Nonlinear Dynamics and Quantum Chaos - An Introduction [2 ed.] 9783031012488, 9783031012495

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Graduate Texts in Physics

Sandro Wimberger

Nonlinear Dynamics and Quantum Chaos An Introduction Second Edition

Graduate Texts in Physics Series Editors Kurt H. Becker, NYU Polytechnic School of Engineering, Brooklyn, NY, USA Jean-Marc Di Meglio, Matière et Systèmes Complexes, Bâtiment Condorcet, Université Paris Diderot, Paris, France Sadri Hassani, Department of Physics, Illinois State University, Normal, IL, USA Morten Hjorth-Jensen, Department of Physics, Blindern, University of Oslo, Oslo, Norway Bill Munro, NTT Basic Research Laboratories, Atsugi, Japan Richard Needs, Cavendish Laboratory, University of Cambridge, Cambridge, UK William T. Rhodes, Department of Computer and Electrical Engineering and Computer Science, Florida Atlantic University, Boca Raton, FL, USA Susan Scott, Australian National University, Acton, Australia H. Eugene Stanley, Center for Polymer Studies, Physics Department, Boston University, Boston, MA, USA Martin Stutzmann, Walter Schottky Institute, Technical University of Munich, Garching, Germany Andreas Wipf, Institute of Theoretical Physics, Friedrich-Schiller-University Jena, Jena, Germany

Graduate Texts in Physics publishes core learning/teaching material for graduateand advanced-level undergraduate courses on topics of current and emerging fields within physics, both pure and applied. These textbooks serve students at the MSor PhD-level and their instructors as comprehensive sources of principles, definitions, derivations, experiments and applications (as relevant) for their mastery and teaching, respectively. International in scope and relevance, the textbooks correspond to course syllabi sufficiently to serve as required reading. Their didactic style, comprehensiveness and coverage of fundamental material also make them suitable as introductions or references for scientists entering, or requiring timely knowledge of, a research field.

Sandro Wimberger

Nonlinear Dynamics and Quantum Chaos An Introduction Second Edition

Sandro Wimberger Department of Mathematical, Physical and Computer Sciences University of Parma Parma, Italy

ISSN 1868-4513 ISSN 1868-4521 (electronic) Graduate Texts in Physics ISBN 978-3-031-01248-8 ISBN 978-3-031-01249-5 (eBook) https://doi.org/10.1007/978-3-031-01249-5 1st edition: © Springer International Publishing Switzerland 2014 2nd edition: © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

I dedicate this book to Atena and Dario, and to the scientific curiosity.

Foreword to the First Edition

The field of nonlinear dynamics and chaos has grown very much over the last few decades and is becoming more and more relevant in different disciplines. Nonlinear dynamics is no longer a field of interest to specialists. A basic knowledge of the main properties of nonlinear systems is required today in a wide range of disciplines from chemistry to economy, from medicine to social sciences. The point is that more than three hundred years after that Newton equations had been formulated and after more than a hundred years of quantum mechanics, a general understanding of the qualitative properties of the solutions of classical and quantum equations should be a common knowledge. The present book is intended for such a purpose. It presents a clear and concise introduction to the field of nonlinear dynamics and chaos, suitable for graduate students in mathematics, physics, chemistry, engineering, and natural sciences in general. Also, students interested in quantum technologies and quantum information will find this book particularly stimulating. This volume covers a wide range of topics usually not found in similar books. Indeed, dissipative and conservative systems are discussed, with more emphasis on the latter ones since, in a sense, they are more fundamental. The problem of the emergence of chaos in the classical world is discussed in detail. The second part of this volume is devoted to the quantum world, which is introduced via a semiclassical approach. Both the stationary aspects of quantum mechanics as well as the time-dependent quantum properties are discussed. This book is therefore a valuable and useful guide for undergraduate and graduate students in natural sciences. It is also very useful as a reference for researchers in the field of classical and quantum chaos. Como, Italy December 2013

Giulio Casati

vii

Preface to the Second Edition

After eight years of its use as a coursebook that has served well its author and, hopefully, some of my colleagues and students, a new edition has been called for. Thanks to numerous hints by my students and colleagues, a substantial number of minor errors, mostly typos, is corrected. In particular, I thank Rémy Dubertrand and Peter Schlagheck for useful corrections, and Jörg Main for his kind review of the first edition. The biggest change is certainly the addition of the new Chap. 4 on dissipative classical systems. This new chapter is relevant in particular to engineers and biological physicists. The former Chap. 4 moved to Chap. 5 correspondingly. Wherever useful, the new chapter has been linked to the rest of the book to guarantee its integration. Special thanks goes to Michele Delvecchio for his help in the preparation of some of the new figures and to Mogens Høgh Jensen for granting the copyright to reprint and providing a color version of Fig. 4.38. I am very grateful to Hisako Niko from Springer Nature for her support and encouragement. Parma, Italy March 2022

Sandro Wimberger

ix

Preface to the First Edition

This book developed from the urgent need of a text for students in their undergraduate and graduate career. While many excellent books on classical chaos as well as on quantum chaos are on the market, only a joint collection of some of them could be proposed to the students from my experience. Here, I try to give a coherent but concise introduction to the subject of classical nonlinear dynamics and quantum chaos on an equal footing, and adapted to a four hour semester course. The stage is set by a brief introduction into the terminology of the physical description of nonintegrable problems. Chapter 2 may as well be seen as part of the introduction. It presents the definition of dynamical systems in general, and useful concepts which are introduced while discussing simple examples of one-dimensional mappings. The core of the book is divided into the two main chapters 3 and 4, which discuss classical and quantum aspects, repsectively. Both chapters are linked wherever possible to stress the connections between classical mechanics, semiclassics, and a pure quantum approach. All the chapters contain problems which help the reader to consolidate the knowledge (hopefully!) gained from this book. Readers will optimally profit from the book if there are familar with the basic concepts of classical and quantum mechanics. The best preparation would be a theory course on classical mechanics, including the Lagrange and Hamiltonian formalism, and any introductory course on quantum theory, may it be theoretical or experimental. This book could never have been prepared without the precious guidance of Tobias Schwaibold, Aldo Rampioni and Christian Caron from Springer. I am very grateful for their support and patience. I acknowledge also the help of my students in preparing this text, in particular Andreas Deuchert, Stephan Burkhardt, Benedikt Probst and Felix Ziegler. Many important comments on the manuscript came from my colleagues in Heidelberg, Heinz Rothe and Michael Schmidt, as well as from Giulio Casati, Oliver Morsch and Vyacheslav Shatokhin, to all of whom I am very grateful. Finally, I thank my teachers in the very subject of the book, Andreas Buchleitner, Detlef Dürr, Italo Guarneri, Shmuel Fishman and Peter Schlagheck, for their continuous support. Heidelberg, Germany December 2013

Sandro Wimberger xi

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Fundamental Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Classical Versus Quantum Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 5 7 7

2 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Evolution Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 One-Dimensional Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Logistic Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 The Dyadic Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Deterministic Random Number Generators . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 11 11 15 17 18 19

3 Nonlinear Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Integrable Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Hamiltonian Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Important Techniques in the Hamiltonian Formalism . . . . . . . . . . . . 3.3.1 Conserved Quantity H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Canonical Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Liouville’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Hamilton-Jacobi Theory and Action-Angle Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Integrable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Poisson Brackets and Constants of Motion . . . . . . . . . . . . . 3.5 Non-Integrable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Perturbation of Low-Dimensional Systems . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Adiabatic Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Principle of Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Canonical Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 One-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 25 26 26 28 30 32 35 38 38 39 40 40 42 46 46 47 xiii

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Contents

3.7.3 Problem of Small Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.7.4 KAM Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.7.5 Example: Two Degrees of Freedom . . . . . . . . . . . . . . . . . . . . 54 3.7.6 Secular Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.8 Transition to Chaos in Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . 59 3.8.1 Surface of Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.8.2 Poincaré-Cartan Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.8.3 Area Preserving Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.8.4 Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.8.5 Poincaré–Birkhoff Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.8.6 Dynamics near Unstable Fixed Points . . . . . . . . . . . . . . . . . . 73 3.8.7 Mixed Regular-Chaotic Phase Space . . . . . . . . . . . . . . . . . . . 76 3.9 Criteria for Local and Global Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.9.1 Ergodocity and Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.9.2 The Maximal Lyapunov Exponent . . . . . . . . . . . . . . . . . . . . . . 86 3.9.3 Numerical Computation of the Maximal Lyapunov Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.9.4 The Lyapunov Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.9.5 Kolmogorov-Sinai Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.9.6 Resonance Overlap Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4 Dissipative Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Fixed Point Scenarios in Two-Dimensional Systems . . . . 4.3 Damped One-Dimensional Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Nonlinear Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Nonlinear Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Poincaré–Bendixson Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Damped Forced Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Driven One-Dimensional Harmonic Oscillator . . . . . . . . . . 4.5.2 Duffing Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Lorenz Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Simple Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Box-Counting Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.3 Examples from Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Bifurcation Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Examples of Pitchfork Bifurcations . . . . . . . . . . . . . . . . . . . . 4.8.2 Tangent Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.3 Transcritical Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 103 105 106 108 109 110 110 112 113 114 115 120 125 125 127 128 129 129 131 131

Contents

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4.8.4 Higher-Order Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.5 Hopf Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Two Routes to Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.1 Landau’s Transition to Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.2 Ruelle–Takens–Newhouse Route to Chaos . . . . . . . . . . . . . . 4.10 Intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Coupled Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11.1 Circle Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11.2 Arnold Tongues and Farey Trees . . . . . . . . . . . . . . . . . . . . . . . 4.11.3 Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11.4 Kuramoto Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Increasing Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

132 132 134 134 135 135 140 140 141 144 145 147 148 150

5 Aspects of Quantum Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introductory Remarks on Quantum Mechanics . . . . . . . . . . . . . . . . . . 5.2 Semiclassical Quantization of Integrable Systems . . . . . . . . . . . . . . . 5.2.1 Bohr–Sommerfeld Quantization . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Wentzel–Kramer–Brillouin–Jeffreys Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Einstein–Brillouin–Keller Quantization . . . . . . . . . . . . . . . . . 5.2.4 Semiclassical Wave Function for Higher-Dimensional Integrable Systems . . . . . . . . . . . . 5.3 Semiclassical Description of Non-Integrable Systems . . . . . . . . . . . 5.3.1 Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Feynman Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Method of Stationary Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Van Vleck Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Semiclassical Green Function . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.6 Gutzwiller’s Trace Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.7 Applications of Semiclassical Theory . . . . . . . . . . . . . . . . . . 5.4 Wave Functions in Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Phase-Space Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Weyl Transform and Wigner Function . . . . . . . . . . . . . . . . . . 5.4.3 Localization Around Classical Phase-Space Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Anderson and Dynamical Localization . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Anderson Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Dynamical Localization in Periodically Driven Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Universal Level Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Level Repulsion: Avoided Crossings . . . . . . . . . . . . . . . . . . .

153 153 155 155 156 168 171 175 175 177 180 181 183 184 189 197 198 199 203 207 207 211 215 215 216

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5.6.2 Level Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 Symmetries and Constants of Motion . . . . . . . . . . . . . . . . . . 5.6.4 Density of States and Unfolding of Spectra . . . . . . . . . . . . . 5.6.5 Nearest-Neighbour Statistics for Integrable Systems . . . . 5.6.6 Nearest-Neighbour Statistics for Chaotic Systems . . . . . . 5.6.7 Gaussian Ensembles of Random Matrices . . . . . . . . . . . . . . 5.6.8 More Sophisticated Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

218 219 221 222 224 227 231 237 238 239 248

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

1

Introduction

Abstract

We set the stage for our discussion of classical and quantum dynamical systems. Important notions are introduced, and similarities and differences of classical and quantum descriptions are sketched.

1.1

Fundamental Terminology c

The notion dynamics derives from the Greek word η δ υναμις ´ , which has the meaning of force or power. From a physical point of view we mean a change of linear momentum according to Newton’s second law F=

dp , dt

(1.1.1)

where the momentum p in the absence of an electromagnetic field is usually just the product of a fixed mass times velocity. If there is no external force and the motion occurs at constant speed one speaks of “kinematics”, whereas if there is a static equilibrium, in the simplest case of zero velocity, one speaks of “statics”. Chaos roots in the ancient Greek word τ o` χ αoς ´ and means chaos in a sense of beginning or primary matter (unformed, empty). In this sense it is used in Greek mythology, where it designates the state of the world before all arranged order (before c ´ μoς, see footnote 27 in [1]). This meaning of chaos the creation of the cosmos o κ oσ is close to the colloquial use of the word today. Here we will focus on deterministic chaos.1 What does deterministic mean here? We mean that each initial condition, consisting of a pair of position and velocity (x0 , v0 ) at some given time t = t0 , of a differential equation is mapped in a unique way onto some final state at time t > t0 : (1.1.2) (x0 , v0 ) −→ (x(t), v(t)).

1 From

the Latin word determinare meaning to determine.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. Wimberger, Nonlinear Dynamics and Quantum Chaos, Graduate Texts in Physics, https://doi.org/10.1007/978-3-031-01249-5_1

1

2

1

Introduction

Since the initial condition determines the dynamics at all times such systems are predictable. Unfortunately, this is true only in a mathematical sense but problematic in practice. One may show that a solution of Newton’s equations of motion (1.1.1) exists and that it is unique. But things change if one has to compute a solution when either (a) the initial condition is known only with a certain accuracy or (b) a computer (which always uses a reduced number system) is needed for the calculation. It turns out that there are differential equations (in fact most of them) whose solutions strongly depend on the initial conditions. When the computation is started with just slightly varied initial data the trajectories may differ substantially (from a point of view of shape and relative distance) already after a short time. Hence in all practical applications these systems turn out to be unpredictable. The weather forecast, for instance, usually breaks down after a few days (chaotic dynamics), whereas the earth has been traveling around the sun in 365.25 days for quite a long time now (quasi-regular dynamics). When one is describing the reality with physical theories the notion of universality plays a crucial role: Ein historisches Kriterium für die Eigenart der Prinzipien kann auch darin bestehen, dass immer wieder in der Geschichte des philosophischen und naturwissenschaftlichen Erkennens der Versuch hervortritt, ihnen die höchste Form der „Universalität“ zuzusprechen, d.h. sie in irgendeiner Form mit dem allgemeinen Kausalsatz selbst zu identifizieren oder aus ihm unmittelbar abzuleiten. Es zeigt sich hierbei stets von neuem, dass und warum eine solche Ableitung nicht gelingen kann – aber die Tendenz zu ihr bleibt nichtsdestoweniger fortbestehen. [Ernst Cassirer: Determinismus und Indeterminismus in der modernen Physik. Historische und systematische Studien zum Kausalproblem. Gesammelte Werke Bd. 19, Herausgegeben von Birgit Recki, Meiner, 2004, S. 69f]

Translation: A historical criterion for the specific character of principles may also be found in the fact that in the development of philosophical and scientific knowledge the attempt is made repeatedly to ascribe to them the highest form of “universality” – that is, to identify them in some way with the general causal principle or to derive them immediately from it. In these attempts it becomes evident again and again why such derivation cannot succeed – but the tendency to attempt it nevertheless continues unabated. [Ernst Cassirer: Determinism and indeterminism in modern physics: historical and systematic studies of the problem of causality. Yale University Press, 1956, p. 55]

Universality in physics describes the phenomenon that certain properties in mechanical systems are independent of the details of the system. Based on this similar qualitative or quantitative behaviour, it is possible to group these systems into universality classes. These contain different systems that show the same universal behaviour (described by the so-called universal quantities). As an example we mention a statistical system close to a phase transition. Its behaviour is described by a critical exponent that is actually the same for a large number of thermodynamical systems. Therefore the critical exponent constitutes a universal quantity in order to

1.1 Fundamental Terminology

3

Fig. 1.1 Ways of categorising dynamical systems

classify phase transitions. For further information we refer to statistical mechanics, which is excellently explained e.g. in [2]. Universality may show up in physics in many ways. Figure 1.1 summarises what we mean by it in the present context of dynamical systems. For classical Hamiltonian problems, the various classes of dynamics will be introduced and discussed in Chap. 3. The number of degrees of freedom n and the number of conserved quantities s determine the possible dynamics of a physical system. As space dimension d one usually has d = 1, 2, 3. In the Hamiltonian description, see Sect. 3.2, this leads to a phase space of dimension d P = 2 × d = 2, 4, 6 per particle. Hence, the phase space of N particles moving in three space dimensions has the dimension d P = 2 × 3N = 6N .

(1.1.3)

Thereby n = 3N is the number of degrees of freedom and the factor of two in the formula arises because each point particle is described by a position and a generalized momentum. In Sect. 3.4 we will see that for s = n the motion is always regular. We give two examples: • One particle in one space dimension: If the system is conservative (no explicit time-dependence) the energy is conserved. Since there is only one degree of freedom the motion will always be regular, see Sect. 3.4. • One particle in two space dimensions: If the energy is conserved, we have one constant of motion. For the motion to be integrable we would need yet another conserved quantity such as, for example, the momentum in one direction, or the angular momentum. An example where this is not the case is a particle moving on a billiard table with defocusing boundaries (accordingly the motion of typical orbits is chaotic). On the other hand, if the table has the shape of a circle the dynamics will be regular again (rotational symmetry then leads to angular momentum conservation).

4

1

1.2

Introduction

Complexity

Many physical systems are complex systems. As their name already says, complex systems consist of a large number of constituents or of coupled degrees of freedom. The root of the work is the ancient Greek π λ´εκειν which, together with the latin cum, means to knit together or to intertwine different parts. Complex systems appear in every day life, e.g. as traffic jams, and in many disciplines besides physics such as biology and chemistry [3]. Being more than just the sum of their parts, complex systems may show emergent phenomena, see for instance the wonderful article by P. W. Anderson [4]. In this book we use the notion of complexity in a restricted sense of physical phenomena which are described by finite systems of coupled differential equations. For classical systems, these equations are typically nonlinear ordinary differential equations that are hard to solve by hand. In quantum theory one has to deal with the even harder problem of partial differential equations in general. Two simple examples of classical mechanics, for which the corresponding nonlinear equations of motions can be solved analytically, are • A pendulum in the gravity field: Let the angle Θ describe the deflection; the equation of motion reads: ¨ Θ(t) +

g sin(Θ(t)) = 0, L

(1.2.1)

where g is the gravitational acceleration and L is the length of the pendulum. • The two-body Kepler problem: The relative coordinate r of the two centre of masses follows Newton’s equation of motion r¨ +

γM r = 0. r3

(1.2.2)

Here γ is the gravitational constant and M = m 1 + m 2 denotes the sum of the two masses [5–7]. Of course, these systems are not examples of complex systems. Over the last 150 years there has been a huge interest and progress in the area of nonlinear dynamics of strongly coupled systems. As an important example, we mention the study of our solar system, that means of a classical many-body problem with gravitational interaction. Involved and extensive computations have been carried out in order to find fixed points of the highly nontrivial dynamics (e.g. Lagrangian points [8,9]). Questions of predictability and stability of the motion of asteroids or other objects, which might hit the earth, are of such importance that many research programs exist just for this purpose, see e.g. [10]. After the solvable two-body case the three-body Kepler problem is the next model in order of increasing complexity. Already in the nineteenth century Henri Poincaré investigated its dynamics in great detail. He first showed strong hints for the nonintegrability of the global problem. All proofs of its non-integrability are in general

1.3 Classical Versus Quantum Dynamics

5

not simple and often rely on restrictive notions of non-integrability [11]. Possible singularities arise from three-body collisions that complicate the analytical treatment, see e.g. [11–13]. Being one of the first examples of possibly highly complicated dynamics that has been investigated so extensively, its study marked the beginning of the field of nonlinear and chaotic dynamics. General N -body Kepler problems in classical or in quantum mechanics can only be treated with perturbative methods, whose application range is typically restricted. Numerical techniques may be applied too. Computer simulations are indeed a standard tool nowadays. They are heavily used also in this book, in particular in Chaps. 2–4, as well as for the examples in Chap. 5. Numerical simulations are very well suited, for example, • for “Experimental mathematics”, • for computing solutions in a certain (possibly large) parameter range (over finite times), • in order to find qualitative results which sometimes lead to a better understanding or even new theoretical methods. The reliability of computational techniques (with unavoidable round-off errors) for chaotic systems is mathematically guaranteed by the so-called shadowing theorem for hyperbolic systems, discussed, for instance, in [14,15].

1.3

Classical Versus Quantum Dynamics

Complex systems are equally important in the microscopic world, where the laws of quantum mechanics apply. Here a strong coupling between the degrees of freedom leads to the breakdown of approximations that use separable wave functions.2 Results for strongly coupled or strongly distorted systems have been achieved, for instance, in • Atomic physics: Rydberg atoms in time-dependent strong electromagnetic fields [16], • Solid state physics: strongly interacting many-particle systems [17], • Complex atomic nuclei (consisting of many nucleons which are heavily interacting): For them a universal description was introduced based on ensembles of random matrices [18,19]. In this book we treat classical and quantum mechanical systems. So let us outline here already some relations between the two fields of physics:

wave function Ψ (x1 , x2 ) is called separable if it is of the form Ψ (x1 , x2 ) = Φ1 (x1 )Φ2 (x2 ). Formally this is true only if either the particles denoted by 1 and 2 are non-interacting and distinguishable or the potential is separable in the two degrees of freedom x1 and x2 of a single-particle problem.

2A

6

1

Introduction

• The semiclassical limit:  → 0. – The correspondence between classical and quantum dynamics is good if the dimensionless quantity (Action of the classical path)/(2π ) is a large number. This implies a large number of quantum levels per unit Planck cell, i.e. a high density of levels (or states) in the semiclassical picture, c.f. Sect. 5.2.2. The short-hand notation  → 0 implies exactly this approximation. – The approximation just discussed introduces a natural time scale (which depends on the effective ). Hence, one has to be careful when treating timedependent systems because the two limits  → 0 and t → ∞ do not commute in general. A simultaneous limit procedure may be nevertheless possible under certain circumstances, see e.g. [20]. • Decoherence: The destruction of superpositions. – Describes the crossover from quantum to classical mechanics. – Attention: The semiclassical limit and the phenomenon of decoherence [21] are not equivalent. An important open question: What happens if decoherence and semiclassics apply simultaneously? We will, however, not treat decohering quantum systems but restrict ourselves to classical Hamiltonian problems and their quantum counterparts (see e.g. [22] for the treatment of dissipative dynamical quantum systems). Purely classical dissipative systems are partly treated in Chap. 2 and more extendedly in Chap. 4. • Statistical comparison: In classical mechanics the motion of particles is described by phase-space trajectories and the description of a statistical ensemble uses probability distributions on phase space, which may mimic the evolution of a quantum mechanical wave packet of comparable size to some extent. In quantum mechanics one can also define a quasi-probability distribution on the phase space by, e.g., Wigner functions, see Sect. 5.4.2. It is then possible to compare the effects of classical and quantum mechanics by comparing these two functions. It turns out, for example, that non-classical states can be characterized by the fact that the Wigner function takes negative values (which is not possible for a probability distribution in the strict mathematical sense). Chaos is a phenomenon that is present in many fields of classical physics (e.g. in dynamics of stars and fluids and in biophysical models [14,15,23,24]). Since quantum mechanics is the more fundamental theory we can ask ourselves whether there is chaotic motion in quantum systems as well. A key ingredient of the chaotic phenomenology is the sensitive dependence of the time evolution upon the initial conditions. The Schrödinger equation is a linear wave equation, implying also a linear time evolution. The consequence of linearity is that a small distortion of the initial conditions leads only to a small and constant change in the wave function at all times (see Sect. 5.1). Certainly, this is not what we mean when talking about “quantum chaos”. But how is then chaos created in the classical limit and is there another notion of quantum chaos? In general, a rigorous definition of quantum chaos,

References

7

or “quantum chaology” (Michael Berry, [25,26]) is much more difficult than defining chaos in classical systems. Usually, one computes corresponding properties of a classical and a quantum mechanical system and identifies connections. When treating billiard systems (particles moving on a billiard table), for example, one finds a more pronounced amplitude of the wave function (and therefore a more pronounced density) in regions where the classical system has a periodic orbit, c.f. Sect. 5.4.3.3. This is the semiclassical approach presented in the first part of Chap. 5. It is also possible to characterize quantum mechanical systems by the analysis of their eigenspectra. This approach uses a statistical description based on ensembles of random matrices, whose properties are compared with the ones of real quantum systems. Both approaches may be reconciled, as shown in Sect. 5.6, where we discuss the quantum spectra for regular and chaotic systems in some detail.

Problems 1.1 Try to think about possible ways to quantify the properties of a complex system. Think, in particular, about the dynamics of simple physical problems, such as a billiard ball moving in a plane and confined by reflecting hard walls. What is the origin of complicated dynamical behaviour?

References 1. Humboldt, A.V.: Kosmos. Entwurf einer physikalischen Weltbeschreibung. Cotta, Stuttgart (1845) 2. Schwabl, F.: Statistische Mechanik. Springer, Heidelberg (2004) 3. Mallamace, F., Stanley, H.E. (eds.): The physics of complex systems. In: Proceedings of the International School of Physics ‘E. Fermi’, vol. Course XIX. IOS Press, Amsterdam (1996) 4. Anderson, P.W.: Science 177(4047), 393 (1972) 5. Scheck, F.: Mechanics: from Newton’s Laws to Deterministic Chaos. Springer, Heidelberg (2007) 6. Rebhan, E.: Theoretische Physik: Mechanik. Spektrum Akademischer Verlag, Heidelberg (2006) 7. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989) 8. Finn, J.M.: Classical Mechanics. Jones and Burtlett, Boston (2010) 9. Murray, C.D.: Solar System Dynamics. Cambridge University Press, Cambridge (2012) 10. Near-Earth Object Program – NASA. http://neo.jpl.nasa.gov 11. Chenciner, A.: Scholarpedia 2(10), 2011 (2007) 12. Henkel, M.: (2002). arXiv:physics/0203001 13. Wang, Q.D.: Celest. Mech. Dyn. Astron. 50, 73 (1991) 14. Lichtenberg, A.J., Lieberman, M.A.: Regular and Chaotic Dynamics. Springer, Berlin (1992) 15. Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (2002) 16. Buchleitner, A., Delande, D., Zakrzewski, J.: Phys. Rep. 368(5), 409 (2002) 17. Mahan, G.D.: Many Particle Physics (Physics of Solids and Liquids). Kluwer Academic/Plenum Publishers, New York (2000) 18. Guhr, T., Müller-Gröling, A., Weidenmüller, H.A.: Phys. Rep. 299(4–6), 189 (1998) 19. Bohigas, O., Giannoni, M.J., Schmit, C.: Phys. Rev. Lett. 52, 1 (1984) 20. Abb, M., Guarneri, I., Wimberger, S.: Phys. Rev. E 80, 035206 (2009)

8

1

Introduction

21. Hornberger, K.: In: Buchleitner, A., Viviescas, C., Tiersch, M. (eds.) Entanglement and Decoherence. Lecture Notes in Physics, vol. 768. Springer, Berlin (2009) 22. Haake, F.: Quantum Signatures of Chaos (Springer Series in Synergetics). Springer, Berlin (2010) 23. Strogatz, S.H.: Nonlinear Dynamics and Chaos. Westview Press, Boston (1994) 24. Thompson, J.M.T., Stewart, H.B.: Nonlinear Dynamics and Chaos. Wiley, Chichester (2002) 25. Berry, M.V.: Proc. R. Soc. Lond. A: Math. Phys. Sci. 413(1844), 183 (1987) 26. Berry, M.V.: Phys. Scr. 40(3), 335 (1989)

2

Dynamical Systems

Abstract

Dynamical systems are formally defined—may they be classical or quantum. We introduce important concepts for the analysis of classical nonlinear systems. In order to focus on the essential questions, this chapter restricts to one-dimensional discrete maps as relatively simple examples of dynamical systems. Chapter 4 will extend the discussion of dissipative dynamical systems, in particular, to more physical applications in two or more dimensions.

2.1

Evolution Law

A dynamical system is given by a set of states Ω and an evolution law telling us how to propagate these states in (discrete or continuous) time. Let us assume that the propagation law is homogeneous in time. That means it depends on the initial state but not on the initial time. Mathematically speaking a dynamical system is given by a one-parameter flow or map T : G × Ω → Ω, (g, h ∈ G, ω ∈ Ω) → T g (ω) ∈ Ω,

(2.1.1)

such that the composition is given by T g ◦ T h = T g+h .

(2.1.2)

For a discrete dynamical system we have G = N or G = Z (discrete time steps) and for a continuous dynamical system G = R+ or G = R (continuous time). The set Ω contains all possible states. It describes the physical reality one wants to model and is called the phase space. From an algebraic point of view the above defined maps form a semi-group that operates on Ω. If the map T is invertible for all t ∈ G this

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. Wimberger, Nonlinear Dynamics and Quantum Chaos, Graduate Texts in Physics, https://doi.org/10.1007/978-3-031-01249-5_2

9

10

2

Dynamical Systems

structure extends to a group1 and we say that the dynamical system is invertible. The sets T t (ω) t∈G define orbits or trajectories of the map. They can be discrete or continuous, finite or infinite. The above definitions are quite abstract and therefore we give a few examples: 1. The simplest example is a discrete dynamical system defined by an iterated map. Let f be a map of the interval Ω onto itself. We define T n = f ◦ f ◦ ... ◦ f , G = N.   

(2.1.3)

n times

If the map f is invertible so is the dynamical system, and we can extend time to G = Z. Two concrete examples of such discrete maps will be given in Sect. 2.2. 2. An example from classical physics is the motion of N particles in three space dimensions. The dynamics are governed by Newton’s equations of motion for the vector of positions x(t) m x¨ (t) = F(x(t)).

(2.1.4)

Defining the composite vector y(t) = (y1 (t) ≡ x(t), y2 (t) ≡ x˙ (t)) and f (y(t)) = (y1 (t), F(y1 (t))/m), we obtain y˙ (t) = f (y(t)).

