Regular and Chaotic Motions in Dynamic Systems NATO ASI Series Advanced Science Institutes Series A series presenting the results of activities sponsored by the NA TO SCience Committee, which aims at the dissemination of advanced scientific and technological knowledge, with a view to strengthening links between scientific communities. The series is published by an international board of publishers in conjunction with the NATO Scientific Affairs Division A Life Sciences Plenum Publishing Corporation B Physics New York and London C Mathematical D. Reidel Publishing Company and Physical Sciences Dordrecht, Boston, and Lancaster D Behavioral and Social Sciences Martinus Nijhoff Publishers E Engineering and The Hague, Boston, and Lancaster Materials Sciences F Computer and Systems Sciences Springer-Verlag G Ecological Sciences Berlin, Heidelberg, New York, and Tokyo Recent Volumes in this Series Volume 114-Energy Transfer Processes in Condensed Matter edited by Baldassare Di Bartolo Volume 115-Progress in Gauge Field Theory edited by G.'t Hooft, A. Jaffe, H. Lehmann, P. K. Mitter, I. M. Singer, and R. Stora Volume 116-Nonequilibrium Cooperative Phenomena in Physics and Related Fields edited by Manuel G. Velarde Volume 117-Moment Formation In Solids edited by W. J. L. Buyers Volume 118-Regular and Chaotic Motions in _-Dynamic Systems edited by G. Velo and A. S. Wightman Volume 119-Analytical Laser Spectroscopy edited by S. Martellucci and A. N. Chester Volume 120-Chaotic Behavior in Quantum Systems: Theory and Applications edited by Giulio Casati Series B: Physics Regular and Chaotic Motions in Dynamic Systems Edited by G. Vela Institute of Physics University of Bologna Bologna, Italy and A. S. Wightman Princeton University Princeton, New Jersey Plenum Press New York and London Published in cooperation with NATO Scientific Affairs Division Proceedings of the Fifth International School of Mathematical Physics and NATO Advanced Study Institute on Regular and Chaotic Motions in Dynamic Systems, held July 2-14, 1983, at the Ettore Majorana Center for Scientific Culture, Erice, Sicily, Italy Library of Congress Cataloging in Publication Data International School of Mathematical Physics (5th: 1983: Ettore Majorana Inter- national Centre for Scientific Culture) Regular and chaotic motions in dynamic systems. (NATO ASI series. Series B, Physics; v 118) "Proceedinas of the Fifth International School of Mathematical Physics and NATO Advanced Study Institute on Regular and Chaotic Motions in DynamiC Systems held July 2-14, 1983, at the Ettore Majorana Center for Scientific Culture, Erice, Sicily, Italy". "Published in cooperation with NATO Scientific Affairs Division." Bibliography: p. Includes index. 1. Dynamics-Congresses. 2. Chaotic behavior in systems-Congresses. 3. Mathematical physics-Congresses. I. Velo, G.II. Wightman, A. S. III. NATO Ad- vanced Study Institute on Regular and Chaotic Motions in Dynamic Systems (1983: Ettore Majorana International Centre for Scientific Culture). IV. Title. V. Series. QC133.158 1983 531'.11 84-26369 ISBN-13: 978-1-4684-1223-9 e-ISBN-13: 978-1-4684-1221-5 DOl: 10.1007/978-1-4684-1221-5 ©1985 Plenum Press, New York Softcover reprint of the hardcover 1s t edition 1985 A Division of Plenum Publishing Corporation 233 Spring Street, New York, N.Y. 10013 All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher PREFACE The fifth International School ~ Mathematical Physics was held at the Ettore Majorana Centro della Culture Scientifica, Erice, Sicily, 2 to 14 July 1983. The present volume collects lecture notes on the session which was devoted to'Regular and Chaotic Motions in Dynamlcal Systems. The School was a NATO Advanced Study Institute sponsored by the Italian Ministry of Public Education, the Italian Ministry of Scientific and Technological Research and the Regional Sicilian Government. Many of the fundamental problems of this subject go back to Poincare and have been recognized in recent years as being of basic importance in a variety of physical contexts: stability of orbits in accelerators, and in plasma and galactic dynamics, occurrence of chaotic motions in the excitations of solids, etc. This period of intense interest on the part of physicists followed nearly a half a century of neglect in which research in the subject was almost entirely carried out by mathematicians. It is an in- dication of the difficulty of some of the problems involved that even after a century we do not have anything like a satisfactory solution. The lectures at the school offered a survey of the present state of the theory of dynamical systems with emphasis on the fundamental mathematical problems involved. We hope that the present volume of proceedings will be useful to a wide circle of readers who may wish to study the fundamentals and go on to research in the subject. With this in mind we have included a selected bibliography of books and reviews which the participants found helpful as well as a brief bibliography for four seminars which were held in addition to the main lecture series. There were sixty-one participants from sixteen countries. G. Velo and A.S. Wightman Directors of the School v CONTENTS Introduction to the Problems 1 A.S. Wightman Applications of Scaling Ideas to Dynamics L.P. Kadanoff Lecture I. Roads to Chaos: Complex Behavior from Simple Systems • • 27 II. From Periodic Motion to Unbounded Chaos: Investigations of the Simple Pendulum •• 45 III. The Mechanics of the Renorma1ization Group 60 IV. Escape Rates and Strange Repe1lors • 63 Introduction to Hyperbolic Sets 73 O.E. Lanford III Topics in Conservative Dynamics 103 S. Newhouse Classical Mechanics and Renorma1ization Group 185 G. Ga11avotti Measures Invariant Under Mappings of the Unit Interval. 233 P. Collet and J.-P. Eckmann Integrable Dynamical Systems • • • • • • • • • • • • • • 267 E. Trubowitz Appendix (Seminars) Iteration of Polynomials of Degree 2, Iterations of Polynomial-like Mappings 293 A. Douady Boundary of the Stability Domain around the Origin for Chirikov's Standard Mapping ••• 295 G. Dome vii viii CONTENTS Incommensurate Structures in Solid State Physics and Their Connection with Twist Mappings • • • • 296 S. Aubry Julia Sets - Orthogonal Polynomials Physical Interpretations and Applications. 