Introduction to Nonlinear Oscillations

Since 2001 Vladimir Nekorkin is Head of the Laboratory of Dynamics of Nonequilibrium Media at the Novgorod State University. His expertise is in the areas of the dynamics of nonlinear systems, neurodynamics, nonlinear waves, and bifurcation theory.

"The main concepts, ideas, and tools from nonlinear dynamics are introduced and explained in a very clear manner combining qualitative explanations, illustrations, as well as rigorous definitions." (Zentralblatt MATH 2016)"The experience of the author in teaching the subject of the book shows up in the didactical, concise and accessible fashion he conveys the contents...This book will then be a valuable asset as a textbook for introductory courses on nonlinear dynamics, or as a tool for self-study for those who are interested in understanding oscillatory systems for technical or scientific reasons." (Mathematical Reviews/MathSciNet 11/05/2017)

• Preface XI1 Introduction to the Theory of Oscillations 11.1 General Features of the Theory of Oscillations 11.2 Dynamical Systems 21.2.1 Types of Trajectories 31.2.2 Dynamical Systems with Continuous Time 31.2.3 Dynamical Systems with Discrete Time 41.2.4 Dissipative Dynamical Systems 51.3 Attractors 61.4 Structural Stability of Dynamical Systems 71.5 Control Questions and Exercises 82 One-Dimensional Dynamics 112.1 Qualitative Approach 112.2 Rough Equilibria 132.3 Bifurcations of Equilibria 142.3.1 Saddle-node Bifurcation 142.3.2 The Concept of the Normal Form 152.3.3 Transcritical Bifurcation 162.3.4 Pitchfork Bifurcation 172.4 Systems on the Circle 182.5 Control Questions and Exercises 193 Stability of Equilibria. A Classification of Equilibria of Two-Dimensional Linear Systems 213.1 Definition of the Stability of Equilibria 223.2 Classification of Equilibria of Linear Systems on the Plane 243.2.1 Real Roots 253.2.2 Complex Roots 293.2.3 Oscillations of two-dimensional linear systems 303.2.4 Two-parameter Bifurcation Diagram 303.3 Control Questions and Exercises 334 Analysis of the Stability of Equilibria of Multidimensional Nonlinear Systems 354.1 Linearization Method 354.2 The Routh–Hurwitz Stability Criterion 364.3 The Second Lyapunov Method 384.4 Hyperbolic Equilibria ofThree-Dimensional Systems 414.4.1 Real Roots 414.4.2 Complex Roots 434.4.3 The Equilibria ofThree-Dimensional Nonlinear Systems 454.4.4 Two-Parameter Bifurcation Diagram 464.5 Control Questions and Exercises 495 Linear and Nonlinear Oscillators 535.1 The Dynamics of a Linear Oscillator 535.1.1 Harmonic Oscillator 545.1.2 Linear Oscillator with Losses 575.1.3 Linear Oscillator with “Negative” Damping 605.2 Dynamics of a Nonlinear Oscillator 615.2.1 Conservative Nonlinear Oscillator 615.2.2 Nonlinear Oscillator with Dissipation 685.3 Control Questions and Exercises 696 Basic Properties of Maps 716.1 Point Maps as Models of Discrete Systems 716.2 Poincaré Map 726.3 Fixed Points 756.4 One-Dimensional Linear Maps 776.5 Two-Dimensional Linear Maps 796.5.1 Real Multipliers 796.5.2 Complex Multipliers 826.6 One-Dimensional Nonlinear Maps: Some Notions and Examples 846.7 Control Questions and Exercises 877 Limit Cycles 897.1 Isolated and Nonisolated Periodic Trajectories. Definition of a Limit Cycle 897.2 Orbital Stability. Stable and Unstable Limit Cycles 917.2.1 Definition of Orbital Stability 917.2.2 Characteristics of Limit Cycles 927.3 Rotational