现代动力系统理论导论 [Introduction to the Modern Theory of Dynamical Systems]

现代动力系统理论导论 [Introduction to the Modern Theory of Dynamical Systems] pdf epub mobi txt 电子书 下载 2025

[美] 卡托克(Katok A.) 著
图书标签:
  • 动力系统
  • 非线性动力学
  • 混沌理论
  • 数学建模
  • 微分方程
  • 拓扑学
  • 相空间
  • 稳定性分析
  • bifurcations
  • ergodic theory
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出版社: 世界图书出版公司
ISBN:9787510032929
版次:1
商品编码:10888247
包装:平装
外文名称:Introduction to the Modern Theory of Dynamical Systems
开本:16开
出版时间:2011-04-01
用纸:胶版纸
页数:802

具体描述

内容简介

this book provides the first self-contained comprehensive exposition of the theory of dynamical systems as a core mathematical discipline closely intertwined with most of the main areas of mathematics. the authors introduce and rigorously develop the theory while providing researchers interested in applications with fundamental tools and paradigms.
the book begins with a discussion of several elementary but fundamental examples. these are used to formulate a program for the general study of asymptotic properties and to introduce the principal theoretical concepts and methods. the main theme of the second part of the book is the interplay between local analysis near individual orbits and the global complexity of the orbit structure. the third and fourth parts develop in depth the theories of !ow-dimensional dynamical systems and hyperbolic dynamical systems.
the book is aimed at students and researchers in mathematics at all levels from ad-vanced undergraduate up. scientists and engineers working in applied dynamics, non-linear science, and chaos will also find many fresh insights in this concrete and clear presentation. it contains more than four hundred systematic exercises.

目录

preface
0. introduction
1. principal branches of dynamics
2. flows, vector fields, differential equations
3. time-one map, section, suspension
4. linearization and localization

part 1examples and fundamental concepts
1. firstexamples
1. maps with stable asymptotic behavior contracting maps; stability of contractions; increasing interval maps
2. linear maps
3. rotations of the circle
4. translations on the torus
5. linear flow on the torus and completely integrable systems
6. gradient flows
7. expanding maps
8. hyperbolic toral automorphisms
9. symbolic dynamical systems sequence spaces; the shift transformation; topological markov chains;
the perron-frobenius operator for positive matrices

2. equivalence, classification, andinvariants
1. smooth conjugacy and moduli for maps equivalence and moduli; local analytic linearization; various types of moduli
2. smooth conjugacy and time change for flows
3. topological conjugacy, factors, and structural stability
4. topological classification of expanding maps on a circle expanding maps; conjugacy via coding; the fixed-point method
5. coding, horseshoes, and markov partitions markov partitions; quadratic maps; horseshoes; coding of the toral automor- phism
6. stability of hyperbolic total automorphisms
7. the fast-converging iteration method (newton method) for the conjugacy problem methods for finding conjugacies; construction of the iteration process
8. the poincare-siegel theorem
9. cocycles and cohomological equations

3. principalclassesofasymptotictopologicalinvariants
1. growth of orbits
periodic orbits and the-function; topological entropy; volume growth; topo-logical complexity: growth in the fundamental group; homological growth
2. examples of calculation of topological entropy
isometries; gradient flows; expanding maps; shifts and topological markov chains; the hyperbolic toral automorphism; finiteness of entropy of lipschitz maps; expansive maps
3. recurrence properties

4.statistical behavior of orbits and introduction to ergodic theory
1. asymptotic distribution and statistical behavior of orbits
asymptotic distribution, invariant measures; existence of invariant measures;the birkhoff ergodic theorem; existence of symptotic distribution; ergod-icity and unique ergodicity; statistical behavior and recurrence; measure-theoretic somorphism and factors
2. examples of ergodicity; mixing
rotations; extensions of rotations; expanding maps; mixing; hyperbolic total automorphisms; symbolic systems
3. measure-theoretic entropy
entropy and conditional entropy of partitions; entropy of a measure-preserving transformation; properties of entropy
4. examples of calculation of measure-theoretic entropy
rotations and translations; expanding maps; bernoulli and markov measures;hyperbolic total automorphisms
5. the variational principle

