## Chaos: A Program Collection for the PCIthasbeenthirteenyearssincetheappearanceofthe?rsteditionofthisbook, 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, scientists and even nonacademic people, in particular because of the beauty of computer generated images appearing frequently in this ?eld. Quite generally, computers are an ideal tool for exploring and demonstr- 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 st- dard.Evenmoreimportant,thoseprogramswouldnotproperlyoperateunder recent versions of the Windows operating system. In addition, there is an - 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 somemorerecentdevelopmentsinthe?eldofchaoticdynamics.Finallyanew program treating the important class of two-dimensional discrete (‘kicked’) systems has been added and described in Chap.13. |

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### Contents

1 | |

5 | |

13 Literature on Chaotic Dynamics | 8 |

Nonlinear Dynamics and Deterministic Chaos | 11 |

21 Deterministic Chaos | 12 |

22 Hamiltonian Systems | 13 |

222 Poincare Sections | 16 |

223 The KAM Theorem | 18 |

The Duffing Oscillator | 157 |

82 Numerical Techniques | 161 |

84 Computer Experiments | 168 |

Resonances and Bistability | 171 |

844 Coexisting Limit Cycles and Strange Attractors | 174 |

845 Suggestions for Additional Experiments | 176 |

85 Suggestions for Further Studies | 181 |

References | 183 |

224 Homoclinic Points | 20 |

23 Dissipative Dynamical Systems | 22 |

231 Attractors | 24 |

232 Routes to Chaos | 26 |

24 Special Topics | 27 |

241 The PoincareBirkhoff Theorem | 28 |

242 Continued Fractions | 29 |

243 The Lyapunov Exponent | 32 |

244 Fixed Points of OneDimensional Maps | 35 |

245 Fixed Points of TwoDimensional Maps | 38 |

246 Bifurcations | 44 |

References | 45 |

Billiard Systems | 47 |

31 Deformations of a Circle Billiard | 50 |

32 Numerical Techniques | 53 |

33 Interacting with the Program | 54 |

34 Computer Experiments | 58 |

342 Zooming In | 60 |

343 Sensitivity and Determinism | 61 |

344 Suggestions for Additional Experiments | 63 |

35 Suggestions for Further Studies | 66 |

References | 67 |

Gravitational Billiards The Wedge | 69 |

41 The Poincare Mapping | 70 |

42 Interacting with the Program | 75 |

43 Computer Experiments | 77 |

432 Bifurcation Phenomena | 81 |

433 Plane Filling Wedge Billiards | 86 |

434 Suggestions for Additional Experiments | 88 |

44 Suggestions for Further Studies | 89 |

45 Real Experiments and Empirical Evidence | 90 |

The Double Pendulum | 91 |

52 Numerical Algorithms | 93 |

54 Computer Experiments | 98 |

542 Dynamics of the Double Pendulum | 102 |

543 Destruction of Invariant Curves | 107 |

544 Suggestions for Additional Experiments | 110 |

55 Real Experiments and Empirical Evidence | 111 |

References | 113 |

Chaotic Scattering | 114 |

61 Scattering Off Three Disks | 117 |

62 Numerical Techniques | 121 |

64 Computer Experiments | 124 |

642 Tree Organization of ThreeDisk Collisions | 127 |

643 Unstable Periodic Orbits | 129 |

644 Fractal Singularity Structure | 131 |

645 Suggestions for Additional Experiments | 133 |

65 Suggestions for Further Studies | 135 |

66 Real Experiments and Empirical Evidence | 136 |

Fermi Acceleration | 137 |

71 Fermi Mapping | 138 |

72 Interacting with the Program | 139 |

73 Computer Experiments | 142 |

732 KAM Curves and Stochastic Acceleration | 144 |

733 Fixed Points and Linear Stability | 146 |

734 Absolute Barriers | 148 |

735 Suggestions for Additional Experiments | 150 |

74 Suggestions for Further Studies | 154 |

References | 155 |

Feigenbaum Scenario | 185 |

91 OneDimensional Maps | 186 |

92 Interacting with the Program | 188 |

93 Computer Experiments | 191 |

932 The Chaotic Regime | 195 |

933 Lyapunov Exponents | 199 |

934 The Tent Map | 200 |

935 Suggestions for Additional Experiments | 202 |

94 Suggestions for Further Studies | 206 |

95 Real Experiments and Empirical Evidence | 208 |

References | 209 |

Nonlinear Electronic Circuits | 211 |

102 Numerical Techniques | 214 |

103 Interacting with the Program | 215 |

104 Computer Experiments | 220 |

1042 PeriodDoubling | 221 |

1043 Return Map | 225 |

1044 Suggestions for Additional Experiments | 226 |

105 Real Experiments and Empirical Evidence | 229 |

References | 230 |

Mandelbrot and Julia Sets | 231 |

112 Numerical Methods | 235 |

113 Interacting with the Program | 236 |

114 Computer Experiments | 242 |

1142 Zooming into the Mandelbrot Set | 244 |

1143 General TwoDimensional Quadratic Mappings | 245 |

1144 Suggestions for Additional Experiments | 249 |

115 Suggestions for Further Studies | 251 |

116 Real Experiments and Empirical Evidence | 252 |

References | 253 |

Ordinary Differential Equations | 255 |

121 Numerical Techniques | 256 |

123 Computer Experiments | 268 |

1232 A Simple Hopf Bifurcation | 270 |

1233 The Duffing Oscillator Revisited | 273 |

1234 Hills Equation | 275 |

1235 The Lorenz Attractor | 281 |

1236 The Rossler Attractor | 284 |

1237 The HenonHeiles System | 285 |

1238 Suggestions for Additional Experiments | 288 |

124 Suggestions for Further Studies | 293 |

References | 298 |

Kicked Systems | 301 |

131 Interacting with the Program | 303 |

132 Computer Experiments | 307 |

1322 The Kicked Quartic Oscillator | 309 |

1323 The Kicked Quartic Oscillator with Damping | 311 |

1324 The Henon Map | 312 |

1325 Suggestions for Additional Experiments | 313 |

133 Real Experiments and Empirical Evidence | 316 |

System Requirements and Program Installation | 319 |

A21 Windows Operating System | 320 |

A3 Programs | 321 |

General Remarks on Using the Programs | 323 |

B2 Input of Mathematical Expressions | 325 |

Glossary | 327 |

335 | |

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### Common terms and phrases

amplitude angle area-preserving behavior bifurcation diagram billiard boundary curve Cantor set Chaos chaotic dynamics chaotic motion chaotic region Chap color Computer Experiments constant coordinate space detail differential equations discussed disk displayed double pendulum Duffing oscillator dynamical systems elliptic explore Feigenbaum Feigenbaum constant Fermi fractal dimension frequency function Hamiltonian systems Hénon hyperbolic fixed points impact parameter initial conditions integrable interval invariant curves iterated map iterated points Julia-set kicked left mouse button limit cycle linear logistic map Lorenz Lyapunov exponent magnification Mandelbrot map Mandelbrot-set menu nonlinear dynamics numerical experiments ode example period-doubling bifurcations period-one period-two periodic orbits perturbation phase space Phys plot Poincaré map Poincaré section preset parameters quantum ratio rotation scattering Sect sequence shown in Fig shows solution stability islands stable fixed point strange attractor structure studied torus trajectory two-dimensional variables velocity wall oscillation window zero

### References to this book

Discrete Dynamical Systems and Difference Equations with Mathematica Mustafa R.S. Kulenovic,Orlando Merino No preview available - 2002 |