高維非線性系統的隱藏吸引子（英文）

《高維非線性系統的隱藏吸引子（英文）》是2017年科學出版社出版的圖書，作者是魏周超（Zhouchao Wei）、張偉（Wei Zhang）、姚明輝（Minghui Yao）。

基本介紹

• 中文名：高維非線性系統的隱藏吸引子（英文）
• 別名：Hidden Attractors in High Dimensional Nonlinear Systems
• 作者：魏周超、張偉、姚明輝
• 出版社：科學出版社
• 出版時間：2017年12月01日
• 頁數：198 頁
• 定價：89 元
• 開本：32 開
• 裝幀：平裝
• ISBN：9787030547224 

內容簡介

真實的動力系統幾乎都含有各種各樣的非線性因素，諸如機械系統中的間隙、乾摩擦，結構系統中的材料彈塑性、構件大變形，控制系統中的元器件飽和特性、變結構控制策略等。實踐中，人們經常試圖用線性模型來替代實際的非線性系統，以方便地獲得其動力學行為的某種逼近。然而，被忽略的非線性因素常常會在分析和計算中引起無法接受的誤差，使得線性逼近成為一場徒勞。特別對於系統的長時間歷程動力學問題，有時即使略去很微弱的非線性因素，也會在分析和計算中出現本質性的錯誤。

因此，人們很早就開始關注非線性系統的動力學問題。早期研究可追溯到1673年Huygens對單擺大幅擺動非等時性的觀察。從19世紀末起，Poincare，Lyapunov，Birkhoff，Andronov，Arnold和Smale等數學家和力學家相繼對非線性動力系統的理論進行了奠基性研究，Duffing，vanderPol，Lorenz，Ueda等物理學家和工程師則在實驗和數值模擬中獲得了許多啟示性發現。他們的傑出貢獻相輔相成，形成了分岔、混沌、分形的理論框架，使非線性動力學在20世紀70年代成為一門重要的前沿學科，並促進了非線性科學的形成和發展。

近20年來，非線性動力學在理論和套用兩個方面均取得了很大進展。這促使越來越多的學者基於非線性動力學觀點來思考問題，採用非線性動力學理論和方法，對工程科學、生命科學、社會科學等領域中的非線性系統建立數學模型，預測其長期的動力學行為，揭示內在的規律性，提出改善系統品質的控制策略。一系列成功的實踐使人們認識到：許多過去無法解決的難題源於系統的非線性，而解決難題的關鍵在於對問題所呈現的分岔、混沌、分形、孤立子等複雜非線性動力學現象具有正確的認識和理解。

近年來，非線性動力學理論和方法正從低維向高維乃至無窮維發展。伴隨著計算機代數、數值模擬和圖形技術的進步，非線性動力學所處理的問題規模和難度不斷提高，已逐步接近一些實際系統。在工程科學界，以往研究人員對於非線性問題繞道而行的現象正在發生變化。人們不僅力求深入分析非線性對系統動力學的影響，使系統和產品的動態設計、加工、運行與控制滿足日益提高的運行速度和精度需求，而且開始探索利用分岔、混沌等非線性現象造福人類。

