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Index
Cover
Related Titles
Title Page
Copyright
Contents
Preface
1 Introduction to the Theory of Oscillations
1.1 General Features of the Theory of Oscillations
1.2 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
2.4 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
3.3 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
4.5 Control Questions and Exercises
5 Linear and Nonlinear Oscillators
5.1 The Dynamics of a Linear Oscillator
5.2 Dynamics of a Nonlinear Oscillator
5.3 Control Questions and Exercises
6 Basic Properties of Maps
6.1 Point Maps as Models of Discrete Systems
6.2 Poincaré Map
6.3 Fixed Points
6.4 One-Dimensional Linear Maps
6.5 Two-Dimensional Linear Maps
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
7.3 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
8.4 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
9.3 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
10.4 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
11.4 Control Questions and Exercises
12 The Van der Pol Method. Self-Sustained Oscillations and Truncated Systems
12.1 The Notion of Asymptotic Methods
12.2 Self-Sustained Oscillations and Self-Oscillatory Systems
12.3 Control Questions and Exercises
13 Forced Oscillations of a Linear Oscillator
13.1 Dynamics of the System and the Global Poincaré 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
14.3 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
15.2 The Poincaré Map and Synchronous Regime
15.3 Amplitude-Frequency Characteristic
15.4 Control Questions and Exercises
16 Parametric Oscillations
16.1 The Floquet Theory
16.2 Basic Regimes of Linear Parametric Systems
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
End User License Agreement
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