The Method of Normal Forms, Second Edition

Contents......Page 8
Preface......Page 12
Introduction......Page 14
1.1 Introduction......Page 20
1.2 Duf.ng Equation......Page 22
1.3 Rayleigh Equation......Page 26
1.4 Duf.ng–Rayleigh–van der Pol Equation......Page 28
1.5 An Oscillator with Quadratic and Cubic Nonlinearities......Page 30
1.6 A General System with Quadratic and Cubic Nonlinearities......Page 35
1.7 The van der Pol Oscillator......Page 37
1.8 Exercises 1.8.1......Page 40
2.1 Introduction......Page 44
2.2 A Two-Dimensional System with Diagonal Linear Part......Page 47
2.3 A Two-Dimensional System with a Nonsemisimple Linear Form......Page 52
2.4 An......Page 53
2.5 A Two-Dimensional System with Purely Imaginary Eigenvalues......Page 55
2.6 A Two-Dimensional System with Zero Eigenvalues......Page 61
2.7 A Three-Dimensional System with Zero and Two Purely Imaginary Eigenvalues......Page 65
2.8 The Mathieu Equation......Page 67
2.9 Exercises 2.9.1......Page 70
3.1 Linear Maps......Page 74
3.2 Nonlinear Maps......Page 79
3.3 Center-Manifold Reduction......Page 85
3.4 Local Bifurcations......Page 89
3.5 Exercises 3.5.1......Page 104
4.1 Linear Systems......Page 110
4.2 Fixed Points of Nonlinear Systems......Page 113
4.3 Center-Manifold Reduction......Page 116
4.4 Local Bifurcations of Fixed Points......Page 120
4.5 Normal Forms of Static Bifurcations......Page 130
0......Page 132
4.6 Normal Form of Hopf Bifurcation......Page 150
4.7 Exercises 4.7.1......Page 159
5.1 Primary Resonance......Page 174
5.2 Subharmonic Resonance of Order One-Third......Page 177
5.3 Superharmonic Resonance of Order Three......Page 180
5.4 An Alternate Approach......Page 182
5.5 Exercises 5.5.1......Page 185
5.5.2......Page 186
6.1 Introduction......Page 188
6.2 Primary Resonance......Page 189
6.3 Subharmonic Resonance of Order One-Half......Page 191
6.4 Superharmonic Resonance of Order Two......Page 193
6.5 Subharmonic Resonance of Order One-Third......Page 195
7.1 The Mathieu Equation......Page 200
7.2 Multiple-Degree-of-Freedom Systems......Page 204
7.3 Linear Systems Having Repeated Frequencies......Page 208
7.4 Gyroscopic Systems......Page 218
7.5 A Nonlinear Single-Degree-of-Freedom System......Page 221
7.6 Exercises 7.6.1......Page 225
2......Page 226
7.6.2......Page 227
7.6.3......Page 228
7.6.4......Page 229
8.1 Nongyroscopic Systems......Page 230
2......Page 234
8.2 Gyroscopic Systems......Page 238
1/2......Page 241
8.3 Two Linearly Coupled Oscillators......Page 242
8.4.2......Page 245
8.4.9......Page 246
8.4.11......Page 247
9.1 Nongyroscopic Systems......Page 248
9.2 Gyroscopic Systems......Page 262
10.1 Introduction......Page 270
10.2 The Case of No Internal Resonance......Page 275
10.3 The Case of Three-to-One Internal Resonance......Page 276
10.4 The Case of One-to-One Internal Resonance......Page 277
10.5 The Case of Two-to-One Internal Resonance......Page 279
10.6 Method of Multiple Scales......Page 280
10.7 Generalized Method of Averaging......Page 289
10.8 A Nonsemisimple One-to-One Internal Resonance......Page 292
10.9.2......Page 298
11.1 AScalarEquation......Page 300
11.2 A Single-Degree-of-Freedom System......Page 308
11.3 A Three-Dimensional System......Page 317
11.4 Crane Control with Time-Delayed Feedback......Page 324
11.5 Exercises 11.5.1......Page 326
References......Page 328
Further Reading......Page 332
Index......Page 338