5.systems with smooth invar1ant measures and more examples
1. existence of smooth invariant measures
the smooth measure class; the perron-frobenius operator and divergence;criteria for existence of smooth invariant measures; absolutely continuous invariant measures for expanding maps; the moser theorem
2. examples of newtonian systems
the newton equation; free particle motion on the torus; the mathematical pendulum; central forces
3. lagrangian mechanics
uniqueness in the configuration space; the lagrange equation; lagrangian systems; geodesic flows; the legendre transform
4. examples of geodesic flows manifolds with many symmetries; the sphere and the toms; isometrics of the hyperbolic plane; geodesics of the hyperbolic plane; compact factors; the dynamics of the geodesic flow on compact hyperbolic surfaces
5. hamiltonian systems symplectic geometry; cotangent bundles; hamiltonian vector fields and flows;poisson brackets; integrable systems
6. contact systems hamiltonian systems preserving a 1-form; contact forms
7. algebraic dynamics: homogeneous and afline systems part 2local analysis and orbit growth

6.local hyperbolic theory and its applications
1. introduction
2. stable and unstable manifolds
hyperbolic periodic orbits; exponential splitting; the hadamard-perron the-orem; proof of the hadamard-perron theorem; the inclination lemma
3. local stability of a hyperbolic periodic point
the hartman-grobman theorem; local structural stability
4. hyperbolic sets
definition and invariant cones; stable and unstable manifolds; closing lemma and periodic orbits; locally maximal hyperbolic sets
5. homoclinic points and horseshoes
general horseshoes; homoclinic points; horseshoes near homoclinic poi
6. local smooth linearization and normal forms
jets, formal power series, and smooth equivalence; general formal analysis; the hyperbolic smooth case

7.transversality and genericity
1. generic properties of dynamical systems
residual sets and sets of first category; hyperbolicity and genericity
2. genericity of systems with hyperbolic periodic points
transverse fixed points; the kupka-smale theorem
3. nontransversality and bifurcations
structurally stable bifurcations; hopf bifurcations
4. the theorem of artin and mazur

8.orbitgrowtharisingfromtopology
1. topological and fundamental-group entropies
2. a survey of degree theory
motivation; the degree of circle maps; two definitions of degree for smooth maps; the topological definition of degree
3. degree and topological entropy
4. index theory for an isolated fixed point
5. the role of smoothness: the shub-sullivan theorem
6. the lefschetz fixed-point formula and applications
7. nielsen theory and periodic points for toral maps

9.variational aspects of dynamics
1. critical points of functions, morse theory, and dynamics
2. the billiard problem
3. twist maps
definition and examples; the generating function; extensions; birkhoff peri-odic orbits; global minimality of birkhoff periodic orbits
4. variational description of lagrangian systems
5. local theory and the exponential map
6. minimal geodesics
7. minimal geodesics on compact surfaces
part 3low-dimensional phenomena

10. introduction: what is low-dimensional dynamics?
motivation; the intermediate value property and conformality; vet low-dimensional and low-dimensional systems; areas of !ow-dimensional dynamics

11.homeomorphismsofthecircle
1. rotation number
2. the poincare classification rational rotation number; irrational rotation number; orbit types and mea-surable classification

12. circle diffeomorphisms
1. the denjoy theorem
2. the denjoy example
3. local analytic conjugacies for diophantine rotation number
4. invariant measures and regularity of conjugacies
5. an example with singular conjugacy
6. fast-approximation methods
conjugacies of intermediate regularity; smooth cocycles with wild cobound-aries
7. ergodicity with respect to lebesgue measure

