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 Dvnamical 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
3.2.1.1 Roots λ1 and λ2 of the Same Sign 26
3.2.1.2 The Roots λ1 and λ2 with Different Signs 27
3.2.1.3 The Roots λ1 and λ2 are Multiples of λ1=λ2=λ 28
3.2.2 Complex Roots 29
3.2.3 Oscillations of two-dimensionallinear 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 of Three-Dimensional Systems 41
4.4.1 Real Roots 41
4.4.1.1 Roots λi of One Sign 41
4.4.1.2 Roots λi of Different Signs 42
4.4.2 Complex Roots 43
4.4.2.1 Real Parts of the Roots λi of One Sign 44
4.4.2.2 Real Parts of Roots λi of Different Signs 45
4.4.3 The Equilibria of Three-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
6.5.1.1 The Stable Node Fixed Point 80
6.5.1.2 The Unstable Node Fixed Point 81
6.5.1.3 The Saddle Fixed Point 82
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 in Three-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
11.3.2.1 Absorbing Domain 141
11.3.2.2 Equilibria and Their Local Properties 142
11.3.2.3 The Lyapunov Function 144
11.3.2.4 Contactless Curves and Control Channels for Separatrices 146
11.3.2.5 Homoclinic Orbits and Their Bifurcations 150
11.3.2.6 Limit Cycles and the Bifurcation Diagram 153
11.3.2.7 I-V Curve ofthe Junction 156
11.4 Control Questions and Exercises 158
12 The Van der Pol Method.Self-Sustained Oscillations and Truncated Systems 159
12.1 The Notion of Asymptotic Methods 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
16.2.2.1 Pendulum Oscillations in the Conservative Case 220
16.2.2.2 Pendulum Oscillations with the Losses Taken into Account 223
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