準(zhǔn)混沌沖擊振子

內(nèi)容概要

　　《準(zhǔn)混沌沖擊振子：重正化符號(hào)動(dòng)力學(xué)及運(yùn)動(dòng)遷移現(xiàn)象（英文版）》介紹了準(zhǔn)混沌運(yùn)動(dòng)研究的最新進(jìn)展，討論了動(dòng)力系統(tǒng)中有序運(yùn)動(dòng)與無序運(yùn)動(dòng)交界處的復(fù)雜的動(dòng)力學(xué)分支行為。準(zhǔn)混沌運(yùn)動(dòng)是由具有自相似結(jié)構(gòu)的穩(wěn)定運(yùn)動(dòng)島鄰域附近運(yùn)動(dòng)軌跡的吸引性來刻畫的，并且其相空間的位移是隨時(shí)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(zhǔn)混沌運(yùn)動(dòng)在物理學(xué)和數(shù)學(xué)上的規(guī)則性和復(fù)雜性。
　　《準(zhǔn)混沌沖擊振子：重正化符號(hào)動(dòng)力學(xué)及運(yùn)動(dòng)遷移現(xiàn)象（英文版）》包含了目前文獻(xiàn)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(xué)、應(yīng)用數(shù)學(xué)的研究生和學(xué)者以及非線性動(dòng)力學(xué)的專家對(duì)準(zhǔn)混沌運(yùn)動(dòng)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(xué)物理系教授，非線性動(dòng)力系統(tǒng)領(lǐng)域知名科學(xué)家，長期專注于一維沖擊振子的動(dòng)力學(xué)行為研究并取得了豐碩的成果，其中包括：在低維混沌和準(zhǔn)混沌哈密頓系統(tǒng)中的運(yùn)動(dòng)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(xué)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(quán)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(zhǔn)混沌沖擊振子:重正化、符號(hào)動(dòng)力學(xué)及運(yùn)動(dòng)遷移現(xiàn)象(英文版)》包含了目前文獻(xiàn)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(xué)、應(yīng)用數(shù)學(xué)的研究生和學(xué)者以及非線性動(dòng)力學(xué)的專家對(duì)準(zhǔn)混沌運(yùn)動(dòng)研究的極大興趣，是一本難得的教科書或參考書。

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