內容大鋼
本書的重點是基於向量場和一元二次函數的非線性動力學。本書從不同視角研究非線性動力學和二次動力系統的分岔。二維動力系統是非線性動力學中最簡單的動力系統之一，但二維二次系統中平衡點和流的局部與全局結構有助於我們理解其他非線性動力系統，這也是解決希爾伯特第十六問題的關鍵一步。本書詳細探論了二維二次系統可能存在的奇異動力學問題；介紹了二維繫統中平衡態和一維流的動力學；討論了鞍形匯和鞍形源分岔，給出了鞍形中心分岔；提出了無限平衡態是非線性系統的開關分岔；從第一類積分流形出發，發展了鞍焦點網路，並給出了鞍、源和匯網路。
本書可作為動力系統和控制專業的參考書，適合數學、機械和電氣工程領域的研究人員、學生和工程師閱讀參考。
目錄
1 Two-Dimensional Linear Dynamical Systems
1.1 Constant Vector Fields
1.2 Linear Vector Fields with a Single Variable
1.3 Variable-Independent Linear Vector Fields
1.4 Variable-Crossing Linear Vector Fields
1.5 Two Linear-Bivariate Vector Fields
Reference
2 Single-Variable Quadratic Systems with a Self-Univariate Quadratic Vector Field
2.1 Constant and Self-Univariate Quadratic Vector Fields
2.1.1 Self-Univariate Quadratic Systems with a Constant
2.1.2 Singular Flows and Bifurcations
2.2 Linear and Self-Univariate Quadratic Vector Fields
2.2.1 Linear and Self-Univariate Quadratic Systems
2.2.2 Flow Switching and Appearing Bifurcations
2.3 Single-Variable Quadratic Systems with a Self-Uni-variate Vector Field
2.3.1 Variable-Crossing and Self-Univariate Quadratic
2.4 Singular Dynamics and Bifurcations
Reference
3 Single-Variable Quadratic Systems with a Non-Self-Univariate Quadratic Vector Field
3.1 Constant and Non-Self-Univariate Quadratic Vector Fields
3.1.1 Non-Self-Univariate Quadratic Systems with a Constant Vector Field
3.1.2 Singular Flows and Bifurcations
3.2 Linear and Non-Self-Univariate Quadratic Vector Fields
3.2.1 Linear and Non-Self-Univariate Quadratic Systems
3.2.2 Flow Switching and Appearing Bifurcations
3.3 With a Non-Self-Univariate Quadratic Vector Field
3.3.1 Quadratic Systems with a Non-Self-Univariate Vector Field
3.3.2 Singular Dynamics and Bifurcations
Reference
4 Variable-Independent Quadratic Dynamics
4.1 Constant and Variable-Independent Quadratic Vector Fields
4.2 Variable-Independent, Linear and Quadratic Vector Fields
4.2.1 Variable-Independent, Linear and Quadratic Systems
4.2.2 Saddle-Node Bifurcations and Global Dynamics
4.3 Two Variable-Independent Univariate Quadratic Vector Fields
4.3.1 Two Variable-Independent Quadratic Global Dynamics
4.3.2 Singularity and Bifurcations
Reference
5 Variable-Crossing Univariate Quadratic Systems
5.1 Constant and Variable-Crossing Univariate Vector Fields
5.2 Linear and Quadratic Variable-Crossing Vector Fields
5.2.1 Linear and Quadratic Variable-Crossing Systems
5.2.2 Bifurcations and Limit Cycles
5.3 Two Variable-Crossing Univariate Quadratic Vector Fields
5.3.1 Two Variable-Crossing Univariate Quadratic Systems
5.3.2 Bifurcations and Global Dynamics
Reference
6 Two-Univariate Product Quadratic Systems
6.1 Two-Univariate Product Quadratic Dynamics
6.2 Dynamics for Two-Univariate-Product Quadratic Systems
6.2.1 With a Constant Vector Field
6.2.2 With an Independent-Variable Linear Vector Field
6.2.3 With a Variable-Crossing Linear Vector Field
6.2.4 Two-Univariate Product Quadratic Vector Fields
6.2.5 Switching Bifurcations
Reference
7 Product-Bivariate Quadratic Systems with a Self-Univariate Quadratic Vector Field
7.1 Product-Bivariate and Self-Univariate Quadratic Dynamics
7.2 Singularity, Bifurcations and Global Dynamics
7.2.1 Saddle-Sink and Saddle-Source Bifurcations
7.2.2 Up-Down and Down-Up Upper-Saddles and Lower-Saddles
7.2.3 Simple Equilibriums with Hyperbolic Flows
7.2.4 Infinite-Equilibriums and Switching Bifurcations
Reference
8 Product-Bivariate Quadratic Systems with a Non-Self-Univariate Quadratic Vector Field
8.1 Product-Bivariate and Non-Self-Univariate Dynamics
8.2 Singularity, Bifurcations and Global Dynamics
8.2.1 Saddle-Center Appearing Bifurcations
8.2.2 Saddle-Saddle and Center-Center Bifurcations
8.2.3 Saddle and Center Flows with Hyperbolic Flows
8.2.4 Switching Bifurcations
Reference
Index
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