Introduction to Nonlinear Oscillations

The book is devoted to the theory of nonlinear oscillations. Under consideration is a wide range of oscillations, including free oscillations, self-exciting oscillations, driven oscillations and parametric oscillations. Described are the methods to study existence and stability of these oscillations. Systematically represented is the theory of bifurcations for one-dimensional and two-dimensional dynamical systems, which is in the base of these methods. 

At the foundation of the new book are lectures on a general course in the theory of oscillations, which were taught by the author for more than twenty years at the Faculty of Radiophysics of Nizhny Novgorod State University,Russia.

Preface 

1 Introduction to the Theory of Oscillations 

1.1 General Features of the Theory of Oscillations 

1.2 Dynamical Systems 

1.2.1 Types of Trajectories 

1.2.2 Dynamical Systems with Continuous Time 

1.2.3 Dynamical Systems with Discrete Time 

1.2.4 Dissipative Dynamical Systems 

1.3 Attractors 

1.4 Structural Stability of Dynamical Systems 

1.5 Control Questions and Exercises 

2 One—Dimensional Dynamics 

2.1 Qualitative Approach 

2.2 Rough Equilibria 

2.3 Bifurcations of Equilibria 

2.3.1 Saddle—node Bifurcation 

2.3.2 The Concept of the Normal Form 

2.3.3 Transcritical Bifurcation 

2.3.4 Pitchfork Bifurcation 

2.4 Systems on the Circle 

2.5 Control Questions and Exercises 

3 Stability of Equilibria.A Classification of Equilibria of Two—Dimensional Linear Systems 

3.1 Definition of the Stability of Equilibria 

3.2 Classification of Equilibria of Linear Systems on the Plane 

3.2.1 Real Roots 

3.2.1.1 Roots λ1 and λ2 of the Same Sign 

3.2.1.2 The Roots λ1 and λ2 with Different Signs 

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

3.2.2 Complex Roots 

3.2.3 Oscillations of two—dimensionallinear systems 

3.2.4 Two—parameter Bifurcation Diagram 

3.3 Control Questions and Exercises 

4 Analysis of the Stability of Equilibria of Multidimensional Nonlinear Systems 

4.1 Linearization Method 

4.2 The Routh—Hurwitz Stability Criterion 

4.3 The Second Lyapunov Method 

4.4 Hyperbolic Equilibria of Three—Dimensional Systems 

4.4.1 Real Roots 

4.4.1.1 Roots λi of One Sign 

4.4.1.2 Roots λi of Different Signs 

4.4.2 Complex Roots 

4.4.2.1 Real Parts of the Roots λi of One Sign 

4.4.2.2 Real Parts of Roots λi of Different Signs 

4.4.3 The Equilibria of Ihree—Dimensional Nonlinear Systems 

4.4.4 Two—Parameter Bifurcation Diagram 

4.5 Control Questions and Exercises 

5 Linear and Nonlinear Oscillators 

5.1 The Dynamics of a Linear Oscillator 

5.1.1 Harmonic Oscillator 

5.1.2 Linear Oscillator with Losses 

5.1.3 Linear Oscillator with "Negative" Damping 

5.2 Dynamics of a Nonlinear Oscillator 

5.2.1 Conservative Nonlinear Oscillator 

5.2.2 Nonlinear Oscillator with Dissipation 

5.3 Control Questions and Exercises 

6 Basic Properties of Maps 

6.1 Point Maps as Models of Discrete Systems 

6.2 Poincare Map 

6.3 Fixed Points 

6.4 One—PDimensional Linear Maps 

6.5 Two—Dimensional Linear Maps 

6.5.1 Real Multipliers 

6.5.1.1 The Stable Node Fixed Point 

6.5.1.2 The Unstable Node Fixed Point 

6.5.1.3 The Saddle Fixed Point 

6.5.2 Complex MultiDliers 

6.6 One—Dimensional Nonlinear Maps： Some Notions and Examples 

6.7 Control Questions and Exercises 

7 Limit Cycles 

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

7.2 Orbital Stability.Stable and Unstable Limit Cycles 

7.2.1 Definition of Orbital Stability 

7.2.2 Characteristics of Limit Cycles 

7.3 Rotational and Librational Limit Cycles 

7.4 Rough Limit Cycles in Three—Dimensional Space 