13. twist maps
1. the regularity lemma
2. existence of aubry-mather sets and homoclinic orbits
aubry-mather sets; invariant circles and regions of instability
3. action functionals, minimal and ordered orbits
minimal action; minimal orbits; average action and minimal measures; stable sets for aubry-mather sets
4. orbits homoclinic to aubry-mather sets
5. nonexisience of invariant circles and localization of aubry-mather sets

14.flowsonsurfacesandrelateddynamicalsystems
1. poincare-bendixson theory
the poincare-bendixson theorem; existence of transversals
2. fixed-point-free flows on the torus
global transversals; area-preserving flows
3. minimal sets
4. new phenomena
the cherry flow; linear flow on the octagon
5. interval exchange transformations
definitions and rigid intervals; coding; structure of orbit closures; invariant measures; minimal nonuniquely ergodic interval exchanges
6. application to flows and billiards
classification of orbits; parallel flows and billiards in polygons
7. generalizations of rotation number
rotation vectors for flows on the torus; asymptotic cycles; fundamental class and smooth classification of area-preserving flows

15.continuousmapsoftheinterval
1. markov covers and partitions
2. entropy, periodic orbits, and horseshoes
3. the sharkovsky theorem
4. maps with zero topological entropy
5. the kneading theory
6. the tent model

16.smoothmapsoftheinterval
1. the structure of hyperbolic repellers
2. hyperbolic sets for smooth maps
3. continuity of entropy
4. full families of unimodal maps part 4hyperbolic dynamical systems

17.surveyofexamples
1. the smale attractor
2. the da (derived from anosov) map and the plykin attractor
the da map; the plykln attractor
3. expanding maps and anosov automorphisms of nilmanifolds
4. definitions and basic properties of hyperbolic sets for flows
5. geodesic flows on surfaces of constant negative curvature
6. geodesic flows on compact riemannian manifolds with negative sectional curvature
7. geodesic flows on rank-one symmetric spaces
8. hyperbolic julia sets in the complex plane rational maps of the riemann sphere; holomorphic dynamics

18.topologicalpropertiesofhyperbolicsets
1. shadowing of pseudo-orbits
2. stability of hyperbolic sets and markov approximation
3. spectral decomposition and specification
spectral decomposition for maps; spectral decomposition for flows; specifica- tion
4. local product structure
5. density and growth of periodic orbits
6. global classification of anosov diffeomorphisms on tori
7. markov partitions

19. metric structure of hyperbolic sets
1. holder structures
the invariant class of hsider-continuons functions; hslder continuity of conju-gacies; hslder continuity of orbit equivalence for flows; hslder continuity and differentiability of the unstable distribution; hslder continuity of the jacobian
2. cohomological equations over hyperbolic dynamical systems
the livschitz theorem; smooth invariant measures for anosov diffeomor-phisms; time change and orbit equivalence for hyperbolic flows; equivalence of torus extensions

20.equilibriumstatesandsmoothinvariantmeasures
1. bowen measure
2. pressure and the variational principle
3. uniqueness and classification of equilibrium states
uniqueness of equilibrium states; classification of equilibrium states
4. smooth invariant measures
properties of smooth invariant measures; smooth classification of anosov dif-feomorphisms on the torus; smooth classification of contact anosov flows on 3-manifolds
5. margulis measure
6. multiplicative asymptotic for growth of periodic points
local product flow boxes; the multiplicative asymptotic of orbit growth supplement
s. dynamical systems with nonuniformly hyperbolic behavior byanatolekatokandleonardomendoza
1. introduction
2. lyapunov exponents
cocycles over dynamical systems; examples of cocycles; the multiplicative ergodic theorem; osedelec-pesin e-reduction theorem; the rue!!e inequality
3. regular neighborhoods
existence of regular neighborhoods; hyperbolic points, admissible manifolds, and the graph transform
4. hyperbolic measures
preliminaries; the closing lemma; the shadowing lemma; pseudo-markov covers; the livschitz theorem
5. entropy and dynamics of hyperbolic measures
hyperbolic measures and hyperbolic periodic points; continuous measures and transverse homoclinic points; the spectral decomposition theorem; entropy,horseshoes, and periodic points for hyperbolic measures
appendix
a. background material
1. basic topology
topological spaces; homotopy theory; metric spaces
2. functional analysis
3. differentiable manifolds
differentiable manifolds; tensor bundles; exterior calculus; transversality
4. differential geometry
5. topology and geometry of surfaces
6. measure theory
basic notions; measure and topology
7. homology theory
8. locally compact groups and lie groups
notes
hintsandanswerstotheexercises
references
index