(2.1.5)

To make the connection between this formulation and the definition of dynamical systems we write the solution as T t (y(0)) = y(t). We obtain the solution in one step but also in two steps if we insert the end of the first trajectory as an initial condition into the second trajectory,2 i.e. T s+t (y(0)) = T t (T s (y(0))) = (T t ◦ T s )(y(0)). Note that x(t) is an element of the configuration space R3N while y(t) is an element of the phase space Ω = R6N . Equation (2.1.5) is equivalent to the Hamiltonian formulation of the problem. Motivated by our general definition (2.1.1) and (2.1.2) we see that this is actually the natural way to treat classical dynamics. We will therefore use the formalism of Hamiltonian mechanics (Sect. 3.2), operating in phase space rather than configuration space, throughout the Chap. 3.

1 Translational

invariance of the flow with respect to the time parameter g is only given if the generator of the dynamics is itself time-independent. Any classical problem can formally be made time-independent, see Sect. 3.3.1.2; hence the property of a translationally invariant group is always obeyed in this generalized sense. The quantum evolution for periodically time-dependent systems can also be cast in a similar way using a theorem of Floquet [1]. For generally time-dependent quantum systems, time ordering [2] must be used to formally write down the evolution law. 2 A formal proof is found in [3] based on the fact that every point in phase space has a unique time evolution.

2.2 One-Dimensional Maps

11

3. The time evolution of a quantum mechanical spinless particle in three space dimensions with time-independent hermitian Hamiltonian Hˆ . Here Ψ0 ∈ L 2 (R3 ) = Ω and ˆ

T t (Ψ0 ) = Uˆ (t)Ψ0 = e−i H t/ Ψ0 .

(2.1.6)

The so-defined dynamical system is obviously invertible, with T −t = Uˆ † (t). 4. Non-invertible quantum dynamics: A reduced quantum mechanical system (a subsystem of a larger one) can—under certain conditions—be described by a master equation for the density operator of the system

 i  ˙ˆ ˆ + Lˆ ρ(t) ˆ . ρ(t) = − Hˆ , ρ(t) 

(2.1.7)

In this case the time evolution has only the property of a semigroup. With the above equation it is possible to model dissipation and decoherence by the Lindblad operator Lˆ , whilst the coherent evolution is induced by the first term on the right hand side of the equation. For more information see, e.g., [4–6]. Replacing the density operator by a classical density distribution in phase space one may model the corresponding quantum evolution to some extent. On the classical level one then has to deal with a Fokker–Planck equation for phase-space densities describing irreversible motion [7].

2.2

One-Dimensional Maps

In the following we discuss two seemingly simple discrete dynamical systems. Those are not Hamiltonian systems but one-dimensional mappings, i.e. the phase space is just one-dimensional. Yet, it will turn out that some general concepts can be easily introduced with the help of such maps, e.g. fixed points and their stability or periodic orbits. They have also the great advantage that much can be shown rigorously for them [8,9]. Therefore, one-dimensional discrete maps form one of the bases for the mathematical theory of dynamical systems, see for instance [10].

2.2.1

The Logistic Map

The logistic map is a versatile and well understood example of a discrete dynamical map which was introduced in 1838 by Pierre Francois Verhulst as a mathematical model for demographic evolution [11]. Its nonlinear iteration equation is given by the formula yn+1 = Ryn (M − yn ),

(2.2.1)

where yn is a population at time n, R ≥ 0 is a growth rate and M is an upper bound for the population. The population at the next time step is proportional to

12

2

Dynamical Systems

the growth rate times the population (which alone would lead to exponential growth for R > 1/M) and to the available resources assumed to be given by M − yn . The system’s evolution is relatively simple for R < 1/M having the asymptotic solution limn→∞ yn = 0 (extinct population) for all initial conditions. For general values of the growth rate, the system shows a surprisingly complicated dynamical behaviour. Most interestingly, in some parameter regimes the motion becomes chaotic, which means that the population yn strongly depends on the initial condition y0 . Additionally, the system might not converge to an asymptotic value or show non-periodic behaviour. Let us now analyse the logistic map in more detail. In order to obtain the standard form of the logistic map we rescale the variable describing the population xn = yn /M and write the time step as the application of the evolution law T leading to T (x) = r x (1 − x), for x ∈ Ω = [0, 1].

(2.2.2)

Here r = M R. If we want T to map the interval Ω onto itself we have to choose r ∈ [0, 4]. First, we look for fixed points of the map T , that means points for which T (x ∗ ) = x ∗

⇔

x ∗ = r x ∗ (1 − x ∗ ).

(2.2.3)

holds. The above equation has two solutions:

∗ x1,1

⎧ ⎪ ⎨an attractive fixed point for = 0 is an indifferent fixed point for ⎪ ⎩ a repulsive fixed point for

r 1

(2.2.4)

and ∗ x1,2

⎧ ⎪ ⎨an attractive fixed point for 1 =1− is an indifferent fixed point for ⎪ r ⎩ a repulsive fixed point for

1 1. Assume, for instance, the two second-order fixed points x2,1 2,2 ∗ we find T (T (x ∗ )) = T (x ∗ ) = x ∗ . This behaviour is found we apply T 2 on x2,1 2,1 2,2 2,1 for all fixed points of order p > 1. Let x ∗p,i , i = 1, ..., p, be the p fixed points of order p (the number of fixed points always equals the order). One finds ∗ = x2,(1,2)

T p (x ∗p,1 ) = T (T (...(x ∗p,1 )...) = T (T (...(x ∗p,2 )...) = ... = T (x ∗p, p ) = x ∗p,1 .       p times

p−1 times

(2.2.8) Hence, if one fixed point of order p is given, the p − 1 other fixed points of that order can be computed by applying T repeatedly. We illustrate the described phenomenon in Fig. 2.2 for p = 2. The two fixed points of second order are mapped onto each other by T .

14

2

Dynamical Systems

1

T (x)

x2∗

x1∗

0

x1∗

0

x2∗

1

x

Fig. 2.2 T maps x1∗ onto x2∗ and vice versa. Accordingly, both x1∗ and x2∗ are fixed points of T 2 = T ◦ T . The plot was made for the parameter r = 3.214, giving x1∗ ≈ 0.5078 and x2∗ ≈ 0.8033 1 0.9 0.8 0.7

0.5

x

400

0.6

0.4

period 1

0.3

period 2

0.2

period 4

0.1 0

1

1.5

2

2.5

3

3.5

4

r Fig. 2.3 Bifurcation diagram of the logistic map. For each value on the abscissa, the value xn=400 ≈ x∞ , i.e. after 400 iterations of the map, for random initial data is represented on the ordinate. The route to chaos goes along a cascade of accumulating bifurcation points. Up to r < 3.5 stable fixed points of order 1, 2 and 4 are shown. Above r∞ ≈ 3.57 chaos develops as motivated in the text

The dynamical behaviour of the logistic map is summarized in its bifurcation diagram, see Fig. 2.3. For each value of r on the abscissa, the value of x is plotted after 400 iterations for random initial data. It turns out that outside the chaotic regime the asymptotic evolution does not depend on the initial conditions. For r < 1, the motion ∗ = 0. For 1 < r < 3, it goes to x ∗ = 1 − 1 for converges towards the fixed point x1,1 1,2 r almost every x0 . At r = 3 is a bifurcation point. For larger values of the growth rate,

2.2 One-Dimensional Maps

15

∗ the asymptotic dynamics converge towards the two fixed points of second order, x2,1 ∗ and x2,2 (defining a periodic orbit). This phenomenon is known as period doubling. In Fig. 2.3 we see both solutions, because x400 is plotted for many initial points. Note that, in the limit of many iterations, the dynamics do not converge to one of the fixed 3 The two fixed points of points, but the evolution jumps between them for all times. √ order two split again into two new ones at r = 1 + 6. This scheme repeats itself infinitely often while the distance between the bifurcation points decreases rapidly. It can be shown [8,13] that the ratios of the lengths between two subsequent bifurcation points approach a limiting value

lim δk =

k→∞

rk − rk−1 = 4.669201 . . . , rk+1 − rk

(2.2.9)

known as the Feigenbaum constant that is believed to be transcendental [14]. For most r beyond the critical value r∞ = 3.569945... (known as accumulation point) the system becomes chaotic, which means that the asymptotic evolution will not converge towards periodic orbits any more.4 Here the values x400 (for many random initial conditions) cover quasi-uniformly the whole phase space Ω = [0, 1]. Note that in this regime the motion strongly depends on the initial state x0 . Nevertheless there exist values 4 ≥ r > r∞ for which new attractive fixed points (of order 3, 5, 6, 7 ...) appear, see [9]. A period-3 cycle and its transition from stability to instability is treated later on in Sect. 4.10 in more detail. The route to chaos via a cascade of accumulating bifurcation points (period doublings) is a general phenomenon and not limited to the reported example of the logistic map. Chapter 4 will extend the discussion of this and other roads to chaos. Examples of so-called mixed Hamiltonian systems showing bifurcations of initially stable resonance islands with increasing perturbation are discussed in Sect. 3.8.7.

2.2.2

The Dyadic Map

Another important example of a discrete dynamical map showing chaotic behaviour is the dyadic map, also known as Bernoulli shift. It is defined by  T (x) =

3 When

2x 2x − 1

for 0 ≤ x < 21 . for 21 ≤ x ≤ 1

(2.2.10)

we say that the dynamics (for the initial condition x0 ) converge towards a periodic orbit with p elements (or towards the fixed point x ∗ of order p, which is an element of the periodic orbit) we mean that limn→∞ T np (x0 ) = x ∗ . 4 Also in the chaotic regime there exist fixed points but they are not attractive. Since at each point of a period doubling the fixed point does not vanish but only loses the property of attraction, the fixed points in the chaotic regime form a dense set (yet of Lebesgue measure zero in the interval [0, 1]).

16

2

Dynamical Systems

T (x) 1

0

1/2

1

x

Fig. 2.4 The dyadic map. As indicated by the arrows, the map is not one-to-one

Its phase space is the unit interval Ω = [0, 1]. When we represent the numbers x ∈ Ω ∞ xi 2−i , xi ∈ {0, 1}, the map can be interpreted in the binary representation x = i=1 as a shift of its binary digits: x = 0.x1 x2 x3 x4 . . . → T (x) = 0.x2 x3 x4 . . .

(2.2.11)

This explains also the name (Bernoulli) shift map. The dyadic map is displayed in Fig. 2.4. The dynamics of the dyadic map can be summarized as follows. If the initial condition is irrational, the motion will be non-periodic. Note that this is true for almost all5 initial values. For x0 ∈ Q, the evolved value converges towards zero if the binary representation of x0 is non-periodic and hence finite, or towards a periodic orbit if the representation shows periodicity. Hence, as for the logistic map, the fixed points in the chaotic regime form a set of Lebesgue measure zero. Being such a simple system the dyadic map can be solved exactly. The logistic map for r = 4 can be mapped onto the dyadic map. This topological conjugation6 is best shown by substituting in the logistic map xn =

5 The

1 (1 − cos(2π yn )) . 2

(2.2.12)

measure theoretical notion “for almost all” means for all but a set of Lebesgue measure zero [15]. 6 Two functions f and g are said to be topologically conjugate if there exists a homeomorphism h (continuous and invertible) that conjugates one into the other, in formulas g = h −1 ◦ f ◦ h. This is important in the theory of dynamical systems because the same must hold for the iterated system gn = h −1 ◦ f n ◦ h. Hence if one can solve one system, the solution of the other one follows immediately.

2.2 One-Dimensional Maps

17

Iterating then gives 1 {1 − cos (2π yn+1 )} = 4xn (1 − xn ) = {1 − cos(2π yn )} {1 + cos(2π yn )} 2 (2.2.13) 1 1 = {1 − cos(4π yn )} = {1 − cos(2π yn+1 )} . (2.2.14) 2 2

xn+1 =

From the last equality it follows that yn+1 = 2yn mod 1, which is just the dyadic map. As a consequence, we see that the logistic map for r = 4 does not tend towards an attractor since clearly the Bernoulli shift does not do so either. Exploiting the connection to the logistic map, the solution of the simpler dyadic map can be used to compute an analytical solution of the dynamics of the logistic map for r = 4 as well. It reads xn+1 = sin2 (2n Θ π ),

(2.2.15)

√ with Θ = π1 sin−1 ( x0 ) for the initial condition x0 , see [16,17]. Analyzing the dyadic map, it is particularly simple to see why the forecast of chaotic systems is so difficult. If one starts with a number that is only precisely known up to m digits, in the binary representation all information and therewith the predictability of the model is lost after m iterations. As a consequence, we can easily compute the rate of spreading of initially close points. This rate is known as Lyapunov exponent and defined by     1  xn (x0 ) − xn (x0 )  1  2−m+n  = lim lim ln  ln σ = lim lim  n→∞ m→∞ n  2−m  = ln 2. n→∞ m→∞ n 2−m (2.2.16) Here the initial conditions x0 and x0 differ only in the mth digit in (2.2.11). σ is called exponent since it characterizes the speed of exponential spreading as time evolves. We will discuss Lyapunov exponents in more detail in Sect. 3.9.

2.2.3

Deterministic Random Number Generators

A somewhat surprising application of chaotic maps is the deterministic generation of so-called pseudo-random numbers. Since the motion of a chaotic map depends sensitively on the initial conditions, a different series of numbers is generated for different initial values. If one does not start the dynamics at one of the fixed points (which form a set of measure zero anyhow), these series will neither be periodic nor be converging to a single point. Unfortunately, this scenario does not work on a computer, which needs to rely on a finite set of numbers,7 and therefore necessarily

7 This

set is normally composed exclusively of rational numbers, leading e.g. to non-chaotic behaviour for the dyadic map.

18

2

Dynamical Systems

produces periodic orbits at some stage. Nevertheless, so-called linear congruential generators, as generalizations of the dyadic map, can be used as low-quality random number generators: an+1 = N1 an mod N2 , an+1 bn+1 = , N2

(2.2.17) (2.2.18)

where N1 , N2 , an ∈ N [18]. This procedure generates “uncorrelated” and uniformly distributed numbers bn (n ∈ N) in the unit interval, but the constants N1 and N2 should be chosen very carefully in order to maximize the periodicity as well as to minimize correlations between subsequent numbers [19]. Other chaotic maps or coupled sets of them may also be used as pseudo-random number generators, see e.g. [20–25] for recent references on this subject. While such algorithms are generally very fast and easy to implement, they tend to have many problems (see Chap. 7.1 of [19]) and today other methods like the Mersenne Twister [26] have widely replaced them due to their much larger period (219937 − 1, not a mistake!) and the availability of efficient implementations [27]. It should be stressed that the performance of these random number generators needs to be analyzed using number theoretical and statistical tools, not the ones presented in this chapter. More information on random number generators based on computer algorithms can be found e.g. in [28,29]. As a final remark, we note that the best random number generators are hardware random number generator that are build on nature itself [30], e.g. on the physical process of a radioactive decay that occurs at truly random times in a large ensemble of non-correlated unstable particles [30]. Therewith, we conclude the examination of one-dimensional discrete maps which have been introduced as toy models to exemplify chaotic behaviour. In the next chapter we come to “real” physical applications of classical mechanics. Since even systems with one degree of freedom have a two-dimensional phase space and a continuous time evolution a priori, we expect a more complicated dynamical behaviour for them. In the next chapter, we will restrict ourselves exclusively to so-called Hamiltonian systems without friction or dissipation, whose time evolution is invertible. Chapter 4 then comes back to the discussion of some typical examples and applications of dissipative classical systems that can be compared to their conservative counterparts treated in Chap. 3.

Problems 2.1 Prove the following theorem: Let T (x) be a dynamical map, as introduced in Sect. 2.1, continuous and differentiable with respect to x. For a given fixed point x ∗ we have: If the map’s derivative with respect to x is T  (x)|x=x ∗  < 1, then x ∗ is attractive. 2.2 Find the fixed points of order two of the logistic map from (2.2.2) with r as a free parameter. Check also the stability of the fixed points found.

References

19

2.3 Given the following map 

T (x) = r 1 − (2x − 1)4 , for x ∈ Ω = [0, 1],

(2.2.19)

plot some iterations of the map for fixed r , and find (i) a maximum of fourth order and (ii) decide which range for the parameter r is reasonable. 2.4 Find all 1st and 2nd order fixed points of the dyadic shift map from (2.2.10). 2.5 The dyadic map can be generalised to maps with other slope parameters, e.g. to a triadic map of the following form  T (x) =

3x 3 2 (1 − x)

for 0 ≤ x < 13 . for 13 ≤ x ≤ 1

(2.2.20)

Determine the fixed points of this map and check their stability using the criterion from Problem 2.1 applied to the discrete evolution. Find also a cycle of period two and check its stability.

References 1. Shirley, J.H.: Phys. Rev. 138, B979 (1965) 2. Sakurai, J.J.: Modern Quantum Mechanics. Addison-Wesley Publishing Company, Reading (1994) 3. Teschl, G.: Ordinary Differential Equations and Dynamical Systems (Graduate Studies in Mathematics). American Mathematical Society, Providence (2012) 4. Hornberger, K.: In: Buchleitner, A., Viviescas, C., Tiersch, M. (eds.) Entanglement and Decoherence. Lecture Notes in Physics, vol. 768. Springer, Berlin (2009) 5. Breuer, P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, Oxford (2007) 6. Walls, D.F., Milburn, G.J.: Quantum Optics. Springer, Berlin (2008) 7. Risken, H.: The Fokker–Planck Equation: Methods of Solutions and Applications. Springer Series in Synergetics. Springer, Heidelberg (1996) 8. Schuster, H.G.: Deterministic Chaos. VCH, Weinheim (1988) 9. Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (2002) 10. Metzler, W.: Nichtlineare Dynamik und Chaos. Teubner, Stuttgart (1998) 11. Verhulst, P.F.: Corr. Math. Phys. 10, 113 (1838) 12. Weisstein, E.W.: Logistic map. MathWorld – A Wolfram Web Resource. http://mathworld. wolfram.com/LogisticMap.html 13. Feigenbaum, M.J.: J. Stat. Phys. 21, 669 (1979) 14. Briggs, K.: Feigenbaum scaling in discrete dynamical systems. Ph.D. thesis, University of Melbourne, Melbourne (1997) 15. Walter, W.: Analysis 2. Springer, Berlin (2002) 16. Little, M., Heesch, D.: J. Differ. Equ. Appl. 10(11), 949 (2004) 17. Schröder, E.: Math. Ann. 3, 296 (1870) 18. Castiglione, P., Falcioni, M., Lesne, A., Vulpiani, A.: Chaos and Coarse Graining in Statistical Mechanics. Cambridge University Press, Cambridge (2008)

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19. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes 3rd Edition: the Art of Scientific Computing, 3rd edn. Cambridge University Press, New York (2007) 20. Falcioni, M., Palatella, L., Pigolotti, S., Vulpiani, A.: Phys. Rev. E. 72, 016220 (2005) 21. François, M., Grosges, T., Barchiesi, D., Erra, R.: Commun. Nonlinear Sci. Numer. Simul. 19(4), 887 (2014) 22. Farsana, F., Gopakumar, K.: Procedia Comput. Sci. 93, 816 (2016). Proceedings of the 6th International Conference on Advances in Computing and Communications 23. Murillo-Escobar, M.A., Cruz-Hernández, C., Cardoza-Avendaño, L., Méndez-Ramires, R.: Nonlinear Dyn. 87(1), 407 (2017) 24. Sahari, M.L., Boukemara, I.: Nonlinear Dyn. 94(1), 723 (2018) 25. Elmanfaloty, R.A., Abou-Bakr, E.: Chaos, Solitons Fractals 118, 134 (2019) 26. Matsumoto, M., Nishimura, T.: ACM Trans. Model. Comput. Simul. 8(1), 3 (1998) 27. Saito, M., Matsumoto, M.: In: Keller, A., Heinrich, S., Niederreiter, H. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2006, pp. 607–622. Springer, Berlin (2008) 28. Knuth, D.E.: The Art of Computer Programming, Volume 2: Seminumerical Algorithms, 3rd edn. Addison-Wesley, Boston (1997) 29. L’Ecuyer, P.: In: Gentle, J., Härdle, W., Mori, Y. (eds.) Handbook of Computational Statistics. Springer Handbooks of Computational Statistics, pp. 35–71. Springer, Berlin (2012) 30. Herrero-Collantes, M., Garcia-Escartin, J.C.: Rev. Mod. Phys. 89, 015004 (2017)

3

Nonlinear Hamiltonian Systems

Abstract

In this chapter we investigate the dynamics of classical nonlinear Hamiltonian systems, which—a priori—are examples of continuous dynamical systems. As in the discrete case (see examples in Chap. 2), we are interested in the classification of their dynamics. After a short review of the basic concepts of Hamiltonian mechanics, we define integrability (and therewith regular motion) in Sect. 3.4. Nonintegrability is then discussed in Sect. 3.5. The addition of small non-integrable parts to the Hamiltonian function (Sects. 3.6.1 and 3.7) leads us to the formal theory of canonical perturbations, which turns out to be a highly valuable technique for the treatment of systems with one degree of freedom and shows profound difficulties when applied to realistic systems with more degrees of freedom. We will interpret these problems as the seeds of chaotic motion in general. A key result for the understanding of the transition from regular to chaotic motion is the KAM theorem (Sect. 3.7.4), which assures the stability in nonlinear systems that are not integrable but behave approximately like integrable ones. Within the framework of the surface of section technique, chaotic motion is discussed from a phenomenological point of view in Sect. 3.8. More quantitative measures of local and global chaos are finally presented in Sect. 3.9.

3.1

Integrable Examples

The equations of motion of point particles in classical mechanics are ordinary differential equations. In this section we give examples where these equations can be solved with the help of constants of motion. The systems discussed here are all onedimensional and hence one conserved quantity (as for example the energy) suffices to integrate the equations of motion for one particle and to obtain explicit solutions. For simplicity we set the particle mass m = 1.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. Wimberger, Nonlinear Dynamics and Quantum Chaos, Graduate Texts in Physics, https://doi.org/10.1007/978-3-031-01249-5_3

21

22

3

Nonlinear Hamiltonian Systems

As a first example, we analyze the harmonic oscillator. Its defining equation is linear and reads x¨ + ω2 x = 0.

(3.1.1)

The solution is found with the help of the ansatz x(t) = Aeλt + Be−λt , with constant coefficients A and B and an imaginary λ. However, there is a more systematic way of solving the problem. Therefore we transform to a system of two first-order differential equations: x˙ = y,

(3.1.2)

y˙ = −ω x,

(3.1.3)

y y˙ = −ω2 x x˙   d y2 ω2 x 2 . ⇔0= + dt 2 2

(3.1.4)

2

which implies that

(3.1.5)

We have found a constant of motion which is nothing but the total energy. It can be used to solve one of the two equations y2 ω2 x 2 + 2  2 ⇒ y = x˙ = 2E − ω2 x 2 . E=

In the next step we use the relation 

d t(x) dx

(3.1.6) (3.1.7)

= x(t) ˙ −1 :



dx √ 2E − ω2 x 2   1 xω + t0 . ⇔ t = arcsin √ ω 2E dt =

(3.1.8) (3.1.9)

The solution of the original problem thus reads √ 2E x(t) = sin(ω(t − t0 )). ω

(3.1.10)

Furthermore, we can compute the period of the motion by solving √ x˙ = 0 ⇒ x = ±

2E 2π ⇒T = . ω ω

(3.1.11)

Linear systems of differential equations can often be solved by a simple ansatz. One example are atoms located at the sites of a one-dimensional lattice which couple

3.1 Integrable Examples

23

via a linear force to their nearest neighbours (as if there were harmonic springs between them). The system is used as a model for phononic degrees of freedom in a crystal [1]. Here we are specifically interested in nonlinear systems, e.g. with a force which is a nonlinear function of position. Nonlinearity leads to complex coupled Hamilton’s equations of motion, and is hence a seed for realising chaotic motion. We give now three examples of integrable systems with a nonlinear force. (a) First we investigate a system with a more general force that includes the case of the harmonic oscillator x¨ = F(x),

F(x) = a + bx + cx 2 + d x 3 ,

(3.1.12)

with real coefficients a, b, c, d (possible higher orders in x are rarely needed). We solve the problem in analogy to the linear system above: x˙ = y, y˙ = F(x).

(3.1.13) (3.1.14)

Again we find a constant of motion (that we will denote by E because it is again the total energy)    1 2 1 3 1 4 d y2 = 0. (3.1.15) − ax + bx + cx + d x y y˙ = F(x)x˙ ⇔ dt 2 2 3 4 This can be integrated   dx   dt =  2 ax + 21 bx 2 + 13 cx 3 + 41 d x 4 + E

(3.1.16)

and we obtain the so-called elliptic integrals as solutions, see [2,3]. (b) The force is given by a power law: F(x) = −an|x|n−1

(3.1.17)

and derives from the potential V (x) = a|x|n . We chose a > 0 and n > 1. Following the same derivation we arrive at the integral   dx   dt = (3.1.18) . 2 E − a |x|n The integral can be solved with a few mathematical tricks to compute the oscillation period  1  T 2 2π E n Γ ( n1 ) T = dt = , (3.1.19) n E a Γ ( 21 + n1 ) 0 using the Γ function [2,3].

24

3

Nonlinear Hamiltonian Systems

(c) The classical pendulum, see (1.2.1), obeys the equation Θ¨ = −

g sin(Θ) . L

(3.1.20)

For small angles, the sine function can be approximated by the angle itself and one recovers the harmonic oscillator. For the general pendulum one finds 1 2 g cos(Θ) Θ˙ − , 2 L

E= which leads to





(3.1.21)

dΘ . √ 2(E + g/L cos(Θ))

dt =

(3.1.22)

The period can be computed as for the harmonic oscillator: Θ˙ = 0 ⇒ Θmax where E = −g/L cos(Θmax ).

(3.1.23)

It reads  T =

T

dt = 4

0

L 2g



Θmax

√

0

dΘ . cos(Θ) − cos(Θmax )

(3.1.24)

We apply the transformation cos(Θ) = 1 − 2k 2 sin2 (ϕ) with   EL 1 1 EL 1+ k = sin(Θmax /2) = sin arccos(− g ) = 2 2 g 

(3.1.25)

and use cos(Θmin ) = 1 − 2k 2 to obtain

T =4

L g



π/2 0

dϕ

 =4 1 − k 2 sin2 (ϕ)

L K (k). g

(3.1.26)

The function K (k) is the complete elliptic integral of the first kind [2]. When we approximate cos(x) = 1 − x 2 /2 the period for small k is given by

T (k ≈ 0) = 2π

L g



Θ2 1 + max + · · · 16

 .

(3.1.27)

The first term is just the period of the harmonic oscillator. The second term gives a correction, which already depends on the energy. Hence, T will in general be a function of E, a property typical of nonlinear systems. This implies that resonant driving at constant ω is not possible over long times. Additionally, the period at the crossover from libration to rotation where Θ = π and E = g/L ⇔

3.2 Hamiltonian Formalism

25

k = 1 diverges, T (k = 1) = ∞. Physically speaking it takes infinitely long to go through this separatrix.1

3.2

Hamiltonian Formalism

The Lagrangian formalism in configuration space is well-suited for solving practical ˙ t) problems (in particular with constraints). It uses the Lagrangian function L (q, q, where the dimension of the coordinate vector q equals the numbers of degrees of freedom N [4,5]. For some problems, it is more useful to turn to a phase-space description via a Legendre transform introducing the Hamiltonian H (q, p, t) = p · q − L . Here pi = ∂ L /∂ q˙i are the generalized momenta. When we introduce the notation z T = (q, p), the Hamiltonian equations of motion read2   0 I z˙ (t) = J · ∇ H (z, t), J = , (3.2.1) −I 0 with I being the N -dimensional identity matrix. They form a set of 2N first-order ordinary differential equations. The antisymmetric matrix J is also called symplectic. The symplectic property distinguishes the Hamiltonian formalism, see e.g. [6] for more details. For us, it suffices to note that both variable sets q and p are independent and equally important. The Hamiltonian formalism is well suited for • The description of non-integrable systems, • Perturbation theory, • The comparison to optics and to nonrelativistic quantum mechanics in the framework of semiclassical methods (for the latter see Sects. 5.2 and 5.3). Since the Hamiltonian is the total energy of the system, it can easily be given in explicit form. Here are some simple examples: 1. A point mass in an external potential: H=

p2 + V (q). 2m

(3.2.2)

The equations of motion are equivalent to Newton’s second law m q¨ = −∂ V /∂q and read p ∂H ∂V ∂H = , p˙ = − =− . (3.2.3) q˙ = ∂p m ∂q ∂q

1 A separatrix is a boundary in phase space separating two regions of qualitatively different motions, see Fig. 3.1 below for the pendulum. 2 To simplify the notation, vector quantities are not always specified in boldface. From the local context it should be clear that the corresponding symbols represent N -dimensional vectors.

26

3

Nonlinear Hamiltonian Systems

2. The one-dimensional harmonic oscillator, see (3.1.1), H=

p2 1 + mω2 q 2 . 2m 2

(3.2.4)

The equations of motion are q˙ =

p , m

p˙ = −mω2 q,

(3.2.5)

with solution q(t) = q0 sin(ωt + ϕ0 ),

p(t) = mωq0 cos(ωt + ϕ0 ).

(3.2.6)

The solution obeys the defining equation for an ellipse in the plane spanned by q and p for all times q(t)2 p(t)2 + = 1, (3.2.7) 2 (q0 mω)2 q0 showing that there are only closed (oscillatory) orbits possible in phase space. 3. A point-mass pendulum in a gravity field, see (3.1.20), H=

pϑ2 − mgL cos(q). 2m L 2

(3.2.8)

The equations of motion are q˙ =

pϑ m L2

and

p˙ ϑ = −mgL sin(q).

(3.2.9)

The phase-space trajectories for different total energies are shown in Fig. 3.1. Using the identification ϑ = q mod 2π , the angle ϑ and the angular momentum pϑ form a canonical pair of variables, see Sect. 3.3.2.

3.3

Important Techniques in the Hamiltonian Formalism

3.3.1

Conserved Quantity H

3.3.1.1 Autonomous Systems A system that is not explicitly time-dependent is called autonomous. For such systems the form of the differential equations implies that every point in phase space (q, p) has a unique time evolution.3 The Hamiltonian H (q, p) = E does not depend explicitly on time and is a constant of the motion.

is due to the fact that the velocity field of (q, p), which is given by J · ∇ H (q, p) in (3.2.1), does not depend on time [7].

