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Introduction to Nonlinear Oscillations (非线性振动理论导引) (英文版)

作者：
Vladimir I. Nekorkin
定价：
69.00 元
版面字数：
230千字
开本：
16开
装帧形式：
精装
页数：
253
最新
印次时间：
2015年1月
ISBN：
978-7-04-042131-6
物料号：
42131-00
出版时间：
2015年4月
读者对象：
学术著作
一级分类：
自然科学
二级分类：
物理
三级分类：
物理学交叉学科

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.

  • 目录
    • 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