 29 2.4.3 The Lyapunov Exponent ........................... 32 2.4.4 Fixed Points of One-Dimensional Maps............... 35 2.4.5 Fixed Points of Two-Dimensional Maps .............. 38 2.4.6 Bifurcations ...................................... 44 References .................................................. 45 3 Billiard Systems ........................................... 47 3.1 Deformations of a Circle Billiard .......................... 50 3.2 Numerical Techniques.................................... 53 3.3 Interacting with the Program ............................. 54 3.4 Computer Experiments................................... 58 3.4.1 From Regularity to Chaos .......................... 58 3.4.2 Zooming In....................................... 60 XII Contents 3.4.3 Sensitivity and Determinism ........................ 61 3.4.4 Suggestions for Additional Experiments .............. 63 3.5 Suggestions for Further Studies............................ 66 3.6 Real Experiments and Empirical Evidence.................. 66 References .................................................. 67 4 Gravitational Billiards: The Wedge ........................ 69 4.1 The Poincar´e Mapping ................................... 70 4.2 Interacting with the Program ............................. 75 4.3 Computer Experiments................................... 77 4.3.1 Periodic Motion and Phase Space Organization ....... 77 4.3.2 Bifurcation Phenomena ............................ 81 4.3.3 ‘Plane Filling’ Wedge Billiards ...................... 86 4.3.4 Suggestions for Additional Experiments .............. 88 4.4 Suggestions for Further Studies............................ 89 4.5 Real Experiments and Empirical Evidence.................. 90 References .................................................. 90 5 The Double Pendulum..................................... 91 5.1 Equations of Motion ..................................... 91 5.2 Numerical Algorithms.................................... 93 5.3 Interacting with the Program ............................. 93 5.4 Computer Experiments................................... 98 5.4.1 Different Types of Motion .......................... 98 5.4.2 Dynamics of the Double Pendulum ..................102 5.4.3 Destruction of Invariant Curves .....................107 5.4.4 Suggestions for Additional Experiments ..............110 5.5 Real Experiments and Empirical Evidence..................111 References ..................................................113 6 Chaotic Scattering.........................................115 6.1 Scattering off Three Disks ................................117 6.2 Numerical Techniques....................................121 6.3 Interacting with the Program .............................121 6.4 Computer Experiments...................................124 6.4.1 Scattering Functions and Two-Disk Collisions.........124 6.4.2 Tree Organization of Three-Disk Collisions ...........127 6.4.3 Unstable Periodic Orbits ...........................129 6.4.4 Fractal Singularity Structure........................131 6.4.5 Suggestions for Additional Experiments ..............133 6.5 Suggestions for Further Studies............................135 6.6 Real Experiments and Empirical Evidence..................136 References ..................................................136
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