3 This

27

p

ϑ

3.3 Important Techniques in the Hamiltonian Formalism

separatrix −π

0

π

q

Fig. 3.1 Phase-space trajectories of the pendulum. We see two different kinds of dynamical behaviour. Closed curves around the origin represent oscillations, where the position variable is restricted to −π < q < π . In contrast, curves for which pϑ = 0 holds for all times describe rotations of the pendulum. The points in phase space separating the two qualitatively different types of motion form the separatrix (marked by the arrow). In phase space we have two trajectories intersecting at (q, pϑ ) = (±π, 0) moving along the separatrix

3.3.1.2 Explicitly Time-Dependent Systems The energy of the system E(t) = H (q(t), p(t), t) is generally time-dependent. The reduction to a time-independent (autonomous) system is yet always possible, see e.g. [8] and [9] for more details. For this, we introduce the extended phase space spanned by the new variables P = (P0 ≡ −E, p) = P (τ ),

Q = (Q0 ≡ t, q) = Q (τ ).

(3.3.1)

With the Hamiltonian H (P , Q ) = H (q, p, t) − E(t) the generalized equations of motion read dP dQ ∂H ∂H , . (3.3.2) =− = dτ ∂Q dτ ∂P From dQ0 dt ∂H ∂H ⇔ = =− =1 (3.3.3) dτ ∂ P0 dτ ∂E we can infer that τ = t + const. which leads to dP0 ∂H =− dτ ∂ Q0

⇔

dE ∂H = . dτ ∂t

(3.3.4)

The number of independent variables is thus raised by one. This is why onedimensional systems described by a time-dependent Hamiltonian, such as H = p2 2m + V (x)[1 + cos(ωt)], are sometimes referred to as having “one-and-a-half degrees of freedom”. Going to the extended phase-space defined by (3.3.1) formally reduces all time-dependent problems to time-independent ones. For this reason, the following sections usually treat the time-independent case without loss of generality.

28

3

3.3.2

Nonlinear Hamiltonian Systems

Canonical Transformations

The choice of the generalized coordinates q is not restricted. Physically speaking it is possible to use N arbitrary quantities which describe the system’s location in configuration space in a unique way. The structure of the Lagrangian equations does not depend on this choice. If the problem is given in a certain set of coordinates it can be useful to change to another set of coordinates, in which the formulation of the problem is simpler in some way. In general, new coordinates are of the form4 Q i = Q i (q, t), for i = 1, . . . , N . Since in the Hamiltonian theory coordinates and momenta are treated as independent variables, one can enlarge the class of possible coordinate transformations and change the 2N coordinates (q, p) to (Q(q, p, t), P(q, p, t)). This enlargement is one of the main features of Hamiltonian mechanics. In order to preserve the symplectic structure of the equations of motion, ∂ H

, Q˙ i = ∂ Pi



∂H P˙i = − , ∂ Qi

(3.3.5)

with a new Hamiltonian H , the coordinate transformations have to satisfy certain conditions. Such transformations are then called canonical transformations. Usually, one arrives at Hamilton’s equations of motion via the principle of extremal action. This principle is then valid for both, the Hamiltonian H in the old coordinates and H in the new ones. This gives [4–6]  0 = δL = δ



 ( pq˙ − H ) − P Q˙ − H dt = δ



dF dt, dt

(3.3.6)

where δ denotes the variation of the corresponding function. This relation directly implies that ∂F dF ∂F ˙ ∂F (q, Q, t) = pq˙ − P Q˙ + (H − H ) = q˙ + Q+ . dt ∂q ∂Q ∂t

(3.3.7)

The function F generates the canonical transformation and its Legendre transforms with respect to the coordinates (q, Q) immediately give the three other possible generating functions which connect the old and new coordinates [5,6]. In the following, we implicitly use the extended phase-space picture of the last section and drop all explicit time dependencies. Equation (3.3.6) can then be cast in the form of total differentials [10]  δ ( pdq − PdQ) = 0, (3.3.8)

4 From a mathematical point of view

ible) for all t.

Q(q, t) has to be a diffeomorphism (differentiable and invert-

3.3 Important Techniques in the Hamiltonian Formalism

29

which implies dF = pdq − PdQ .

(3.3.9)

Here dF denotes the total differential of the generating function F(q, Q). Total differentials have the property that their contribution to a closed curve integral in (simply connected) phase space vanishes [10]. Any canonical transformation is induced by such a generating function. From (3.3.9) we obtain in the same way as above in (3.3.7) pi =

∂F , ∂qi

Pi = −

∂F . ∂ Qi

(3.3.10)

For a given function F these equations connect the old and the new variables. For some applications it can be more convenient to express the generator as a function of q and P. This can be done by a Legendre transform d(F + P Q) = pdq + Qd P.

(3.3.11)

The expression in brackets on the left hand side is a function of q and P which we denote by S. Using the new generator, the equations connecting the old and the new coordinates read ∂S ∂S , Qi = . (3.3.12) pi = ∂qi ∂ Pi When doing practical computations, the first part of (3.3.12) is used to calculate the new Pi by inverting the function, the second one to determine the new coordinates Q i . Because the notion of a canonical transformation is quite general, the original sense of coordinates and momenta may get lost (just think of the transformation which interchanges momenta and coordinates, see below). Therefore the variables q and p are often simply referred to as canonically conjugate variables. For a given set of new conjugate variables (Q, P), the transformation (q, p) → (Q, P) is canonical if (and only if) the following three conditions are fulfilled with respect to the derivative in the old and the new variables5 {Q i , Q k } = 0, {Pi , Pk } = 0, {Q i , Pk } = δi,k .

(3.3.13)

   ∂ f ∂g ∂ f ∂g denotes the Poisson The expression { f , g} = { f , g}q, p = k ∂q − ∂ p ∂ p ∂q k k k k bracket. These formal conditions are discussed in many textbooks on classical mechanics, see for example [4–6,11]. Two examples for canonical transformations are:

5 As for the coordinates, the transformation from the old to the new variables has to be a diffeomor-

phism, see previous footnote.

30

3

(b)

(a)

Nonlinear Hamiltonian Systems

p

P

p (P,Q)

(p,q) A(γ )

(p,q)

γ

γ

q q

Q

Fig. 3.2 A canonical transformations can be interpreted in two ways. In the active interpretation a closed curve is transformed leaving the action (given by the encircled area) invariant (a), whereas in the passive interpretation the coordinates (the axes) change (b)

1. The function S(q, P) = q P leads to the new coordinates Qi =

∂S = qi , ∂ Pi

pi =

∂S = Pi ∂qi

(3.3.14)

∂F = −qi . ∂ Qi

(3.3.15)

and therefore generates the identity. 2. When defining F(q, Q) = q Q, we find pi =

∂F = Qi , ∂qi

Pi = −

The so-defined canonical transformation interchanges coordinates and momenta. We note that a canonical transformation can be interpreted in two ways (see Fig. 3.2). To visualize the two possibilities we show how a closed curve γ in a two-dimensional phase space changes. The area encircled by γ is proportional to the action of the curve. In the active picture, the new coordinates are viewed as functions of the old coordinates (in other words we still use the old coordinate system) and γ transforms to the new curve A(γ ) (A(q, p) = (Q, P) defines the canonical transformation) encircling a deformed area of the same size. In the passive picture instead the coordinate system is changed while γ preserves its form. This latter view motivates the following theorem.

3.3.3

Liouville’s Theorem

Liouville’s theorem states that the time evolution of a Hamiltonian system conserves the phase-space volume. A simple way to see this is to show that canonical transformations are volume preserving in phase space. Since every time evolution can be

3.3 Important Techniques in the Hamiltonian Formalism

31

expressed as a canonical transformation the same has to be true for the time evolution. Another proof is based on the fact that the Hamiltonian flow is divergence free. We will come back to this idea in Sect. 3.4. A more formal proof is given in the appendix at the end of this chapter. Let Ω be an arbitrary phase-space volume, (q, p) the set of old coordinates, and A the transformation from the old coordinates to the new coordinates (Q, P). We want to show that    ! dqd p = dQd P = D(q, p) dqd p, (3.3.16) Ω

Ω

A(Ω)

where D denotes the functional determinant of the coordinate transformation. In order to have equality we need D = 1. Doing a few straight forward transformations we obtain D= =

∂(Q 1 , . . . , Q N , P1 , . . . , PN ) ∂(q1 , . . . , q N , p1 , . . . , p N )

(3.3.17)

∂(Q 1 ,...,Q N ,P1 ,...,PN ) ∂(q1 ,...,q N ,P1 ,...,PN ) ∂(q1 ,...,q N , p1 ,..., p N ) ∂(q1 ,...,q N ,P1 ,...,PN )

(3.3.18)



=

∂(Q 1 ,...,Q N )  ∂(q1 ,...,q N )  P=const.



∂( p1 ,..., p N )  ∂(P1 ,...,PN ) q=const.

.

(3.3.19)

Now we express the canonical transformation with the generating function S(q, P), see Sect. 3.3.2. Inserting the relations of (3.3.12) into the above formulas we find ∂ Qi ∂2 S = , ∂qk ∂qk ∂ Pi

∂ pi ∂2 S = . ∂ Pk ∂qi ∂ Pk

(3.3.20)

We note that the determinants in the numerator and in the denominator of (3.3.19) equal each other because they can be brought into the same form by interchanging rows and columns. After having shown that canonical transformations preserve the phase-space volume we demonstrate that the temporal evolution itself is generated by canonical transformations. Let us assume we are given the solution of the Hamiltonian equations of motion. We define the old and the new coordinates by q0 = q, p0 = p and Q(q, t) = q(t), P( p, t) = p(t). To show that the transformation defined in this way is canonical we explicitly give a generating function that fulfills (3.3.7). It reads  F (q, Q, t) = −

Q

pdq ,

(3.3.21)

q

where the integral goes along the trajectory that solves the Hamiltonian equations of motion for the initial condition (q, p). For the sake of clarity we have explicitly

32

3

Nonlinear Hamiltonian Systems

Fig. 3.3 For the harmonic oscillator the phase-space evolution of an ensemble of initial conditions (shaded area) is a rotation with constant period T

p

T/4

q

T/4

mentioned the time dependence of the generator F . The momentum is assumed to be a function of the trajectory q(t ), 0 ≤ t ≤ t. We note that from a physical point of view the generating function is [apart from the minus sign in (3.3.21)] the action as a function of the starting point and the endpoint of the given trajectory. To check whether we thus obtain the correct initial and final momentum p and P we calculate ∂F = pi , ∂qi

∂F = −Pi . ∂ Qi

(3.3.22)

For the computation of p we have to differentiate the integral of (3.3.21) with respect to its starting point and find ∂∂qFi = pi (q) = pi (q0 ) = pi (t = 0). Differentiation with

respect to the endpoint results in ∂∂ F Q i = − pi (Q) = − pi (q(t)) = − pi (t). Hence F gives the correct relations which concludes our proof. Figure 3.3 gives an example of how a phase-space volume is transformed in time.

3.3.4

Hamilton-Jacobi Theory and Action-Angle Variables

In Sect. 3.3.2, we have briefly reviewed canonical transformations which are an important tool. Now we are going to use them to solve the equations of motion by trying to find new variables in such a way that the Hamiltonian becomes trivial. Assume the Hamiltonian does not depend any more on the spatial coordinates, i.e. H = H (P). In this case the equations of motion are of the form

∂H P˙i = − = 0, ∂ Qi

∂ H

Q˙ i = . ∂ Pi

(3.3.23)

From the first equation, we conclude that the Pi are all constants of motion. Therefore, the right hand side of the second equation is constant in time. The solution for the coordinates therefore reads Q i (t) = Q i (0) + t

∂ H

. ∂ Pi

(3.3.24)

3.3 Important Techniques in the Hamiltonian Formalism

33

To reach this goal we work with an ansatz for the transformation based on the ∂S into the original Hamiltonian and assuming generator S(q, P). Inserting pi = ∂q i that it equals a Hamiltonian of the above form, we arrive at the Hamilton–Jacobi equation for the generating function S   ∂S ∂S = H (P) = E = const. H q1 , . . . , q N , ,..., (3.3.25) ∂q1 ∂q N Without loss of generality we assumed energy conservation, see Sect. 3.3.1.2. To understand the meaning of the generating function its total differential is computed:   ∂S ∂S dqi + d Pi = pi dqi . (3.3.26) dS = ∂qi ∂ Pi i

i

Since the new momenta P are constants of motion (d P = 0), S can be obtained by integration    pi (q )dqi = p(q )dq + const. (3.3.27) S(q, P) = i

Therefore the generator S differs from the action function only by a constant. This gives the following interesting analogy: while the indefinite integral (3.3.27) determines the solution of the Hamilton–Jacobi equation, its definite counterpart is the basis of Hamilton’s principle that leads to Hamilton’s equations [8]. The Hamilton–Jacobi equation generally gives a single nonlinear partial differential equation of first order which determines S as a function of its 2N arguments. In contrast, the original Hamilton’s equations of motion are 2N ordinary differential equations of first order. While the number of equations is reduced, the Hamilton– Jacobi equation is difficult to solve in general. However, the Hamilton–Jacobi theory is very important in the classification of the dynamics (see below) and for the understanding of the correspondence between classical and quantum mechanics (see Sect. 5.2.2).

3.3.4.1 Special Case: One-Dimensional Systems As an example, we discuss the motion of a point particle on a line, i.e. in a onedimensional configuration space, and in a time-independent potential. The corresponding Hamiltonian is conservative and of the form H ( p, x) =

p2 + V (x). 2m

(3.3.28)

If (θ, I ) denote the new coordinates and S(x, I ) denotes the generating function, the Hamilton–Jacobi equation reads  H

∂S ,x ∂x



1 = 2m



∂S ∂x

2

+ V (x) = H (I ).

(3.3.29)

34

3

Nonlinear Hamiltonian Systems

Note that H (I ) = E(I ) is a constant of the motion. Solving for p = ∂∂ xS , see (3.3.12), allows us to integrate the equation. The result is the following expression  ∂ S(x, I ) = ± 2m (H (I ) − V (x)) ∂x  x ⇒ S(x, I ) = 2m (H (I ) − V (x ))dx ,

(3.3.30) (3.3.31)

x1

where x1 is the left turning point of the motion, see Fig. 3.4 below. Using the formal solution, the new coordinate can be computed θ=

∂S = ∂I



x



x1

dx

2 m

∂ H

. (H (I ) − V (x )) ∂ I

(3.3.32)

A confined system shows periodic behaviour which means that θ is periodic. Therefore we define θ (x2 , I ) = π , where we assume x2 to be the right turning point of the motion. The relation between H and I is calculated in the following way: ∂I = ∂ H



∂ H

∂I

−1

=

1 π



x2 x1



dx 2 m

(H (I ) − V (x))

.

(3.3.33)

Integrating (3.3.33) over E = H , one obtains a relation for I (E), I (E) =

1 π



x2 x1

  1 p dx, 2m (E(I ) − V (x))dx = 2π

(3.3.34)

where we have denoted H = E to stress its physical meaning. The contour integral is over one period T . Hence I is the action of a trajectory with energy E. This is also the reason why (I , θ ) are called action-angle variables. As the action I is fixed, the time evolution of the angle θ is given by periodic oscillations θ=

2π t + const. T

(3.3.35)

The period can be computed from the equation ω(I ) =

∂ H

2π = . T ∂I

(3.3.36)

Hence the solution of the one-dimensional problem is given up to the integral (3.3.34). For an illustration of the action-angle variables see Fig. 3.4. But solving the Hamilton-Jacobi equation is simple only in two cases: • In one-dimensional systems as presented above.

3.4 Integrable Systems

35

Fig. 3.4 Motion of a particle in a one-dimensional potential. a shows the potential and the energydependent turning points x1 and x2 . b represents the motion schematically in phase space. The radius of the circle is determined by the action variable I [2π I is the area within the circle, see (3.3.34)] while the position on the circle is given by the angle variable θ. The turning points correspond to θ = 0 and θ = π

• In separable systems. We call the variables q = {qi }i separable if the total action of the classical system can be written as a sum of independent actions for each variable qi S(q, P) =

N 

Si (qi , P).

(3.3.37)

i=1

This form of the total action effectively reduces the problem to the one-dimensional case for each index i = 1, . . . , N separately. In all other cases the calculation of the action variables is a difficult task, since one must solve an implicit partial differential equation for the unknown new momenta. Apart from that, the Hamilton–Jacobi theory is used as the starting point for canonical perturbation theory. It is also one of the bridges to quantum mechanics: The path-integral formulation uses the action as a functional of the trajectory. In the classical limit it is possible to make a stationary phase approximation which then essentially yields the Hamilton–Jacobi equation for the action, see Sect. 5.2.2.

3.4

Integrable Systems

The definition of integrability is the following: Let N be the number of degrees of freedom and let s be the number of constants of motion. If s = N the system is called integrable. Thereby the constants of motion ci (q, p) have to be in   involution and independent. Here in involution means that they fulfill the relations ci , c j = 0 for all i, j, with the Poisson bracket defined in Sect. 3.3.2. The second condition states that the differential 1-forms dci (q, p) are linearly independent at each point of the set MC = {(q, p) : ci (q, p) = Ci for i = 1 . . . N }. The independence of the

36

3

Nonlinear Hamiltonian Systems

γ2

γ1

Fig. 3.5 Example for a basis of cycles on a two-dimensional torus (having a doughnut-like shape). While γ2 encloses the “hole” of the torus, γ1 encircles the tube forming it. Both curves γ1 and γ2 cannot be contracted to a point, they are so-called irreducible curves

constants of motion ensures that they actually characterize the dynamics in the entire accessible phase space. If a Hamiltonian system is integrable, its equations of motion can be solved by quadratures,6 just in the spirit of the previous section. If N constants of motion exist, the canonical transformation defined by (q, p) → (θ, I ) with Ii = Ci ,

(3.4.1)

gives a solution of the Hamilton–Jacobi equation. Therefore another equivalent definition of integrability is that a solution of the Hamilton–Jacobi equation can be found with the angle-action variables (θ, I ). In the last section we have introduced action-angle variables for an integrable onedimensional system. It turns out that this concept can indeed be extended to higherdimensional integrable systems. Mathematically speaking, one can show that the set MC is diffeomorphic to the N -dimensional torus7 if it is compact and connected.8 The action-angle variables can then be introduced as follows. Let θi , i = 1, . . . , N be the angle variables. We assume they are scaled such that θi ∈ [0, 2π ). Taking a

6 The notion quadrature means the application of basic operations such as addition, multiplication, the computation of inverse functions and the calculation of integrals along a path. 7 That the motion is bound to the topological structure of a torus, has to do with a theorem on the existence of continuous vector fields on tori, known as Hairy Ball theorem [12], which follows from the Poincaré–Hopf theorem for such fields on differentiable manifolds. 8 The set M is not compact if the motion is not restricted to a finite area in phase space. Examples C for such unbounded motion are the free particle or a particle in a linear potential (Wannier-Stark problem [13]).

3.4 Integrable Systems

37

basis of cycles on the torus γi , i = 1, . . . , n one can define action variables by  1 p(I , θi )dq(I , θi ). (3.4.2) Ii = 2π γi A basis of cycles is given if the increase of the coordinate θi on the cycle γ j is equal to 2π if i = j and 0 if i = j. For the two dimensional torus (see Fig. 3.5), such a basis is given by a trajectory encircling the tube of the torus and another one going around it. Generally speaking, the tori are defined as the following sets of phase-space points T ≡ {( p, q) ∈ phase space : p = p(I , θ), q = q(I , θ) for fixed action I and θi ∈ [0, 2π )},

(3.4.3) with periodic variables in the angles: q(I ; . . . , θi + 2π, . . .) = q(I ; . . . , θi , . . .) and p(I ; . . . , θi + 2π, . . .) = p(I ; . . . , θi , . . .). The generating function connects the old and the new variables by θi = ∂ S/∂ Ii . The solution of the equations of motion for an integrable system, see (3.3.24), is then given in the possibly high-dimensional phase space parametrized by the action-angle variables (I , θ ): θ (t) = θ0 + ω(I )t,

ω(I ) =

∂ H

= const. ∂I

(3.4.4)

By construction, the action variables Ii are the constants of the motion. The fact that the components of θ are angle-like, and called therefore cyclic variables in the Lagrange formalism, does not necessarily mean that the actual motion in phase space is periodic. This is only true in a system with one spatial dimension where the motion is along a simple circle. On a doughnut (a two-dimensional torus) this may no longer be the case since the periodicities in the motion of the two angles may differ. A general (non-periodic) motion on a torus is called conditionally periodic. Once we have found the cyclic variables {θi }i , the new generating function can be the identical transformation, see (3.3.14), S(θ, α) =

N 

Si (θi , α) =

i=1

N 

θi αi ,

(3.4.5)

i=1

with the constants of motion Ii = αi . This means that, in the angle-action variables of the Hamilton–Jacobi solution, the integrable problem is trivially separable, see (3.3.37). The following table summarizes the dimensionality of the important objects in phase space for integrable systems where a torus structure exists: The reduction of Phase space Energy surface Torus

2N 2 N−1 N

2N first-order differential equations to only N constants of the motion is remarkable. The origin of this reduction is the symplectic structure in the phase-space variables. For a careful derivation of what we have shortly summarized please see [6].

38

3.4.1

3

Nonlinear Hamiltonian Systems

Examples

In conservative systems the energy is a constant of motion. Therefore, conservative systems of one particle in one space dimension are always integrable. Another class of integrable systems are separable systems, see (3.3.37). The simplest example of a separable system is a Hamiltonian that can be written as a sum over N independent Hamiltonians (non-interacting degrees of freedom). Conserved quantities are then simply the independent energies. Another example is the two-body Kepler problem (the motion of two point particles in free space that interact via gravity). Here the conserved quantities are six: the energy, the three components of the total momentum and two quantities coming from angular momentum conservation (usually the square L2 and one of its projections, e.g., L z ). This can be best seen by using the conservation of the centre-of-mass momentum to reduce the system to an effective one-body problem [6]. Then the remaining conserved quantities mentioned above are three independent constants of motion for this one-body problem.

3.4.2

Poisson Brackets and Constants of Motion

The Poisson brackets, which we used to define the notion of two constants of motion being in involution, can also be applied to compute the time evolution of a function f (q, p, t) in phase space df ∂f = { f , H} + . (3.4.6) dt ∂t Assume f is not explicitly time-dependent, then in order to be a constant of motion it has to satisfy {H , f } = 0. If we take for f a given density in phase space, i.e. f = ρ(q, p, t), we immediately find the following continuity equation   ∂ρ(q, p, t) q˙ 0= · ∇q, p ρ(q, p, t) + . (3.4.7) p˙ ∂t We used Hamilton’s equations and Liouville’s theorem stating that the total time derivative of any phase-space density is zero. Consequently, the Hamiltonian flow in phase space corresponds to the one of an incompressible fluid. This idea is applied e.g. in [6,14] to prove Liouville’s theorem based on the fact that Hamilton’s flow is free of any divergence in phase space. To see this we write Hamilton’s equations of motion as z˙ (t) = J · ∇ H (z) (see Sect. 3.2) and compute     0 I  ∇q H (q, p) (3.4.8) div z˙ (t) = ∇q , ∇ p ∇p −I 0   = ∇q · ∇ p − ∇ p · ∇q H (q, p) = 0. Inserting q and p into (3.4.6) one also arrives at a compact formulation of Hamilton’s equations of motion: {qi , H } = q˙i ,

{ pi , H } = p˙ i .

(3.4.9)

3.5 Non-Integrable Systems

39

We note that (3.4.6) is formally analogous to the Heisenberg equation in quantum mechanics. Many textbooks use this formal similarity as a connecting bridge between classical Hamiltonian systems and their quantum mechanical counterparts, see for instance [15].

3.5

Non-Integrable Systems

A system is non-integrable if there are less independent constants of motion than degrees of freedom. The most famous example is the three-body Kepler problem whose Hamiltonian is given by (we assume equal masses m for simplicity) H=

 1  2 γ m2 γ m2 γ m2 p1 + p22 + p23 − − − . 2m |r1 − r2 | |r2 − r3 | |r3 − r1 |

(3.5.1)

As in (1.2.2), γ is the gravitational constant. The system has three coordinates each of them with three components. Therefore we have nine degrees of freedom. The number of independent conserved quantities is, as for the two-body Kepler problem, just six. So-called restricted versions of the three-body Kepler problem are much studied in the context of planetary motion, see e.g. [16]. Since these models assume restrictions to generality, e.g. that two of the bodies move in circular coplanar orbits around their common centre-of-mass and the third body does not back-act onto the other two (approximation of a zero mass body), approximate solutions for a system composed of, for instance, sun, earth, and moon may be found [16]. In 1899 Poincaré showed that it is impossible to compute even approximate longtime solutions for the N -body Kepler problem if N ≥ 3 [17]. Small errors grow very fast during the temporal evolution for most initial conditions. At this stage, it seems that there are only a few integrable systems, for which computations can be done, and that long-time predictions in all other cases are nearly a hopeless task. Fortunately, it turns out that this is not the whole story. In 1954 Kolmogorov [18] and in 1963 Arnold [19,20] and Moser [21,22] were able to show that there are systems which are not integrable, but nevertheless remain predictable in some sense. Assume a Hamiltonian describing an integrable system that (in a sense that needs to be clarified) is distorted just weakly by a perturbation which makes the system non-integrable. Then the solution of the distorted system is very close to the solution of the integrable system. This statement, which we will treat in more detail in Sect. 3.7.4, is known as KAM theorem and justifies the application of perturbation theory. Prior to that, as an additional motivation, we treat special cases of perturbed systems in the following section.

40

3

Fig. 3.6 In one-dimensional systems the action is an adiabatic invariant. Therefore the area that is enclosed by a phase-space trajectory is changing only weakly when the parameter λ is varied

Nonlinear Hamiltonian Systems

p

λ = 0

λ =0

q

3.6

Perturbation of Low-Dimensional Systems

3.6.1

Adiabatic Invariants

We consider a system whose Hamiltonian depends continuously on a parameter λ. As an example one may think of a pendulum with length λ, or of two particles whose interaction is varied with λ. Suppose that the parameter changes slowly with time. Then (if we think of the pendulum) the amplitude of the oscillation will be a function of the length of the pendulum. If we very slowly increase its length and then very slowly decrease it to the original value, the amplitude will be the same again at the end of the process. Furthermore, we will observe that the ratio between the energy and the frequency changes very little although the energy and the frequency themselves may change a lot. It turns out that this is a general phenomenon. Quantities which change just a bit under slow variations of a parameter are called adiabatic invariants. A strict mathematical definition of an adiabatic invariant can be found in [6]. In the following, we want to show that in one-dimensional systems the action  1 pdq is an adiabatic invariant under a slow change of the parameter9 I (E, λ) = 2π λ. A consequence of this statement is that the area enclosed by a phase-space trajectory changes only slightly when the parameter is varied, see Fig. 3.6. The proof follows the one given in [5]. When λ is varying slowly with time the system is no longer autonomous. By slowly we mean that the change of λ is small during one period of the motion, i.e. T dλ dt  λ. The energy and therewith the period T (E) of the motion change (the trajectories in phase space are not strictly closed any more). Since the variation of λ is small, the variation of the energy E˙ ≡ ddtE will be small as well. If E˙ is averaged over one period of the motion, possible short time oscillations are “smeared out” and one finds a good measure for the temporal evolution of the system’s energy by

9

There can be more than one such parameter but for simplicity we assume only one. This result is strictly true only if the angular frequency ω(I , λ) never equals zero, see [6] and Sect. 3.7.2.

3.6 Perturbation of Low-Dimensional Systems

computing ˙ T = 1  E T



T



0

41

  T 1 ∂ H (t) ∂H dt ≈ λ˙ (t) dt. ∂t T ∂λ 0

(3.6.1)

The period T is the one obtained by assuming that λ is constant. Since the change of λ is nearly constant we take λ˙ out of the time integration and obtain an integral over a closed trajectory. Using this fact, it can be expressed as a closed line integral over the coordinate. We use q˙ = ∂∂Hp and find dq ∂H dq = ⇒ dt = ∂ H . dt ∂p ∂p

(3.6.2)

Applying this expression, the period is written as 

T

T =

 dt =

0

dq ∂H ∂p

(3.6.3)

and we obtain from (3.6.1) the following formula for the average energy change 

−1 ∂H dq ∂p .  −1 ∂H dq ∂p

∂H ∂λ

˙ T ≈ λ˙  E 



(3.6.4)

As already mentioned, the integration has to be done along a closed trajectory and hence for constant λ. Therefore the energy is also constant along that trajectory. The momentum p = p(q, E, λ) is a function of the coordinate, the energy and the parameter λ. At this point, it is sufficient to let p (and not as well q) depend on λ. This is because we assume that the closed curve p(q) has a sufficiently regular structure such that when cutting it into two (or more) pieces it can be expressed as a graph p(q). The assumption is justified because the trajectory is distorted only slightly switching on the perturbation λ. Since we know that the energy must be constant on a closed trajectory, differentiating the equation H ( p, q, λ) = E with respect to λ results in   ∂H ∂p ∂H ∂p ∂H ∂H . (3.6.5) + =0⇔ =− ∂λ ∂ p ∂λ ∂λ ∂λ ∂ p Inserting the expression into (3.6.4) we find  ˙ T ≈ −λ˙   E or

  0≈

∂p ∂λ dq , ∂p dq ∂E

 ∂ p dλ ∂ p  dE  dq. + ∂ E dt T ∂λ dt

(3.6.6)

(3.6.7)

42

3

Nonlinear Hamiltonian Systems

But this implies

d    dI  pdq = 2π ≈ 0. (3.6.8) T dt dt T The result shows that, in the range of validity of the approximations used, the action remains invariant when the parameter λ is adiabatically varied.

3.6.2

Principle of Averaging

We want to present two examples in which the idea of averaging over one period of a periodic drive is used (see previous subsection). We therefore treat systems that have two different time scales. The Hamiltonians shall be of the form H ( p, x, t) = H0 ( p, x) + V (x) sin(ωt),

(3.6.9)

where the time scale of the unperturbed (and integrable) system H0 is assumed to be much longer than 2π/ω. A valuable description will be obtained by averaging over the short-time behaviour. The idea to average over the fast time scale is also known as principle of averaging, which turns out to be a very useful tool for practical applications.