and Librational Limit Cycles 947.4 Rough Limit Cycles inThree-Dimensional Space 947.5 The Bendixson–Dulac Criterion 967.6 Control Questions and Exercises 988 Basic Bifurcations of Equilibria in the Plane 1018.1 Bifurcation Conditions 1018.2 Saddle-Node Bifurcation 1028.3 The Andronov–Hopf Bifurcation 1048.3.1 The First Lyapunov Coefficient is Negative 1058.3.2 The First Lyapunov Coefficient is Positive 1068.3.3 “Soft” and “Hard” Generation of Periodic Oscillations 1078.4 Stability Loss Delay for the Dynamic Andronov–Hopf Bifurcation 1088.5 Control Questions and Exercises 1109 Bifurcations of Limit Cycles. Saddle Homoclinic Bifurcation 1139.1 Saddle-node Bifurcation of Limit Cycles 1139.2 Saddle Homoclinic Bifurcation 1179.2.1 Map in the Vicinity of the Homoclinic Trajectory 1179.2.2 Librational and Rotational Homoclinic Trajectories 1219.3 Control Questions and Exercises 12210 The Saddle-Node Homoclinic Bifurcation. Dynamics of Slow–Fast Systems in the Plane 12310.1 Homoclinic Trajectory 12310.2 Final Remarks on Bifurcations of Systems in the Plane 12610.3 Dynamics of a Slow-Fast System 12710.3.1 Slow and Fast Motions 12810.3.2 Systems with a Single Relaxation 12910.3.3 Relaxational Oscillations 13010.4 Control Questions and Exercises 13311 Dynamics of a Superconducting Josephson Junction 13711.1 Stationary and Nonstationary Effects 13711.2 Equivalent Circuit of the Junction 13911.3 Dynamics of the Model 14011.3.1 Conservative Case 14011.3.2 Dissipative Case 14111.4 Control Questions and Exercises 15812 The Van der PolMethod. Self-Sustained Oscillations and Truncated Systems 15912.1 The Notion of AsymptoticMethods 15912.1.1 Reducing the System to the General Form 16012.1.2 Averaged (Truncated) System 16012.1.3 Averaging and Structurally Stable Phase Portraits 16112.2 Self-Sustained Oscillations and Self-Oscillatory Systems 16212.2.1 Dynamics of the Simplest Model of a Pendulum Clock 16312.2.2 Self-Sustained Oscillations in the System with an Active Element 16612.3 Control Questions and Exercises 17313 Forced Oscillations of a Linear Oscillator 17513.1 Dynamics of the System and the Global Poincaré Map 17513.2 Resonance Curve 18013.3 Control Questions and Exercises 18314 Forced Oscillations in Weakly Nonlinear Systems with One Degree of Freedom 18514.1 Reduction of a System to the Standard Form 18514.2 Resonance in a Nonlinear Oscillator 18714.2.1 Dynamics of the System of Truncated Equations 18814.2.2 Forced Oscillations and Resonance Curves 19214.3 Forced Oscillation Regime 19414.4 Control Questions and Exercises 19515 Forced Synchronization of a Self-Oscillatory System with a Periodic External Force 19715.1 Dynamics of a Truncated System 19815.1.1 Dynamics in the Absence of Detuning 20215.1.2 Dynamics with Detuning 20315.2 The Poincaré Map and Synchronous Regime 20515.3 Amplitude-Frequency Characteristic 20715.4 Control Questions and Exercises 20816 Parametric Oscillations 20916.1 The Floquet Theory 21016.1.1 General Solution 21016.1.2 Period Map 21316.1.3 Stability of Zero Solution 21416.2 Basic Regimes of Linear Parametric Systems 21616.2.1 Parametric Oscillations and Parametric Resonance 21716.2.2 Parametric Oscillations of a Pendulum 22016.3 Pendulum Dynamics with a Vibrating Suspension Point 22816.4 Oscillations of a Linear Oscillator with Slowly Variable Frequency 23017 Answers to Selected Exercises 233Bibliography 245Index 247