Renormalization in Area-Preserving Maps

By: Robert S Mackay

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Hardcover | 1 December 1992

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326 Pages

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This book is adapted and revised from 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. How they work and much of their dynamics are described in this book. The asymptotically universal structure is found on small scales in phase-space and long time-scales. The key to understanding it is renormalisation, that is, looking at a system on successively smaller phase-space and longer time scales. Having presented this idea, the author briefly surveys the use of the idea of renormalisation in physics. The renormalisation picture is then presented as the key to understanding the transition from regular to chaotic motion in area-preserving maps. Although written ten years ago, the subject matter continues to interest many today. This updated version will be useful to both researchers and students.

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Non-Fiction Science Physics Classical Mathematics Engineering & Technology Technology in General Maths for Engineers Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Science in General Maths for Scientists

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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

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