微分方程動(dòng)態(tài)系統(tǒng)和混沌導(dǎo)論

內(nèi)容概要

30年來(lái)，動(dòng)力系統(tǒng)的數(shù)學(xué)理論與應(yīng)用有了很大發(fā)展。30多年前還沒(méi)有高速的臺(tái)式計(jì)算機(jī)和計(jì)算機(jī)圖像，“混沌”一詞也沒(méi)有在數(shù)學(xué)界使用，而對(duì)于微分方程與動(dòng)力系統(tǒng)的研究興趣主要僅限于數(shù)學(xué)界中比較小的范圍。到今天，處處有計(jì)算機(jī)，求微分方程近似解的軟件包已得到廣泛運(yùn)用，使人們從圖形中就能看到結(jié)果。對(duì)于非線性微分方程的分析已為廣大學(xué)者所接受，一些復(fù)雜的動(dòng)力學(xué)行為，如馬蹄映射、同宿軌、Lorenz系統(tǒng)中揭示出來(lái)的復(fù)雜現(xiàn)象，以及數(shù)學(xué)方面的分析，使學(xué)者們確信簡(jiǎn)單的穩(wěn)定運(yùn)動(dòng)，如平衡態(tài)和周期解己不總是微分方程解的最重要的行為，而混沌現(xiàn)象揭示出來(lái)的美妙性態(tài)正促使各個(gè)領(lǐng)域的科學(xué)家與工程師細(xì)心關(guān)注在他們自己領(lǐng)域中提出的重要的微分方程及其混沌特性。動(dòng)力系統(tǒng)現(xiàn)象在今天已出現(xiàn)在幾乎每個(gè)科學(xué)領(lǐng)域中，從化學(xué)中的振蕩Belousov-Zhabotinsky反應(yīng)到電子工程中的混沌Chua電路，從天體力學(xué)中的復(fù)雜運(yùn)動(dòng)到生態(tài)系統(tǒng)中的分岔。

作者簡(jiǎn)介

作者：(美)赫希 (美)斯梅爾

書(shū)籍目錄

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 FamilyCHAPTER 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

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《微分方程動(dòng)態(tài)系統(tǒng)和混沌導(dǎo)論》這次修訂本新增加一名作者，即著名的混沌理論專(zhuān)家德瓦尼（R.Devaney），內(nèi)容也有很大改動(dòng)。

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