Nonlinear dynamics and Chaos: with applications to physics, biology, chemistry, and engineering

Cover......Page 1
Contents......Page 4
Preface......Page 8
1.0 Chaos, Fractals, and Dynamics......Page 11
1.1 Capsule History of Dynamics......Page 12
1.2 The Importance of Being Nonlinear......Page 14
1.3 A Dynamical View of the World......Page 19
Part I. One-Dimensional Flows......Page 22
2.0 Introduction......Page 23
2.1 A Geometric Way of Thinking......Page 24
2.2 Fixed Points and Stability......Page 26
2.3 Population Growth......Page 29
2.4 Linear Stability Analysis......Page 32
2.5 Existence and Uniqueness......Page 34
2.6 Impossibility of Oscillations......Page 36
2.7 Potentials......Page 38
2.8 Solving Equations on the Computer......Page 40
Exercises......Page 44
3.0 Introduction......Page 52
3.1 Saddle-Node Bifurcation......Page 53
3.2 Transcritical Bifurcation......Page 58
3.3 Laser Threshold......Page 61
3.4 Pitchfork Bifurcation......Page 63
3.5 Overdamped Bead on a Rotating Hoop......Page 69
3.6 Imperfect Bifurcations and Catastrophes......Page 77
3.7 Insect Outbreak......Page 81
Exercises......Page 87
4.1 Examples and Definitions......Page 101
4.2 Uniform Oscillator......Page 103
4.3 Nonuniform Oscillator......Page 104
4.4 Overdamped Pendulum......Page 109
4.5 Fireflies......Page 111
4.6 Superconducting Josephson Junctions......Page 114
Exercises......Page 121
Part II. Two-Dimensional Flows......Page 128
5.1 Definitions and Examples......Page 129
5.2 Classification of Linear Systems......Page 135
5.3 Love Affairs......Page 144
Exercises......Page 146
6.1 Phase Portraits......Page 151
6.2 Existence, Uniqueness, and Topological Consequences......Page 154
6.3 Fixed Points and Linearization......Page 156
6.4 Rabbits versus Sheep......Page 161
6.5 Conservative Systems......Page 165
6.6 Reversible Systems......Page 169
6.7 Pendulum......Page 174
6.8 Index Theory......Page 180
Exercises......Page 187
7.0 Introduction......Page 202
7.1 Examples......Page 203
7.2 Ruling Out Closed Orbits......Page 205
7.3 Poincaré-Bendixson Theorem......Page 209
7.4 Liénard Systems......Page 216
7.5 Relaxation Oscillators......Page 217
7.6 Weakly Nonlinear Oscillators......Page 221
Exercises......Page 233
8.1 Saddle-Node, Transcritical, and Pitchfork Bifurcations......Page 247
8.2 Hopf Bifurcations......Page 254
8.3 Oscillating Chemical Reactions......Page 260
8.4 Global Bifurcations of Cycles......Page 266
8.5 Hysteresis in the Driven Pendulum and Josephson Junction......Page 271
8.6 Coupled Oscillators and Quasiperiodicity......Page 279
8.7 Poincaré Maps......Page 288
Exercises......Page 294
Part III. Chaos......Page 308
9.0 Introduction......Page 309
9.1 A Chaotic Waterwheel......Page 310
9.2 Simple Properties of the Lorenz Equations......Page 319
9.3 Chaos on a Strange Attractor......Page 325
9.4 Lorenz Map......Page 334
9.5 Exploring Parameter Space......Page 338
9.6 Using Chaos to Send Secret Messages......Page 343
Exercises......Page 349
10.0 Introduction......Page 356
10.1 Fixed Points and Cobwebs......Page 357
10.2 Logistic Map; Numerics......Page 361
10.3 Logistic Map: Analysis......Page 365
10.4 Periodic Windows......Page 369
10.5 Liapunov Exponent......Page 374
10.6 Universality and Experiments......Page 377
10.7 Renormalization......Page 387
Exercises......Page 396
11.0 Introduction......Page 406
11.1 Countable and Uncountable Sets......Page 407
11.2 Cantor Set......Page 409
11.3 Dimension of Self-Similar Fractals......Page 412
11.4 Box Dimension......Page 417
11.5 Pointwise and Correlation Dimensions......Page 419
Exercises......Page 424
12.1 The Simplest Examples......Page 430
12.2 Hénon Map......Page 436
12.3 Rössler System......Page 441
12.4 Chemical Chaos and Attractor Reconstruction......Page 444
12.5 Forced Double-Well Oscillator......Page 448
Exercises......Page 455
Answers to Selected Exercises......Page 461
References......Page 471
Author Index......Page 481
Subject Index......Page 484
Plate: 1 Spiral waves of chemical activity in a shallow dish of the Belousov-Zhabotinsky reaction......Page 283
Plate: 2 Divergence of nearby trajectories of the Lorenz attractor......Page 284
Plate: 3 Fractal basin boundaries for the periodicaliy forced double-well oscillator......Page 285
Plate: 4 Maps of the short-term behavior of tiie periodically forced double-well osciilator......Page 286
Back Cover......Page 505