《微分方程動態系統和混沌導論》是2007年2月1日世界圖書出版公司出版的圖書，作者是M. W. Hirsch, S. Smale, R. L. Devaney。
基本介紹
- 作者：M. W. Hirsch, S. Smale, R. L. Devaney
- ISBN：9787506282819
- 頁數：417 頁
- 定價：55.00元
- 出版社：世界圖書出版公司
- 出版時間：2007年2月1日
- 裝幀：平裝
- 副標題：微分方程動態系統和混沌導論
- 叢書： 經典英文數學教材系列
內容介紹,作者介紹,作品目錄,
內容介紹
30年來，動力系統的數學理論與套用有了很大發展。30多年前還沒有高速的台式計算機和計算機圖像，“混沌”一詞也沒有在數學界使用，而對於微分方程與動力系統的研究興趣主要僅限於數學界中比較小的範圍。到今天，處處有計算機，求微分方程近似解的軟體包已得到廣泛運用，使人們從圖形中就能看到結果。對於非線性微分方程的分析已為廣大學者所接受，一些複雜的動力學行為，如馬蹄映射、同宿軌、Lorenz系統中揭示出來的複雜現象，以及數學方面的分析，使學者們確信簡單的穩定運動，如平衡態和周期解己不總是微分方程解的最重要的行為，而混沌現象揭示出來的美妙性態正促使各個領域的科學家與工程師細心關注在他們自己領域中提出的重要的微分方程及其混沌特性。動力系統現象在今天已出現在幾乎每個科學領域中，從化學中的振盪Belousov-Zhabotinsky反應到電子工程中的混沌Chua電路，從天體力學中的複雜運動到生態系統中的分岔。
本書是30年前世界著名的動力系統專家赫希（M. Hirsch）和斯梅爾（S. Smale）合著的“Differential Equations, Dynamical Systems and Linear Algebra”一書的修訂本，原書初版後被許多高校作為動力系統入門的標準教材，多年來在國際上產生較大影響。這次修訂本新增加一名作者，即著名的混沌理論專家德瓦尼（R. Devaney）。
30年來，動力系統的數學理論與套用有了很大發展，動力系統現象已出現在幾乎每個科學領域中，從化學中的振盪Belousov-Zhabotinsky反應到電子工程中的混沌Chua電路，從天體力學中的複雜運動到生態系統中的分岔。
這樣，作為一部微分方程與動力系統的教材來，有著比上世紀70年代更加廣泛多樣的讀者群體。因而與初版相比較，本書做了以下幾方面較大的改動：
1．線性代數內容做了壓縮。去掉了在抽象的線性空間及賦范空間的相應推廣。不再包含關於n階矩陣約化為標準型的複雜證明，改為論述不高於4階的矩陣。
2．詳細討論了Lorenz吸引子、Shil’nikov系統和雙卷吸引子的混沌特性。
3．新增許多套用實例，原有實例也做了更新。
4．新增若於章節討論離散動力系統。
5．主要討論光滑系統，因而簡化了許多定理的假設。
本書由三編組成：第一編論述微分方程的線性系統及一階非線性方程；第二編是本書的核心，集中討論以二維為主的非線性系統及其在各個方面的套用；第三編處理高維系統，特彆強調在平面系統中不會出現的混沌特性，以及通過離散化系統來研究這類性質的基本方法。
本書作為微分方程和動力系統方面的教材有著較廣泛的適用性，其讀者對象不僅是理工科大學數學、套用數學及相關專業的師生，而且也適用於相關領域的科技工作者。
作者介紹
Smale是當代大師級的數學家，Hirsch也在頂級數學家之列。
作品目錄
CHAPTER 1 First-Order Equations
1.1 The Simplest Example
1.2 The Logistic Population Model
1.3 Constant Harvesting and Bifurcations
1.4 Periodic Harvesting and Periodic Solutions
1.5 Computing the Poincard Map
1.6 Exploration：A Two-Parameter Family
CHAPTER 2 Planar Linear Systems
2.1 Second-Order Differential Equations
2.2 Planar Systems
2.3 Preliminaries from Algebra
2.4 Planar Linear Systems
2.5 Eigenvalues and Eigenvectors
2.6 Solving Linear Systems
2.7 The Linearity Principle
CHAPTER 3 Phase Portraits for Planar Systems
3.1 Real Distinct Eigenvalues
3.2 Complex Eigenvalues
3.3 Repeated Eigenvalues
3.4 Changing Coordinates
CHAPTER 4 Classification of Planar Systems
4.1 The Trace-Determinant Plane
4.2 Dynamical Classification
4.3 Exploration：A 3D Parameter Space
CHAPTER 5 Higher Dimensional Linear Algebra
5.1 Preliminaries from Linear Algebra
5.2 Eigenvalues and Eigenvectors
5.3 Complex Eigenvalues
5.4 Bases and Subspaces
5.5 Repeated Eigenvalues
5.6 Genericity
CHAPTER 6 Higher Dimensional Linear Systems
6.1 Distinct Eigenvalues
6.2 Harmonic Oscillators
6.3 Repeated Eigenvalues
6.4 The Exponential of a Matrix
6.5 Nonautonomous Linear Systems
CHAPTER 7 Nonlinear Systems
7.1 Dynamical Systems
7.2 The Existence and Uniqueness Theorem
7.3 Continuous Dependence of Solutions
7.4 The Variational Equation
7.5 Exploration：Numerical Methods
CHAPTER 8 Equilibria in Nonlinear Systems
8.1 Some Nustrative Examples
8.2 Nonlinear Sinks and Sources
8.3 Saddles