3.6.2.1 Free Particle with Dipole Coupling Let us start with the simple and exactly solvable example of a free particle of mass m = 1 driven by a time-dependent linear force. The system’s Hamiltonian reads H (x, p, t) =

p2 + F x sin(ωt), 2

(3.6.10)

which results in the following equations of motion: x(t) ˙ = p(t),

p(t) ˙ = −F sin(ωt).

(3.6.11)

Since the force is linear the equations can easily be solved. Solutions are of the form x(t) = x f (t) + ξ(t),

p(t) = p0 + η(t),

(3.6.12)

with x f (t) = x0 + p0 t, ξ(t) = ωF2 sin(ωt), and η(t) = ωF cos(ωt). They consist of two parts, one for the free motion and another one describing the short-time oscillations due to the dipole coupling proportional to x, see Fig. 3.7. In the limit of very fast driving (ω → ∞) the system behaves like a free particle because it cannot follow the force. A reasonable effective description in the regime of fast driving is obtained by averaging the solution over one period of the perturbation which will lead to an effective static Hamiltonian [see (3.6.21)]. Finally we note that the amplitude of the momentum correction η is by one order of magnitude in the perturbation parameter ω−1 larger than that of the amplitude of the spatial correction ξ .

3.6 Perturbation of Low-Dimensional Systems Fig. 3.7 Fast driving leads to small oscillations ξ(t) around the free motion x f (t)

43 x(t )

x(t )

x f (t )

ξ (t ) t

3.6.2.2 General Case In the general case we have H ( p, x, t) = H0 ( p, x) + V (x) sin(ωt). As above, we make the following ansatz to separate the slowly varying part and the fast part of the solution [23] x(t) = x(t) ¯ + ξ(t),

p(t) = p(t) ¯ + η(t).

(3.6.13)

The quantities with overbar describe the slowly varying and the Greek variables the fast oscillating parts. The latter are assumed to satisfy ξ T = ηT = 0 for T = 2π/ω (ω is the angular frequency of the time-dependent part of the Hamiltonian). We now derive effective equations of motion by expanding the Hamiltonian in η and ξ , thereby assuming |ηn | ∼ |ξ n−1 | as motivated in the previous subsection.10 Starting from x˙ = ∂∂Hp = ∂∂Hp0 we find (by Taylor expansion) η2 ∂ 3 H0 ∂ 2 H0 ∂ H0 + . +η x˙¯ + ξ˙ ≈ ∂ p¯ ∂ p¯ 2 2 ∂ p¯ 3

(3.6.14)

The same can be done with the momentum. Application of p˙ = − ∂∂Hx results in ∂ H0 ∂ 2 H0 p˙¯ + η˙ ≈ − −η ∂ x¯ ∂ x∂ ¯ p¯ ∂V η2 ∂ 3 H0 ∂2V − ξ sin(ωt) − . sin(ωt) − ∂ x¯ ∂ x¯ 2 2 ∂ x∂ ¯ p¯ 2

(3.6.15)

These equations are averaged over one period. Using ξ T = ηT = 0 we arrive at ∂ H0 1 ∂ 3 H0 2 η T x˙¯ = + ∂ p¯ 2 ∂ p¯ 3 1 ∂ 3 H0 2 ∂ H0 ∂2V ξ sin(ωt) − η T . p˙¯ = − − T ∂ x¯ ∂ x¯ 2 2 ∂ x∂ ¯ p¯ 2

(3.6.16)

10 This ansatz implicitly takes account of the symplectic structure of the variables η and ξ , i.e. of the area preservation conserved also by the perturbation. The proper and consistent way of doing perturbation theory is formally introduced in the next section.

44

3

Nonlinear Hamiltonian Systems

Substituting the latter equations back into (3.6.14) and (3.6.15) we obtain the equations of motion for the fast oscillating variables ∂ 2 H0 , ∂ p¯ 2 ∂V η˙ ≈ − sin(ωt), ∂ x¯ ξ˙ ≈ η

(3.6.17) (3.6.18)

in first order and zeroth order in |η| for ξ˙ and η, ˙ respectively. The solutions are then readily found and read 1 ∂ V ∂ 2 H0 sin(ωt), ω2 ∂ x¯ ∂ p¯ 2 1 ∂V η≈ cos(ωt). ω ∂ x¯ ξ≈

(3.6.19)

As expected they describe small and fast oscillations around (x(t), ¯ p(t)). ¯ Using (3.6.19) and (3.6.16), it is possible to derive an averaged system such that the equations of motion for the slowly varying parts read ∂ H¯ , x˙¯ = ∂ p¯

∂ H¯ p˙¯ = − . ∂ x¯

(3.6.20)

In order to satisfy equation (3.6.20), the averaged Hamiltonian H¯ needs to take the form   1 ∂ V 2 ∂ 2 H0 H¯ ( p, ¯ x) ¯ = H0 ( p, ¯ x) ¯ + , 4ω2 ∂ x¯ ∂ p¯ 2   p¯ 2 1 ∂V 2 = ¯ + . + V0 (x) 2 4ω2 ∂ x¯   

(3.6.21)

≡Veff (x) ¯

In the final step we assumed the special form H0 ( p, x) = p 2 /2 + V0 (x) for the unperturbed Hamiltonian. We note that the effective potential may be trapping even if the original potential is not. Assume for example V (x) = x 3 . The potential is— strictly speaking—not trapping because it has no minimum. The effective potential has, however, a shape proportional to x¯ 4 . From a physical point of view the time-dependent term proportional to x 3 changes its sign so quickly that the particle effectively feels a quasi-static confining potential. An important application of this behaviour is the trapping of charged particles like ions in time-dependent electric fields [24,25].

3.6 Perturbation of Low-Dimensional Systems

45

3.6.2.3 Gauß Averaging In what follows, we express—in the spirit of Sect. 3.6.1—the above principle of averaging as a statement for the change of action arising from perturbations. We assume a one-dimensional conservative system and choose angle-action variables (θ, I ). The Hamiltonian function is of the form H0 (I ) and the equations of motion read ∂ H0 . (3.6.22) I˙ = 0, θ˙ = ω(I ), ω(I ) = ∂I Next we add a small perturbation to the Hamiltonian (  1) H = H0 (I ) +  H1 (I , θ ).

(3.6.23)

Since H0 is independent of θ , the new equations of motion read ∂ H1 I˙ = − ≡  f (I , θ ), ∂θ

θ˙ = ω(I ) + 

∂ H1 ≡ ω(I ) + g(I , θ ). ∂I

(3.6.24)

The averaging principle (which was already used by Gauß for estimating how planets perturb each other’s motion) states that the given system can be approximated by an averaged one [6,7]. By J we denote a new (averaged) action variable and define it via the equation 1 J˙ =  f¯(I ) with f¯(I ) ≡ 2π



2π

f (I , θ ) dθ.

(3.6.25)

0

We note that this equation is simpler because it does (by construction) not depend on the angle any more. The time evolution of the new action variable reads J (t) = J (0) + t f¯(I ) = I (0) + t f¯(I ). Let us estimate the difference between the approximate and the true solution in first order11 in :  t f (I , θ (t) = θ0 + ωt)dt J (t) − I (t) ≈ t f¯ −  0  t

 = − f (I , θ0 + ωt) − f¯(I ) dt

(3.6.26)

0

 =− ω



θ0 +ωt

θ0

 f (I , θ ) − f¯(I ) dθ.

 The function f (I , θ ) − f¯(I ) is trivially 2π -periodic in θ . Hence the integral over it gives a finite number that we call C. Then the following estimate holds |J (t) − I (t)| ≤

11 Note

 C. ω

that we can assume in (3.6.26) θ˙ ≈ ω within this approximation.

(3.6.27)

46

3

Nonlinear Hamiltonian Systems

Fig. 3.8 The figure shows the full solution I (t) and the approximate (averaged) solution J (t), c.f. the previous Fig. 3.7

I(t)

J(t)

t

Equation (3.6.27) tells us that there is a continuous connection between the perturbed and the unperturbed system. Accordingly, for small perturbations the two systems behave similarly, and for  → 0 they give the same solution.12 For an illustration of the full and the approximate solution see Fig. 3.8. The averaging principle expressed by (3.6.27) gives the essential idea for the more formal and more general theory of perturbations introduced in the next section.

3.7

Canonical Perturbation Theory

Perturbation theory is an important concept in many areas of physics. If a system differs from a solvable one only by a small perturbation we can hope to describe the full system by the solvable one plus a small change. The computation of this small change often has to be done carefully, and it is a common phenomenon that the perturbative expansion obtained suffers from difficulties (e.g. a divergence, see Sect. 3.7.6). Nevertheless also in these cases the first few terms often give a hint on what happens. In classical mechanics there is an elegant concept of how to perform perturbative calculations in a systematic manner, known as canonical perturbation theory. The concept is exemplified in Sect. 3.7.2 for conservative systems of one degree of freedom only. For systems of two or more degrees of freedom profound difficulties arise, which, in a sense, can be understood as the seeds of chaotic motion (Sect. 3.7.3).

3.7.1

Statement of the Problem

As in the previous section, we assume a Hamiltonian function H0 describing an integrable system that is altered by a small perturbation  H1 , with   1. We further assume that angle-action variables (θ, I ) can be used to solve the unperturbed system. The Hamiltonian of the full system thus reads H (θ, I ) = H0 (I ) +  H1 (θ, I ).

12 C.f.

footnote in Sect. 3.6.1.

(3.7.1)

3.7 Canonical Perturbation Theory

47

A solution of the problem would be obtained if one finds a new set of angle-action variables (ϕ, J ) such that the Hamiltonian in the new coordinates has the form H = K (J ). Using perturbation theory one could expand the new variables in terms of the old ones. Usually, this is not done because one must be careful to preserve the symplectic structure in the variables ϕ and J . This is also the reason why we used an expansion linear in ξ and quadratic in η in Sect. 3.6.2.2. A convenient way to take care of this is to expand the generating function. Therefore, we write S(θ, J ) = J · θ + S1 (θ, J ).

(3.7.2)

The first term generates the identity while the second term is meant to describe the perturbation and needs to be determined.

3.7.2

One-Dimensional Case

We start with the case of a one-dimensional motion and follow the analysis from [26], where more details can be found. The canonical transformation (θ, I ) → (ϕ, J ) defined by the generating function S(θ, J ) can be written as I =

∂S ∂ S1 =J+ , ∂θ ∂θ

ϕ=

∂S ∂ S1 =θ+ . ∂J ∂J

(3.7.3)

In order to preserve the periodicity of ϕ, the perturbative part of the generating function needs to be periodic too, i.e. S1 (θ, J ) = S1 (θ + 2π, J ). In canonical perturbation theory, S1 is expanded in orders of  S1 (θ, J ) =  S1(1) (θ, J ) +  2 S1(2) (θ, J ) + O( 3 ).

(3.7.4)

Next we insert the expansion into the Hamiltonian function and obtain the Hamilton– Jacobi equation, see (3.3.25),  (1) (2) ∂ S1 2 ∂ S1 3 K (J ) = H0 J +  + + O( ) ∂θ ∂θ   (1) (2) ∂ S1 2 ∂ S1 3 +  H1 θ, J +  + + O( ) , ∂θ ∂θ 

(3.7.5)

where, for the sake of clarity, we have labeled the new Hamiltonian by K . In order to get an expansion in  also the Hamiltonian functions H0 and H1 are expanded in Taylor series.

48

3

Nonlinear Hamiltonian Systems

! (1) (2) ∂ S1 2 ∂ S1 3  + + O( ) ∂θ ∂θ !2 (1) (2) ∂ S1 2 ∂ S1 3  + + O( ) ∂θ ∂θ

∂ H0 K (J ) = H0 (J ) + ∂J +

1 ∂ 2 H0 2 ∂ J2

∂ H1 +  H1 (J , θ ) +  ∂J

(3.7.6)

! (1) (2) ∂ S1 2 ∂ S1 3  + + O( ) + O( 4 ). ∂θ ∂θ

Sorting the different powers of  we find ! ∂ S1(1) K (J ) = H0 (J ) +  ω0 (J ) (3.7.7) + H1 (θ, J ) ∂θ ⎫ ⎧  (1) 2 (2) (1) ⎬ ⎨ ∂ S ∂ S ∂ S ∂ω (J ) ∂ H 1 0 1 1 1 + O( 3 ), +  2 ω0 (J ) 1 + + ⎩ ∂θ 2 ∂J ∂θ ∂ J ∂θ ⎭ where we have introduced the angular frequency ω0 (J ) = the left-hand side of (3.7.5)

∂ H0 (J ) ∂J .

We expand also

K (J ) = K 0 (J ) +  K 1 (J ) +  2 K 2 (J ) + O( 3 ),

(3.7.8)

such that we can equate the same powers of . Thus we find O( 0 ) : H0 (J ) = K 0 (J ), O( 1 ) : ω0 (J )

(3.7.9)

∂ S1(1)

+ H1 (θ, J ) = K 1 (J ),  (1) 2 (1) ∂ S1(2) ∂ H1 ∂ S1 1 ∂ω0 (J ) ∂ S1 2 O( ) : ω0 (J ) + + = K 2 (J ). ∂θ 2 ∂J ∂θ ∂ J ∂θ ∂θ

(3.7.10) (3.7.11)

As it should be the new Hamiltonian equals H0 in zeroth order. To compute the next correction to the energy we exploit the periodicity of the motion in the old angle variable θ and average (3.7.10) over it. Since S1 is periodic in θ , the average value of ∂ S1 /∂θ over one period has to vanish. For the first-order energy correction we then find  2π 1 K 1 (J ) = H 1 (J ) ≡ H1 (J , θ ) dθ. (3.7.12) 2π 0 The result for K 1 is used to compute S1(1) (1)

∂ S1 ∂θ

=

1 [K 1 (J ) − H1 (J , θ )] . ω0 (J )

(3.7.13)

3.7 Canonical Perturbation Theory

49

We note that the right-hand side of equation (3.7.13) is just the difference between the perturbation H1 (J , θ ) and its average K 1 (J ). The latter is subtracted to render the first-order correction periodic in θ , as initially required for S1 (θ, J ). The generating function can be used to compute the new action-angle variables in first order in : (1)

ϕ =θ +

∂ S1 (θ, J ) , ∂J

(1)

J = I −

∂ S1 (θ, J ) . ∂θ

(3.7.14)

Using (3.7.8) we obtain for the frequency corrected to first order ω(J ) = ω0 (J ) + 

∂ K 1 (J ) . ∂J

(3.7.15)

The fact that the perturbation induces a shift of the unperturbed frequency is sometimes called “frequency pulling”. The perturbed motion consists of the unperturbed motion plus an O() oscillatory correction. A potential source of difficulties is the frequency ω0 (J ) which may be close or equal to zero. This can occur for a motion near a separatrix (remember the pendulum problem from Sect. 3.1, where the frequency goes to zero when approaching the separatrix), and in higher dimensions generally at nonlinear resonances (see next subsection). The computation of the second-order correction goes along the same line. First we average (3.7.11) over θ and obtain  (1) 2 (1) ∂ H1 (θ, J ) ∂ S1 (θ, J ) 1 ∂ω0 (J ) ∂ S1 (θ, J ) + K 2 (J ) = . 2 ∂J ∂θ ∂J ∂θ

(3.7.16)

The solution for the energy correction is used to compute the next correction to the generating function ⎡ ⎤  (1) 2 (2) (1) ∂ S1 (θ, J ) 1 ∂ω0 (J ) ∂ S1 ∂ H1 ∂ S1 ⎦ 1 ⎣ − = K 2 (J ) − . (3.7.17) ∂θ ω0 (J ) 2 ∂J ∂θ ∂ J ∂θ The corresponding terms for the action-angle variables and the frequency can be calculated as above. We have seen that calculations become simpler when we average over the original variable θ . Nevertheless, the non-averaged part of the Hamiltonian function enters the computation again when we calculate the correction to the generating function. Here terms like K 1 − H1 appear, which is nothing else than the average of H1 minus the function itself. That also means that, when doing canonical perturbation theory, it is helpful to separate the mean motion from a small oscillatory motion. This is the more formal justification of the method of Gauß averaging introduced in Sect. 3.6.2.3.

50

3.7.3

3

Nonlinear Hamiltonian Systems

Problem of Small Divisors

When applying canonical perturbation theory to systems with more than one degree of freedom, the derivation is a bit less transparent than in the previous subsection. The generalization of (3.7.7) reads , (1) (3.7.18) K (J ) = H0 (J ) +  ω0 (J ) · ∇θ S1 + H1 (θ, J ) ⎧ ⎫ (1) (1) ⎨ ⎬  ∂ S ∂ S ∂ω (J ) 1 0i 1 1 +  2 ω0 (J ) · ∇θ S1(2) + + ∇ J H1 · ∇θ S1(1) ⎩ ⎭ 2 ∂ Jj ∂θi ∂θ j i, j

+ O( ), 3

with frequencies ω0 (J ) = ∇ J H0 (J ). Difficulties occur when we want to calculate the first-order correction to the generating function. The first two equations for the energy correction read O( 0 ) : H0 (J ) = K 0 (J ), O( 1 ) :

(1) ω0 (J ) · ∇θ S1 (θ,

(3.7.19) J ) + H1 (θ, J ) = K 1 (J ).

(3.7.20)

To compute the first-order energy correction we average (3.7.20) over the angles θi , i = 1, . . . , N : K 1 (J ) = H 1 (J ) =

1 (2π ) N



2π



2π

dθ1 . . .

0

dθ N H1 (θ, J ).

(3.7.21)

0

Using the expression for K 1 we solve for the first-order correction of the generating function which reads (1) ω0 (J ) · ∇θ S1 (θ, J ) = K 1 (J ) − H1 (θ, J ).

(3.7.22)

Since ω0 (J ) is a vector we cannot just divide by it. To derive a formal solution we exploit the periodicity of S1 and H1 and expand both into Fourier series: S1(1) (θ, J ) =



(1) ik·θ S1,k e ,

H1 (θ, J ) =

k=0



H1,k eik·θ .

(3.7.23)

k

The first summation excludes the k = 0 mode since, by definition, S1 is periodic in θ implying that there is no constant term in its Fourier expansion [26]. Furthermore, (1) K 1 (J ) ≡ H1,0 , the constant term. Inserting the ansatz into (3.7.22) we solve for S1 and find  H1,k eik·θ S1(1) (θ, J ) = i . (3.7.24) k · ω0 (J ) k=0

3.7 Canonical Perturbation Theory

51

While in the one-dimensional case the only source of problems was the single frequency ω0 (J ), we now have a term of the form k · ω0 (J ) in the denominator where ω0 (J ) is a vector with N components each depending on J , and k is summed over Z N . But if k · ω0 (J ) = 0 (resonance condition) the sum will obviously diverge. This is always the case if there exist ω0i and ω0 j such that ωω00ij ∈ Q for all i = j. Hence, the set of frequencies for which the perturbative expansion diverges lies dense in the set of all possible frequencies. Furthermore, even if this does not occur it is always possible to find a k ∈ Z N such that k · ω0 (J ) is arbitrarily small. This phenomenon is known as the problem of small divisors [17,18,20,21]. Despite these difficulties, results which are valid for some finite time can be computed by brute force truncations of the perturbation series.

3.7.4

KAM Theorem

We have just seen that the addition of even smallest non-integrable perturbations can lead to difficult analytic problems. For a long time it was an open question whether such perturbations would immediately render an integrable system chaotic. One can reformulate this question and ask whether there is a smooth or an immediate transition (in the strength of the perturbation) from regular to chaotic motion when a non-integrable perturbation is added to an integrable system. This problem was resolved in the early 1960s when Arnold and Moser proved a theorem that had previously been suggested by Kolmogorov. This theorem (today known as KAM theorem) assures the existence of regular structures under small perturbations. For their proof Arnold and Moser used a technique known as superconvergent perturbation theory. The basic idea of this method is to compute the change of the coordinates successively in first-order perturbation theory. That means a correction is computed in first order and the result is used to do again first-order perturbation theory. If this is applied infinitely often one obtains a perturbation series procedure α of the form α cα  (2 ) that is quadratically convergent in the deviation . Although the very quick convergence in some way outstrips the divergences found in traditional perturbation theory, it produces very complicated expressions. However, for mathematical proofs, such series are a very useful tool. The idea of superconvergent perturbation theory is explained with the help of Newton’s algorithm for finding the root of an equation, which is based on a similar iterative construction. Given a function f we are interested in the solution of the equation f (x) = 0. Using Newton’s technique we start with an initial value x0 . The next value x1 is found by taking the intersection of the tangent of f at x0 with the abscissa, see Fig. 3.9. When successively iterating this procedure one may find the root. Newton’s algorithm leads to the following iterative equation xn+1 = xn −

f (xn ) . f (xn )

(3.7.25)

52

3

Fig. 3.9 The Newton procedure uses an initial guess x0 and the tangent of f (x0 ) to obtain an estimate x1 for the root. Iterating this procedure results in a very fast convergence rate

Nonlinear Hamiltonian Systems

f (x)

x1

x0

x

The change of the approximate result in the n-th step is given by n = xn − xn−1 = −

f (xn−1 ) . f (xn−1 )

(3.7.26)

In order to compute the convergence rate, we estimate n+1 in terms of n . Therefore f is expressed as 1 f (xn ) ≈ f (xn−1 ) + (xn − xn−1 ) f (xn−1 ) + (xn − xn−1 )2 f

(xn−1 ) + · · · 2 1 (3.7.27) = f (xn−1 ) + n f (xn−1 ) + n2 f

(xn−1 ) + · · · 2 The same can be done with the derivative of f 1 f (xn ) ≈ f (xn−1 ) + n f

(xn−1 ) + n2 f

(xn−1 ) + · · · 2

(3.7.28)

The wanted relation is computed in the following way f (xn−1 ) + n f (xn−1 ) + 21 n2 f

(xn−1 ) + · · · f (xn ) ≈ − f (xn ) f (xn−1 ) + n f

(xn−1 ) + · · ·   1 2

1

≈ − f (xn−1 ) + n f (xn−1 ) + n f (xn−1 ) + · · ·

2 f (xn−1 )   f

(xn−1 ) × 1 − n

+ ··· f (xn−1 ) f (xn−1 ) f

(xn−1 ) 1 2 f

(xn−1 ) f (xn−1 ) − n + n

− n

≈−

f (xn−1 ) f (xn−1 ) f (xn−1 ) 2 f (xn−1 )

n+1 = −

+ · · · = n − n + O(n2 ) = O(n2 ),

(3.7.29)

where we expanded the denominator in the small parameter ∝ n . Hence O(n+1 ) = O(n2 ). This means that the iteration is quadratically convergent and we can write x = x0 +

∞  n=1

cn n =

∞  n=0

n

cn 02 .

(3.7.30)

3.7 Canonical Perturbation Theory

53

The last equation highlights both, the quick convergence and the importance of a good initial guess in order for the method to converge (i.e. 0 must be sufficiently small).

3.7.4.1 Statement of the KAM Theorem KAM theorem: Let H = H0 +  H1 be a Hamiltonian function where H0 describes an integrable system. In the action-angle variables (I , θ ) of H0 the motion lies on tori, see the set T in Sect. 3.4. We assume that the given torus (fixed by the chosen energy of the system) is non-resonant, i.e. ∀k ∈ Z N \{0} : k · ω(I ) = 0,

(3.7.31)

) . Then for sufficiently small  there where the frequencies are defined by ωi = ∂ H∂0I(I i exists a torus for the whole system and new angle-action variables (ϕ, J ) which read

ϕ = θ + g(J , θ ),

J = I +  f (J , θ ).

(3.7.32)

This means that the new torus parametrized by ϕ is “near” the old one and that the old and the new torus transform smoothly into each other. The frequencies on the new torus are again given by ω(I ). For the proof several conditions are required [27–29]: • Sufficient nonlinearity or nondegeneracy condition: The nonlinearity has to be strong enough to make the frequencies linearly independent over some domain in J (please keep in mind that J and ω are vectors in general).  det

 ∂ω(J )  = 0.  ∂J J ≈I

(3.7.33)

A system for which (3.7.33) is satisfied is generally called non-degenerate. • Sufficient regularity: A smoothness condition for the perturbation (sufficient number of continuous derivatives of H1 ) has to hold. This condition guarantees that the Fourier coefficients of expansions as in (3.7.23) decrease sufficiently fast. For more details consult [28,30]. • Diophantine condition: The frequency vector ω is “badly” approximated by rationals, which is quantified by ∀k ∈ Z N \{0} : |k · ω(I )| ≥

γ , |k|τ

(3.7.34)

where γ is the Diophantine constant (with physical unit of a frequency) that depends only on , on the magnitude of the perturbation H1 and on some measure of the nonlinearity of the unperturbed Hamiltonian H0 . The Diophantine exponent τ depends on the number of degrees of freedom and the regularity properties of H1 .

54

3

Nonlinear Hamiltonian Systems

Let us discuss the range of application of the theorem. For a more detailed discussion consult the appendix on Kolmogorov’s theorem in [6]. The tori of an integrable system are defined by the action variables Ii , i = 1, . . . , N . Another way to distinguish them is by their frequencies (every torus has its own frequency vector ω(I )). Assume we are given a frequency vector that satisfies the resonant condition, which means that ∃ k ∈ Z N \{0} : k · ω(I ) = 0. This relation can only be fulfilled if the frequencies are rationally related. But since the irrational numbers lie dense in R we can find a new frequency vector ω with only rationally unrelated components in the neighbourhood of ω(I ). Hence the set of non-resonant tori lies dense in the set of all tori. The same holds for the set of resonant tori. More importantly, it can be shown that the Lebesgue measure of the union of all resonant tori of the unperturbed non-degenerate system is equal to zero. Hence, the non-resonant condition and the non-degeneracy condition “restrict” the tori for which the KAM theorem can be applied to a set of Lebesgue measure one. However, there is still the Diophantine condition, which is a stronger assumption. It restricts the set of tori for which the KAM theorem can be applied to a set of Lebesgue measure nearly one in the sense that the measure of the complement of their union is small when the perturbation is small. In the next section we will illustrate the Diophantine condition for systems with two degrees of freedom.

3.7.5

Example: Two Degrees of Freedom

For two degrees of freedom the non-resonant condition for ω1,2 = 0 reads ω1 k 1 + ω2 k 2  = 0

⇔

ω1 k2 = − . ω2 k1

(3.7.35)

Hence the frequencies are not allowed to be rationally related. A stronger restriction whose meaning is not so easily accessible is the Diophantine condition. In our case here it reads γ ∀k ∈ Z2 \{0} : |k · ω(I )| ≥ τ , (3.7.36) |k| for appropriate Diophantine constant γ and exponent τ . The condition refers to the question how good irrational numbers can be approximated by rational numbers. The better − kk21 approximates ωω21 , the smaller is the left hand side of (3.7.36). On one hand every irrational number can be approximated by rational numbers in arbitrary precision (the rational numbers lie dense in the real numbers). On the other hand one may need large k1 and k2 , which make the right hand side of (3.7.36) smaller as both k1 and k2 have to increase. Hence, we are confronted with an interplay of these two effects. The KAM theorem states the existence of a dimensionless function γ¯ () > 0, which goes to zero as  → 0, and gives the following condition for the stability of a

3.7 Canonical Perturbation Theory

55

torus of a two-dimensional system:    ω1 k2  γ¯ ()   ω − k  > k 2.5 , ∀ k1 , k2 ∈ Z (k1 = 0), 2 1 1

(3.7.37)

see [26,31]. In general very little is known about γ¯ . To satisfy (3.7.37) one needs fractions of frequencies which are difficult to approximate by rational numbers. The question how good an irrational number can be approximated by rational numbers can be answered with the help of the theory of continued fractions, see [31] and references therein. For rational numbers, the continued fraction expansion13 is finite. For non-rational numbers it is infinite. π , for instance, is well approximated already by π =3+

1 7+

1 1 15+ ...

≈

355 = 3.141592. 15

(3.7.38)

The reason is that the numbers 7 and 15 appearing in the expansion are rather large making the convergence to the true value fast. For the same reasoning, it turns out that the worst approximated number is the golden mean μ, which is defined by √ 5−1 1 = . (3.7.39) μ= 1 2 1+ 1 1+ 1+...

Here, the continued fraction expansion consists just of 1’s, hence the convergence of finite approximations of the series is worst. As an aside, we mention that μ−1 = √ ω1 5+1 ±1 are most stable with 2 . Hence, the tori whose frequencies satisfy ω2 = μ respect to perturbations.

3.7.6

Secular Perturbation Theory

We have now learnt that the KAM theorem does not apply for resonances occurring in the unperturbed system. In this case one may use instead secular14 or resonant perturbation theory. We introduce the method for the special and most transparent case of an unperturbed system with one degree of freedom and a time-periodic perturbation. For a more general discussion the reader may consult [27]. We assume that the unperturbed Hamiltonian function H0 can be solved with action-angle variables and write the full Hamiltonian as H (I , θ, t) = H0 (I ) +  H1 (I , θ, Ωt),

13 Number

(3.7.40)

theory shows that the continued fraction representation is unique [32]. term “secular” originates in the Latin word saeculum or saecula implying that the corresponding slow-motion part, e.g. in the case of planets, may take a long time, even centuries, in contrast to the fast oscillatory motion that is usually averaged over. 14 The

56

3

Nonlinear Hamiltonian Systems

where H1 is assumed to be periodic in time, H1 (I , θ, Ωt + 2π ) = H1 (I , θ, Ωt). As long as  is small the system has two different time scales. The internal time scale of H0 given by the rotational motion along a given torus of the unperturbed problem with frequency ω0 = ∂∂HI0 . The second one is defined by the driving frequency Ω. The goal of this section is to make an approximate calculation near to resonance conditions on the two frequencies, that means for r ω0 = sΩ,

r , s ∈ Z.