目錄

Contents

Preface

Chapter 1 Basic geometrical point of view of dynamical systems 1

1.1 Self-excited and hidden attractors 1

1.2 Hidden oscillations in Hilbert's 16th problem and applied models 3

1.3 The main contents of this book 6

Reference 9

Chapter 2 Hidden attractors without equilibria 12

2.1 Hidden chaos without equilibria in three-dimensional autonomous system 12

2.1.1 The proposed system 13

2.1.2 Forming mechanism of the new chaotic attractors 17

2.1.3 Conclusion 22

2.2 Hidden hyperchaos without equilibria in four-dimensional autonomous system 23

2.2.1 The hyperchaotic system from generalized di.usionless Lorenz equations 25

2.2.2 Dynamical structure of the proposed hyperchaotic system 29

2.3 Conclusion 35

Reference 35

Chapter 3 Hidden hyperchaotic attractors in a modi-ed Lorenz-Stenflo system 39

3.1 Introduction 39

3.2 The hyperchaotic system from Lorenz-Stenflo system 40

3.2.1 Formulation of the system 40

3.2.2 Hidden hyperchaotic attractors with only one stable equilibrium 42

3.2.3 Non-equivalence with existing hyperchaotic systems 45

3.3 Some basic properties and bifurcation analysis 45

3.3.1 Symmetry and invariance and dissipativity 45

3.3.2 Equilibria and stability 46

3.3.3 Bifurcation analysis 48

3.4 The ultimate bound and positively invariant set 52

3.4.1 Four dimensional hyperelliptic estimate of the ultimate bound and positively invariant set 52

3.4.2 Two dimensional cylindrical estimate of the ultimate bound and positively invariant set 55

3.5 Conclusion 57

Reference 60

Chapter 4 Hidden attractors， multiple limit cycles and boundedness in the generalized 4D Rabinovich system 63

4.1 Introduction 63

4.2 The proposed system and hidden hyperchaos 65

4.2.1 Formulation of the system 65

4.2.2 Hidden hyperchaotic attractors with a unique stable equilibrium 66

4.2.3 Initial conditions and coexisting attractors 69

4.3 Generation of hidden attractors 70

4.4 Local bifurcation in the generalized hyperchaotic Rabinovich system 71

4.4.1 Equilibrium and stability 71

4.4.2 Hopf bifurcation 72

4.5 Boundedness of motion for the hyperchaotic system 76

4.6 Conclusion 79

Reference 80

Chapter 5 On the periodic orbit bifurcating from one single non-hyperbolic equibrium 84

5.1 Introduction 84

5.2 The proposed system and chaotic attractors 85

5.3 The averaging theory for periodic orbits 88

5.4 Statements of the main results 89

5.5 Conclusion 95

Reference 95

Chapter 6 Hidden attractors and dynamical behaviors in an extended Rikitake system 99

6.1 Introduction 99

6.2 Existence of equilibria 100

6.3 Hidden attractors that arise from stable equilibria 102

6.3.1 Coexistence of stable equilibria and hidden attractor 103

6.3.2 Finding hidden attractors by a simple linear transformation 104

6.4 Hopf bifurcation analysis 106

6.5 Dynamics analysis at in-nity 110

6.6 Conclusion 114

Reference 115

Chapter 7 Hidden chaotic regions and complex dynamics in 3D homopolar disc dynamo 118

7.1 Introduction 118

7.2 Description of the self-exciting homopolar disc dynamo and related problems 120

7.3 Study of hidden attractors from a simple linear transformation 125

7.4 Study of hidden attractors from Hopf bifurcation 127

7.4.1 An outline of the Hopf bifurcation methods 127

7.4.2 Hopf bifurcation analysis 129

7.4.3 Hidden attractors and numerical simulations 131

7.4.4 Unstable periodic orbits 131

7.5 Existence of homoclinic orbits 134

7.6 In-nity dynamics by Poincar.e compacti-cation 137

7.7 Conclusion 143

Reference 144

Chapter 8 Hidden hyperchaos and circuit application in 5D homopolar disc dynamo 147

8.1 Introduction 147

8.2 5D hyperchaotic self-exciting homopolar disc dynamo 148

8.3 Hidden attractors and multistability 152

8.3.1 Hidden attractors with two stable equilibria 153

8.3.2 Coexistence of point， periodic， quasi-periodic and hidden chaotic attractors 156

8.4 Electronic circuit implementation of the 5D hyperchaotic system 161

8.5 Conclusion 163

Reference 164

Chapter 9 Bifurcation and circuit realization for delayed system with hidden attractors 168

9.1 Introduction 168

9.2 Hopf bifurcation analysis with multiple delays 170

9.2.1 Stability of equilibrium 171

9.2.2 Existence of Hopf bifurcation with 72

9.2.3 Existence of Hopf bifurcation with 75

9.3 Direction， stability and numerical results of Hopf bifurcation with 77

9.3.1 Direction of Hopf bifurcations and stability of the bifurcating periodic orbits 177

9.3.2 Numerical simulations 186

9.4 Circuit implementation of the multiple time-delay system 190

9.5 Conclusion 192

Reference 193

Index 196