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
8 Basic Bifurcations of Equilibria in the Plane
9 Bifurcations of Limit Cycles.Saddle Homoclinic Bifurcation
10 The Saddle-Node Homoclinic Bifurcation.Dynamics of Slow-Fast Systems in the Plane
11 Dynamics of a Superconducting Josephson Junction
12 The Van der Pol Method.Self-Sustained Oscillations and Truncated Systems
13 Forced Oscillations of a Linear Oscillator
14 Forced Oscillations in Weakly Nonlinear Systems with One Degree of Freedom
15 Forced Synchronization of a Self-Oscillatory System with a Periodic External Force
16 Parametric Oscillations
17 Answers to Selected Exercises
Bibliography
Index