(3.7.41)

In the following, we use a time-independent Hamiltonian H in the extended phase space (Sect. 3.3) which equivalently describes the same system. It is given by H = H − E and can be written as H = H0 (I ) + Ω J +  H1 (I , θ, ϕ),

(3.7.42)

E . Obviously, we have where we have introduced the variables ϕ = Ωt and J = − Ω to solve the new system for H = 0. In the next step we transform into the coordinate system that is rotating with θ¯ (t) = θ + ω0 t. The generating function of this transformation reads S( I¯, θ, J¯, ϕ) = θ I¯ + ϕ J¯ − rs ϕ I¯, where I¯, θ¯ , J¯ and ϕ¯ denote the new coordinates. The relation between the old and the new coordinates is given by

I =

∂S = I¯, ∂θ

θ¯ =

s ∂S =θ− ϕ ¯ r ∂I

(3.7.43)

and ∂S = ϕ. (3.7.44) ∂ J¯ ¯ is a constant, since θ˙¯ = ω0 − s Ω = 0 due to We note that on the resonance θ(t) r equation (3.7.41). When we express the Hamiltonian function in the new coordinates it reads  s  s H = H0 ( I¯) + Ω J¯ − I¯ +  H1 ( I¯, θ¯ + ϕ, ¯ ϕ). ¯ (3.7.45) r r In order to compute an approximate solution of the full system we make two approximations, one concerning H0 and one concerning H1 . The first one is a second-order approximation of H0 ( I¯) in the vicinity of the resonance which reads J=

s ∂S = J¯ − I¯, ∂ϕ r

  H0 ( I¯) ≈ H0 ( I¯0 ) + ω0 ( I¯0 ) I¯ − I¯0 + where

1 ¯ m0 ( I )

=

∂ 2 H0 ¯ ( I ). ∂ I¯2

ϕ¯ =

 2 1 I¯ − I¯0 , 2m 0 ( I¯0 )

(3.7.46)

Using the resonance condition Ω rs = ω0 , we write

s 1  ¯ ¯ 2 H0 ( I¯) − Ω I¯ ≈ H0 ( I¯0 ) − ω0 I¯0 + I − I0 r 2m 0

(3.7.47)

3.7 Canonical Perturbation Theory

57

and find that in zeroth order of  I¯ − I¯0 dθ¯ ∂H ≈ , = ¯ dt m0 ∂I

dϕ¯ ∂H = Ω. = dt ∂ J¯

(3.7.48)

The second approximation is an average of H1 over r periods of the variable ϕ¯ which is known under the name adiabatic perturbation theory (c.f. also Sect. 3.6) and leads to an effective Hamiltonian ¯ J¯) = H0 ( I¯) − ω0 I¯ + Ω J¯ +  H 1 ( I¯, θ), ¯ H ( I¯, θ,  2πr 1 s ¯ = H 1 ( I¯, θ) H1 ( I¯, θ¯ + ϕ, ¯ ϕ) ¯ dϕ. ¯ 2πr 0 r

(3.7.49) (3.7.50)

According to our initial assumptions, H1 is 2π -periodic Hence, it in ϕ as well as θ . i(mθ +nϕ) . H (I )e can be expanded into a Fourier series as H1 (I , θ, ϕ) = ∞ nm m,n=−∞ Expressed in the new variables this reads ∞  s s ¯ Hnm ( I¯)eim θ ei ( r m+n )ϕ¯ . H1 ( I¯, θ¯ + ϕ, ¯ ϕ) ¯ = r m,n=−∞

(3.7.51)

Averaging (3.7.51) over ϕ¯ leads to H¯ 1 ( I¯, θ¯ ) =

∞ 

¯ Hnm ( I¯)eim θ δsm,−r n

m,n=−∞

= H00 ( I¯) +

∞ .  l=1 ∞ 

= H00 ( I¯) + 2

¯

¯

Hsl,−rl e−irl θ + H−sl,rl eirl θ Hsl,−rl cos(rl θ¯ ),

/

(3.7.52)

l=1

where for simplicity we assumed that Hn,m = H−n,−m . If this equality did not hold, we would have to deal with an additional (l-dependent) phase factor in the cosine. Now we transform back to the old coordinate system. The energy is given by E = −Ω J¯ and the old Hamiltonian H is therefore given by H = H − Ω J¯. Additionally, we make a relatively crude approximation and neglect all but the constant and the first term (l = 1) of the Fourier expansion of H 1 . The result is an effective Hamiltonian function near the resonance: ¯ H ( I¯, θ¯ ) = H0 ( I¯) − ω0 I¯ +  H 1 ( I¯, θ).

(3.7.53)

58

3 I¯

Nonlinear Hamiltonian Systems Resonance width

−π

+π

Θ¯

Fig. 3.10 The resonance width W  √ is the width of the separatrix in the I -direction and according to (3.7.56) W ≈ 4 2m 0  Hr ,−s ∝ 

¯ = H 1 ( I¯0 , θ) ¯ + O(). Ignoring all terms Since I¯ = I¯0 + O(), we can write H 1 ( I¯, θ) 2 of O( ) and higher results in H ( I¯, θ¯ ) ≈ H0 ( I¯0 ) − ω0 I¯0 +  H00 ( I¯) 2 1 ¯ ¯ + higher orders. + I − I0 + 2 Ht,−s ( I¯) cos(r θ) 2m 0 2 1 ¯ I − I0 + 2 Hr ,−s ( I¯0 ) cos(r θ¯ ) + const( I¯0 ). ≈ 2m 0

(3.7.54) (3.7.55)

When we introduce p = I¯ − I0 and c = 2 Hr ,−s (I0 ), the effective Hamiltonian H in p2 ¯ This is the same Hamiltonian equation (3.7.55) has the form H = 2m + c cos(r θ). 0 as the one of a pendulum in a gravitational field. The central result of this section is thus that the dynamics created by small resonant perturbations can be effectively described by a pendulum approximation:    θ¯ ¯I (θ) ¯ = 2 2m 0  Hr ,−s cos . 2

(3.7.56)

This directly gives an estimate for the resonance width W assuming it equals the width of the separatrix of the pendulum, see Figs. 3.1 and 3.10 and problem 3.3:  W ≈ 4 2m 0  Hr ,−s .

(3.7.57)

√ We highlight that W is proportional to the square root of the perturbation W ∝ . As a possible application we give an example much studied in the literature. We take the Hamiltonian of the hydrogen atom and assume an additional timedependent driving force pointing in the x-direction (originating, for example, from a time-dependent electric field in dipole coupling). H (r, p, t) =

p2 1 + F x cos(Ωt) . − |r| 2

(3.7.58)

3.8 Transition to Chaos in Hamiltonian Systems

59

At time t = 0 the quantum spectrum reads E n (t = 0) = − 2n1 2 . If we choose n ≥ 10 this Hamiltonian can be used to describe a Rydberg atom which starts to behave similar as its classical counterpart. In the classical system resonance islands appear if the orbital period is resonant with the driving period, e.g. for ωKepler ≈ Ω. Also stable quantum wave packets exist corresponding to quantized states on the resonance island in classical phase space, see [33] for details and more references. We will come back to this example in Sect. 5.4.3.3.

3.8

Transition to Chaos in Hamiltonian Systems

In the previous sections we have focused on near-integrable systems. The equations of motion of integrable systems can be solved by quadratures (see Sect. 3.4), the phase-space trajectory is confined to an N -dimensional torus and action-angle variables can be used. In one space dimension with a trapping potential, this leads always to a periodic motion, see Sect. 3.3.4.1, while in higher dimensions there are more possible kinds of dynamics. Since the motion is periodic in each single coordinate (in action-angle variables) the structure of the overall dynamics of higher-dimensional systems is determined by the coordinates’ frequencies. In case of all frequencies being rationally related, i.e. k · ω(I ) = 0 for a tuple of integers k = 0, the motion is periodic. If in contrast this is not true for all frequencies the phase-space trajectory explores the whole torus and the system’s motion is ergodic on the torus, see Sect. 3.9.1. The characterization of chaotic dynamics instead turns out to be much more involved. From the KAM theorem, we know that there exist systems which are “near” to integrable ones and therefore behave (at least for some time) approximately like them. In this section we will be concerned with systems and their phenomenology for which this is no longer true. When analyzing chaotic dynamics (especially by means of numerical simulations) it is important to be able to visualize the results. Therefore we start this section by introducing the surface of section, which is a highly valuable method to visualize and analyze chaotic systems of low dimensionality.

3.8.1

Surface of Sections

The phase space of a Hamiltonian system with N degrees of freedom is 2N dimensional. This makes a visualization of the dynamics in nearly all cases impossible. Already for a conservative system with two degrees of freedom the energy surface is a three dimensional hyperplane and hence a plot of the trajectory is sometimes possible, but difficulties arise in many circumstances. To this end, a valuable technique called the surface of section (or Poincaré surface of section) has been introduced by Poincaré (1892) and Birkhoff (1932). In principle, the technique is applicable to systems with an arbitrary phase-space dimension but its most important application is to conservative systems with two degrees of freedom, for which we introduce the method now. As already mentioned, their energy surface is three-

60

3

(a)

Nonlinear Hamiltonian Systems

(b) p1 p1

q2

q1

q1

Fig. 3.11 The surface of section is a method to visualize the dynamics. a shows a phase-space trajectory reduced to the energy surface with E = const. In (b) we see the corresponding points on the surface of section reduced by one dimension with respect to the original phase space shown in (a)

dimensional. In order to find a two-dimensional object that can easily be plotted on a sheet of paper (or on a computer screen), we choose a plane in phase space and visualize each point where the trajectory passes through the plane, see Fig. 3.11. A possible and often used example for such a plane is the one defined by setting q2 = 0. The surface of section (SOS) is then filled up by the points from the following set   SOS = (q, p) ∈ R4 | q2 = 0, H (q, p) = E .

(3.8.1)

Since the momenta typically appear quadratically in the Hamiltonian H=

p2 p12 + 2 + V (q1 , q2 ) 2m 1 2m 2

(3.8.2)

there is an ambiguity in sign. For fixed energy E, solving for p2 gives at e.g. q2 = 0: 0   1 1 p12 2 − V (q1 , 0) . p2 = ± 2m 2 E − 2m 1

(3.8.3)

For consistency, the SOS must be constructed by considering only trajectories with one fixed sign, e.g. by taking only trajectories which cross the section in Fig. 3.11 from one side (from q2 < 0 towards q2 = 0). A practical and robust way of finding in numerical simulations the crossing point of the continuously time-dependent trajectory with the SOS is described by Hénon in [34]. Using the SOS we can define the so-called Poincaré map that maps a phase-space point on the SOS to the next point on the SOS, see Fig. 3.12. Since the time evolution in Hamiltonian systems is unique the same is true for the Poincaré map. Since it is

3.8 Transition to Chaos in Hamiltonian Systems Fig. 3.12 Every point (q0 , p0 ) on the SOS is mapped by the Poincaré map P onto the next point where the trajectory (starting at (q0 , p0 )) passes through the SOS. This way the surface of section allows one to reduce the three-dimensional continuous time evolution to a two-dimensional and discrete time evolution

61

p1

q1

a discretization of a continuous dynamical system, it has the mathematical structure of a discrete dynamical system (see Sect. 2.1). We note that many properties of the system’s dynamics are encoded in the Poincaré map. Assume, for example, that the system has a periodic orbit for the initial condition (q0 , p0 ) ∈ SOS. Then the corresponding Poincaré map P has a fixed point of some order n for that initial condition, i.e. Pn (q0 , p0 ) = (q0 , p0 ). A simpler construction of the SOS is possible when the system’s Hamiltonian is periodic in one variable. The natural way to define the SOS in these models is by taking snapshots at multiple integers of the period (with fixed energy). Assume q2 is the periodic variable with period Q then the snapshots are taken for q2 = 0, Q, 2Q, . . . and H (q1 , q2 , p1 , p2 ) = E. Since time-dependent Hamiltonian systems can be understood as time-independent systems on an extended phase space the periodic variable can also be the time. Assume a Hamiltonian (with one-and-a-half degrees of freedom) that is periodic in the time variable H (q, p, t) = H (q, p, t + T ). In this case, the Poincaré map is a stroboscopic map which is obtained by simply plotting the points of the trajectory at all multiples of the period:   SOS = (q, p, t) ∈ R3 | t = nT , n ∈ N; H (q, p, t = nT ) = H (q, p, t = 0) , (3.8.4) which is illustrated in Fig. 3.13. It is possible to introduce a surface of section also in higher-dimensional systems. For a conservative system with N degrees of freedom the energy surface is (2N − 1)dimensional and hence the SOS is (2N − 2)-dimensional. Therefore for N > 2 the technique is often not useful anymore to visualize the dynamics. On the other hand if one still uses a two-dimensional, i.e. further reduced SOS, the result is not necessarily meaningful because the trajectory then does not necessarily pass through the chosen plane.

3.8.2

Poincaré-Cartan Theorem

Liouville’s theorem states that the phase-space volume is preserved under time evolution. We want to show that the same is true for the Poincaré map of a two-degrees-

62

3

Nonlinear Hamiltonian Systems

(b)

(a) p

p1

2T

T

3T t

q1

q

Fig. 3.13 A stroboscopic map exploits the periodicity of the Hamiltonian in time. The points on the SOS are shown in (b) and originate from the trajectory shown in (a)

of-freedom conservative system, or also for a system with one-and-a-half degrees of freedom whose Hamiltonian function is time dependent. The area preservation of the Poincaré map is known as Poincaré–Cartan theorem. Exactly this property of area conservation makes the Poincaré map such a valuable tool. To derive the result we make use of the Poincaré–Cartan integral invariant, which we introduce because it is important by itself. Its natural formulation is in the language of differential forms, which can be found e.g. in [6]. As reviewed in Sect. 3.3.2, Hamilton’s principle of extremal action directly leads to the result that the differential forms pdq − H dt

(3.8.5)

are invariant under canonical transformations. Since the Hamiltonian flow can itself be seen as a continuous canonical transformation we obtain   (3.8.6) ( pdq − H dt) = ( pdq − H dt) . γ1

γ2

Here the two closed curves γ1 and γ2 encircle the flow, i.e. the trajectories as they evolve in time in the extended phase space. In the present context the integral above is called Poincaré–Cartan integral invariant. Having the Poincaré–Cartan integral invariant at hand we come back to our original intention, namely the proof of the Poincaré–Cartan theorem, which states that the Poincaré map is area preserving [31]. First we treat the case of a Hamiltonian system with one-and-a-half degrees of freedom. This includes the special case of a periodic system with H (q, p, t + T ) = H (q, p, t). By h t (q0 , p0 ) = (q(t), p(t)) we denote the time evolution of the Hamiltonian system. Let σ0 ⊂ SOS be an area in the (q, p)-plane at t = 0 and let σi = Pi (σ0 ) = h i T (σ0 ) ⊂ SOS be the same area propagated i-times with the Poincaré map, see Fig. 3.14. We have to show that

3.8 Transition to Chaos in Hamiltonian Systems

63

p1

σ0

σ2

σ1

t

q1 Fig. 3.14 All three areas, σ0 , σ1 = P (σ0 ) and σ2 = P (P (σ0 )), have the same size because the Poincaré map is area preserving

|σ0 | = |σi | for all i ∈ N. Since the σi lie in planes of constant time we apply (3.8.6) for dt = 0 and find immediately  ∂σ0

 p1 dq1 =

∂σ1

p1 dq1 ,

(3.8.7)

with ∂σ0 and ∂σ1 denoting the closed curves encircling the areas σ0 and σ1 . The situation is a little more complicated for conservative systems with two degrees of freedom because constructing the SOS is slightly more complicated (see the previous subsection). Assume we start the time evolution with an area σ0 ⊂ SOS at time zero. When the area is propagated in time it forms a tube just as one can see in Fig. 3.14. Now the SOS is not a plane at a certain point in time, but a hyperplane in phase space. Therefore, it is possible that the points of the propagated area σ0 do not pass the SOS at the same time. Mathematically speaking it is not always possible to find a t such that σ1 = P (σ0 ) = h t (σ0 ). That means we have to start the analysis with (3.8.6) for dt = 0. The SOS is defined by the two conditions q2 = 0 and H (q, p) = E. Exploiting this we have 



∂σ0

giving again

H dt =

∂σ1

 ∂σ0

H dt = 0,

(3.8.8)

p1 dq1 ,

(3.8.9)

 p1 dq1 =

∂σ1

where the two closed curves do not have to be at constant time. Formally one may apply Stokes’ theorem now, relating the closed line integral with the area enclosed by

64

3

Nonlinear Hamiltonian Systems

(b)

(a)

p

p

q

q

Fig. 3.15 Shearing as an example of an area preserving map: at t = 0 (a) and after an iteration of the map (b). Here p is kept constant while q is shifted by an amount linear in p and independent of q

the line. The result is the invariance of the areas, i.e. |σ0 | = |σi | valid for all sections i ∈ N, what we wanted to show. In the next section we give some examples of area preserving maps, which can be seen as Poincaré maps of real or fictitious continuous time dynamics.

3.8.3

Area Preserving Maps

One of the simplest examples of an area preserving map is shearing. It keeps p constant and shifts q by an amount linear in p and independent of q, see Fig. 3.15. An important class of area preserving maps that are connected with Hamiltonian systems are twist maps. Assume the dynamics of an integrable system with two degrees of freedom that can be solved with angle-action variables (θ, I ). The motion is confined to a two-dimensional torus parametrized by the two angle variables. In general the Hamiltonian flow on the torus reads θ1 (t) = ω1 t + θ1 (0), θ2 (t) = ω2 t + θ2 (0),

(3.8.10) (3.8.11)

where ωi = ∂ H∂ I(Ii ) . By t2 = 2π ω2 we denote the time for θ2 (t) to complete a circle. Since the SOS is defined by θ2 = q2 = 0 this is also the time after which the curve passes through the SOS again. Hence the Poincaré map is given by P (θ1 , I1 ) = (θ1 + ω1 t2 , I1 ) = (θ1 + 2π α, I1 ), where we have introduced the winding number α = ωω21 , see Fig. 3.16. Poincaré maps derived from perturbed integrable Hamiltonian systems generally have the form of so-called twist maps [27] In+1 = In + f (In+1 , θn ), θn+1 = θn + 2π α(In+1 ) + g(In+1 , θn ),

(3.8.12) (3.8.13)

3.8 Transition to Chaos in Hamiltonian Systems

65

(b)

(a) ω2 ω1

I1

Fig. 3.16 The twist map twists a point on the circle (parametrized by θ1 ) with radius I1 . In (a) both directions of the torus are indicated whereas in (b) the Poincaré map of the initially straight line (with the arrow) is shown as a curved line since the winding number depends on the action variable in general

where f and g are smooth functions and α denotes the winding number. An important non-trivial special case is the standard map, which is also called Chirikov–Taylor map. It describes the dynamics in the SOS of the kicked rotor whose Hamiltonian is formally given by Chirikov and Shepelyansky [35] H (θ, I , t) =

∞  I2 δ(t − n) . + K cos(θ ) 2 n=−∞

(3.8.14)

It is called rotor because the motion is confined to a circle parametrized by the angle θ with angular momentum I . The iteration has the form In+1 = In + K sin(θn ) mod 2π, θn+1 = θn + In+1 mod 2π.

(3.8.15) (3.8.16)

We highlight that θn and In are both periodic variables here. While the angle is always periodic, one may choose not to wind around the action In , what then describes an unbounded system in (angular) momentum space. More details and motivation for the standard map may be found in [27,31,36]. The SOS dynamics of this map are shown in Fig. 3.17. For K = 0 we see a straight movement of the points on the SOS. In case of K = 0 the topology changes. At the upper and the lower edge the straight lines have transformed into wavy curves and in the middle circles have emerged. The structure in the centre of Fig. 3.17b is called a nonlinear resonance island, compare Sect. 3.7.6.

3.8.4

Fixed Points

A fixed point of the Poincaré map P is defined by P (z ∗ ) = z ∗ ,

(3.8.17)

66

(a)

3

Nonlinear Hamiltonian Systems

(b)

Fig. 3.17 a Dynamics of the standard map for K = 0. b For small K = 0.2 the topology changes. The island structure in the middle is called a nonlinear resonance, the straight lines going from θ = 0 to θ = 2π are the stable KAM tori surviving the perturbation characterized by K . The centre of the island is a stable/elliptic fixed point of the motion

for z ∗ = (q ∗ , p ∗ ) ∈ SOS. As in the case of the logistic map (Sect. 2.2.1), the fixed point and therewith the dynamics near the fixed point can be characterized by the derivative of the map at the fixed point. This is due to the fact that in a small neighbourhood of the fixed point the map can be linearly approximated by P (z ∗ + δz) = z ∗ + dP (z ∗ ) · δz +O(δz 2 ).   

(3.8.18)

≡δz

Usually M (z ∗ ) = dP (z ∗ ) is called monodromy matrix or tangent map. It gives the linear relation between δz and δz    ∂ P1 ∗ ∗ ∂ P1 ∗ ∗    (q , p ) ∂ p (q , p ) δq δq . (3.8.19) = ∂∂q P2 ∗ ∗ ∂ P2 ∗ ∗ δ p

δ p ∂q (q , p ) ∂ p (q , p ) As an example we compute the monodromy matrix of the standard map. Following the notation of Sect. 3.8.3 we write it as I = I + K sin(θ ), θ = θ + I .

(3.8.20) (3.8.21)

It is convenient to split the map into two maps P = P2 ◦ P1 . First we perform the iteration in I and keep θ fixed I = I + K sin(θ ), θ = θ.

(3.8.22) (3.8.23)

3.8 Transition to Chaos in Hamiltonian Systems

67

The resulting monodromy matrix is  M1 =

∂I

∂I ∂θ

∂I

∂I

∂θ ∂θ

∂θ





 1 K cos(θ ) = . 0 1

(3.8.24)

In the second step we change θ I = I, θ = θ + I .

(3.8.25) (3.8.26)

The monodromy matrix of P2 reads  M2 =

∂I

∂I ∂θ

∂I

∂I

∂θ ∂θ

∂θ





 10 = , 11

(3.8.27)

which leads to the overall monodromy matrix  M = M2 ◦ M1 =

 1 K cos(θ ) . 1 1 + K cos(θ )

(3.8.28)

±1. In Sect. 3.8.2 we have Since P is area preserving the determinant of M equals  shown that the Poincaré map leaves the integral ∂σ0 pdq invariant. This integral measures not only the surface area of σ0 , but also its orientation, which is preserved as well. We check this property for our example by computing det(M ) = det(M2 )det(M1 ) = 1 · 1 = 1. After this example, we come to the classification of fixed points. In the case of onedimensional maps, see e.g. Sect. 2.2.1, there was only one relevant quantity namely the absolute value of the derivative of the map at the fixed point. For two-dimensional maps we have two directions and hence the nature of the fixed point is determined by the eigenvalues of the monodromy matrix. The characteristic equation for the monodromy matrix reads det(M − λ) = 0,

(3.8.29)

λ2 − λ tr (M ) + det (M ) = 0,

(3.8.30)

which can be written as

where by “tr” we have denoted the trace of the matrix. Using det(M ) = 1 we find the following simple formula for the eigenvalues of M λ1,2 =

1 1 tr(M )2 − 4. tr(M ) ± 2 2

Depending on the value of tr (M ) we distinguish three cases:

(3.8.31)

68

3

Nonlinear Hamiltonian Systems

η

(b)

(a)

η

ξ ξ

Fig. 3.18 Behaviour of the dynamics near an elliptic (a) and an hyperbolic (b) fixed point in normal coordinates (ξ, η), i.e. after transforming to an appropriate principal-axis representation. In the first case, points on a small concentric circle around the fixed point are mapped onto points on the same circle. In the second case, the invariant geometrical object is a hyperbola

1. |tr(M )| < 2: The eigenvalues are two complex numbers that can be written as λ1 = eiβ ,

λ2 = e−iβ ,

(3.8.32)

with 0 < β < π . 2. |tr(M )| > 2: The eigenvalues are two real numbers whose product equals one: λ1 = λ,

λ2 =

1 . λ

(3.8.33)

3. |tr(M )| = 2: The eigenvalues are λ1 = λ2 = ±1. Having the eigenvalues of the monodromy matrix at hand, it is easy to gain an intuition on the dynamics near the fixed point. Therefore, we perform a coordinate transformation to the eigenbasis of M , which is also called principal-axis representation. The linearized Poincaré map then reads           2 ξ ξ λ1 0 ξ ξ = + +O .

η η 0 λ2 η η

(3.8.34)

In the first case [(3.8.32)], the special form of the eigenvalues tells us that there is a basis in which M can be written as a rotation. Using the same β as introduced in (3.8.32) it is possible to write 

 cos(β) − sin(β) M = . sin(β) cos(β)

(3.8.35)

3.8 Transition to Chaos in Hamiltonian Systems

69

Hence, in the direct neighbourhood of z ∗ the Poincaré map P behaves like a rotation around the fixed point, see Fig. 3.18a. The dynamics around such a fixed point are stable in the sense that the orbit {Pn (z)}n∈N remains in the neighbourhood of the fixed point z ∗ for a long time if z lies next to it. In the second case the monodromy matrix takes the form  M =

 λ 0 . 0 1/λ

(3.8.36)

As for the elliptic fixed point there is a geometrical object that stays invariant when the linearized map is applied—a hyperbola, see Fig. 3.18b. If we start with a point on a hyperbola in the neighbourhood of z ∗ and apply the linearized map, we end up on the hyperbola again. For λ < 0, we additionally jump between the two inversion symmetric branches of the hyperbola with every application of the Poincaré map. When moving along the principal axis, one direction leads to stable dynamics while the other direction leads to unstable dynamics. This is because the absolute value of one eigenvalue is always smaller than one (in this direction the fixed point is attractive) while the absolute value of the other eigenvalue is always larger than one (in this direction the fixed point is repulsive). In case of the elliptic fixed point the β , is a measure of how fast points rotate on a circle winding number, defined by α = 2π around the fixed point when M acts on them. Analogously to the winding number we define the Lyapunov exponent for the hyperbolic fixed point, see also Sect. 2.2.2, by σ ≡ ln λ.

(3.8.37)

The Lyapunov exponent and its connection to the stability of trajectories will be discussed in detail in Sect. 3.9.2. The third case (λ1 = λ2 = ±1) is special in the sense that it marks the transition between the stable and the unstable case. Such a fixed point is called parabolic, and it is said to be marginally stable. The monodromy matrix can be written as (λ1 = 1) 

10 M = c1

 (3.8.38)

with a real positive constant c.15 The invariant curves of this map describe a shearing along a straight line with constant q, see Fig. 3.19. In case of q = 0, no evolution occurs and hence the p-axis is a line of fixed points. If the eigenvalue equals −1 then the map additionally flips the sign of q.

Toeplitz matrix cannot be diagonalised for c = 0 but only be brought into Jordan normal form with a single 2 × 2 block. Its only eigenvector (0, 1) has indeed eigenvalue 1 that can be easily checked.

15 This

70

3

Nonlinear Hamiltonian Systems

Fig. 3.19 A parabolic fixed point leads to shearing

3.8.5

Poincaré–Birkhoff Theorem

In Sect. 3.7.4 we have introduced the KAM theorem which assures the existence of invariant tori with irrational frequency ratios and therewith the stability of conditionally periodic phase-space trajectories under small perturbations. Here we treat the tori and phase-space curves for which the KAM theorem is not applicable. Hence this section is about the fate of tori with rational frequency ratios, in other words curves with rational winding number, when a small perturbations is added to the Hamiltonian. The presentation qualitatively follows [26]. To keep things simple we assume a conservative Hamiltonian system with two degrees of freedom and apply the surface of section technique. The Hamiltonian function is given by an integrable part that is assumed to be solved by action-angle variables (I1 , I2 , θ1 , θ2 ) plus a small perturbation H (I1 , I2 , θ1 , θ2 ) = H0 (I1 , I2 ) +  H1 (I1 , I2 , θ1 , θ2 ).

(3.8.39)

The unperturbed motion governed by H0 is constrained to a two-dimensional torus in the four-dimensional phase space. The corresponding Poincaré map is a twist map (see Sect. 3.8.3) and reads θ1 = θ1 + 2π α(I1 ), I1 = I1 ,

(3.8.40) (3.8.41)

where the primes indicate the iterated variables and α = ωω21 is the winding number.16 The KAM theorem tells us that for a sufficiently small perturbation  H1 the phasespace curves with irrational winding number α are only slightly perturbed as long as the non-degeneracy condition from (3.7.33) is satisfied. We are now going to show how a phase-space trajectory with a rational winding number behaves when the perturbation is switched on.

winding number α depends via the frequencies on both actions I1 and I2 but we skip the dependence on I2 in the following since I2 is fixed for the Poincaré map, c.f. (3.8.1).

16 The

3.8 Transition to Chaos in Hamiltonian Systems

(a)

71

(b) + +

R −

−

R

Fig. 3.20 a We see the dynamics of the unperturbed system. The curve C (dashed line) has a rational winding number α = nr . The two neighbouring curves C − and C + have an irrational winding number and rotate anticlockwise and clockwise with respect to C when Pn is applied. b In the perturbed system C is destroyed while the other two curves are distorted only a little. The curves R and R = Pn R can be used to analyze the fixed point structure of Pn

Let C ∈ SOS be a circle that is invariant under the application of the Poincaré map P of the unperturbed system, i.e. P (C ) = C . In the following we are going to call such curves invariant curves. Additionally, we assume that the dynamics of the points on C are described by a rational winding number α(I1 ) =

r , n

r , n coprime integers.

