現代動力係統理論導論 [Introduction to the Modern Theory of Dynamical Systems] pdf epub mobi txt 電子書 下載 2025

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[美] 卡托剋（Katok A.） 著

齣版社： 世界圖書齣版公司

ISBN：9787510032929

版次：1

商品編碼：10888247

包裝：平裝

外文名稱：Introduction to the Modern Theory of Dynamical Systems

開本：16開

齣版時間：2011-04-01

用紙：膠版紙

頁數：802

內容簡介

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.

目錄

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
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