常微分方程理论中的几何方法 第2版

出版时间：2004-11  出版社：世界图书出版公司  作者：V.I.ARNOLD  页数：351  
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内容概要

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).

书籍目录

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 DenseChapter 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 FoliationChapter 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 TheoremChapter 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 PointsSamples of Examination ProblemsIndex

用户评论 (总计3条)

•   阿诺写的常微分方程第二册,有和代数几何的联系
•   知道有第二册，但从没买到，超星也没有，现在终于看到了！好书经典
•   老板叫买.确实是本好书。建议您一定对微分流形很熟悉的情况下阅读,会好些....