(3.8.42)

By construction all points on C are fixed points of the n times iterated Poincaré map Pn . Now let us assume that the winding number α depends smoothly on I1 . Then there exist two curves C + and C − with irrational winding numbers in a small neighborhood of C which lie on either side of it, see Fig. 3.20a. When we also assume that α(I1 ), for instance, is increasing smoothly with I1 (using the convention of Fig. 3.16) then, relative to C , C + rotates clockwise and C − rotates counterclockwise when Pn is applied. Next we consider the perturbed system with Poincaré map P  . Since C has a rational winding number it is already destroyed for weak perturbations, whereas the nearby curves C + and C − are only slightly distorted since their winding number is irrational. By definition the distorted curves are invariant curves of P  . Additionally, for sufficiently small perturbations their relative twist is preserved under the application of Pn . Because of this relative twist there must be one point between them for each radius whose angle is conserved when Pn is applied. The union of these points also forms a curve which we denote by R. Because of the angle preservation property of the points of R, any fixed point z ∗ of Pn will lie on R. Hence, if we define the curve R = Pn R the set of fixed points of Pn is given by F(Pn ) = R ∩ R , see Fig. 3.20b. Simple geometry tells us that if there are no tangent points of R and R the fixed points always come in pairs (“what goes in has to come out” - a closed curve that enters another closed curve has to leave it at another point). This result is called

72

(a)

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Nonlinear Hamiltonian Systems

(b)

Fig. 3.21 a To distinguish between elliptic and hyperbolic fixed points the flow-lines (defined by the action of Pn ) near the fixed point are analyzed. The fixed points are found to alternate between being elliptic and hyperbolic. b Around an elliptic (stable) fixed point the trajectories oscillate, whilst they diverge away from a hyperbolic (unstable) fixed point. Convergence is only obtained when the stable manifold is hit exactly, c.f. Sect. 3.8.6. Note that in comparison with the previous figure the sense of rotation of C ± changes assuming here that α(I1 ) decreases smoothly with I1

the Poincaré–Birkhoff theorem: For any curve C of an unperturbed system with rational winding number α = nr and r, n coprime (whose points are all fixed points of Pn ), there will remain only an even number of fixed points 2kn (k = 1, 2, 3, . . .) under perturbation (that means fixed points of the map Pn ). The exact number of 2kn fixed points can be explained as follows. We already mentioned that there are in general 2k intersections of the curves R and R . Each of these intersections leads to a fixed point of order n. But as we have shown in Sect. 2.2.1 for one-dimensional maps (around (2.2.8)) there are always n fixed points of order n and hence 2kn fixed points overall. Following the flow lines (resulting from the action of Pn ) near the fixed point it is easy to distinguish between an elliptic and a hyperbolic fixed point, see Fig. 3.21a. Whether a fixed point is elliptic or hyperbolic alternates from fixed point to fixed point. Hence in the perturbed system we have kn stable and kn unstable fixed points. Around the stable ones there are again invariant curves (see. Fig. 3.21b) which we treat in the same way as the original curves. A successive application of the KAM theorem and the Poincaré-Birkhoff theorem therefore results in a self-similar fixed point structure which is becoming more and more complicated with increasing perturbation. After having understood the behaviour of the dynamics in the neighbourhood of a resonant trajectory we characterize the motion near unstable fixed points in more detail. We will see the Poincaré–Birkhoff theorem at work in the following subsections when discussing the seeds of chaotic motion.

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3.8.6

73

Dynamics near Unstable Fixed Points

As we have seen in Sect. 3.8.4 stable (elliptic) fixed points locally lead to invariant curves in the SOS that have the shape of circles centred around the fixed point. This structure can be compared with the phase space of the pendulum, see Fig. 3.1, and hence the dynamics near such fixed points are small oscillations. The dynamical behaviour near unstable (hyperbolic) fixed points is more complicated. In Sect. 3.8.4 we learned that the monodromy matrix M at an unstable fixed point has two eigenvalues which read 1 (3.8.43) λ− = λ, λ+ = , λ where without loss of generality we assume |λ| > 1. Therefore, the eigenvector v+ belonging to λ+ indicates the directions (±v+ ) for which the fixed point acts attractively while ±v− gives the directions for which the fixed point acts repulsively. In general, a hyperbolic fixed point can be characterized by two invariant curves which are the stable (or ingoing) manifold V+ and the unstable (or outgoing) manifold V− defined by 4 3 ∗ (3.8.44) V± ≡ z ∈ SOS | lim Pn (z) = z . n→±∞

Negative n means that we iterate backwards in time. As before, z ∗ denotes the fixed point. The same definition is used to examine the continuous dynamics h t (z) in the whole phase space Ω. One just has to replace z ∈ SOS by z ∈ Ω and limn→±∞ Pn (z) by limt→±∞ h t (z). Of course, in this case there can be more than two directions (phase-space dimensions). In order to illustrate the definition, two examples of integrable systems are discussed first: 1. Double Well: The first example is the dynamics of a point particle with mass m = 1 in a double well potential given by the Hamiltonian function H=

p2 + (q 2 − 1)2 . 2

(3.8.45)

The system has three fixed points from which two are stable and one is unstable, see Fig. 3.22. The two stable fixed points correspond to the two minima (s) of the potential in configuration space, while the unstable fixed point corresponds to the maximum (u). For all fixed points p = 0 holds. In our example, the stable and the unstable manifolds V+ and V− coincide and form a separatrix between the two types of motion. 2. Pendulum: As a second example we treat a point-mass pendulum (with mass m = 1) in a (scaled) gravity field described by the Hamiltonian H=

p2 − cos(q). 2

(3.8.46)

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3

Fig. 3.22 a The double well potential with the two minima (s) and the unstable maximum (u). b The stable and the unstable manifold V+ and V− coincide and form the separatrix

Nonlinear Hamiltonian Systems

(a)

V ( q) ( u)

( s)

( s)

q

(b)

p

V+

V− q

Due to the periodicity in q, the system has an infinite number of stable and unstable fixed points corresponding to the maxima and the minima of the potential, see Fig. 3.23. Here the unstable manifold of one unstable fixed point equals the stable manifold of the two neighbouring unstable fixed points. For instance, the unstable manifold for q = π is identical to the stable manifolds for q = −π and q = 3π . The phenomenon that V+ and V− form a separatrix is not just present for the particle in the double well potential and for the pendulum, but it is a general phenomenon for integrable systems. The two examples have special features because both are integrable systems. The separatrix is highly unstable with respect to non-integrable perturbations. Hence, for non-integrable systems the smooth joining or the equality of stable and unstable manifolds is not possible [26,37]. Even though the generic situation is far more complicated, there are some rules that need to be obeyed. It is, for example not possible that V+ or V− intersect with themselves, yet V± may intersect with V∓ . When the stable and the unstable manifolds from the same fixed point or from fixed points of the same family17 intersect, we call them homoclinic points. If this is not the case and the intersecting manifolds originate from different fixed points, or fixed points of different families, the intersections are called heteroclinic points.

the map Pn has a fixed point z ∗ there will always be n − 1 other fixed points of Pn given by Pi (z ∗ ) (i = 1, . . . , n − 1), see Sect. 2.2.1. We say these fixed points belong to the same family. 17 If

3.8 Transition to Chaos in Hamiltonian Systems

75

p

V+ V−

V+ V− q

Fig. 3.23 Phase space of the pendulum including stable (V+ ) and unstable (V− ) manifolds. The unstable manifold of each fixed point equals the stable manifolds of its two neighbouring fixed points

By definition V± are invariant curves, that means P (V± ) = V± . Using this property, it is easy to show that if z ∈ SOS is a homoclinic point then Pn (z), n ∈ N is a homoclinic point as well. Additionally, we have z ∈ V± which means that limn→±∞ Pn (z) = z ∗ . From this we conclude that either z = z ∗ or {Pn (z)}n∈N is a sequence of homoclinic points with accumulation point z ∗ . Let us assume the situation shown in Fig. 3.24a where z = z ∗ . We see a hyperbolic fixed point z ∗ and its stable and unstable manifold V+ and V− which intersect at the homoclinic point z. We also see two points z ∈ V+ and z

∈ V− slightly behind z (with respect to the direction indicated by the application of P ). Because P is pushing every point on V± in a well defined direction, it preserves the order of the points P (z), P (z ) and P (z

). The only possible way to keep this order and to take care of P (z) being a homoclinic point is by constructing a loop, see Fig. 3.24b. The enclosed hatched areas must be the same because of area preservation. Because of the continuity of the Poincaré map, the first homoclinic point z actually implies two more intersections whereby the second one is its own picture P (z), as shown in Fig. 3.24b. It is clear that the distance between the homoclinic points decreases as we come closer to z ∗ . Therefore, the curve has to extend much more in the direction perpendicular to V+ in order to preserve the area. This behaviour is shown in Fig. 3.24c. The result is a highly complicated structure of infinitely many homoclinic points and enclosed areas. Another way to examine chaotic dynamics is the propagation of a whole line element. Here, each point of the line element corresponds to a single initial condition. In many cases this method is more appropriate because it is easier to compute the propagation of many initial conditions than to compute the stable and the unstable manifold of a fixed point. Starting with such a line element in the neighbourhood of a fixed point one finds two different structures depending on whether the fixed point is stable (elliptic) or unstable (hyperbolic). Since in the first case the Poincaré map is a twist map (Sect. 3.8.4) the line element forms a tightly curling structure looking like a

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Fig. 3.24 If z is a homoclinic point then there exist infinitely many other intersections (which are again homoclinic points). a shows a homoclinic point z and nearby points z and z

on the stable and unstable manifold, respectively. The manifolds V+ and V− have to intersect again and again. b Because of the continuity and the property of area conservation of the Poincaré map, the points z

, z, and z now on the same unstable manifold map onto the points P (z

), P (z), and P (z ), as shown. In particular, the hatched areas are mapped onto each other and are equal in size. Iterating this procedure one obtains more complex pictures as sketched in (c)

whorl. In the case of a hyperbolic fixed point, the line element stretches exponentially fast and flails backwards and forwards. The resulting structures has been termed a tendril, see Fig. 3.25. More pictures of whorls and tendrils can be found in [26,37]. The Poincaré–Birkhoff theorem assures the existence of a hierarchy of stable and unstable fixed points at all scales where all the unstable fixed points locally lead to chaotic structures of the just mentioned form. Therefore, chaotic motion is present in a self-similar manner on all scales of the system. This behaviour and what happens for even larger perturbations is summarized in the next subsection.

3.8.7

Mixed Regular-Chaotic Phase Space

In the previous two subsections the formation of chaos in nearly integrable systems was discussed. We have shown that hierarchies of stable and unstable fixed points

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77

Fig. 3.25 Phase-space structure with elliptic (red dots) and hyperbolic (blue crosses) fixed points. Due to the Poincaré–Birkhoff theorem the fixed points always come in pairs. Additionally, we see a typical structure that arises when a line element is propagated close to a hyperbolic fixed point—a tendril (green thick structure)

and therefore of chaotic structures arise near resonant unperturbed orbits. In the neighbourhood of hyperbolic fixed points phase-space trajectories show a strange behaviour and form tendrils. When regular structures (invariant curves) gradually disappear, chaotic structures (stochastically distributed looking points in the SOS) appear, and in some parameter regime both structures coexist. In this subsection we give a brief account of the transition from regular to chaotic dynamics. Formal criteria and concepts for the classification of chaotic motion are then introduced in the next section. We assume a physical system governed by the Hamiltonian function H (q, p) = H0 (q, p) +  H1 (q, p),

(3.8.47)

where H0 is assumed to be integrable and H1 is a non-integrable perturbation, whose strength is given by  (in other words H0 and H1 have approximately the same magnitude). Depending on  several regimes can be distinguished. We partly illustrate the corresponding change of phase space by the standard map. 1.  → 0: In this limit the KAM theorem (Sect. 3.7.4) is applicable and one finds quasi-regular dynamics which are similar to the one given by H0 . The phase space mostly consists of invariant tori. The union of all destroyed tori has a measure of order . 2.  small: We are in the regime that is described by the Poincaré-Birkhoff theorem (Sect. 3.8.5). Close to tori with rational winding number, chains of hyperbolic fixed points become visible. These fixed points locally lead to chaotic structures (Sect. 3.8.6). 3.  intermediate: The chaotic layers grow, merge and macroscopic chaotic structures form. Curves with a highly irrational winding number (Sect. 3.7.4) encircle islands

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Fig. 3.26 SOS of the standard map (Sect. 3.8.4) for K = 1. Regular structures can be found even on very small scales

of regularity in the chaotic sea. The phase space is now dominated by self-similar structures, see Fig. 3.26. 4.  large: In case of very strong perturbations even the islands of regular structures become smaller and smaller as the dynamics become more and more unstable there. A phenomenon that can be observed is the bifurcation of stable fixed points—a stable fixed point transforms into two stable and an unstable fixed point, see Fig. 3.27, just what happens at the period doubling transition of the logistic map (see Fig. 2.3). 5.  → ∞: In numerical calculations no regular structures are visible anymore. Nevertheless, there may be such structures on very small scales. In general there is no guarantee for global chaos in the sense that really all (fixed) points have become unstable.

3.8.7.1 The Hénon and Heiles Problem As a preparation for Sect. 3.9 and as an additional example for the discussed mechanisms, we introduce the Hénon and Heiles model. It was proposed by Hénon and Heiles as a simplified model for the motion of a star in a gravity field [38] and is

3.8 Transition to Chaos in Hamiltonian Systems

79

(a)

(b)

(c)

(d)

Fig. 3.27 Part of the SOS of the standard map (Sect. 3.8.4) for different kicking strengths K . When K is increased the main (stable) fixed point bifurcates (a–d). First it stretches (a–b), afterwards a double island structure is developed via a pitchfork bifurcation of the previously stable fixed point (c–d). For very strong kicking strengths the two islands (stable fixed points) drift apart (d) and finally disappear for K → ∞

defined by the Hamiltonian function H=

 1  y3 1 2 px + p 2y + x 2 + y2 + x 2 y − . 2 2 3

(3.8.48)

The third term of the Hamiltonian couples the motion in the x and the y direction and the last term introduces a nonlinear force. The only conserved quantity is the energy E. Hence, the solutions lie on a three-dimensional energy surface in a fourdimensional phase space. We restrict ourselves to situations where E < 1/6, since in this case the motion remains bounded. In order to visualize the dynamics the surface of section technique (Sect. 3.8.2) is used.

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Numerical results of the dynamics of the Hénon and Heiles System are shown in Fig. 3.28. As explained in Sect. 3.8.1, we plot SOSs in the plane ( p y , y) keeping fixed another variable, and typically one chooses x = 0 in this case with px > 0 (fixing the direction in which the SOS is crossed) [39]. For low energies (Fig. 3.28a, E = 0.01)   3 the potential U (x, y) = 21 x 2 + y 2 + x 2 y − y3 is nearly rotationally symmetric, the angular momentum is approximately conserved and the system is therefore quasi-integrable. With increasing energy (Fig. 3.28b, E = 0.045) the phase-space trajectories explore regions where the potential is no longer rotationally symmetric. Nevertheless, there are still mostly invariant curves. Even Fig. 3.28c (E = 0.08) still looks regular on a macroscopic scale, but when zooming in (Fig. 3.28f) we see how chaotic structures have developed in the neighbourhood of the separatrix. For E = 0.115 (Fig. 3.28d) the phase space shows regular and macroscopic chaotic structures of similar size. If the energy is increased further (E = 0.15), the system becomes dominantly chaotic and we mostly see stochastically distributed points, see Fig. 3.28e.

3.9

Criteria for Local and Global Chaos

In Sect. 3.4 we have defined integrable systems. For such systems the phase space consists of invariant tori. The phase-space trajectories lie on tori and the motion is conditionally periodic in most cases. Hence, we know how regular phase-space structures look like and we have tools at hand, such as the criterion for integrability and the KAM theorem, to quantify regular structures. Additionally, the Poincaré– Birkhoff theorem tells us how these structures are destroyed when a non-integrable perturbation is added to the Hamiltonian. What we do not have yet, are criteria for local or global chaos when the perturbation is not small any more. In this section, we close this gap and present a number of techniques that have been developed to characterize chaotic motion. Some of these methods are more stringent than others, and not all of them are easy to apply in practice. We note that there is a great interest in such criteria since widespread chaos enhances transport processes, and a knowledge of the stability properties can be used to engineer transport in phase space, see e.g. [40–47]. First we present two strong criteria: • Lyapunov exponents > 0: The idea behind this criterion is that in the chaotic regime the distance between two trajectories belonging to nearby initial conditions increases exponentially with time. We are going to treat Lyapunov exponents in detail in Sects. 3.9.2 and 3.9.3. • Kolmogorov–Sinai entropy > 0: The Kolmogorov–Sinai entropy is a concept that is defined via the deformation of sets in phase space under time evolution. Similar to the Lyapunov exponents it makes use of the fact that the distance of nearby trajectories increases exponentially in the chaotic regime. Therefore it is no great surprise that it can be expressed as a sum over phase-space averaged Lyapunov exponents, see Sects. 3.9.5. If the Kolmogorov–Sinai entropy is larger than zero,

3.9 Criteria for Local and Global Chaos

81

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 3.28 Surfaces of sections for the Hénon and Heiles System for different energies E (a–e). f shows the details inside the red square of (c). We note that chaotic structures grow fast around unstable fixed points, whilst the surroundings of stable fixed points are largely preserved

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Nonlinear Hamiltonian Systems

the system has the mixing property (Sects. 3.9.1), from which it then follows that it is ergodic. The criteria mentioned above are strong but often difficult to prove or even to compute for a given system. Many more chaos criteria exist though. We mention just a small selection of weaker criteria that may work in some cases and may not work so well in others. The following criteria are quite intuitive: • Resonance overlap criterion: The resonance overlap criterion exploits the fact that when different resonances overlap there cannot be any KAM tori in between them anymore, see Sects. 3.9.6. The perturbation strength for which this happens is used as an approximate definition for the onset of global chaos. Lacking a strict justification and becoming computationally very demanding in its refined versions [27] the overlap criterion is mostly used to give rough estimates. • Decay of correlations: The behaviour of long-time correlations is a measure for the stability of a trajectory. One example for such a correlation is the autocorrelation function defined by C(τ ) = lim

T →∞

1 T



T /2

−T /2

f ∗ (t + τ ) f (t) dt

(3.9.1)

for any dynamical quantity f . Often C is averaged over the phase space to become a global instead of a local measure. Whether a system is chaotic or not manifests itself in the decay of the correlation with time [27]. Typically, chaotic systems show a fast correlation decay, while regular and mixed ones lead to a slower decay, possibly garnished by regular recurrences from orbits within a resonance island. More information and rigorous statements for hyperbolic systems may be found in [48]. • Power spectrum: The power spectrum of a dynamical quantity f is defined by 2     1 T /2 −iωt  f (t) e dt  . S(ω) =  lim T →∞ T −T /2

(3.9.2)

It is connected with the autocorrelation function via the Wiener-Khinchin theorem [48,49]. As discussed in [26,48,49], the power spectrum consists of single lines for regular systems, whilst one expects a broad distribution of frequencies for chaotic ones. • Strong exchange of energy between the degrees of freedom: A strong energy exchange between different degrees of freedom can be an indicator for chaotic motion. This happens, for instance, in the double pendulum moving in a plane, see e.g. [39,50].

3.9 Criteria for Local and Global Chaos

3.9.1

83

Ergodocity and Mixing

In this section we introduce two concepts to characterize the behaviour of dynamical systems—ergodicity and mixing. Ergodicity is a concept that is of great importance in statistical physics, but should be well distinguished from chaotic motion. It is a necessary condition for chaos but even integrable systems can be ergodic. The property of mixing is stronger and a nearly sufficient condition for the existence of chaos. For further information on these topics consult [14,29,48]. We assume a dynamical system defined in Sect. 2.1 with phase space Ω and time evolution T . The dynamics may be discrete or continuous but we assume that T is invertible. Possible applications are the continuous dynamics of a Hamiltonian system (T t = h t ) or the discrete dynamics in a SOS governed by the Poincaré map (T n = Pn ). We also assume a measure18 ν for which ν(Ω) = 1 holds and that is invariant under the time evolution. This means that for all measurable sets A ⊆ Ω we have   (3.9.3) ν T t (A) = ν (A) . Using these definitions we introduce the notion of an ergodic system: Definition (Ergodicity): A dynamical system is called ergodic if for all T t invariant sets A ⊆ Ω one has either ν(A) = 0 or ν(A) = 1. The set A is called T t invariant if T t (A) = A. From a physical point of view this tells us that in ergodic systems there are no other sets of non-zero measure that are invariant under time evolution but the whole phase space Ω. If a dynamical system is ergodic it is possible to interchange time averages and ensemble averages. Let us express this with formulas. We assume a for almost every (with respect to ν)19 function f ∈ L 1 (Ω, ν) with values in R. Then t−1 1 ω ∈ Ω the function f¯(ω) = limt→∞ t n=0 f (T n (ω)) exists and the relation f¯ =

1 ν(Ω)

 Ω

f dν

(3.9.4)

holds. Since the right hand side of (3.9.4) does not depend on an initial condition the same has to be true for f¯. Hence the time average  depend on the choice  does not of ω. This is only possible if the (forward) orbit T t (ω) t≥0 in some sense fills the whole phase space. We have defined f¯ for a discrete dynamical system. In the continuous case the sums are replaced 5 t by integrals. In case of a Hamiltonian system the definition reads f¯ = limt→∞ 1t 0 f (q(t ), p(t )) dt . As invariant measure one usually chooses the Lebesgue measure μ of f 5 5 and hence the phase-space average 5 1 1 |Ω| f (q, p) dμ(q, p) = f (q, p) dqd p, where = dqd p. reads μ(Ω) Ω Ω |Ω| Ω The property that the time average equals the ensemble average can also be used to define ergodicity.

18 For 19 For

a short introduction to measure theory consult, e.g. [51]. almost every ω ∈ Ω means for all ω except for a set of measure zero.

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Ergodicity obviously does not imply chaotic motion. Assume, for example, an integrable system that is solved by action-angle variables. If the frequencies ωi are not rationally related the motion is conditionally periodic and explores the whole torus20 defined by the action variables Ii . Therefore, such a system with more than one degree of freedom is ergodic on the torus, which defines the allowed phase space (energy surface) (see Sect. 3.4). The concept of ergodicity is of great importance in statistical mechanics where the ergodic-hypothesis (hypothetical equality of time and ensemble average) is used to derive, e.g., the micro-canonical ensemble [52]. A first and important result of ergodic theory is the famous recurrence theorem by Poincaré [14]. It is valid for Hamiltonian flows in a finite phase space Ω, and says: Poincaré recurrence theorem: Let A ⊆ Ω be a measurable subset. Then the set   / A (3.9.5) S = a ∈ A | ∃ t : ∀t > t , T t (a) ∈ has measure zero, i.e. ν(S) = 0. The theorem tells us that every point ω ∈ Ω (except for a set of points whose union has measure zero) will come arbitrarily close to itself again when it is propagated in time. This phenomenon is called Poincaré recurrence. Let us come to the definition of mixing which is a stronger property than ergodicity. It can be shown that a mixing system is ergodic but ergodicity does not necessarily imply mixing. The idea for the definition comes from everyday life. Assume we have a cocktail shaker with initially 20% rum and 80% coke (the example is taken from [27]). At t = 0 the two fluids are not well mixed because we have just put them together. When stirring the shaker for some time the rum and the coke will be well mixed and every volume of the resulting fluid will contain approximately 20% rum. More mathematically thinking, we could say that if we stir the shaker infinitely often, there is reasonable hope that in a dynamical system modelling the physical situation of the shaker this will be true for any arbitrarily small volume. Definition (Mixing): A measure preserving dynamical system is said to be (strongly) mixing if for any two measurable sets A, B ⊆ Ω one has   (3.9.6) lim ν A ∩ T t (B) = ν(A)ν(B). t→∞

To understand the definition we come back to the example with the shaker. Let A be the region in phase space that is originally occupied by rum (ν(A) = 0.2) and let B be the region originally occupied by coke (ν(B) = 0.8). Then the fraction pr of rum in the region initially filled with coke after an infinite time of stirring is given by   limt→∞ ν A ∩ T t (B) pr = . (3.9.7) ν(B)

20 To

be more precise the points of the trajectory z(t) are dense on the torus [6].

3.9 Criteria for Local and Global Chaos

85

If the system is mixed we have pr = 0.2, and hence pr = ν(A). Bearing this physical picture in mind, it is reasonable that mixing implies ergodicity. Why the other direction does not hold can be seen for the example used above. Assume an integrable Hamiltonian system whose motion is confined to a two-dimensional torus with frequencies that are not rationally related. The motion on the torus is diffeomorphic to the motion on a square with periodic boundary conditions. Assume now the time evolution of two little squares inside the large square. Because of the simple structure of the time evolution the squares do not intersect or change their shape. This is because each point in the original square is propagated with a constant velocity vector. Hence, the system does not have the property of mixing although it is ergodic. We mention that there is another possible way to define mixing that finds application in the mathematical theory of dynamical systems. Let f (T t (ω)) be a timedependent and square-integrable observable of our system with f ∈ L 2 (Ω, ν). One may define an operator Uˆ (called Koopman operator) by Uˆ f (ω) = f (T ω). One can show that if the system is mixing the Koopman operator Uˆ has a purely continuous spectrum [29,48,53]. The Koopman operator (and its adjoint/inverse) can be seen as classical analogue of the quantum evolution operator (see also the appendix of Chap. 5). In contrast to ergodicity, mixing is a necessary and a nearly sufficient criterion for global chaos (complete hyperbolicity) [29] in the whole phase space or in a subset of it. As an example for a system that has the property of mixing we give Arnold’s cat map [27]. Let Ω be the unit square. The time evolution is given by the map       11 x x = , mod 1. (3.9.8) y

12 y    ≡T

The monodromy matrix of this map is given by T itself. It is straight forward to check that det(T ) = 1 and hence the cat map is area and orientation preserving. From the characteristic equation det(T − λI ) = 0 we compute the eigenvalues of T which are given by √ 3± 5 λ± = . (3.9.9) 2 In the direction of λ− the map is contracting and in the direction of λ+ it is expanding. Hence, all points in the phase space are hyperbolic points and therefore the system is called completely hyperbolic. This tells us, in particular, that also all the fixed points of the maps T n (n ∈ N) are hyperbolic. In Sect. 3.9.2 we will investigate Lyapunov exponents. The Lyapunov exponents of the cat map are particularly simple to compute and serve as a good example. Taking the definition from Sect. 3.8.4 (σ = ln(λ)) we find σ± = ln(λ± ). The complete hyperbolicity is stronger than the mixing property and actually implies true chaoticity in the sense of exponential growth of deviations with time [27,29,48]. The cat map would result from a Poincaré map of a free particle where a harmonic potential ∝ θ 2 is instantaneously switched on and off periodically in time (with

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Fig. 3.29 The Sinai billiard is a particle moving in a quadratic box with a circular scatterer in the middle. Two trajectories are shown, one of them is periodic (red line with two arrows, corresponding to the simplest unstable periodic orbit), the other one is not and represents a chaotic trajectory (black line with only one arrow). An experimental realization in the semiclassical regime is shown in Fig. 5.16

periodic boundary conditions for both variables (θ, p)). Standard examples of chaotic Hamiltonian systems with existing physical realizations are: • Standard map: The system still shows tiny local regular structures even for large kicking strength K (for the definition of the standard map see Sect. 3.8.4). Hence, it is not globally hyperbolic and its “chaoticity” is not strictly proven. • Sinai billiard: It represents a particle moving in a quadratic box with a circular scatterer in the middle, see Fig. 3.29. It can be shown that its dynamics are globally chaotic in the sense of exponential instability [54,55].21

3.9.2

The Maximal Lyapunov Exponent

One of the most striking characteristics of chaotic dynamics is the exponential separation of nearby trajectories. In systems with a bounded phase space this certainly happens only until the borders are reached. From the Poincaré recurrence theorem we also know that two trajectories get close to each other again after some time. Nevertheless, the exponential growth of perturbations or errors in the initial condition (for some time) turns out to be a well applicable indicator for chaotic motion. Lyapunov exponents are used to quantify this process. We start considering the Lyapunov exponent for Poincaré maps. We assume the dynamics of a Hamiltonian system on the SOS governed by the Poincaré map P . By z ∗ ∈ SOS we denote a hyperbolic fixed point. We start with the computation of the monodromy matrix Mn (z ∗ ) of the n-fold iterated

21 Interested

readers may check the following references and websites offering easy-access numerical implementations of billiard systems: [56,57].

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87

Poincaré map leads to 6nPn at the fixed ∗ point. Straight ∗forward differentiation M (Pn−i+1 (z )), where M (z ) = dP (z ∗ ) is the monodromy dPn (z ∗ ) = i=1 matrix of P (z ∗ ). Using the fixed point property P (z ∗ ) = z ∗ we find Mn (z ∗ ) = M n (z ∗ ).