前言/序言



现代动力系统理论导论 [Introduction to the Modern Theory of Dynamical Systems] 图书简介 本书旨在为读者提供一个全面、深入且严谨的现代动力系统理论的入门。在物理学、工程学、生物学乃至经济学等众多科学领域中,我们经常需要描述系统随时间演化的规律。动力系统理论正是研究这些时间演化过程的数学框架。本书并非仅仅停留在对经典可积系统或简单线性系统的探讨,而是聚焦于那些展现出复杂、非线性行为的现代理论核心概念。 本书的结构设计旨在引导初学者逐步掌握从基础拓扑概念到复杂的混沌现象分析所需的数学工具和直觉。我们认为,理解动力系统的精髓,必须建立在扎实的数学基础之上,尤其是微分拓扑和测度论的初步概念。 第一部分:基础概念与拓扑动力系统 在本书的开篇,我们将从最基础的层面构建动力系统的数学模型。我们从连续时间系统(常微分方程)和离散时间系统(映射)入手,定义了相空间、轨迹和流的概念。不同于传统的教科书可能将线性系统置于中心,本书强调的是拓扑结构在动力系统分类中的决定性作用。 我们将详细介绍拓扑动力系统的框架。这意味着我们关注的是系统行为在连续形变下保持不变的性质,而非仅仅依赖于具体的坐标表示。这包括对同胚、共轭(拓扑共轭)的深入讨论。通过引入庞加莱截面的概念,我们将高维流的问题转化为低维映射的问题,这是分析复杂系统行为的关键技术之一。 我们还会详细探讨紧凑性和极限集理论。极限集,特别是吸引子(Attractors)和排斥子(Repellers),是理解系统长期行为的核心。本书会用大量的篇幅来解释拉格朗日(Lagrange)稳定性和庞加莱不变集原理,为后续的稳定性分析打下坚实的基础。 第二部分:微分动力系统与稳定性理论 在建立了拓扑框架后,我们将转向更具体的、基于光滑微分方程的系统。这一部分是连接理论与实际工程应用的桥梁。我们将详细分析线性系统的解的结构,包括特征值分析、稳定/不稳定流形的概念。 对于非线性系统,雅可比矩阵的线性化方法是分析平衡点的局部稳定性的首要工具。然而,现代动力系统理论的精髓在于处理线性化失败的情况——即中心流形理论和规范型理论。本书将以严谨的方式推导和应用这些理论,展示如何将复杂系统的行为简化为其在低维中心流形上的动力学,从而揭示分岔(Bifurcation)的内在机制。 分岔理论是理解系统定性变化的关键。我们将系统地介绍鞍结分岔、超临界/次临界霍普夫分岔等一系列经典分岔,并展示它们如何从参数空间中的一个点引发全局拓扑结构的改变。这些分析将依赖于范畴导向的稳定性分析,确保我们理解稳定性概念的深刻内涵。 第三部分:非线性与混沌动力学 本书的后半部分,我们将步入现代动力系统研究中最具吸引力和挑战性的领域——非线性动力学与混沌。我们将不再满足于简单的稳定平衡点或周期振荡,而是致力于理解那些对初始条件极端敏感的系统行为。 首先,我们将严格定义混沌,区分其拓扑定义和度量定义。我们将深入探讨庞加莱-霍普夫定理,以及指数分离作为混沌的内在特征。 为了量化复杂性,本书引入了李雅普诺夫指数(Lyapunov Exponents)。我们将详细阐述如何计算和解释最大李雅普诺夫指数,以及它在区分可积、准周期和混沌行为中的作用。 对于离散系统,拓扑熵和延拓性的概念被用来衡量系统对信息的处理能力。我们将分析洛伦兹吸引子、鲁洛吸引子等经典的奇异吸引子的拓扑结构,尽管不直接深入到具体的物理模型推导,但我们会分析其分形性质和基本结构。 分形几何与测度在现代动力系统中的地位不言而喻。我们将引入豪斯多夫维数和关联维数的概念,用以描述吸引子的复杂几何结构。特别地,对于拓扑混合性和遍历性的讨论,将使读者理解为什么某些系统虽然看似随机,却依然服从严格的确定性规则。 第四部分:保守系统与可积性 虽然本书的核心是现代非线性理论,但对哈密顿系统的讨论是不可或缺的。我们将介绍泊松括号、李维尔定理以及卡尔曼-艾恩霍芬定理(Kailen–Ehrenfest Theorem)在保守系统中的应用。 我们将区分可积系统和非可积系统。卡尔曼-艾恩霍芬定理为我们理解KAM理论(Kolmogorov–Arnold–Moser Theory)提供了背景。KAM理论是现代数学物理中关于微小摄动如何保留大部分积分结构的基石。本书将以定性的方式阐述KAM定理的意义,即小扰动下,大部分原有的准周期运动会得以保留,只有在共振区域才会出现混沌的萌芽。 总结 《现代动力系统理论导论》旨在为研究生和高年级本科生提供一个坚实的基础,让他们能够从拓扑学的角度理解系统的长期行为,掌握分析非线性、处理分岔、并量化混沌的数学工具。本书的难度适中,强调概念的深度而非计算的繁琐,目标是培养读者对复杂系统内在规律的洞察力,为进一步深入研究如随机性、延迟动力学或特定工程应用打下坚实的理论根基。全书配有大量的图示和启发性的习题,以促进理论与实践的结合。

