非线性振动理论导引 英文PDF|Epub|txt|kindle电子书版本网盘下载

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1 Introduction to the Theory of Oscillations1

1.1 General Features of the Theory of Oscillations1

1.2 Dynamical Systems2

1.2.1 Types of Trajectories3

1.2.2 Dynamical Systems with Continuous Time3

1.2.3 Dynamical Systems with Discrete Time4

1.2.4 Dissipative Dynamical Systems5

1.3 Attractors6

1.4 Structural Stability of Dvnamical Systems7

1.5 Control Questions and Exercises8

2 One-Dimensional Dynamics11

2.1 Qualitative Approach11

2.2 Rough Equilibria13

2.3 Bifurcations of Equilibria14

2.3.1 Saddle-node Bifurcation14

2.3.2 The Concept of the Normal Form15

2.3.3 Transcritical Bifurcation16

2.3.4 Pitchfork Bifurcation17

2.4 Systems on the Circle18

2.5 Control Questions and Exercises19

3 Stability of Equilibria.A Classification of Equilibria of Two-Dimensional Linear Systems21

3.1 Definition of the Stability of Equilibria22

3.2 Classification of Equilibria of Linear Systems on the Plane24

3.2.1 Real Roots25

3.2.1.1 Roots λ1 and λ2 of the Same Sign26

3.2.1.2 The Roots λ1 and λ2 with Different Signs27

3.2.1.3 The Roots λ1 and λ2 are Multiples of λ1=λ2=λ28

3.2.2 Complex Roots29

3.2.3 Oscillations of two-dimensionallinear systems30

3.2.4 Two-parameter Bifurcation Diagram30

3.3 Control Questions and Exercises33

4 Analysis of the Stability of Equilibria of Multidimensional Nonlinear Systems35

4.1 Linearization Method35

4.2 The Routh-Hurwitz Stability Criterion36

4.3 The Second Lyapunov Method38

4.4 Hyperbolic Equilibria of Three-Dimensional Systems41

4.4.1 Real Roots41

4.4.1.1 Roots λi of One Sign41

4.4.1.2 Roots λi of Different Signs42

4.4.2 Complex Roots43

4.4.2.1 Real Parts of the Roots λi of One Sign44

4.4.2.2 Real Parts of Roots λi of Different Signs45

4.4.3 The Equilibria of Three-Dimensional Nonlinear Systems45

4.4.4 Two-Parameter Bifurcation Diagram46

4.5 Control Questions and Exercises49

5 Linear and Nonlinear Oscillators53

5.1 The Dynamics of a Linear Oscillator53

5.1.1 Harmonic Oscillator54

5.1.2 Linear Oscillator with Losses57

5.1.3 Linear Oscillator with"Negative"Damping60

5.2 Dynamics of a Nonlinear Oscillator61

5.2.1 Conservative Nonlinear Oscillator61

5.2.2 Nonlinear Oscillator with Dissipation68

5.3 Control Questions and Exercises69

6 Basic Properties of Maps71

6.1 Point Maps as Models of Discrete Systems71

6.2 Poincaré Map72

6.3 Fixed Points75

6.4 One-Dimensional Linear Maps77

6.5 Two-Dimensional Linear Maps79

6.5.1 Real Multipliers79

6.5.1.1 The Stable Node Fixed Point80

6.5.1.2 The Unstable Node Fixed Point81

6.5.1.3 The Saddle Fixed Point82

6.5.2 Complex Multipliers82

6.6 One-Dimensional Nonlinear Maps：Some Notions and Examples84

6.7 Control Questions and Exercises87

7 Limit Cycles89

7.1 Isolated and Nonisolated Periodic Trajectories.Definition of a Limit Cycle89

7.2 Orbital Stability.Stable and Unstable Limit Cycles91

7.2.1 Definition of Orbital Stability91

7.2.2 Characteristics of Limit Cycles92

7.3 Rotational and Librational Limit Cycles94

7.4 Rough Limit Cycles in Three-Dimensional Space94

7.5 The Bendixson-Dulac Criterion96

7.6 Control Questions and Exercises98

8 Basic Bifurcations of Equilibria in the Plane101

8.1 Bifurcation Conditions101

8.2 Saddle-Node Bifurcation102

8.3 The Andronov-Hopf Bifurcation104

8.3.1 The First Lyapunov Coefficient is Negative105

8.3.2 The First Lyapunov Coefficient is Positive106