Now let λ > 1 and therefore read

1 λ

(3.9.10)

< 1 be the eigenvalues of M (z ∗ ). The eigenvalues of M n (z ∗ )

1 . (3.9.11) λn The result can be used to illustrate the dynamics near unstable fixed points. Starting with the initial condition z ∗ the dynamics are trivial and we find z n = Pn (z ∗ ) = z ∗ . Now we start the dynamics with a slightly distorted initial condition ζ = z ∗ + δz. The perturbation δz is assumed to be small, and hence we linearize the dynamics around the orbit z n = z ∗ . This allows us to compute an approximate solution of the dynamics which reads ζn = Pn (ζ ) = z ∗ + Mn (z ∗ ) · δz + O(δz 2 ). Inserting the eigenvalues of Mn (z ∗ ) and changing the basis of the exponentials we find λ1 = λn ,

λ2 =

ζn = z ∗ + en ln(λ) δz 1 + e−n ln(λ) δz 2 + O(δz 2 ),

(3.9.12)

where δz 1 is the component of δz pointing in the direction of the first eigenvector of M and δz 2 the component pointing in the direction of the second eigenvector. In the contracting direction the dynamics converge exponentially fast towards the fixed point while in the expanding direction the distance to the fixed point and therewith to the original orbit increases exponentially. As we know from Sect. 3.8.4, the two quantities ± ln(λ) are the Lyapunov exponents of P at the point z ∗ . In the example above, we have seen that the dynamics near an unstable fixed point can be described with the two Lyapunov exponents at the fixed point. After a few applications of the Poincaré map P the distance of ζn to the orbit z n = z ∗ is approximately given by en ln(λ) δz 1 (the other contribution is exponentially small). Hence, when characterizing the dynamics, there is a special interest in computing the largest Lyapunov exponent. Extending the example to a general situation one defines the maximal Lyapunov exponent by   1 Pn (z + δz) − Pn (z) , ln n→∞ δz→0 n δz

σ (z) ≡ lim lim

(3.9.13)

where  ·  denotes the euclidean norm. Why does this definition make any sense? The term Pn (z + δz) − Pn (z) can be written as Mn (z) · δz + O(δz 2 ). Let us look at the matrix n1 Mn (z). By λ± (n) we denote its larger/smaller eigenvalue and by v± (n) the corresponding eigenvectors. Assume further that λ± = limn→∞ λ± (n) and v± = limn→∞ v± (n) exist [26]. Then for almost every δz the Lyapunov exponent is given by σ = ln(λ+ ) (because of the limits the large eigenvalue eventually dominates). Only if one chooses the perturbation exactly in the direction of the stable direction, δz ∝ v− , the outcome will be σ = ln(λ− ). What we have just shown can be applied

88

3

Nonlinear Hamiltonian Systems

to the practical computation of the maximal Lyapunov exponent at a hyperbolic fixed point. It turns out that in the chaotic regime the maximal Lyapunov exponent is (nearly) independent of the initial condition z. We motivate this by noting that the expression in the logarithm of (3.9.13) can be written as Pn (z + δz) − Pn (z) 7 Pi (z + δz) − Pi (z) = , δz Pi−1 (z + δz) − Pi−1 (z) n

(3.9.14)

i=1

where P0 (z) ≡ z. As above we use the notation z n = Pn (z). Since we are interested in the limit δz → 0 we make a linear approximation and write Pi (z + δz) = Pi (z) + Mi (z) · δz + O(δz 2 ). Additionally, we use Mi (z) = M ◦ Mi−1 (z). Then (3.9.14) can be written as n n 7 7 Mi (z) · δz + O(δz 2 ) M ◦ Mi−1 (z) · δz + O(δz 2 ) = Mi−1 (z) · δz + O(δz 2 ) Mi−1 (z) · δz + O(δz 2 ) i=1

(3.9.15)

i=1

≈

n 7

λ(z i−1 ),

i=1

where λ(z i−1 ) denotes the larger of the two eigenvalues of M (z i−1 ). With every iteration the orbit z i follows the direction of the largest eigenvalue of M a little farther (if the eigenvalues are of different magnitude), and hence the approximation is good for small δz as long as i is not too small. The Lyapunov exponent therefore reads n−1 1 ln [λ(z i )] . n→∞ n

σ (z) ≈ lim

(3.9.16)

i=0

We note that this formula has the structure of a time average. If we assume the system to be ergodic in some region in the SOS that we denote by Ωe , we find  1 σ ≈ ln [λ(z)] dz. (3.9.17) |Ωe | Ωe In this approximation the maximal Lyapunov exponent is given by an average over some phase-space region of the maximal eigenvalue of M (z). If this approximation for σ is good, it can only depend weakly on the initial condition z. For a continuous time evolution given by h t of an Hamiltonian system the maximal Lyapunov exponent is defined analogously:   1 h t (z + δz) − h t (z) . (3.9.18) σ (z) ≡ lim lim ln t→∞ δz→0 t δz Assume we are given the Lyapunov exponent of the full dynamics. The theorem of Abramov tells us how to relate this exponent to the one computed for the dynamics in the SOS [27].

3.9 Criteria for Local and Global Chaos

89

Theorem of Abramov: Let σc be the maximal Lyapunov exponent of the full dynamics and let τ be the mean return time onto the SOS. Then the maximal Lyapunov exponent σSOS of the dynamics in the SOS is given by σ S O S = τ σc . This can be seen as follows. Let tn denote the time in which the continuous trajectory propagates from z n−1 ∈ SOS to z n ∈ SOS. We write n1 = t1n tnn ≡ t1n τn and define the mean return time onto the SOS by τ = limn→∞ τn . Insertion into the definitions leads to     τn 1 h tn (z + δz) − h tn (z) Pn (z + δz) − Pn (z) = . (3.9.19) ln ln n δz tn δz In the limit n → ∞ and δz → 0 this gives the above theorem.

3.9.3

Numerical Computation of the Maximal Lyapunov Exponent

We present here a numerical method to compute the maximal Lyapunov exponent of a Hamiltonian system. The time evolution follows from z˙ (t) = J · ∇ H (z(t)) ≡ F(z(t)),

(3.9.20)

see (3.2.1). For the computation of the Lyapunov exponent we compare a trajectory z(t) starting from the initial value z 0 and a nearby trajectory starting at z 0 + h 0 with time evolution z(t) + h(t). The distance between the two trajectories is given by d(t) = h(t). Using this notation the maximal Lyapunov exponent is given by   1 d(t) . ln t→∞ d(0)→0 t d(0)

σ = lim

lim

(3.9.21)

If d(t) increases exponentially we may reach the boundary of the phase space during the numerical computation. Additionally, there is the risk of computer overflow because of very large numbers. The method we present here tries to avoid these problems. We start with d(0) = h 0  which is assumed to be normalized to one. Then we propagate h 0 along a time interval τ and obtain h(τ ) = h 1 and therewith d1 = h 1 . The vector h 1 is normalized to one again and afterward propagated by τ . With the result h 2 we compute d2 . Going on like this, the sequence di for i = 1, . . . , k is calculated, see Fig. 3.30. In analogy with (3.9.21) we define σk =

k 1  ln(di ). kτ

(3.9.22)

i=1

It can be shown that if τ is small enough the limit limk→∞ σk exists and gives the maximal Lyapunov exponent σ [26]. What remains is the question how to propagate h i in time. Since τ and h 0  will be small numbers we can linearize the dynamics.

90

3

Fig. 3.30 Computation of the maximal Lyapunov exponent. After each time step τ the distance di is renormalized to one

Nonlinear Hamiltonian Systems

di+ 3

di+ 2 d(0) di+ 1

d(0)

zi+ 3 zi+ 2

zi+ 1

z˜i d(0)

d(0)

zi = ( pi , qi )

˙ = F(z(t)) + dF(z(t)) · h(t) + The linearized equations of motions read z˙ (t) + h(t) O(h(t)2 ) and lead to the formula ˙ ≈ dF(z(t)) · h(t). h(t)

(3.9.23)

For the numerical calculation this is written as δh i ≈ dF(z i ) · h i τ , where z i = z(iτ ) and h i+1 = h i + δh i . The trajectory z(t) is practically computed with a stable and precise numerical method [58].

3.9.4

The Lyapunov Spectrum

In Sect. 3.9.2 and in our examples we have seen that it is reasonable to define not only one Lyapunov exponent but one for each phase-space dimension. Such a full set of Lyapunov exponents is called the Lyapunov spectrum. Using the definition of the maximal Lyapunov exponent for the two-dimensional Poincaré map (3.9.13), the Lyapunov spectrum is defined by   1 Pn (z + hv± ) − Pn (z) ln n→∞ h→0 n |h| = ln(λ± ).

σ± (z) ≡ lim lim

(3.9.24)

The quantities λ± and v± are defined in Sect. 3.9.2. σ+ = σ is the maximal Lyapunov exponent and σ− is a measure for the rate of contraction of the Poincaré map in the direction perpendicular to v+ (v+ points in the direction in which P is maximally expanding). In the neighbourhood of a hyperbolic fixed point the Lyapunov spectrum contains the same information as the eigenvalues of the monodromy matrix, see Sects. 3.8.4 and 3.9.2. In the case of the continuous dynamics given by h t the Lyapunov spectrum is defined in a similar way. As for the Poincaré map, there exist distinct directions vi such that 2N (dimensionality of the phase space) Lyapunov exponents can be defined by   1 h t (z + hvi ) − h t (z) . (3.9.25) σi (z) ≡ lim lim ln t→∞ h→0 t |h|

3.9 Criteria for Local and Global Chaos

91 growing A

shrinking A

λ1

preserving A

λ2

Fig. 3.31 In order to calculate the Lyapunov spectrum an m-dimensional volume is propagated in tangent space and its distortion is measured in every direction. This is shown in the figure in case of N = 2 and of energy conservation (E =const.). Consequently, there are m = 2N − 1 = 3 Lyapunov exponents

Since the time evolution 2N h t is phase-space volume preserving the exponents σi satσi = 0. In general the Lyapunov spectrum has the following isfy the relation i=1 structure σ1 ≥ σ2 ≥ · · · ≥ σ N = 0 = −σ N ≥ · · · ≥ −σ2 ≥ −σ1 .

(3.9.26)

This relation changes slightly if the energy is conserved. Because the energy-surface is (2N − 1)-dimensional there are only 2N − 1 Lyapunov exponents. Equation (3.9.26) then holds with one exponent less (−σ N has to be left away). The vector vn corresponding to σ N = 0 points in the direction of the Hamiltonian flow. This is because perturbations can only grow linearly in the direction of the flow. We note that there may be more than one (or two) exponents equal to zero. It can easily be shown, for example, that for integrable systems one has σi = 0 for all i. The numerical computation of the full Lyapunov spectrum is more sophisticated than the computation of the maximal exponent, but in principle follows the same line of reasoning. More details on the presented method can be found in [59–61]. To compute the maximal Lyapunov exponent we have to calculate the time evolution of the distance of two nearby initial conditions (which is given by the norm of a vector), see Sect. 3.9.3. The natural generalization of this method is to propagate an m-dimensional volume in the tangent space and to measure its distortion in all directions. This is illustrated in Fig. 3.31. If there are no conserved quantities m = 2N ; in case of energy conservation we have m = 2N − 1. For the propagation we choose a set of pairwise orthogonal vectors ωi ∈ Rm , i = 1, . . . , p with 1 ≤ p ≤ m that span a p-dimensional parallelepiped ω1 × ω2 × · · · × ω p (the edges of the parallelepiped are given by the vectors ωi ). By V p we denote its volume   (3.9.27) V p = ω1 × ω2 × · · · × ω p  .

92

3

Nonlinear Hamiltonian Systems

For the computation we need a quantity called the p-th order Lyapunov exponent which is defined by   V p (t) 1 , (3.9.28) σ p (V p ) = lim ln t→∞ t V p (0)   where we have used the notation V p (t) = h t (ω1 ) × h t (ω2 ) × · · · × h t (ω p ). It can be shown that σ p in the chaotic regime is almost independent of V p (like the Lyapunov exponents are nearly independent of the choice of z and δz, see Sect. 3.9.2). Its relation to the Lyapunov spectrum is given by σp =

p 

σi ,

(3.9.29)

i=1

p where i=1 σi denotes the sum over the p largest Lyapunov exponents. Hence, if we want to compute m the full spectrum we have to successively compute σ p for σi = 0). p = 1, . . . , m − 1 ( i=1 We note that during the numerical computation the parallelepiped ω1 × ω2 × · · · × ω p will be stretched much more in some directions than in others. Therefore, numerical problems occur when very small angles cannot be resolved anymore. The solution of this problem is not only to renormalize successively the size of the volume (as we did in the computation of the maximal Lyapunov exponent), but also to orthogonalize the vectors ωi after each time interval τ . During this procedure, we have to take care that the orthogonalized vectors span the same p-dimensional subspace as the original ones.

3.9.5

Kolmogorov-Sinai Entropy

The Kolmogorov–Sinai (KS) entropy is a concept, which was originally developed within information theory and is applied to the theory of dynamical systems [31,48, 62]. For the definition we need a measure preserving dynamical system as defined in Sect. 3.9.1. The time evolution is assumed to be discrete and invertible. By ξ = {C1 , . . . , Ck } we denote a finite partition of the phase space Ω into k pairwise disjoint and measurable sets. We assume that such a partition exists for all k ∈ N. To shorten the writing we define the function E (x) = x ln(x) for x ∈ R+ . The ξ -dependent function h is defined by h(ξ ) = lim

n→∞



  E ν Ci0 ∩ T −1 (Ci1 ) ∩ · · · ∩ T −n (Cin ) ,

(3.9.30)

i 0 ,...,i n

for i  = 1, . . . , k and  = 0, . . . , n and the measure ν introduced in Sect. 3.9.1. To obtain the KS entropy h KS we have to take the supremum of h(ξ ) over all possible partitions ξ h KS = sup [h(ξ )] . ξ

(3.9.31)

3.9 Criteria for Local and Global Chaos

93

Let us interpret this definition. When wepropagate the partition ξ backwards in time we obtain a new partition ξ(−1) = T −1 (C1 ), . . . , T −1 (Ck ) . Now assume the of the two partitions ξ and ξ(−1) which is defined by ξ ∧ ξ(−1) =  refinement Ci ∩ T −1 (C j ) | i, j = 1, . . . , k; ν(Ci ∩ T −1 (C j )) > 0 . Typically each set B ∈ ξ ∧ ξ(−1) has a smaller measure than one of the Ci . In the next step one takes the refinement of ξ , ξ(−1) and ξ(−2) and so on. The Kolmogorov-Sinai entropy h KS quantifies the decrease of the measure of sets belonging to the successive refinements. It is only positive if there is an exponential decay in the average measure of the elements of the refinements in each step. Therefore, it is no big surprise that h is related to the Lyapunov exponents of the system [27]. The relation reads

h KS

1 = ν(Ω)

⎡



⎣

Ω



⎤ σi (ω)⎦ dν(ω),

(3.9.32)

σi (ω)>0

where the sum is taken over all positive Lyapunov exponents. For a chaotic system with just two degrees of freedom this simplifies to h KS = σ+ .

(3.9.33)

Here σ+ denotes the maximal Lyapunov exponent. Systems for which almost every connected phase-space region has a positive KS entropy are called K-systems. All K-systems have the property of mixing and therefore they are ergodic as well. For Hamiltonian systems, a positive KS entropy is a strong criterion for chaotic dynamics. Nevertheless, the application of this criterion is generally difficult because it is hard to show that a given system is actually a K-system. Two examples of K-systems are the Sinai billiard (Sect. 3.9.1) and the related Lorentz gas model [48,63,64].

3.9.6

Resonance Overlap Criterion

One of the great challenges in the field of nonlinear dynamics and chaos is to find criteria for the onset of global chaos. In a two-dimensional system, global transport is possible only without KAM tori. A criterion that is used to give analytical (and numerical) predictions for the onset of global chaos was developed by Chirikov and others. It is known as resonance overlap criterion or just overlap criterion. It is very intuitive but lacks a strict justification and accurate predictions since the method— when refined to give more accurate results—quickly becomes computationally very demanding. Nevertheless, it is used as a qualitative criterion to give rough estimates. For more information on the topic consult [27] and references therein. During its development the resonance overlap criterion has been applied mostly to the standard map which is the Poincaré map of the kicked rotor (it works especially well for this system). We will also follow this path and introduce the method by

94

3

Nonlinear Hamiltonian Systems

applying it to the kicked rotor. The Hamiltonian of the kicked rotor is given by H (I , θ, t) =

∞  I2 δ(t − n). + K cos(θ ) 2 n=−∞

(3.9.34)

We recall the standard map which has been introduced in Sect. 3.8.3 and reads In+1 = In + K sin(θn ) mod 2π, θn+1 = θn + In+1 mod 2π.

(3.9.35) (3.9.36)

It is a stroboscopic map evaluated at times tn = n ∈ Z when the system is kicked [31]. In the vicinity of regions with regular structures (as for example near the first-order fixed point of the standard map at (I1 , θ1 ) = (0, 0)) there are two frequencies in the system. One is given by the frequency of the time-dependent part of the Hamiltonian ω1 = 2π and the other one (ω2 ) by the dynamics in the (q, p)-plane. The frequency ω2 is difficult to compute but obviously exists, since there are invariant closed curves in the SOS (at least for not too large values of K , see Fig. 3.17). The resonance centred around (I1 , θ1 ) in the SOS is called the 1:1 resonance. This is because (I1 , θ1 ) is a first-order fixed point of the standard map and hence the condition ω1 = ω2 holds. The resonances centred around the two second-order fixed points are called 2:1 resonances because at the these points one has 2ω1 = ω2 . Resonances should not be confused with KAM tori which in case of the standard map create continuous curves in the SOS extending from θ = 0 to θ = 2π , see Fig. 3.17 and [31] for more figures. This structure can easily be understood when we look at the unperturbed and therefore integrable system. For K = 0, the invariant curves in the SOS are straight lines describing the free motion (with constant velocity) of the rotor in configuration space, see Fig. 3.17. Numerically, three different regimes can be identified: • K → 0: The system is nearly integrable. • K  K c ≈ 0.972: The last KAM torus, which divides the phase space into two non-connected regions, is destroyed. • K  4: The linear stability of the 1:1 resonance is destroyed and bifurcations appear. This value is a good approximation for the onset of global chaos in the standard map. The resonance overlap criterion estimates the critical kicking strength K c at which the last KAM torus is destroyed. As explained above, KAM tori of the kicked rotor create invariant curves in the SOS going from θ = 0 to θ = 2π . These curves lie in between the resonances and therefore separate them. When K is increased, the resonances grow and successively new resonances are created. Assume now this happens until the borders of the different resonances touch each other. Obviously, there will be no more KAM curves between them anymore. The strategy of the overlap criterion is to estimate the width of the resonances and, with this information, the

3.9 Criteria for Local and Global Chaos

95

value of the kicking strength K for which they overlap. This result is then taken as an approximation to K c . • Overlap of the 1:1 resonance with itself: In the simplest approximation we estimate at which kicking strength the 1:1 resonance in the SOS overlaps with itself. This is possible because the phase space is 2π -periodic in both variables (I and θ ). To estimate the width of the island we use lowest order secular pertur√ bation theory (pendulum approximation, see Sect. 3.7.6) and find ΔI1 ≈ 4 K . The resonance overlaps with itself when ΔI1 equals the length of the SOS in the I -direction (2π ). Equating these two values we find Kc ≈

π2 ≈ 2.47. 4

(3.9.37)

As mentioned above the numerical result is K c ≈ 0.972, and hence we obviously need to improve the analytical estimate. • Overlap of the 1:1 and the 2:1 resonance: To be more accurate we take into account also the two 2:1 resonances and compute the value of K for which they start to overlap with the 1:1 resonance. For the calculation of the width of the 2:1 resonance we have to do second-order secular perturbation theory and find ΔI1 + ΔI2 = 2π , the defining ΔI2 ≈ K . When we use the overlap condition √ equation for K c can be derived. It reads 2 K c + K2c = π and has the solution K c ≈ 1.46.

(3.9.38)

We have obtained a better result but we need to improve it further. • Overlap of the 1:1 and the 2:1 resonance plus width of separatrix: Refining the approximation in a third step we take into account also the width of the separatrix. This is done in [27] and results in K c ≈ 1.2.

(3.9.39)

This result is again better than the previous one but still above the numerically obtained value. Further improvements become very complicated and computationally demanding. But at least for the standard map renormalization techniques accurately predict K c [27]. The resonance overlap criterion and the described approximation method is illustrated in Fig. 3.32.

Appendix Another Proof of Liouville’s Theorem. Using the notation from (3.2.1), we denote the time-evolution by h t (z) = z(t). Liouville’s theorem then states that |Ω| = |h t (Ω)|

96

3

Nonlinear Hamiltonian Systems

I

Δ I2

Δ I1 2π

θ

Fig. 3.32 The left figure shows a schematic illustration of the resonance overlap criterion: To find an approximation for the critical kicking parameter K c at which the last KAM-torus vanishes the widths of the islands corresponding to the 1:1 and the 2:1 resonances are calculated. To improve the estimation of K c the width of the separatrix has to be taken into account. The separatrix region is described by the dotted (black) lines. The figure on the right shows the phase space of the kicked rotor for K = 1 as a direct application of the criterion

for any set Ω in the phase space. By |Ω| we denote the volume of Ω. An explicit formula for the time-dependent phase-space volume reads  ∂h t (z) |h t (Ω)| = dz. dz = det ∂z h t (Ω) Ω 





(3.9.40)

To prove Liouville’s theorem we will show that d|h tdt(Ω)| = 0. Since the derivative of a function is its unique linear approximation we expand the determinant in the integral of (3.9.40) in first order in t. When we write the time evolution as h t (z) = z + h˙ t (z)t + O(t 2 ) the integrand reads  ∂ h˙ t (z) 2 det I + + O(t ) ∂z 

(I is the identity matrix). To expand also the determinant we make use of the relation det(I + At) = 1 + tr (A) t + O(t 2 ) which holds for any matrix A and any real number t. In our case the trace reads   ˙ ∂ h t (z) = div h˙ t (z) tr ∂z

3.9 Criteria for Local and Global Chaos

which leads to

 |h t (Ω)| =

Ω

97

  1 + div h˙ t (z) · t + O(t 2 ) dz.

(3.9.41)

The derivate of this expression with respect to time is given by d |h t (Ω)| = dt

 Ω

div h˙ t (z) dz = 0,

(3.9.42)

which concludes our proof, c.f. Sect. 3.4.2.

Problems 3.1 Given the Hamiltonian H (x, p) = p 2 /2 + V (x), with V (x) = a|x|n and a > 0, n > 1, from (3.1.17), for a particle of unit mass, determine the period of oscillations as a function of a and the energy E. 3.2 Nonlinear oscillator. A particle of unit mass moves in one-dimension in the potential V (x) = ω2 x 2 /2 − ax 3 /3, with a > 0: (a) Prove that the Hamiltonian is a constant of the motion. (b) Sketch the potential V (x) as a function of x. (c) Sketch the flow of trajectories in phase space ( p, x). Locate any hyperbolic (unstable) and elliptic (stable) fixed point. Draw any separatrix of the motion. 3.3 Pendulum. Show that the area √enclosed by the separatrix of the pendulum, with H = p 2 /2 − k cos(θ ), equals 16 k. Deduce from this result the maximal action for librating motion. 3.4 Symplectic matrices. A real 2N × 2N matrix is called symplectic if:  MT J M = J

with

J=

 0 −1 , 1 0

with the identity N × N matrix 1. Prove now the following statements: (a) (b) (c) (d)

If M1 and M2 are symplectic then M1 M2 is symplectic. If λ is an eigenvalue of a symplectic matrix then also 1/λ. M is symplectic if and only if M is invertible with M −1 = −J M T J . 2 × 2 matrices have the property: M symplectic ⇔ det M = 1.

(3.9.43)

98

3

Nonlinear Hamiltonian Systems

3.5 Canonical transformation. (a) Is the following transformation of phase-space coordinates ( p, q) to (Q, P) canonical Q = q cos( p) , P = q sin( p) ?

(3.9.44)

(b) Can the transformation in (a) be made canonical by introducing the parameters (ν, μ) ∈ Q such that Q = q ν cos(μ p) , P = q ν sin(μ p) ?

(3.9.45)

How must (ν, μ) be chosen in order to guarantee that the transformation is indeed canonical? 3.6 Separable Motion. Find the solution of the Hamilton-Jacobi equation for the motion of a particle of unit mass in three dimensions governed by the Stark Hamiltonian: H (|r| = r , p) =

1 p2 − + F z. 2 r

You can solve this problem step by step: (a) Using parabolic coordinates (ξ, η, φ) obtained fromthe cylindrical coordinates √ 2 2 (ρ,  φ, z) via ξ = r + z and η = r − z, with r = ρ + z and ρ = ξ η = x 2 + y 2 , show first that pφ = ξ ηφ˙ , pη =

1 1 (ξ + η)η˙ , pξ = (ξ + η)ξ˙ . 4η 4ξ

(b) Now show that H (η, ξ, φ; pη , pξ , pφ ) = 2

ξ pξ2 + η pη2 ξ +η

+

pφ2 2ηξ

−

2 F + (ξ − η) . ξ +η 2

(c) Finally, write down the time-independent Hamilton–Jacobi equation H˜



∂S ∂S ∂S , , ; ξ, η, φ ∂ξ ∂η ∂φ

 = E,

and solve it with the separation ansatz for S = S1 (φ) + S2 (ξ ) + S3 (η).

3.9 Criteria for Local and Global Chaos

99

3.7 Canonical perturbation theory. Find the generating function and the new Hamiltonian in second-order perturbation theory for the Hamiltonian of Problem 3.2 with a ≡ −3, i.e., for p 2 + ω2 x 2 + x 3 . 2

H = H0 (x, p) +  H1 (x) = 3.8 Continued fractions.

(a) Compute the following two infinite periodic continued fractions 1+

1+

1 2+

(3.9.46)

1 2+

1 1 2+ ...

1 1+

.

1 2+

(3.9.47)

1 1+

1 1 2+ 1+...

√ √ √ 6, 8, and 10. (b) Determine the continued fraction expansions of the numbers √ (c) For which natural number n ∈ N does n have a continued fraction expansion with period one of the following form: √ n=m+

1 k+

1

,

(3.9.48)

k+ 1 1 k+ ...

for k, m ∈ N? 3.9 Kicked Rotor I. Given the time-dependent Hamiltonian H ( p, q, t) = H0 ( p) + V (q)T

∞ 

δ(t − nT ) ,

n=−∞

how must one choose H0 ( p), V (q) and T such that the following mapping corresponds to the standard map of (3.8.16):         pn+1 pn p(t = nT − δ) pn

→ with = limδ→0+ . qn qn+1 qn q(t = nT − δ) 3.10 Kicked rotor II—chaos and stochasticity. Let us use the following form of the standard map p → p = p + K sin(q) q → q = q + p .

(3.9.49)

100

3

Nonlinear Hamiltonian Systems

In this problem we do not take the modulus operation in the evolution of momenta p (in the evolution of q it might as well be taken, why?). (a) For K = 10, compute numerically how the average energy p 2 /2 increases with the number of iterations of the map. To do so define ca. 1000 initial points at p = 0, with equidistant values q ∈ [1.0001, 1.0002, . . . , 1.1], and average the energy over these initial conditions. What kind of stochastic-like motion do you observe? (b) Calculate analytically how, for large K # 1, the energy p 2 /2 increases on average with the number of iterations of the map. You should express pn as a function of p0 and q0 , q1 , . . . , qn−1 . Use the fact that the iterated values of sin(q) are in good approximation uncorrelated for large | p|.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

Kittel, C.: Introduction to Solid State Physics. Wiley, New York (2005) Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972) Gradshteyn, I., Ryzhik, I.: Table of Integrals. Academic Press, Series and Products (1980) Rebhan, E.: Theoretische Physik: Mechanik. Spektrum Akademischer, Heidelberg (2006) Landau, L.D., Lifschitz, E.M.: Course in Theoretical Physics I. Mechanics. Pergamon Press, Oxford (1960) Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989) Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, New York (1988) Goldstein, H., Poole, C.P., Safko, J.L.: Klassische Mechanik. Wiley-VCH, Weinheim (2006) Struckmeier, J.: J. Phys. A: Math. Gen. 38(6), 1257 (2005) Forster, O.: Analysis. Vieweg + Teubner (2009) Scheck, F.: Mechanics: From Newton’s Laws to Deterministic Chaos. Springer, Heidelberg (2007) Abraham, R., Marsden, J.E., Ratiu, R.: Manifolds, Tensor Analysis, and Applications, 2nd edn. Springer, Berlin (1988) Glück, M., Kolovsky, A.R., Korsch, H.J.: Phys. Rep. 366(3), 103 (2002) Dürr, D.: Bohmsche Mechanik als Grundlage der Quantenmechanik. Springer, Berlin (2001) Schwabl, F.: Quantum Mechanics. Springer, Berlin (2005) Murray, C.D.: Solar System Dynamics. Cambridge University Press, Cambridge (2012) Poincaré, H.: Les Méthodes nouvelles de la méchanique céleste. Gauthier-Villars (1899) Kolmogorov, A.N.: Dokl. Akad. Nauk. SSR 98, 527 (1954) Arnold, V.I.: Russ. Math. Surv. 18, 13 (1963) Arnold, V.I.: Dokl. Akad. Nauk. SSSR 156, 9 (1964) Moser, J.K.: Nach. Akad. Wiss. Göttingen, Math. Phys. Kl. II 1, 1 (1962) Moser, J.K.: Math. Ann. 169, 136 (1967) Percival, I.C., Richards, D.: Introduction to Dynamics. Cambridge University Press, Cambridge (1982) Dehmelt, H.G.: Adv. At. Mol. Phys. 3, 53 (1967) Paul, W.: Rev. Mod. Phys. 62, 531 (1990) Tabor, M.: Chaos and Integrability in Nonlinear Dynamics. Wiley, New York (1989) Lichtenberg, A.J., Lieberman, M.A.: Regular and Chaotic Dynamics. Springer, Berlin (1992)

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28. Moser, J.K.: Stable and Random Motion in Dynamical Systems. Princton University Press (2001) 29. Arnold, V.I., Avez, A.: Ergodic Problems of Classical Mechanics. W. A. Benjamin, New York/Amsterdam (1968) 30. Chierchia, L., Mather, J.N.: Scholarpedia 5(9), 2123 (2010) 31. Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (2002) 32. Hardy, G., Wright, E.: An Introduction to the Theory of Numbers. Oxford University Press, Oxford (2008) 33. Buchleitner, A., Delande, D., Zakrzewski, J.: Phys. Rep. 368(5), 409 (2002) 34. Hénon, M.: Phys. D: Nonlinear Phenom. 5(2), 412 (1982) 35. Chirikov, B., Shepelyansky, D.: Scholarpedia 3(3), 3550 (2008) 36. Chirikov, B.V.: Phys. Rep. 52(5), 263 (1979) 37. Berry, M.V.: Topics in nonlinear mechanics. In: Jorna, S. (ed.) American Institute of Physics Conference Proceedings, no. 46, pp. 19–120 (1978) 38. Hénon, M., Heiles, C.: Astron. J. 69, 73 (1964) 39. Korsch, H.J., Jodl, H.J., Hartmann, T.: Chaos - A Program Collection for the PC. Springer, Berlin (2008) 40. Schanz, H., Otto, M.F., Ketzmerick, R., Dittrich, T.: Phys. Rev. Lett. 87, 070601 (2001) 41. Fishman, S., Guarneri, I., Rebuzzini, L.: Phys. Rev. Lett. 89, 084101 (2002) 42. Buchleitner, A., d’Arcy, M.B., Fishman, S., Gardiner, S.A., Guarneri, I., Ma, Z.Y., Rebuzzini, L., Summy, G.S.: Phys. Rev. Lett. 96, 164101 (2006) 43. Wang, L., Benenti, G., Casati, G., Li, B.: Phys. Rev. Lett. 99, 244101 (2007) 44. Sadgrove, M., Schell, T., Nakagawa, K., Wimberger, S.: Phys. Rev. A 87, 013631 (2013) 45. Schell, T., Sadgrove, M., Nakagawa, K., Wimberger, S.: Fluct. Noise Lett. 12, 1302004 (2013) 46. Klages, R., Dorfman, J.R.: Phys. Rev. Lett. 74, 387 (1995) 47. Sato, Y., Klages, R.: Phys. Rev. Lett. 122, 174101 (2019) 48. Gaspard, P.: Chaos. Scattering and Statistical Mechanics. Cambridge University Press, Cambridge (1998) 49. Schuster, H.G.: Deterministic Chaos. VCH, Weinheim (1988) 50. Weisstein, E.W.: Double pendulum. In: MathWorld—A Wolfram Web Resource. http:// mathworld.wolfram.com/DoublePendulum.html 51. Lieb, E.H., Loss, M.: Analysis. American Mathematical Society (2001) 52. Landau, L.D., Lifschitz, E.M.: Course in Theoretical Physics V. Statistical Physics. ButterworthHeinemann, Oxford (1990) 53. Lasota, A., Mackey, M.C.: Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics. Applied Mathematical Sciences, Springer, New York (1994) 54. Sinai, Y.G.: Uspekhi Mat. Nauk. 25:2, 141 (1970) 55. Chernov, N., Marhavian, R.: Chaotic Billiards. American Mathematical Society (2006) 56. JuliaDynamics. Dynamical billiards. Open source software for nonlinear dynamics and chaos. https://juliadynamics.github.io/JuliaDynamics 57. Porter, M.: Billiard simulator. A GUI to simulate billiard systems on Matlab. https://it. mathworks.com/matlabcentral/fileexchange/10692-billiard-simulator 58. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes 3rd Edition: The Art of Scientific Computing, 3rd edn. Cambridge University Press, New York (2007) 59. Benettin, G., Froeschle, C., Schneidecker, J.P.: Phys. Rev. A 19, 2454 (1979) 60. Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.M.: Meccanica 15, 9 (1980) 61. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Phys. D: Nonlinear Phenom. 16(3), 285 (1985) 62. Sinai, Y.: Scholarpedia 4(3), 2034 (2009) 63. van Beijeren, H., Latz, A., Dorfman, J.R.: Phys. Rev. E 57, 4077 (1998) 64. Bunimovich, L., Burago, D., Chernov, N., Cohen, E., Dettmann, C., Dorfman, J., Ferleger, S., Hirschl, R., Kononenko, A., Lebowitz, J., Liverani, C., Murphy, T., Piasecki, J., Posch, H., Simanyi, N., Sinai, Y., Szasz, D., Tel, T., van Beijeren, H., van Zon, R., Vollmer, J., Young, L.: Hard Ball Systems and the Lorentz Gas. Springer, Berlin (2001)

4

Dissipative Systems

Abstract

Up to now we have focused on Hamiltonian systems obeying Liouville’s theorem of volume conservation of the phase-space flow. In the first few sections we now come back to Chap. 2 and extend the discussion on more realistic dissipative systems. These are characterized by new features, both of stability (motion towards regular attractors) and chaos (motion towards chaotic attractors). Bifurcation scenarios are revisited and systematically discussed in Sect. 4.8. Intermittent fluctuations close to bifurcations are presented as an alternative route to chaos in Sect. 4.9. Section 4.10 presents basic models of synchronization, before we conclude the chapter in Sect. 4.11.

