準混沌沖擊振子

內(nèi)容概要

　　《準混沌沖擊振子：重正化符號動力學及運動遷移現(xiàn)象（英文版）》介紹了準混沌運動研究的最新進展，討論了動力系統(tǒng)中有序運動與無序運動交界處的復雜的動力學分支行為。準混沌運動是由具有自相似結(jié)構(gòu)的穩(wěn)定運動島鄰域附近運動軌跡的吸引性來刻畫的，并且其相空間的位移是隨時間的冪指數(shù)而漸近增加的。本專著全面、系統(tǒng)、自成體系地研究了一維經(jīng)典沖擊振子模型，并以完美的形式展示了準混沌運動在物理學和數(shù)學上的規(guī)則性和復雜性。
　　《準混沌沖擊振子：重正化符號動力學及運動遷移現(xiàn)象（英文版）》包含了目前文獻中很多不曾涉及的新內(nèi)容和新結(jié)果，它將激發(fā)物理學、應用數(shù)學的研究生和學者以及非線性動力學的專家對準混沌運動研究的極大興趣，是一本難得的教科書或參考書。
　　John H.
Lowenstein為紐約大學物理系教授，非線性動力系統(tǒng)領域知名科學家，長期專注于一維沖擊振子的動力學行為研究并取得了豐碩的成果，其中包括：在低維混沌和準混沌哈密頓系統(tǒng)中的運動遷移現(xiàn)象，區(qū)間及多邊形分段等距自相似結(jié)構(gòu)的數(shù)學理論。

書籍目錄

1 Introduction
1.1 Kicked oscillators
1.2 Poincare sections
1.3 Crystalline symmetry
1.4 Stochastic webs
1.5 Normal and anomalous diffusive behavior
1.6 The sawtooth web map
1.7 Renormalizability
1.8 Long-time asymptotics
1.9 Linking local and global behavior
1.10 Organization of the book
References
2 Renormalizability of the Local Map
2.1 Heuristic approach to renormalizability
2.1.1 Generalized rotations
2.1.2 Natural return map tree
2.1.3 Examples
2.2 Quadratic piecewise isometries
2.2.1 Arithmetic preliminaries
2.2.2 Domains
2.2.3 Geometric transformations on domains
2.2.4 Scaling sequences
2.2.5 Periodic orbits
2.2.6 Recursive tiling
2.2.7 Computer-assisted proofs
2.3 Three quadratic models
2.3.1 Modell
2.3.2 Modelll
2.3.3 Model III
2.4 Proofofrenormalizability
2.5 Structure of the discontinuity set
2.5.1 Modell
2.5.2 Modellll
2.6 More general renormalization
2.7 The π/7 model
References
3 Symbolic Dynanucs
3.1 Symbolic representation of the residual set
3.1.1 Hierarchical symbol strings
3.1.2 Eventually periodic codes
3.1.3 Simplified codes for quadratic models
3.2 Dynamical updating of codes
3.3 Admissibility
3.3.1 Quadratic example
3.3.2 Models I, II, and III
3.3.3 Cubic example
3.4 Minimality
References
4 Dimensions and Measures
4.1 Hausdorff dimension and Hausdorff measure
4.2 Construction of the measure
4.3 Simplification for quadratic irrational
4.4 A complicated example: Model II
4.5 Discontinuity set in Model III
4.6 Multifractal residual set of the π/7 model
4.7 Asymptotic factorization
4.8 Telescoping
4.9 Unique ergodicity for each ∑（i）
4.10 Multifractal spectrum of recurrence time dimensions
4.10.1 Auxiliary measures and dimensions
4.10.2 Simpler calculation of the recurrence time dimensions
4.10.3 Recurrence time spectrum for the π/7 model
References
5 Global Dynanucs
5.1 Global expansivity
5.1.1 Lifting the return map PK （O）
5.1.2 Lifting the higher-level return maps
5.2 Long-time asymptotics
5.3 Quadratic examples
5.4 Cubic examples
5.4.1 Orbits in the （O,k,6∞） sectors
……
6 Transport
7 Hamiltonian Round-Off
Appendix A Data Tables
Appendix B The Codometer
Index
Color Figure Index

章節(jié)摘錄

版權頁：   插圖：   It is not immediately obvious that choosing λ to be a low-degree algebraic inte-ger should help our search for dynamical self-similarity （beyond the restriction that it places on the denominator of the rotation number）. Of course, it is well known that the lowest-degree algebraic integers, solutions of quadratic equations, enjoy algebraic self-similarity in their continued.fraction expansions. Moreover, for one-dimensional maps analogous to piecewise isometries, namely the interval exchange transformations, one has a powerful theorem of Boshernitzan and Carroll （1997） es-tablishing their renormalizability for quadratic irrational parameters. Unfortunately,no comparable theorem for two-dimensional PWI's has been proved. However, for two-dimensional PWI's, the renormalizability of an important class of models with quadratic irrational λ has been rigorously established by Kouptsov et al. （2002） us-ing computer assisted proofs. It is here that the true advantage of the restriction to low-degree algebraic numbers makes itself felt: it makes it possible to use com-puter software to perform exact calculations on specific models, most of which have exceedingly complicated multi-level return map structures, thereby verifying impor-tant properties of each model and, by exhaustion, the entire class. Before examining three particularly interesting models from the class of PWI's of the square with rational rotation numbers and quadratic irrational parameters, it will be useful to illustrate how the systematic search for renormalizable return map structure succeeds in a particularly simple example. The contrast with the λ = 1/2 case will be striking.

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《準混沌沖擊振子:重正化、符號動力學及運動遷移現(xiàn)象(英文版)》包含了目前文獻中很多不曾涉及的新內(nèi)容和新結(jié)果，它將激發(fā)物理學、應用數(shù)學的研究生和學者以及非線性動力學的專家對準混沌運動研究的極大興趣，是一本難得的教科書或參考書。

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