Introduction to Nonlinear Oscillations

A systematic outline of the basic theory of oscillations, combining several tools in a single textbook. The author explains fundamental ideas and methods, while equally aiming to teach students the techniques of solving specific (practical) or more complex problems. 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 two-dimensional 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 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