非线性振动理论导引

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

《非线性振动理论导引（英文版）》基于多种方法和方式，系统地介绍了振动理论的数学基础。其目的是通过解释这些最本质的振动理沦思想和方法，传授给读者解决实际和复杂问题的能。此外，《非线性振动理论导引（英文版）》还介绍了现代非线性动力学的基本概念、理论及表达方式，从而领引读者深入讨论一维和二维情况下的稳定性理论和分岔理论。并日.介绍张弛振荡理论和近似方法，并更侧重映射方法讨论及其应用。
　　《非线性振动理论导引（英文版）》配备练习和解答，可作为物理和工程等业学习振动理论教材，也可供计算神经科学相关专业的学生或科学家参考。.
　　VladimirⅠ.Nekorkin是俄罗斯Nizhny Novgorod州立大学的数学和物理教授。现任俄岁斯科学院应用物理所非线性动力学系主任。研究领域为：非线性系统动力学、神经动力学、非线性波和分岔理论。