用户评价

评分

最近我入手了这本《现代动力系统理论导论》,不得不说,它是一部极具挑战性但也极其 rewarding 的作品。我一直对非线性动力学领域充满了兴趣,尤其关注混沌现象背后的数学原理。这本书在这方面简直是我的“解药”。作者在探讨蝴蝶效应和分形几何时,那种将微小扰动放大到宏观尺度,以及在看似混乱中发现隐藏秩序的能力,让我拍案叫绝。我特别喜欢书中关于吸引子概念的阐述,无论是奇特的吸引子还是极限环,都展示了动力系统如何趋向稳定或周期性的状态。阅读过程中,我常常停下来,反复咀嚼作者对不同类型吸引子之间关系的分析,以及它们是如何影响系统的长期行为的。这本书不仅提供了理论框架,更像是为读者提供了一套探索复杂系统行为的“工具箱”。那些详细的例证和数学推导,虽然需要耐心和专注,但一旦理解,便能体会到那种智力上的豁然开朗。我发现,通过阅读这本书,我对于日常生活中遇到的许多看似难以预测的现象,如天气变化、金融市场的波动,都有了新的理解角度。

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这部关于现代动力学系统理论的著作,我早就听说过它的声名远扬,但直到最近才有机会真正翻阅。初次接触,最直观的感受是它那庞杂却又井然有序的知识体系。我尤其欣赏作者在开篇部分对于“系统”这一核心概念的阐释,那绝非简单的定义堆砌,而是深入浅出地剖析了其内在的演化逻辑和普遍性。读到关于映射的迭代和不动点的讨论时,仿佛打开了一扇通往奇妙数学世界的大门,那些看似抽象的概念,在作者的笔下变得生动且富有洞察力。书中对于周期轨道的分类和稳定性分析的部分,更是将理论的严谨性展现得淋漓尽致。我能够想象,对于那些刚刚踏入这个领域的研究者来说,这无疑是一本奠定坚实基础的宝典。作者在语言的运用上也颇为讲究,既有严密的数学表述,又不乏引人入胜的类比和解释,使得即便是复杂的概念,也能够被逐步消化。我想,对于任何对理解事物演化规律充满好奇的人来说,这本书都将是一次令人难忘的智力冒险,它不仅仅是理论的堆砌,更是一种思考方式的启迪。

