Introduction to Nonlinear Oscillations

Preface XI

1 Introduction to the Theory of Oscillations 1

1.1 General Features of the Theory of Oscillations 1

1.2 Dynamical Systems 2

1.2.1 Types of Trajectories 3

1.2.2 Dynamical Systems with Continuous Time 3

1.2.3 Dynamical Systems with Discrete Time 4

1.2.4 Dissipative Dynamical Systems 5

1.3 Attractors 6

1.4 Structural Stability of Dynamical Systems 7

1.5 Control Questions and Exercises 8

2 One-Dimensional Dynamics 11

2.1 Qualitative Approach 11

2.2 Rough Equilibria 13

2.3 Bifurcations of Equilibria 14

2.3.1 Saddle-node Bifurcation 14

2.3.2 The Concept of the Normal Form 15

2.3.3 Transcritical Bifurcation 16

2.3.4 Pitchfork Bifurcation 17

2.4 Systems on the Circle 18

2.5 Control Questions and Exercises 19

3 Stability of Equilibria. A Classification of Equilibria of Two-Dimensional Linear Systems 21

3.1 Definition of the Stability of Equilibria 22

3.2 Classification of Equilibria of Linear Systems on the Plane 24

3.2.1 Real Roots 25

3.2.2 Complex Roots 29

3.2.3 Oscillations of two-dimensional linear systems 30

3.2.4 Two-parameter Bifurcation Diagram 30

3.3 Control Questions and Exercises 33

4 Analysis of the Stability of Equilibria of Multidimensional Nonlinear Systems 35

4.1 Linearization Method 35

4.2 The Routh–Hurwitz Stability Criterion 36

4.3 The Second Lyapunov Method 38

4.4 Hyperbolic Equilibria ofThree-Dimensional Systems 41

4.4.1 Real Roots 41

4.4.2 Complex Roots 43

4.4.3 The Equilibria ofThree-Dimensional Nonlinear Systems 45

4.4.4 Two-Parameter Bifurcation Diagram 46

4.5 Control Questions and Exercises 49

5 Linear and Nonlinear Oscillators 53

5.1 The Dynamics of a Linear Oscillator 53

5.1.1 Harmonic Oscillator 54

5.1.2 Linear Oscillator with Losses 57

5.1.3 Linear Oscillator with “Negative” Damping 60

5.2 Dynamics of a Nonlinear Oscillator 61

5.2.1 Conservative Nonlinear Oscillator 61

5.2.2 Nonlinear Oscillator with Dissipation 68

5.3 Control Questions and Exercises 69

6 Basic Properties of Maps 71

6.1 Point Maps as Models of Discrete Systems 71

6.2 Poincaré Map 72

6.3 Fixed Points 75

6.4 One-Dimensional Linear Maps 77

6.5 Two-Dimensional Linear Maps 79

6.5.1 Real Multipliers 79

6.5.2 Complex Multipliers 82

6.6 One-Dimensional Nonlinear Maps: Some Notions and Examples 84

6.7 Control Questions and Exercises 87

7 Limit Cycles 89

7.1 Isolated and Nonisolated Periodic Trajectories. Definition of a Limit Cycle 89

7.2 Orbital Stability. Stable and Unstable Limit Cycles 91

7.2.1 Definition of Orbital Stability 91

7.2.2 Characteristics of Limit Cycles 92

7.3 Rotational and Librational Limit Cycles 94

7.4 Rough Limit Cycles inThree-Dimensional Space 94

7.5 The Bendixson–Dulac Criterion 96

7.6 Control Questions and Exercises 98

8 Basic Bifurcations of Equilibria in the Plane 101

8.1 Bifurcation Conditions 101

8.2 Saddle-Node Bifurcation 102

8.3 The Andronov–Hopf Bifurcation 104

8.3.1 The First Lyapunov Coefficient is Negative 105

8.3.2 The First Lyapunov Coefficient is Positive 106

8.3.3 “Soft” and “Hard” Generation of Periodic Oscillations 107

8.4 Stability Loss Delay for the Dynamic Andronov–Hopf Bifurcation 108

8.5 Control Questions and Exercises 110

9 Bifurcations of Limit Cycles. Saddle Homoclinic Bifurcation 113

9.1 Saddle-node Bifurcation of Limit Cycles 113

9.2 Saddle Homoclinic Bifurcation 117

9.2.1 Map in the Vicinity of the Homoclinic Trajectory 117

9.2.2 Librational and Rotational Homoclinic Trajectories 121

9.3 Control Questions and Exercises 122

10 The Saddle-Node Homoclinic Bifurcation. Dynamics of Slow–Fast Systems in the Plane 123

10.1 Homoclinic Trajectory 123

10.2 Final Remarks on Bifurcations of Systems in the Plane 126

10.3 Dynamics of a Slow-Fast System 127

10.3.1 Slow and Fast Motions 128

10.3.2 Systems with a Single Relaxation 129

10.3.3 Relaxational Oscillations 130

10.4 Control Questions and Exercises 133

11 Dynamics of a Superconducting Josephson Junction 137

11.1 Stationary and Nonstationary Effects 137

11.2 Equivalent Circuit of the Junction 139

11.3 Dynamics of the Model 140

11.3.1 Conservative Case 140

11.3.2 Dissipative Case 141

11.4 Control Questions and Exercises 158

12 The Van der PolMethod. Self-Sustained Oscillations and Truncated Systems 159

12.1 The Notion of AsymptoticMethods 159

12.1.1 Reducing the System to the General Form 160

12.1.2 Averaged (Truncated) System 160

12.1.3 Averaging and Structurally Stable Phase Portraits 161

12.2 Self-Sustained Oscillations and Self-Oscillatory Systems 162

12.2.1 Dynamics of the Simplest Model of a Pendulum Clock 163

12.2.2 Self-Sustained Oscillations in the System with an Active Element 166

12.3 Control Questions and Exercises 173

13 Forced Oscillations of a Linear Oscillator 175

13.1 Dynamics of the System and the Global Poincaré Map 175

13.2 Resonance Curve 180

13.3 Control Questions and Exercises 183

14 Forced Oscillations in Weakly Nonlinear Systems with One Degree of Freedom 185

14.1 Reduction of a System to the Standard Form 185

14.2 Resonance in a Nonlinear Oscillator 187

14.2.1 Dynamics of the System of Truncated Equations 188

14.2.2 Forced Oscillations and Resonance Curves 192

14.3 Forced Oscillation Regime 194

14.4 Control Questions and Exercises 195

15 Forced Synchronization of a Self-Oscillatory System with a Periodic External Force 197

15.1 Dynamics of a Truncated System 198

15.1.1 Dynamics in the Absence of Detuning 202

15.1.2 Dynamics with Detuning 203

15.2 The Poincaré Map and Synchronous Regime 205

15.3 Amplitude-Frequency Characteristic 207

15.4 Control Questions and Exercises 208

16 Parametric Oscillations 209

16.1 The Floquet Theory 210

16.1.1 General Solution 210

16.1.2 Period Map 213

16.1.3 Stability of Zero Solution 214

16.2 Basic Regimes of Linear Parametric Systems 216

16.2.1 Parametric Oscillations and Parametric Resonance 217

16.2.2 Parametric Oscillations of a Pendulum 220

16.3 Pendulum Dynamics with a Vibrating Suspension Point 228

16.4 Oscillations of a Linear Oscillator with Slowly Variable Frequency 230

17 Answers to Selected Exercises 233

Bibliography 245

Index 247

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.