8.3.3 "Soft"and"Hard"Generation of Periodic Oscillations107

8.4 Stability Loss Delay for the Dynamic Andronov-Hopf Bifurcation108

8.5 Control Questions and Exercises110

9 Bifurcations of Limit Cycles.Saddle Homoclinic Bifurcation113

9.1 Saddle-node Bifurcation of Limit Cycles113

9.2 Saddle Homoclinic Bifurcation117

9.2.1 Map in the Vicinity of the Homoclinic Trajectory117

9.2.2 Librational and Rotational Homoclinic Trajectories121

9.3 Control Questions and Exercises122

10 The Saddle-Node Homoclinic Bifurcation.Dynamics of Slow-Fast Systems in the Plane123

10.1 Homoclinic Trajectory123

10.2 Final Remarks on Bifurcations of Systems in the Plane126

10.3 Dynamics of a Slow-Fast System127

10.3.1 Slow and Fast Motions128

10.3.2 Systems with a Single Relaxation129

10.3.3 Relaxational Oscillations130

10.4 Control Questions and Exercises133

11 Dynamics of a Superconducting Josephson Junction137

11.1 Stationary and Nonstationary Effects137

11.2 Equivalent Circuit of the Junction139

11.3 Dynamics of the Model140

11.3.1 Conservative Case140

11.3.2 Dissipative Case141

11.3.2.1 Absorbing Domain141

11.3.2.2 Equilibria and Their Local Properties142

11.3.2.3 The Lyapunov Function144

11.3.2.4 Contactless Curves and Control Channels for Separatrices146

11.3.2.5 Homoclinic Orbits and Their Bifurcations150

11.3.2.6 Limit Cycles and the Bifurcation Diagram153

11.3.2.7 I-V Curve ofthe Junction156

11.4 Control Questions and Exercises158

12 The Van der Pol Method.Self-Sustained Oscillations and Truncated Systems159

12.1 The Notion of Asymptotic Methods159

12.1.1 Reducing the System to the General Form160

12.1.2 Averaged(Truncated)System160

12.1.3 Averaging and Structurally Stable Phase Portraits161

12.2 Self-Sustained Oscillations and Self-Oscillatory Systems162

12.2.1 Dynamics of the Simplest Model of a Pendulum Clock163

12.2.2 Self-Sustained Oscillations in the System with an Active Element166

12.3 Control Questions and Exercises173

13 Forced Oscillations of a Linear Oscillator175

13.1 Dynamics of the System and the Global Poincaré Map175

13.2 Resonance Curve180

13.3 Control Questions and Exercises183

14 Forced Oscillations in Weakly Nonlinear Systems with One Degree of Freedom185

14.1 Reduction of a System to the Standard Form185

14.2 Resonance in a Nonlinear Oscillator187

14.2.1 Dynamics of the System of Truncated Equations188

14.2.2 Forced Oscillations and Resonance Curves192

14.3 Forced Oscillation Regime194

14.4 Control Questions and Exercises195

15 Forced Synchronization of a Self-Oscillatory System with a Periodic External Force197

15.1 Dynamics of a Truncated System198

15.1.1 Dynamics in the Absence of Detuning202

15.1.2 Dynamics with Detuning203

15.2 The Poincaré Map and Synchronous Regime205

15.3 Amplitude-Frequency Characteristic207

15.4 Control Questions and Exercises208

16 Parametric Oscillations209

16.1 The Floquet Theory210

16.1.1 General Solution210

16.1.2 Period Map213

16.1.3 Stability of Zero Solution214

16.2 Basic Regimes of Linear Parametric Systems216

16.2.1 Parametric Oscillations and Parametric Resonance217

16.2.2 Parametric Oscillations of a Pendulum220

16.2.2.1 Pendulum Oscillations in the Conservative Case220

16.2.2.2 Pendulum Oscillations with the Losses Taken into Account223

16.3 Pendulum Dynamics with a Vibrating Suspension Point228

16.4 Oscillations of a Linear Oscillator with Slowly Variable Frequency230

17 Answers to Selected Exercises233

Bibliography245

Index247