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It's a Nonlinear World
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Preface xi Part I: WORLD OF MATHEMATICS 1 1 World of Nonlinear Systems 1.1 Introduction to Nonlinear ODE Models 1.2 Introduction to Difference Equation Models 1.3 Solving Nonlinear ODEs on the Computer .. 3 4 9 12 2 World of Nonlinear ODEs 2.1 Breakdown of Linear Superposition. 2.2 Some Analytically Solvable Examples . 2.3 Fixed Points and Phase-Plane Analysis .. 2.4 Bifurcations . 2.5 Hysteresis and the Jump Phenomena .. 2.6 Limit Cycles . 2.7 Strange Attractors and Chaos . 2.8 Fractal Dimensions . . . 2.9 Poincare Sections . 2.10 Power Spectrum .. . . 29 30 32 36 44 47 49 53 56 58 59 71 71 73 74 75 77 80 82 84 86 89 Fixed Points of One-Dimensional Maps . Stability Criterion . . . . . . . . . . Cobweb Diagram. . . . . . . .... Period Doubling to Chaos . . . . . . Creating Lorenz Maps. ... . . Lyapunov Exponent . . . . . . . . . Two-Dimensional Maps .. . . . . . Mandelbrot and Julia Sets . Chaos versus Noise . . . . . . Controlling Chaos . 3 World of Nonlinear Maps 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 vii viii CONTENTS 4 World of Solitons 4.1 Korteweg-deVries Solitons .. 4.2 Sine-Gordon Solitons . . . . . . . . . . . . . . . . . . 4.3 Similarity Solutions. . . . . 4.4 Numerical Simulation ... 4.4.1 Finite Difference Approximations .. 4.4.2 The Zabusky-Kruskal Algorithm . . 4.4.3 Method of Characteristics . 4.4.4 Numerical Algorithm for the SGE . 4.4.5 Numerical Stability. . . . . . . . . 4.5 Extension to Cellular Automata . 99 101 104 107 109 109 111 114 115 117 119 Part II: OUR NONLINEAR WORLD 129 5 World of Motion 5.1 Nonlinear Drag or Resistance . 5.2 Nonlinear Lift . 5.3 The Pendulum, Simple and Otherwise 5.3.1 The Simple Pendulum . 5.3.2 Parametric Excitation . 5.3.3 The Rotating Pendulum . . 5.4 Nonlinear Springs . 5.4.1 Lattice Dynamics . 5.5 Hysteresis and Jumps Revisited . 5.6 Precession of Mercury . . . . . . 5.7 Saturn's Rings: A "Toy" Model. 5.8 Hamiltonian Chaos . 131 131 137 141 141 146 147 148 152 153 155 158 161 173 173 176 178 181 184 186 187 The Aerodynamics of Sports Balls . . . . . Bend It Like Beckham ... A Major League Curveball . Golf Ball Trajectory . . . . A Falling Badminton Bird . Car Racing .... Medieval Archery . . . . . . . . . . . 6 World of Sports 6.1 6.2 6.3 6.4 6.5 6.6 6.7 7 World of Electromagnetism 7.1 Nonlinear Electrical Circuits. . . . . . . . . 7.1.1 Nonlinear Inductance . 7.1.2 Nonlinear Capacitance . . . 7.1.3 Chua's Circuit: Piecewise-Linear Negative Resistance 7.1.4 Tunnel Diode Oscillator . 7.1.5 Josephson Junction . 193 193 193 196 197 199 203 CONTENTS ix 7.1.6 SQUID Magnetometer. . . . . . . . . . . . . . . .... . 208 7.2 Nonlinear Optics . . . . . . . . . .. . . . . . . . . . . .. 209 7.2.1 Optical Soliton Propagation. . . . . . . . . . . . . . . 210 7.2.2 The Navier-Stokes Equations . . . . . . .. 213 7.2.3 Stimulated Scattering of Light 216 7.3 The Earth's Magnetic Field . . . . . . . . . . . . . . . . . . . .. 218 7.3.1 Aurora Borealis. . . . . . . . . . . . . . . . 218 7.3.2 The Drifting North Magnetic Pole . . . . . . . . . . . 221 7.3.3 The Geodynamo Origin of the Earth's Magnetic Field 222 8 World of Weather Prediction 8.1 Early History . 8.2 The Barotropic Vorticity Equation 8.3 Some Meteorological Concepts ... 8.4 Modern Numerical Weather Forecasting 9 World of Chemistry 9.1 Chemical Reactions . 9.1.1 Autocatalysis . 9.1.2 Michaelis-Menten Enzyme Kinetics .. 9.1.3 Lotka-Volterra Mechanism 9.2 Chemical Oscillators .. . . 9.2.1 The Oregonator . 9.2.2 The Brusselator . 9.3 Chemical Waves and Patterns . 9.3.1 Target Patterns and Spiral Waves . 9.3.2 Reaction-Diffusion Equations . 9.3.3 How the Leopard Got Its Spots . 10 World of Disease 10.1 Classifying the Spread of Infectious Diseases .. 10.2 Basic Models of Disease Transmission . 10.2.1 The SIS Model . 10.2.2 The SIR Model without Vital Dynamics . 10.2.3 The SIR Model with Vital Dynamics . 10.2.4 Herd Immunity and Vaccination .. . . 10.2.5 Geographic Spread of an Epidemic . . . . . . . . . . . . . . . . . 10.3 Examples of Disease Growth . . 10.3.1 Mad Cow Disease. . . . . . . . 10.3.2 Avascular Tumor Growth . . . . . . . . . . 11 World of War 11.1 The Coevolutionary Arms Race . 11.1.1 The Newt versus the Garter Snake. . . . . . . . . 11.1.2 Biological Arms Race with a Dangerous Prey ..... 235 236 237 239 244 255 255 255 257 261 263 263 266 268 268 269 272 281 282 283 284 286 289 291 292 296 296 300 311 312 312 314 x CONTENTS 11.1.3 The Wild Parsnip and Geographic Mosaic Theory . 11.2 Human Conflict . 11.2.1 Political Complexity: Nonlinear Models of Politics 11.2.2 Richardson Arms Race Model . 11.2.3 Chaos-a Model for the Outbreak of War . 11.2.4 The Dynamics of Warfare: Lanchester Equations 11.2.5 War of the Fire Ants . Bibliography Index 322 323 323 324 326 335 337 345 367 |
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