7.5 The Bendixson— Dulac Criterion 

7.6 Control Questions and Exercises 

8 Basic Bifurcations of Equilibria in the Plane 

8.1 Bifurcation Conditions 

8.2 Saddle—Node Bifurcation 

8.3 The Andronov—Hopf Bifurcation 

8.3.1 The First Lyapunov Coefficient is Negative 

8.3.2 The First Lyapunov Coefficient is Positive 

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

8.4 Stability Loss Delay for the Dynamic Andronov— Hopf Bifurcation 

8.5 Control Questions and Exercises 

9 Bifurcations of Limit Cycles.Saddle Homoclinic Bifurcation 

9.1 Saddle—node Bifurcation of Limit Cycles 

9.2 Saddle Homoclinic Bifurcation 

9.2.1 Map in the Vicinity of the Homoclinic Trajectory 

9.2.2 Librational and Rotational Homoclinic Trajectories 

9.3 Control Questions and Exercises 

10 The Saddle—Node Homoclinic Bifurcation.Dynamics of Slow—Fast Systems in the Plane 

10.1 Homoclinic Trajectory 

10.2 Final Remarks on Bifurcations of Systems in the Plane 

10.3 Dynamics of a Slow—Fast System 

10.3.1 Slow and Fast Motions 

10.3.2 Systems with a Single Relaxation 

10.3.3 Relaxational Oscillations 

10.4 Control Questions and Exercises 

11 Dynamics of a Superconducting Josephson Junction 

11.1 Stationary and Nonstationary Effects 

11.2 Equivalent Circuit of the Junction 

11.3 Dynamics of the Model 

11.3.1 Conservative Case 

11.3.2 Dissipative Case 

11.3.2.1 Absorbing Domain 

11.3.2.2 Equilibria and Their Local Properties 

11.3.2.3 The Lyapunov Function 

11.3.2.4 Contactless Curves and Control Channels for Separatrices 

11.3.2.5 Homoclinic Orbits and Their Bifurcations 

11.3.2.6 Limit Cycles and the Bifurcation Diagram 

11.3.2.7 I—V Curve of the Junction 

11.4 Control Questions and Exercises 

12 The Van der Pol Method.Self—Sustained Oscillations and Truncated Systems 

12.1 The Notion of Asymptotic Methods 

12.1.1 Reducing the System to the General Form 

12.1.2 Averaged （Truncated） System 

12.1.3 Averaging and Structurally Stable Phase Portraits 

12.2 Self—Sustained Oscillations and Self—Oscillatory Systems 

12.2.1 Dynamics of the Simplest Model of a Pendulum Clock 

12.2.2 Self—Sustained Oscillations in the System with an Active Element 

12.3 Control Questions and Exercises 

13 Forced Oscillations of a Linear Oscillator 

13.1 Dynamics of the System and the Global Poincare Map 

13.2 Resonance Curve 

13.3 Control Questions and Exercises 

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

14.1 Reduction of a System to the Standard Form 

14.2 Resonance in a Nonlinear Oscillator 

14.2.1 Dynamics of the System of Truncated Equations 

14.2.2 Forced Oscillations and Resonance Curves 

14.3 Forced Oscillation Regime 

14.4 Control Questions and Exercises 

15 Forced Synchronization of a Self—Oscillatory System with a Periodic External Force 

15.1 Dynamics of a Truncated System 

15.1.1 Dynamics in the Absence of Detuning 

15.1.2 Dynamics with Detuning 

15.2 The Poincare Map and Synchronous Regime 

15.3 Amplitude— Frequency Characteristic 

15.4 Control Questions and Exercises 

16 Parametric Oscillations 

16.1 The Floquet Theory 

16.1.1 General Solution 

16.1.2 Period Map 

16.1.3 Stability of Zero Solution 

16.2 Basic Regimes of Linear Parametric Systems 

16.2.1 Parametric Oscillations and Parametric Resonance 

16.2.2 Parametric Oscillations of a Pendulum 

16.2.2.1 Pendulum Oscillations in the Conservative Case 

16.2.2.2 Pendulum Oscillations with the Losses Taken into Account 

16.3 Pendulum Dynamics with a Vibrating Suspension Point 

16.4 Oscillations of a Linear Oscillator with Slowly Variable Frequency 

17 Answers to Selected Exercises 

Bibliography 

Index