常微分方程理论中的几何方法（英文版）

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作　者：	（）V.I.Arnold著
出版社：	世界图书出版公司北京公司
丛编项：
标　签：	微积分

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ISBN：	9787506271929	出版时间：	2004-01-01	包装：	胶版纸
开本：	23cm	页数：	351页	字数：

内容简介

　　Newton's fundamental discovery， the one which he considered necessary to keep secret and published only in the form of an anagram， consists of the following： Data aequatione quotcunque fluentes quantitae involvente fluxiones invenire et vice' versa. In contemporary mathematical language， this means： "It is useful to solve differential equations". At present， the theory of differential equations represents a vast con- glomerate of a great many ideas and methods of different nature， very useful for many applications and constantly stimulating theoretical in- vestigations in all areas of mathematics. Many of the routes connecting abstract mathematical theories to appli- cations in the natural sciences lead through differential equations. Many topics of the theory of differential equations grew so much that they became disciplines in themselves; problems from the theory of differential equations had great significance in the origins of such disciplines as linear algebra， the theory of Lie groups， functional analysis， quantum mechanics， etc. Consequently， differential equations lie at the basis of scientific mathematical philosophy （Weltanschauung）.

作者简介

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图书目录

Preface to the Second Edition
Preface to the First Edition
Notation
Chapter 1 Special Equations
　1 Differential Equations Invariant under Groups of Symmetries
　2 Resolution of Singularities of Differential Equations
　3 Implicit Equations
　4 Normal Form of an Implicit Differential Equation in the Neighborhood of a Regular，Singular Point
　5 The Stationary Schr6dinger Equation
　6 Geometry of a Second-Order Differential Equation and Geometry of a Pair of Direction Fields in Three-Dimensional Space
Chapter 2 First-Order Partial Differential Equations
　7 Linear and Quasilinear First-Order Partial Differential Equations
　8 The Nonlinear First-Order Partial Differential Equation
　9 A Theorem of Frobenius
Chapter 3 Structural Stability
　10 The Notion of Structural Stability
11 Differential Equation on the Torus
12 Analytic Reduction of Analytic Circle Diffeomorphisms to a Totation
13 introducion to the Hyperbolic Theory
14 Anosov Systems
15 Structurally Stable Systems Are Not Every Where Dense
Chapter 4 Perturbation Theory
16 The Averaging Method
17 Averaging in Single-Frequency Systems
18 Averaging in Systems with Several Frquencies
19 Averaging in Hamiltonian Systems
20 Averaging Invariants
21 Averaging in Seifert's Foliation
Chapter 5 Normal Forms
22 Formal Reduction to Linear Normal Forms
23 The Case of Resonance
24 Poincare and Siegel Domains
25 Normal Form of a Mapping in the Neighborhood of a Fixed Point
26 Normal Form of an Equation with Periodic Coefficients
27 Normal Form of the Neighborhood of an Elliptic Curve
28 Proof of Siegel's Theorem
Chapter 6 Local Bifurcation Theory
29 Families and Deformations
30 Matrices Depending on Parameters and Singularities of the Decrement Diagram
31 Bifurcation of Singular Points of a Vector Field
32 Versal Deformation of Phase Portraits
33 Loss of Stability of an Equilibrium Position
34 Loss of Swtability of Self-Sustained Oscillations
35 Versal Deformation of Equivariant Vector Fields on the Plane
36 Metamorphoses of the Toplolgy at Resonances
37 Classification of Syngular Points
Samples of Examination Problems
Index