8.4 Stability
8.5 Bifurcations
8.6 Exploration：Complex Vector Fields
CHAPTER 9 Global Nonlinear Techniques
9.1 Nullclines
9.2 Stability of Equilibria
9.3 Gradient Systems
9.4 Hamiltonian Systems
9.5 Exploration：The Pendulum with Constant Forcing
CHAPTER 10 Closed Orbits and Limit Sets
10.1 Limit Sets
10.2 Local Sections and Flow Boxes
10.3 The Poincare Map
10.4 Monotone Sequences in Planar Dynamical Systems
10.5 The Poincare-Bendixson Theorem
10.6 Applications of Poincare-Bendixson
10.7 Expl0ration：Chemical Reactions That Oscillate
CHAPTER 11 Applications in Biology
11.1 Infectious Diseases
11.2 Predator/Prey Systems
11.3 Competitive Species
11.4 Exploration：Competition and Harvesting
CHAPTER 12 Applications in Circuit Theory
12.1 An RLC Circuit
12.2 The Lienard Equation
12.3 The van der Pol Equation
12.4 A Hopf Bifurcation
12.5 Exploration：Neurodynamics
CHAPTER 13 Applications in Mechanics
13.1 Newton’S Second Law
13.2 Conservative Systems
13.3 Central Force Fields
13.4 The Newtonian Central Force System
13.5 Kepler’s First Law
13.6 The Two-Body Problem
13.7 Blowing Up the Singularity
13.8 Exploration：Other Central Force Problems
13.9 Exploration：Classical Limits of Quantum Mechanical Systems
CHAPTER 14 The Lorenz System
14.1 Introduction to the Lorenz System
14.2 Elementary Properties of the Lorenz System
14.3 The Lorenz Attractor
14.4 A Model for the Lorenz Attractor
14.5 The Chaotic Attractor
14.6 Exploration：The Rossler Attractor
CHAPTER 15 Discrete Dynamical Systems
15.1 Introduction to Discrete Dynamical Systems
15.2 Bifurcations
15.3 The Discrete Logistic Model
15.4 Chaos
15.5 Symbolic Dynamics
15.6 The Shift Map
15.7 The Cantor Middle-Thirds Set
15.8 Exploration：Cubic Chaos
15.9 Exploration：The Orbit Diagram
CHAPTER 16 Homoclinic Phenomena
16.1 The Shil’nikov System
16.2 The Horseshoe Map
16.3 The Double Scroll Attractor
16.4 Homoclinic Bifurcations
16.5 Exploration：The Chua Circuit
CHAPTER 17 Existence and Uniqueness Revisited
17.1 The Existence and Uniqueness Theorem
17.2 Proof of Existence and Uniqueness
17.3 Continuous Dependence on Initial Conditions
17.4 Extending Solutions
17.5 Nonautonomous Systems
17.6 Differentiability of the Flow
Bibliography
Index
1.1 The Simplest Example
1.2 The Logistic Population Model
1.3 Constant Harvesting and Bifurcations
1.4 Periodic Harvesting and Periodic Solutions
1.5 Computing the Poincard Map
1.6 Exploration：A Two-Parameter Family
CHAPTER 2 Planar Linear Systems
2.1 Second-Order Differential Equations
2.2 Planar Systems
2.3 Preliminaries from Algebra
2.4 Planar Linear Systems
2.5 Eigenvalues and Eigenvectors
2.6 Solving Linear Systems
2.7 The Linearity Principle
CHAPTER 3 Phase Portraits for Planar Systems
3.1 Real Distinct Eigenvalues
3.2 Complex Eigenvalues
3.3 Repeated Eigenvalues
3.4 Changing Coordinates
CHAPTER 4 Classification of Planar Systems
4.1 The Trace-Determinant Plane
4.2 Dynamical Classification