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这是一本极其严谨且富有启发性的著作,它为我打开了理解现代动力学系统理论的一扇大门。我尤其被书中关于“测度”和“熵”在动力系统中的作用的阐释所吸引。作者深入浅出地讲解了如何在相空间中定义测度,以及如何量化系统的混乱程度,即通过信息熵来衡量。这对于理解混沌系统的不可预测性,以及其信息增长的速度,有着至关重要的作用。我花了不少时间去理解书中关于“遍历性”和“混合性”的区别,以及它们如何影响系统的长期统计性质。作者在讲解这些概念时,采用了多种数学工具和方法,使得复杂的理论变得更加清晰易懂。这本书的理论体系非常完整,它不仅涵盖了基础的映射和流,还深入探讨了分岔理论、混沌理论以及它们在各个领域的应用。每一次翻阅,我都能从中学到新的知识,并将其与我之前所学到的知识融会贯通。它是一本真正能够提升读者理论认知水平的书籍,对于任何希望深入研究动力系统的人来说,都绝对值得拥有。

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坦白讲,我最初是被这本书的标题吸引的,我对“现代”和“动力系统”这些词汇充满了好奇。拿到书后,我发现它并非一本“速成”指南,而是一部需要细细品味的学术著作。我特别欣赏作者在介绍一些经典动力系统模型时,所展现出的历史视角和思想演变过程。例如,在讲述庞加莱-贝特朗定理的由来时,我仿佛能感受到数学家们在探索未知时的艰辛与智慧。书中关于“全局分岔”和“局部分岔”的区分,以及不同分岔类型所对应的系统行为的改变,让我对系统的“质变”有了更深刻的理解。我也被作者在分析耦合振子系统时的细致入微所打动,如何从简单的单振子模型过渡到复杂的耦合系统,并分析其中的同步现象。这本书的深度和广度都令人印象深刻,它不仅仅停留在理论的表层,而是深入到动力系统理论的核心问题。我感觉,每一次阅读都能有所新的领悟,就像在不断剥离洋葱的层层外衣,最终触碰到核心的真谛。

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不得不说,这本书的“门槛”确实不低,但一旦跨过去,收获是巨大的。我从这本书中获得的,是一种全新的观察世界的方式。在学习了遍历理论和统计力学之间的联系之后,我才真正理解了微观粒子的无规则运动如何宏观上表现出统计规律。作者对于不可积系统和 KAM 定理的讲解,虽然复杂,但却揭示了保守系统中规则运动如何被微小摄动破坏,从而产生混乱。我特别着迷于书中关于“遍历性”的探讨,它告诉我们,一个系统在足够长的时间内,其状态会覆盖其相空间的所有可能区域。这对于理解统计力学中的许多基本假设至关重要。同时,作者还巧妙地将抽象的数学概念与物理学中的实际问题联系起来,比如在讲解轨道稳定性时,我看到了它在天体轨道力学中的应用。这本书要求读者具备一定的数学基础,但如果你愿意投入时间和精力,它会给你带来超越理论学习本身的价值。它教会我如何去“看”那些肉眼无法直接观测到的系统演化轨迹,以及如何在复杂性中寻找规律。

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此书全面论证了现代动力系统理论,是高年级大学本科和研究生的参考教材。

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同学和老师推荐的,买来读读

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经典的动力系统教材。有兴趣的可以看看

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it seems good

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OK!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

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好书,慢慢看

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经典的动力系统教材。有兴趣的可以看看

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it seems good

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动力系统的经典,推荐入手

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