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
   3.2.2 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
   4.4.2 Complex Roots
   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
   6.5.2 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
  11.4 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
  16.3 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