4.3 Exploration：A 3D Parameter Space
CHAPTER 5 Higher Dimensional Linear Algebra
5.1 Preliminaries from Linear Algebra
5.2 Eigenvalues and Eigenvectors
5.3 Complex Eigenvalues
5.4 Bases and Subspaces
5.5 Repeated Eigenvalues
5.6 Genericity
CHAPTER 6 Higher Dimensional Linear Systems
6.1 Distinct Eigenvalues
6.2 Harmonic Oscillators
6.3 Repeated Eigenvalues
6.4 The Exponential of a Matrix
6.5 Nonautonomous Linear Systems
CHAPTER 7 Nonlinear Systems
7.1 Dynamical Systems
7.2 The Existence and Uniqueness Theorem
7.3 Continuous Dependence of Solutions
7.4 The Variational Equation
7.5 Exploration：Numerical Methods
CHAPTER 8 Equilibria in Nonlinear Systems
8.1 Some Nustrative Examples
8.2 Nonlinear Sinks and Sources
8.3 Saddles
8.4 Stability
8.5 Bifurcations
8.6 Exploration：Complex Vector Fields
CHAPTER 9 Global Nonlinear Techniques
9.1 Nullclines
9.2 Stability of Equilibria
9.3 Gradient Systems
9.4 Hamiltonian Systems
9.5 Exploration：The Pendulum with Constant Forcing
CHAPTER 10 Closed Orbits and Limit Sets
10.1 Limit Sets
10.2 Local Sections and Flow Boxes
10.3 The Poincare Map
10.4 Monotone Sequences in Planar Dynamical Systems
10.5 The Poincare-Bendixson Theorem
10.6 Applications of Poincare-Bendixson
10.7 Expl0ration：Chemical Reactions That Oscillate
CHAPTER 11 Applications in Biology
11.1 Infectious Diseases
11.2 Predator/Prey Systems
11.3 Competitive Species
11.4 Exploration：Competition and Harvesting
CHAPTER 12 Applications in Circuit Theory
12.1 An RLC Circuit
12.2 The Lienard Equation
12.3 The van der Pol Equation
12.4 A Hopf Bifurcation
12.5 Exploration：Neurodynamics
CHAPTER 13 Applications in Mechanics
13.1 Newton’S Second Law
13.2 Conservative Systems
13.3 Central Force Fields
13.4 The Newtonian Central Force System
13.5 Kepler’s First Law
13.6 The Two-Body Problem
13.7 Blowing Up the Singularity
13.8 Exploration：Other Central Force Problems
13.9 Exploration：Classical Limits of Quantum Mechanical Systems
CHAPTER 14 The Lorenz System
14.1 Introduction to the Lorenz System
14.2 Elementary Properties of the Lorenz System
14.3 The Lorenz Attractor
14.4 A Model for the Lorenz Attractor
14.5 The Chaotic Attractor
14.6 Exploration：The Rossler Attractor
CHAPTER 15 Discrete Dynamical Systems
15.1 Introduction to Discrete Dynamical Systems
15.2 Bifurcations
15.3 The Discrete Logistic Model
15.4 Chaos
15.5 Symbolic Dynamics
15.6 The Shift Map
15.7 The Cantor Middle-Thirds Set
15.8 Exploration：Cubic Chaos
15.9 Exploration：The Orbit Diagram
CHAPTER 16 Homoclinic Phenomena
16.1 The Shil’nikov System
16.2 The Horseshoe Map
16.3 The Double Scroll Attractor
16.4 Homoclinic Bifurcations
16.5 Exploration：The Chua Circuit
CHAPTER 17 Existence and Uniqueness Revisited
17.1 The Existence and Uniqueness Theorem
17.2 Proof of Existence and Uniqueness
17.3 Continuous Dependence on Initial Conditions
17.4 Extending Solutions
17.5 Nonautonomous Systems
17.6 Differentiability of the Flow
Bibliography
Index