300 D. Bessis Scaling Laws in Turbulence 303 J.-D. Fournier Index • • . • • • • • • • • • • • • • • • • • • • • • • •• 309 REGULAR AND CHAOTIC MOTIONS IN DYNAMICAL SYSTEMS INTRODUCTION TO THE PROBLEMS A. S. Wightman Departments of Mathematics and Physics Princeton University Princeton, N.J. 08544 USA The purpose of this introduction is twofold; first, to sketch the origin of some of the problems that will be discussed in detail later, and, second, to introduce some of the concepts which will be used. In a subject like analytical mechanics, with such a long history and such hard problems, a little sense of history is both enlightening and consoling. A dynamical system is loosely specified as a system with a state at time ~ given by a point x(t) lying in a phase space, M, and a law of evolution given by an ordinary differential equation (= ODE-)- - dx dt (t) = v(x(t)) (1) Here v is a vector field on the phase space M. M is customarily assumed to be a differentiable manifold such as an open set in n-dimensional Euclidean space. Alternatively, one can consider the dynamical system specified by its set of possible histories, the set of mappings, t ~ x(t), of some time interval a < t < b into the phase space satisfying the ODE (1). When _00 < t < 00 , the solutions are said to define a flow; when 0 < t < 00 a semi- flow. A discrete dynamical system is one in which the time takes integer values. Then the dynamics is given by the iterates of a mapping of the phase space M into itself. If t runs over all the integers, Z, one sometimes speaks of a cascade; if over the positive integers, Z+, of a semi-cascade. Although the extension of these definitions to infinite dimensional M is of obvious 2 A. S. WIGHTMAN physical interest (fluid dynamics!), in what follows, for lack of time, attention will be mainly confined to the finite dimensional case. Poincare's Bequest The analysis of dynamical systems (= analytical mechanics = classical mechanics = rational mechanics) is one of the oldest parts of physics, but, in a sense, the modern period begins with Poincare. It is notorious that the physicists of most of the twentieth century had little appreciation of Poincare's work. Nevertheless, it is his outlook which dominates the field today. To appreciate this, it helps to have been brought up, as I was, on a really old-fashioned version of the subject, say that in E.T. Whittaker's! Treatise on the Analytical Dynamics of Particles and Rigid Bodies. That is a remarkable book, which has some coverage of Poincare's technical results but scarcely a word about his general point of view. Nearly a hundred years later, we find our thinking completely dominated by Poincare's geometric attitude, whether we prefer it in the super-Smalean version of R. Abraham and J. Marsden's Foundations of Mechanics or the proletarian version of V. Arnold's Classical Mechanics. What then did Poincare do to exert all this influence? Here is a little list - far from complete. 1) Qualitative Dynamics Generic behavior of flows as a whole, the classification of phase portraits. 2) Ergodic Theory Probabilistic notions, recurrence theorem. 3) Existence of Periodic Orbits; Detailed Analysis of the Structure of a Flow Near a Periodic Orbit. 4) Bifurcation Theory General ideas for systematic theory; detailed study of rotating fluid with gravitational attraction. First, I will comment briefly on 2). It sounds somewhat anachronistic to call Poincare a pioneer of ergodic theory but there is a sense in which it is true. In that sense, the first theorem of ergodic theory was the invariance of the Liouville measure while the second was Poincare's Recurrence Theorem. By the invariance of the Liouville measure. I refer to the fact that defines a measure on 2n-dimensional phase space invariant under INTRODUCTION TO THE PROBLEMS 3 the flow defined by a Hamiltonian system of differential equations ClH i 1, ... n Clq. ' 1 In modern language, the recurrence theorem can be stated as follows Theorem Let T be a mapping of a phase space M into itself which pre- serves a measure ~ on M: -1 ~(X) = ~(T X) for any measurable subset X of M Suppose ~ is finite i.e. ~(M) < 00 Then, if A is any measurable subset of M, almost every point x of A returns to A infinitely often i.e. for an infinite set of posi- tive integers, n, Tnx £ A. Poincare emphasized that his proof required only the finite- ness and invariance of his measure, although the argument used the language of the theory of incompressible fluids. He had al- ready gone far in the direction of generality in these matters by introducing the general notion of integral invariants. These are invariant integrals of differential forms over subsets of M. Incidentally, for those who may wish to read the original, I should note that Poincare did not call this result a recurrence theorem; he referred to it as stabilite ! la Poisson. You can find it, along with a magistral exposition of his theory of inte- gral invariants in his Prize Memoir which won (21 January 1889) the Prize offered by King Oscar II of Sweden. It is published in Acta Math 13 (1890) 1-270. It is interesting to compareiliis stunningly general result with what was going on in physics at that time. Maxwell and Boltzmann had constructed statistical models of gases leading to quantitative predictions of thermodynamic phenomena, and Boltzmann had pub- lished a proof of the so-called H-Theorem giving a mechanical interpretation of the increase of entropy in accord with the Second Law of Thermodynamics. Boltzmann's proof was greeted with skepticism because of the Recurrence Theorem and the invariance under time inversion of the usual Hamiltonian models. Both Maxwell and Boltzmann made independent efforts to justify statistical pro- cedures on the basis of what Boltzmann called the Ergodic Hypothesis: