Introduction to Nonlinear OscillationsISBN: 9783527413300
264 pages
June 2015

Description
Following an introduction to fundamental notions and concepts of modern nonlinear dynamics, the text goes on to set out the basics of stability theory, as well as bifurcation theory in one and twodimensional cases. Foundations of asymptotic methods and the theory of relaxation oscillations are presented, with much attention paid to a method of mappings and its applications.
With each chapter including exercises and solutions, including computer problems, this book can be used in courses on oscillation theory for physics and engineering students. It also serves as a good reference for students and scientists in computational neuroscience.
Table of Contents
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 OneDimensional Dynamics 11
2.1 Qualitative Approach 11
2.2 Rough Equilibria 13
2.3 Bifurcations of Equilibria 14
2.3.1 Saddlenode 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 TwoDimensional 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 twodimensional linear systems 30
3.2.4 Twoparameter 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 ofThreeDimensional Systems 41
4.4.1 Real Roots 41
4.4.2 Complex Roots 43
4.4.3 The Equilibria ofThreeDimensional Nonlinear Systems 45
4.4.4 TwoParameter 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 OneDimensional Linear Maps 77
6.5 TwoDimensional Linear Maps 79
6.5.1 Real Multipliers 79
6.5.2 Complex Multipliers 82
6.6 OneDimensional 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 inThreeDimensional 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 SaddleNode 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 Saddlenode 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 SaddleNode 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 SlowFast 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. SelfSustained 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 SelfSustained Oscillations and SelfOscillatory Systems 162
12.2.1 Dynamics of the Simplest Model of a Pendulum Clock 163
12.2.2 SelfSustained 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 SelfOscillatory 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 AmplitudeFrequency 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