DISCRETE CHAOS Second Edition WITH APPLICATIONS IN SCIENCE AND ENGINEERING C5920_Discl.indd 1 10/16/07 4:29:59 PM C5920_Discl.indd 2 10/16/07 4:30:00 PM DISCRETE CHAOS Second Edition WITH APPLICATIONS IN SCIENCE AND ENGINEERING Saber N. Elaydi Trinity University San Antonio, Texas, U.S.A. C5920_Discl.indd 3 10/16/07 4:30:00 PM MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® soft- ware or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. Maple™ is a trademark of Waterloo Maple Inc. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Preface ix Author Biography xv List of Symbols xvii Foreword xix 1 The Stability of One-Dimensional Maps 1 1.1 Introduction............................ 2 1.2 Mapsvs.DifferenceEquations ................. 2 1.3 Mapsvs.DifferentialEquations................. 4 1.3.1 Euler’sMethod...................... 4 1.3.2 Poincar´eMap....................... 8 1.4 LinearMaps/DifferenceEquations ............... 9 1.5 Fixed (Equilibrium) Points . 14 1.6 Graphical Iteration and Stability . 19 1.7 Criteria for Stability . 25 1.7.1 HyperbolicFixedPoints................. 25 1.7.2 Nonhyperbolic Fixed Points . 28 1.8 Periodic Points and Their Stability . 36 1.9 The Period-Doubling Route to Chaos . 43 1.9.1 FixedPoints....................... 43 1.9.2 2-PeriodicCycles..................... 44 1.9.3 22-PeriodicCycles.................... 46 1.9.4 Beyond μ∞ ........................ 50 1.10Applications............................ 54 1.10.1 Fish Population Modeling . 54 2 Attraction and Bifurcation 61 2.1 Introduction............................ 61 2.2 Basin of Attraction of Fixed Points . 62 2.3 Basin of Attraction of Periodic Orbits . 66 2.4 Singer’sTheorem......................... 70 2.5 Bifurcation ............................ 81 2.6 Sharkovsky’sTheorem...................... 94 2.6.1 Li-YorkeTheorem.................... 94 v vi 2.6.2 A Converse of Sharkovsky’s Theorem . 99 2.7 TheLorenzMap .........................105 2.8 Period-Doubling in the Real World . 108 2.9 Poincar´eSection/Map ......................110 2.9.1 ...............................110 2.9.2 Belousov-Zhabotinskii Chemical Reaction . 111 Appendix ................................115 3 Chaos in One Dimension 119 3.1 Introduction............................119 3.2 Density of the Set of Periodic Points . 120 3.3 Transitivity............................124 3.4 SensitiveDependence.......................129 3.5 DefinitionofChaos........................137 3.6 CantorSets............................144 3.7 SymbolicDynamics........................149 3.8 Conjugacy.............................154 3.9 OtherNotionsofChaos .....................161 3.10 R¨ossler’sAttractor........................163 3.11Saturn’sRings ..........................167 4 Stability of Two-Dimensional Maps 171 4.1 Linear Maps vs. Linear Systems . 171 4.2 Computing An ..........................172 4.3 Fundamental Set of Solutions . 179 4.4 Second-Order Difference Equations . 181 4.5 PhaseSpaceDiagrams......................184 4.6 Stability Notions . 192 4.7 Stability of Linear Systems . 197 4.8 TheTrace-DeterminantPlane..................200 4.8.1 Stability Analysis . 200 4.8.2 Navigating the Trace-Determinant Plane . 204 4.9 Liapunov Functions for Nonlinear Maps . 207 4.10LinearSystemsRevisited ....................215 4.11 Stability via Linearization . 219 4.11.1 The Hartman-Grobman Theorem . 224 4.11.2 The Stable Manifold Theorem . 225 4.12Applications............................228 4.12.1 The Kicked Rotator and the H´enonMap........228 4.12.2 The H´enonMap.....................230 4.12.3 Discrete Epidemic Model for Gonorrhea . 233 4.12.4PerennialGrass......................236 Appendix ................................239 vii 5 Bifurcation and Chaos in Two Dimensions 241 5.1 CenterManifolds.........................241 5.2 Bifurcation ............................248 5.2.1 Eigenvaluesof1or-1..................248 5.2.2 A Pair of Eigenvalues of Modulus 1 - The Neimark- SackerBifurcation....................249 5.3 Hyperbolic Anosov Toral Automorphism . 257 5.4 SymbolicDynamics........................262 5.4.1 Subshifts of Finite Type . 263 5.5 The Horseshoe and H´enonMaps ................268 5.5.1 The H´enonMap.....................272 5.6 A Case Study: The Extinction and Sustainability in Ancient Civilizations . 278 Appendix ................................285 6 Fractals 289 6.1 ExamplesofFractals.......................289 6.2 L-system..............................299 6.3 TheDimensionofaFractal ...................300 6.4 IteratedFunctionSystem ....................314 6.4.1 DeterministicIFS.....................315 6.4.2 The Random Iterated Function System and the Chaos Game............................325 6.5 Mathematical Foundation of Fractals . 330 6.6 The Collage Theorem and Image Compression . 338 7 The Julia and Mandelbrot Sets 341 7.1 Introduction............................341 7.2 Mapping by Functions on the Complex Domain . 342 7.3 The Riemann Sphere . 351 7.4 TheJuliaSet...........................354 7.5 Topological Properties of the Julia Set . 364 7.6 Newton’s Method in the Complex Plane . 371 7.7 TheMandelbrotSet .......................375 7.7.1 Topological Properties . 375 7.7.2 RaysandBulbs......................377 7.7.3 Rotation Numbers and Farey Addition . 380 7.7.4 AccuracyofPictures...................385 Bibliography 389 Answers to Selected Problems 397 Index 414 Preface Preface to the Second Edition The second edition maintains the lucidity of the first edition. Its main feature is the inclusion of many recent results on global stability, bifurcation, chaos, and fractals. The first five chapters of this book include the most comprehensive expo- sition on discrete dynamical systems at the level of advanced undergraduates and beginning graduate students. Notable additions in this book are the L-systems, the trace-determinant stability and bifurcation analysis, the peri- odic structure of the bulbs in the Mandelbrot set, the detailed analysis of the center manifold theory, and new results on global stability. Moreover, new applications to biology, chemistry, and physics were added. The biggest improvement, however, is in technology. A CD of an adapted version of PHASER by Jason Glick and Huseyin Kocac is attached to the back cover of the book. It contains all the material in Chapters 1 and 4. The material for the remaining chapters may be downloaded from http://www.math.miami.edu/∼hk/elaydi/. You can immediately download the CD and start experimenting with PHASER. You may be able to generate all the graphs in the book and use it to solve many problems in the exercises. If you are a Maple or Mathematica user, pro- grams were written by Henrique Oliveira and Rafael Luis and are posted on the Taylor and Francis website. They may be downloaded from “Electronic Products” and then “Downloads and Updates” found in the right pane of the website at http://www.crcpress.com/. I am greatly indebted to Darnum Huang who reviewed thoroughly the first edition and made a plethora of insightful comments and suggestions. He caught numerous typos in the first edition. His contribution to this book is immeasurable. Sincere appreciation goes to Richard Neidinger whose critique led to improvements in several parts of this book. I would like to thank