The physics of chaos in Hamiltonian systems

337 Pages·2007·5.095 MB·English

by  George M. Zaslavsky

THE PHYSICS OF CHAOS
IN HAMILTONIAN SYSTEMS
Second Edition
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THE PHYSICS OF CHAOS
IN HAMILTONIAN SYSTEMS
Second Edition
George M. Zaslavsky
Department of Physics
and
Courant Institute of Mathematical Sciences
New York University, USA
Imperial College Press
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Imperial College Press
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PHYSICS OF CHAOS IN HAMILTONIAN SYSTEMS
Second Edition
Copyright © 2007 by Imperial College Press
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ISBN-13978-1-86094-795-7
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Lakshmi - Phys of Chaos.pmd 1 4/5/2007, 11:32 AM
PREFACE TO THE FIRST EDITION
The term chaos is often used to describe the phenomena in which the
system’s trajectories are sensitive to the slightest changes in initial
conditions. In reality, the properties of such motion resemble those
of random motion. If one restricts oneself to Hamiltonian systems with
area-preserving dynamics, the above de(cid:12)nition of chaos would appear
perplexing. Can a trajectory be chaotic at times and \regular" for
the rest of the time? How does a regular trajectory transform itself
into a chaotic one? And what type of randomness characterises chaotic
dynamics? Fortunately, clear de(cid:12)nitions exist for regular and chaotic
motions. Conditionally periodic motion is an example of regular dy-
namics. The examples of \ideal" chaotic motion refer to the motion on
the negative curvature surface and the so-called Anosov systems. How-
ever, real physical systems or their simpli(cid:12)ed models are very di(cid:11)erent
from the ideal models on chaos. A good example is that of a pendulum
disturbed by a periodic (non-random) force.
Our understanding of chaos is fraught with the following di(cid:14)culties.
The motion known as chaotic occupies a certain area (called stochastic
sea) in the phase space. In ideal chaos, the stochastic sea is occupied
in a uniform manner. This is, however, not the case in real systems
or models. The phase space contains many \islands" which a chaotic
trajectory cannot penetrate. Initially, it had appeared that the e(cid:11)ects
of islands could be easily accounted for by simply changing the phase
volume of the stochastic sea. We now believe that this is not true and
the important properties of chaotic dynamics are in fact determined by
the properties of the motion near the boundary of islands.
v
vi Preface
The di(cid:14)culties in understanding Hamiltonian chaos can also be des-
cribed in an informal way. While regular and chaotic motions possess
some degree of uniformity (monotonity), which is used in their de(cid:12)ni-
tions, real chaotic motion boasts intermediate properties (between
regular and chaotic motions) that have not been accurately de(cid:12)ned and
formulated. Therefore, we can neither use the KAM theory (as the
conditions of non-degeneracy are violated in most cases) nor the Sinai’s
methodof Markov partitions or related techniques (because thecorrela-
tions donot decay exponentially andthe Markov propertyis violated in
some areas) and the estimates of Arnold di(cid:11)usion (since a much faster
di(cid:11)usion takes place).
This book considers many of the di(cid:14)culties described above in
analysing Hamiltonian chaos in real systems. The reader is treated to
the unconventional application of the fractal dimension to space-time
objects, di(cid:11)erent versions of the renormalisation group method, frac-
tional kinetics, and Poincar(cid:19)e recurrences theory as well as the more
traditional applications of the Poincar(cid:19)e and separatrix maps.
Thisbookisusefultothereaderwhohasanundergraduatedegreein
physics. It does not include any methods that are beyond the standard
mathematical physics techniques except for some fractional calculus
which is provided in a special appendix. It is useful to physicists,
engineers and those who are interested in the current problems in chaos
theory and its applications. While mathematicians will not be able
to (cid:12)nd any rigorously proven results here, they will learn about the
challenges of \real physics". Towards that end, we have included many
examplesofnumericalsimulations. Someoftheexampleswereextracted
(and updated) from the author’s previous works. The material covered
is based partly on the course o(cid:11)ered by the author at the Courant
Institute of Mathematical Sciences and the Department of Physics in
New York University. Much of it, however, is new and can be used as
a source of information on the new and emerging directions in modern
chaos theory applied to physical problems.
I would like to thank those who have helped to make this book pos-
sible: Sadrilla Abdullaev, Valentine Afraimovich, Sadruddin Benkadda,
Mark Edelman, Serge Kassibrakis, Jossy Klafter, Leonid Kuznetsov,
Preface vii
Boris Niyazov, Alexander Saichev, Don Stevens, Hank Strauss, Michael
Shlesinger, Harold Weitzner and Roscoe White, all of whom are
co-authors of common publications; Valentine Afraimovich, Leonid
Bunimovich, John Lowenstein, Victor Melnikov, Anatoly Neishtadt,
Yasha Pesin, Vered Rom-Kedar, Michael Shlesinger and Dimitry
Treschev for their numerous discussions; Mark Edelman for his help in
the preparation of the (cid:12)gures; Herman Todorov and Leonid Kuznetsov
for their help in editing the manuscript; and Pat Struse for preparing
the manuscript.
PREFACE TO THE SECOND EDITION
Afterthe(cid:12)rstpublication ofthebookthereweredi(cid:11)erentdevelopments
in the theory and experiments related to a highly complex intermittent
character of chaotic dynamics and to the fractional structure of dy-
namics and kinetics. To re(cid:13)ect these changes, the second edition of the
bookincludes di(cid:11)erent new sections andminor additions in correspond-
ing places. There are extensions of the sections related to Maxwell’s
Demon and billiards. New sections on Ballistic Mode Islands, Rhombic
Billiards, Persistent Fluctuations, and Log-Periodicity can be found. A
new Chapter 12 on weak chaos and pseudochaos is also added.
Pseudochaos is introduced as random dynamics with zero Lyapunov
exponent. As an application of the pseudochaos is the description of
fractional kinetics along the Filamented Surfaces that has numerous
applications in magneto- and hydrodynamics. Appendices have been
considerably modi(cid:12)ed to extend a reference material related to basic
formulas of fractional calculus.
Di(cid:11)erent misprints and typos of the (cid:12)rst edition are eliminated in
the second edition. New sections and references are marked by (*). It is
my great pleasure to thank Mark Edelman for his help in preparing the
material and (cid:12)gures for the second edition.
ix
CONTENTS
PREFACE TO THE FIRST EDITION v
PREFACE TO THE SECOND EDITION ix
1 DISCRETE AND CONTINUOUS MODELS 1
1.1 Coexistence of the Dynamical Order and Chaos 1
1.2 The Standard Map (Kicked Rotator) 3
1.3 The Web-Map (Kicked Oscillator) 8
1.4 Perturbed Pendulum 14
1.5 Perturbed Oscillator 17
1.6 Billiards 19
Conclusions 21
2 SEPARATRIX CHAOS 23
2.1 Nonlinear Resonance and Chain of Islands 23
2.2 Overlapping of Resonances 28
2.3 The Separatrix Map 30
2.4 Stochastic Layer 35
2.5 Hidden Renormalisation Group Near the Separatrix 41
2.6 Renormalisation of Resonances 50
2.7 Stochastic Layer of the Standard Map 51
Conclusions 54
3 THE PHASE SPACE OF CHAOS 56
3.1 Non-universality of the Scenario 56
3.2 Collapsing Islands 63
xi