Renormalization in Area-Preserving Maps

By: R. S. MacKay

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Hardcover | 1 December 1992

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324 Pages

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This is a corrected and annotated version of the author's seminal PhD thesis, in which two forms of asymptotically universal structure were presented and explained for area-preserving maps. Area-preserving maps are the discrete-time analogue of two degree-of-freedom Hamiltonian systems, and much of their dynamics is reviewed in Chapter 1. The asymptotically universal structure is found on small scales in phase-space and long time-scales. Chapter 2 presents a brief survey of the use of the idea of renormalization in physics. The first universal structure, described in Chapter 3, appears at the accumulation of period-doubling sequences, the conservative parallel to the Feigenbaum-Coullet-Tresser universality. The second one appears at the breakup of golden invariant circles, described and explained in Chapter 4. The renormalization picture is presented as the key to understanding the transition from regular to chaotic motion in area-preserving maps.

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Preface
Abstract
Acknowledgements
Introduction to Area Preserving Maps	p. 1
Conservative Systems and Area Preserving Maps	p. 1
Periodic Orbits	p. 27
Invariant Circles	p. 69
Stochastic Behaviour	p. 85
Introduction to Renormalisation	p. 98
Renormalisation in Physics	p. 98
Renormalisation in Dynamical Systems	p. 102
Renormalisation Techniques	p. 111
Period Doubling in Area Preserving Maps	p. 126
Period Doubling Sequences	p. 126
Renormalisation	p. 139
The Universal One Parameter Family	p. 144
Appendices	p. 155
Renormalisation for Invariant Circles	p. 161
Renormalisation	p. 161
Fixed Point Analysis	p. 166
Simple Fixed Point	p. 174
Critical Fixed Point	p. 193
The Universal One Parameter Family	p. 225
Discussion	p. 245
Renormalisation for Maps on a Circle	p. 251
Summary	p. 290
Problems for the future	p. 290
References	p. 294
Supplementary reference list	p. 301
Table of Contents provided by Blackwell. All Rights Reserved.

Renormalization in Area-Preserving Maps

At a Glance

Hardcover

Shipping

How to return your order

Defective items

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