4.1

Introduction

Dissipative dynamical systems do generally not follow Hamiltonian evolution. To illustrate this let us take a classical Hamiltonian of the form H ( p, q) =

p2 + V (q), 2

(4.1.1)

with a potential V (q) depending only on the coordinate q. Newton’s equation of motion then is  ∂V 0 without damping . (4.1.2) = q¨ + ∂q −γ q˙ with damping Hamilton’s version of the dissipative equation would be guessed as q˙ = p

(4.1.3)

∂V p˙ = − − γ p. ∂q

(4.1.4)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. Wimberger, Nonlinear Dynamics and Quantum Chaos, Graduate Texts in Physics, https://doi.org/10.1007/978-3-031-01249-5_4

103

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The parameter γ ≥ 0 denotes the strength of the dissipation. But a fictitious Hamiltonian H˜ which would induce (4.1.4) does not exist for finite γ . This is easily seen by remembering Hamilton’s equation (3.2.1) and by comparing the following two lines of equations:   ∂ ∂ H˜ ! ∂ ( p) ˙ = −γ = − ∂p ∂p ∂q   ∂ ∂ H˜ upper line ! ∂ = −γ . (q) ˙ =0= − ∂q ∂q ∂p

(4.1.5) (4.1.6)

Here, in the last equation, we have used first that (q, p) are independent variables in Hamilton’s formalism, then in the final equation the result from the first line. A direct consequence of the additional damping term with rate γ in (4.1.4) is that, generally, volumes, or areas in two-dimensional phase space, are not any more preserved by the time evolution. In other words, Liouville’s theorem, see Sect. 3.3.3, a nice tool of Hamilton’s mechanics in phase space, is lost in the case of dissipative systems. We assume that our system is described by an ordinary differential equation of integer order n ˙ q, ¨ . . . , q (n−1) ) : R × Rn → R, q (n) = f (t, q, q,

(4.1.7)

where q (n) denotes the nth derivative of q with respect to the variable t (corresponding usually to time in dynamical systems’ theory). As known from the theory of ordinary differential equations [1], such a differential equation can be reduced to n differential equations of first order with x1 ≡ q: ⎛

⎞ x˙1 = x2 (= q (1) ) ⎜ ⎟ x˙1 = x3 (= q (2) ) ⎟. x˙ = F(t, x(t)) ⇔ ⎜ ⎝ ⎠ ... x˙n = f (t, x1 , x2 , . . . , xn−1 )

(4.1.8)

For now we assumed a one-dimensional configuration space, q ∈ R. For higherdimensional configuration spaces, q ∈ Rk , the number of equations multiplies with the integer k > 1, otherwise the same procedure as above applies. An explicit time dependence may be treated in extended phase space, as in Sect. 3.3.1.2, by adding x0 = t with x˙0 = 1 to our vector x in (4.1.8). The concept of a Poincaré surface of section, see Sect. 3.8.1, can be applied in the same manner fixing one of the variables, e.g. x1 = const, and plotting all the others (x2 , x3 , . . . , xn ). This is typically only reasonable for n = 3, for which we obtain a two-dimensional SOS map P : (x2 , x3 ) → (x2 , x3 ),

(4.1.9)

4.2 Fixed Points

105

which now is not necessarily area preserving. Systems are generally called dissipative if any volume element that is enclosed by a finite area in the n-dimensional phase space will contract to zero in the asymptotic time evolution. With the general evolution law, (4.1.8), we may write for the change of a volume element d V in a dissipative system dV = dt

4.2



 dx divF = n

V

dxn V

n ∂ Fi i=1

∂ xi

< 0.

(4.1.10)

Fixed Points

For dissipative systems, we can proceed to analyse fixed points and their stability just as we did in the Hamiltonian case in Sect. 3.8.4. For the general setup defined by the equation of motion (4.1.8) a fixed point x0 is given by the relation !

F(x0 ) = 0.

(4.2.1)

We analyse the stability of the motion around such a fixed point by studying small deviations from this point and the impact on the linearised dynamical evolution. Let us set x(t) = x0 + δx(t). Then the evolution equation (4.1.8) can be expanded into a Taylor series, truncated beyond first order, as follows: δ x˙ (t) = F(x0 + δx(t)) ≈ F(x0 ) +

∂F (x0 )δx j ≡ 0 + D F j (x0 )δx j ≡ D F (x0 )δx(0). ∂x j j

j

(4.2.2) Here D F (x0 ) is the tangent matrix taken at point x0 . Loosely speaking, we may integrate (4.2.2) to see the exponential growth of the deviation with time δx(t) = eD F (x0 ) δx(0) ≡ M δx(0),

(4.2.3)

which defines the monodromy matrix just as in the case of Sect. 3.8.4. For n = 2, we will obtain two eigenvalues of the tangent matrix D F which we will denote σ1 and σ2 . The eigenvalues of the monodromy matrix are then for t ≥ 0 λ1 = eσ1 t λ2 = eσ2 t .

(4.2.4) (4.2.5)

In the case of a real tangent matrix, either both eigenvalues are real or Hermitian conjugate to each other σ1 = σ2∗ . The next subsection summarises all possible scenarios of eigenvalues of the tangent matrix specifically for a two-dimensional dissipative dynamical systems. The importance of all these scenarios lies in the fact that more complex systems may often be locally reduced to a two-dimensional manifold [2]. This is, e.g., possible if a qualitative change when varying a system parameter occurs

106

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only on a subspace of all eigenvectors of the tangent matrix. Then the lower (two)dimensional manifold can be seen as embedded in a higher-dimensional phase space [2]. In this sense, the following scenarios are universal, just as the corresponding bifurcation scenarios reported in Sect. 4.8.

4.2.1

Fixed Point Scenarios in Two-Dimensional Systems

In the following, we classify the various possibilities of eigenvalues (and eigenspaces) of the 2 × 2 tangent matrix introduced in (4.1.8). A more detailed analysis may be found e.g. in [3].

4.2.1.1 Non-degenerate Cases (a) σ1 > σ2 > 0. This case implies for the eigenvalues of the monodromy matrix λ1 > λ2 > 1. It describes the motion around an unstable node. This situation is sketched in Fig. 4.1a. (b) σ1 < σ2 < 0. This implies for the eigenvalues of the monodromy matrix λ1 < λ2 < 1. It describes the motion around a stable node. This situation is sketched in Fig. 4.1b. (c) σ2 < 0 < σ1 . This implies for the eigenvalues of the monodromy matrix λ2 < 1 < λ1 . It describes the motion around an saddle point. This situation is sketched in Fig. 4.1c. (d) σ1 = σ2∗ . This case describes the motion around an unstable focus for Re(σ1 ) > 0 and a stable focus for Re(σ1 ) < 0. Both situations are sketched in Fig. 4.2a, b, respectively.

4.2.1.2 Degenerate Cases In the degenerate case we have only one free parameter. Then there are only two possibilities which are summarised as

Fig. 4.1 Behaviour of the dynamical evolution close to the non-degenerate fixed points mentioned in Sect. 4.2.1.1, corresponding to the cases a to c

4.2 Fixed Points

107

Fig. 4.2 Sketch of the defocussing (a) and focussing (b) dynamics of fixed points mentioned in Sect. 4.2.1.1, corresponding to the case d there

Fig.4.3 Motion around an elliptic fixed point (a) and an unstable (b) or stable (c) star, corresponding to the cases e and f of Sect. 4.2.1.1

(e) σ1 = σ2∗ , with Re(σ1 ) = 0. This case implies for the eigenvalues of the monodromy matrix λ1,2 = e±iα . It describes the motion around a stable elliptic fixed point, just as in the case of conservative two-dimensional maps, see Sect. 3.8.4 and Fig. 3.18a, repeated here in Fig. 4.3a. (f) σ1 = σ2 . This case describes the motion around an unstable star for σ1 > 0 and a stable star for σ1 < 0. Both situations are sketched in Fig. 4.3b, c.

4.2.1.3 Non-diagonalizable Tangent Matrix Finally, we may be in a situation in which the eigenspace of the tangent matrix is only one-dimensional, i.e. both eigenvectors coincide. This happens for matrices of the following forms: (g) σ1 = 1, σ2 = 0 for

DF =

10 00

 .

(4.2.6)

This case implies a motion away from a line of fixed points that is sketched in Fig. 4.4a.

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Fig. 4.4 Motion around the fixed points mentioned in the text, corresponding to the cases g, h and i in Sect. 4.2.1.2. Here, a shows the repulsive line, b the attractive line, whilst c and d show the two cases of an unstable node mentioned in the text

(h) σ1 = −1, σ2 = 0 for

DF =

 −1 0 . 0 0

(4.2.7)

This case implies a motion towards a line of fixed points that is sketched in Fig. 4.4b. (i) σ1 = σ2 = σ for

 σ 1 DF = . (4.2.8) 0σ This case implies a motion close to an unstable node that is sketched in Fig. 4.4d. A special case here is obtained for σ = 0, in which the motion is in one half plane to the right direction, in the other half plane towards the left direction, see Fig. 4.4c.

4.3

Damped One-Dimensional Oscillators

In this section we describe the simplest nonlinear damped oscillator characterized by the following two equivalent versions of the equation(s) of motion: q˙ = p p˙ = −γ p −

 ∂V ∂q

⇔ q¨ = −γ q˙ −

∂V . ∂q

(4.3.1)

4.3 Damped One-Dimensional Oscillators

109

We see that there are two restoring forces present, a conservative one given by the potential V (q) and a dissipative one proportional to the velocity, or momentum in the equations above. In the next subsections we will focus on some specific choices of the potential. We exclusively concentrate here on two-dimensional systems with scalar coordinates (q, p).

4.3.1

Harmonic Oscillator

We start out with the linear harmonic oscillator. It is usually discussed in introductory courses on classical mechanics or the physics of oscillations, and it serves as a basis for what follows here. The harmonic oscillator is characterized by a linear restoring force induced by the potential V (q) = 21 ω02 q 2 . With the standard ansatz q(t) = eλt

(4.3.2)

we obtain from (4.3.1) the following algebraic equation for the parameter λ λ2 + γ λ + ω02 = 0.

(4.3.3)

This quadratic equation is easily solved to get the solutions γ λ± = − ± 2



γ2 − ω02 . 4

(4.3.4)

One may then distinguish the two cases of weak and strong damping. For weak damping, γ < 2ω0 , we may rewrite the square root of the previous equation to see that the solutions oscillate:  γ γ2 λ± = − ± i ω02 − . (4.3.5) 2 4 We observe that the origin ( p, q) = (0, 0) is a stable focus into that the system spirals in the course of the time evolution. In the opposite case of strong damping, γ > 2ω0 , we have λ+ < λ− < 0 and the origin (0, 0) is a stable node, as discussed in Sect. 4.2.1.1. λ± are the eigenvalues of the following evolution matrix defined on the left hand side of (4.3.1):  

 0 1 q q˙ . (4.3.6) = p p˙ −ω02 −γ

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Dissipative Systems

Fig. 4.5 Motion around the stable-node fixed point of the strongly damped harmonic oscillator. E V+ and E V− (dashed lines) sketch the eigenvectors corresponding to the two eigenvalues λ+ and λ− . Lying in the second and forth quadrants p and q have opposite signs for both E V+ and E V− , as described by (4.3.8)

Its eigenvectors, i.e. the directions of the stable and unstable manifold, are approximately connected in this case by ω02 q γ unstable : p = λ− q ≈ −γ q. stable : p = λ+ q ≈ −

(4.3.7) (4.3.8)

Here, we assumed that γ  ω0 . Then in stable case, q is the small component (|λ+ | is large), while it is the large one in the unstable case (|λ− | is small). The dynamics in phase space is sketched in Fig. 4.5.

4.3.2

Nonlinear Oscillators

The vast majority of problems are nonlinear may they originate from such diverse fields of research as biology, chemistry, electronics, or physics. This means that the differential equations representing the models are nonlinear. A nice example is a quadratic oscillator, where in addition to the harmonic restoring force from the previous subsection, we have a second force scaling quadratically with the displacement. The potential is then given by V (q) = 21 ω02 q 2 + 13 βq 3 , inducing the following equation of motion q¨ = −γ q˙ − ω02 q − βq 2 .

(4.3.9)

Figure 4.6 shows the potential V (q) (a) and below the corresponding phase spaces for γ = 0 (b) and finite but small damping 2ω0 > γ > 0 (c). In the latter case of (c), it is clear that we have an attractive fixed point at the origin (0, 0).

4.3.3

Nonlinear Damping

A nonlinearity cannot only enter in the restoring force but also in the damping or amplifying term itself. Popular examples of such cases are electric circuits with

4.3 Damped One-Dimensional Oscillators

111

Fig. 4.6 The nonlinear potential V (q) = 21 ω02 q 2 + 13 βq 3 (a), and the corresponding phase spaces for γ = 0 (b) and 2ω0 > γ > 0 (c)

resistances or amplifiers, see e.g. [4] for an early work. A much studied model is the Van der Pol oscillator characertized by the following differential equation [5] q¨ + γ (q 2 − a 2 )q˙ + ω02 q = 0.

(4.3.10)

From this equation it is clear that we can distinguish two cases in the dynamics: (i) damping for |q| < a and (ii) amplification for |q| > a. The question is then what type of dynamical evolutions are expected and how these depend on the initial conditions. a ≥ 0 is supposed to be a scalar parameter with a unit of a length (as q). Again, the origin (q = 0, p = 0) can be easily identified as a fixed point. A stability analysis with the following ansatz q(t) = q1 eλ+ t + q2 eλ− t ,

(4.3.11)

as in Sect. 4.3.1, leads to  γ a2 (γ a)2 λ± = ± i ω02 − . 2 4

(4.3.12)

These solutions of the characteristic polynomial imply for γ > 0 and γ a/2 > ω0 that the origin is an unstable focus point. Within a certain radius depending on the parameter a in phase space, the trajectories are amplified away from the origin. Outside the same radius, the trajectories are driven inwards by damping on the other hand. The only stable asymptotic solution is then that all trajectories approach this circle, which therefore is called a limit cycle. A pictorial sketch of this scenario is found in Fig. 4.7. We see that the asymptotic, i.e., the limiting dynamics are independent of the precise initial conditions since all the trajectories will eventually always converge to the limit cycle.

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Dissipative Systems

Fig. 4.7 Attraction from inside by amplification and outside by damping towards the attracting limit cycle (red/grey circle)

Fig. 4.8 Sketch of the situation for the Poincaré–Bendixson theorem with a closed and bounded subset R and two trajectories C confined within R, a closed one (red) and an open one (blue)

4.4

Poincaré–Bendixson Theorem

An important theorem for systems with a two-dimensional phase space is the Poincaré–Bendixson theorem [6]. It is the formulation of an extension to the dissipative case of the statement that chaos cannot exist in a two-dimensional phase space for a Hamiltonian system, please cf. Sect. 3.4.1. The Poincaré–Bendixson theorem builds on the following four assumptions: 1. The set R is a closed, bounded subset of the real plane R2 . 2. The two-dimensional system of equations x˙ = F(x) defines a continuously differentiable vector field on an open set that contains R. 3. R does not contain any fixed points. 4. There exists a trajectory C that is confined in the set R, see Fig. 4.8. Poincaré–Bendixson theorem: C is either a closed orbit, or it spirals toward a closed orbit as t → ∞. R always contains such a closed orbit. Proofs of the theorem are found in advanced books on dynamical systems and ordinary differential equations, e.g. in [7–9]. We refrain here from repeating a proof since its content can be explained pictorially very well: The two possible scenarios allowed by the theorem are sketched in Fig. 4.9. There, a sink, or better an asymptotically stable point of equilibrium, is shown on the right into which the a trajectory spirals. On the left, there is an attractive limit cycle to which the evolution converges for large times. The third possibility, to escape asymptotically to infinity, has been excluded by the assumption of a bounded set. Any two-dimensional dissipative dynamical system must hence follow one of this three possibilities. The consequence

4.5 Damped Forced Oscillators

113

Fig. 4.9 The two possible scenarios following from the Poincaré–Bendixson theorem: (Left) attraction to a limit cycle. (Right) attraction to a stable sink. Unbounded trajectories would lie outside the closed and bounded set R that corresponds to the region enclosed by the dashed line in form of an eight

is that true chaotic motion, in the sense of its definitions in Sect. 3.9, is excluded in a two-dimensional phase space. In higher-dimensional systems, with more than two-dimensions in phase space, the Poincaré–Bendixson theorem does not apply. Then, trajectories may indeed wander around in a bounded region without settling eventually down to a fixed point (equilibrium point) or onto a closed orbit (limit cycle). The trajectories may be attracted to a more complex geometric object. The most famous ones are so-called strange attractors [10] that represent fractal sets. The motion is then aperiodic and can indeed be highly sensitive to the initial conditions as required for chaotic motion. The simplest dissipative systems displaying chaotic behaviour are flows in three-dimensional phase space (and their related Poincaré maps in two dimensions, see Sect. 3.8.1). These may have, for instance, the following combination of Lyapunov exponents, cf. Sect. 4.2, σ1 > 0 (expansion), σ1 + σ3 < 0 (to guaranteed the damping) and σ2 = 0 (no stretching or folding along the direction of the flow).

4.5

Damped Forced Oscillators

From the previous section, we know that systems with at least three dimensions are necessary to support chaotic motion. Classes of systems with effectively three dimensions are forced oscillators that are subject to an external time-dependent driving. Here, time acts as the necessary third dimension.

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The simplest case is a driving with a single and constant frequency ω. This system class is described by the following differential equations q¨ + γ q˙ + ∇q V = f cos(ωt),

(4.5.1)

with a constant force amplitude f. We may immediately think of an example in three ˙ force strength F, and dimensions with the coordinates x = (x0 , x1 , x2 ) = (ωt, q, q), the following equations of motion reduced to first order: x˙0 = ω x˙1 = x2

(4.5.2) (4.5.3)

x˙2 = −γ x2 −

∂ V (x1 ) + F cos x0 . ∂ x1

(4.5.4)

In what follows, we will study a famous nonlinear oscillator model in the minimal dimension of phase space showing complex behaviour, the so-called Duffing oscillator. Before we will briefly talk about the much simpler linear driven harmonic oscillator in one-dimension q. Both examples live in a three-dimensional phase space (where the time plays the role of an additional dimension, just as shown in Sect. 3.3.1.2).

4.5.1

Driven One-Dimensional Harmonic Oscillator

For a potential of the form V (q) = ω02 q 2 /2, the force is linear in the coordinate q, and we get the standard driven harmonic oscillator governed by the equation q¨ + γ q˙ + ω02 q = F cos(ωt).

(4.5.5)

This linear differential equation is easily solved by an exponential ansatz with periodic solutions, see also Sect. 4.3.1, q(t) = q± e±iωt ,

(4.5.6)

yielding for the amplitudes q± =

ω02

F . − ω2 ± iγ ω

(4.5.7)

A solution for the inhomogeneous equation (4.5.5) is  qs (t) = Re(q+ eiωt ) = q0 (ω) cos ωt − cot−1



ω02 − ω2 γω

 ,

(4.5.8)

4.5 Damped Forced Oscillators

115

where q0 (ω) ≡ 

F (ω2 − ω02 )2 + γ 2 ω2

,

(4.5.9)

defines a Lorentzian peak around the new resonance frequency shifted by the damping 

to ωres = ω0 1 − γ 2 /ω02 . The general solution q(t) of (4.5.5) is then given by this solution plus the two homogenous solutions [1] of the undriven oscillator from Sect. 4.3.1

   2 2 q(t) = qs (t) + q1 cos( ω − γ 2/4t) + q2 sin( ω − γ 2/4t) e−γ t/2 . (4.5.10) The latter homogenous part is damped to zero for large times t → ∞, and only the inhomogeneous part survives in this limit, as it should be since only the driven part can survive the dissipative damping.

4.5.2

Duffing Oscillator

After the review of the linear driven oscillator, we define now the Duffing model [11] as a standard example of a nonlinear driven oscillator. It is defined by the following equation of motion q¨ + γ q˙ + ω02 q + βq 3 = F sin(ωt).

(4.5.11)

The total potential including the driving can be written as V (q) =

ω02 q 2 βq 4 + − Fq cos(ωt), 2 4

(4.5.12)

where the constants β and F determine the strength of the nonlinear contribution and of the driving force, respectively. Figure 4.10 shows the potential and the phase space for γ = 0 (middle panel) and γ > 0 (lower panel). Physically, the driving may represent a dipole force, proportional to the coordinate x = q, on an atom (or better its internal electronic degrees of freedom) for instance. As previously, the usually positive constant γ characterizes the damping rate. The Duffing model has no general analytical solution [5,12]. The reason is the nonlinearity making it impossible to superpose independent solutions as in the linear case. In the following, we try a qualitative analysis of the Duffing oscillator, trying a superposition ansatz by expanding q(t) into a Fourier series (that seems mathematically justified by assuming solutions oscillating periodically in time): q(t) =

k=∞ k=−∞

qk eikωt ,

(4.5.13)

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Fig. 4.10 (Upper panel): The double well potential from (4.5.12) for F = 0 with the two minima (at q = −1, 1) and the unstable maximum (at q = 0). (Middle panel): Phase-space plot for γ = 0 = F: the stable and the unstable manifolds V+ and V− coincide and form the separatrix, see Fig. 3.22 in Sect. 3.8.6. (Lower panel): Phase-space orbits for positive but still small γ > 0, showing the stable foci in the left and in the right well

with q−k = qk∗ for real coordinates q. The nonlinear term from (4.5.11) expressed using the Fourier amplitudes then reads q(t)3 =



qk1 qk2 qk3 ei(k1 +k2 +k3 )ωt

(4.5.14)

k1 ,k2 ,k3

=

k

qk  qk  qk−k  −k  eikωt ,

(4.5.15)

k  ,k 

where all the sums run from −∞ to ∞ as above. Putting this ansatz into the original equation (4.5.11), we arrive at the nonlinear algebraic equations for the Fourier amplitudes −k 2 ω2 qk + iγ kω + ω02 qk + β

∞

qk  qk  qk−k  −k 

(4.5.16)

k  ,k  =−∞

=

F (δk,1 − δk,−1 ). 2i (4.5.17)

4.5 Damped Forced Oscillators

117

Here, δk,1 represents the Kronecker Delta, giving one for k = 1 and zero otherwise. Periodic driving with a single period ω typically leads in Fourier space to a nearestneighbour coupling as seen on the right hand-side of (4.5.17). We will encounter this phenomenon also in the quantum realm when discussing Floquet systems, see Sect. 5.5.2. In quantum mechanics one calls such systems with only nearest-neighbour coupling tight-binding models, a term which might be known to the reader from simple models for a crystalline solid state [13]. We will come back to such models in Sect. 5.5. Let us first assume that the nonlinear parameter β is small. Then we expect that only the first Fourier amplitudes q1 and q−1 will dominate in the expansions above. For k = 1, we obtain then from (4.5.17) the following approximate relation (−ω2 + iγ ω + ω02 )q1 + β (q1 q1 q−1 + q1 q−1 q1 + q1 q1 q1 + · · · ) =

F . 2i (4.5.18)

that is equivalent to (−ω2 + iγ ω + ω02 + 3β|q1 |2 )q1 =

F , 2i

(4.5.19)

remembering that q−1 = q1∗ . Setting q1 = eiα a/2, with a real phase α that drops out to arrive at the nonlinear relation for the real amplitude a  ω02

3 − ω + βa 2 4 2



2 + (γ ω)

2

a2 = F 2.

(4.5.20)

The latter relation is third order in a 2 , and hence it generally has three solutions for a 2 . These solutions are sketched in Fig. 4.11 around resonance, hence for ω close to ω0 (where “close” is determined by the damping parameter γ , see the previous Sect. 4.5.1). The figure shows a typical hysteresis phenomenon of nonlinear systems at resonance, what is known as bistability. For β = 0, there are two stable branches, one coming from the left and another one coming from the right in parameter space (here by changing the driving frequency ω). Where three solutions coexist, two are stable, whilst the third is unstable as shown in Fig. 4.11. The unstable part can hardly be reached in practice since the system will tend to follow either of the two stable solutions. Which of the two stable solution is selected will depend on the specific initial conditions. Now we look at the Duffing oscillator when the nonlinear parameter β is large. Then the harmonic part of the potential in (4.5.12) is small compared to the dominating nonlinear part. In this case it makes also sense to set the prefactor of the linear part negative, leading to a driven motion in a double well as sketched in Fig. 4.10. The corresponding equation of motion reads q¨ + γ q˙ + κq + βq 3 = F sin(ωt),

(4.5.21)

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Fig. 4.11 Linear for β = 0 (blue dashed line) and nonlinear for β > 0 (red solid/dotted line) response against the control parameter ω. The red solid line shows the stable branch, while the red dotted line shows the unstable branch. The nonlinear response shows a bistability, with three solutions for a fixed value of ω. Two of these solutions are stable (open circles) and one is unstable (cross). Usually, the nonlinear system can be followed by a slow adiabatic change of the control parameter ω along the stable branches only, from the left towards the right on the upper branch and from the right towards the left along the lower branch

with κ < 0 as just motivated. Without driving F = 0, we obtain trajectories moving in a double-well potential, as shown in Fig. 4.10 for the undamped (γ = 0) and the damped (γ > 0) case. For us, it is this latter case with κ < 0 that is of interest since it supports chaotic motion [5,12]. For a driving amplitude F > 0, we analyse now for concreteness a specific set of parameters that are γ = 1/2, β = 1, κ = −1 and driving frequency ω = 1. These parameters give an eigenfrequency of the oscillations in a single well of ω0 = 2. With increasing strength F, we see the following evolution of the dynamics, quite similar to the simpler one-dimensional logistic map studied in Sect. 2.2.1: • 0 < F < 0.3437...: there exist two stable limit cycles in both wells with period T1 = 2π/ω. • F = 0.3437...: the limit cycles are destroyed and new ones appear with doubled period T2 = 2T1 , hence a period doubling occurs just as in the logistic map, see Sect. 2.2.1. • F = 0.355...: the period-2 cycles are destroyed and new ones with period 4 (T4 = 4T1 ) appear. • F = 0.3577...: the period-4 cycles are destroyed and new ones with period 8 (T8 = 8T1 ) appear. This scenario of period doubling continues with increasing F until the critical value F∞ is reached. • F > F∞ = 0.3598...: the trajectories lie on a chaotic attractor. • F∞ < F < 0.3833...: the motion still remains restricted to just one of the two wells. • F > 0.3833...: the motion can jump between the two wells since sufficient energy is pumped into the system by the drive.

4.5 Damped Forced Oscillators

119

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 4.12 Series of phase-space plots for the Duffing oscillator and the parameters γ = 1/2, β = 1, κ = −1, ω = 1. For F = 0 we only have a sink in each of the two wells (a); for F = 0.3 one observes convergence versus a unique limit cycle (b); for F = 0.35 (c) and F = 0.357 (d), and F = 0.37 (e) we observe convergence towards two, four, or eight different limit cycles, respectively; for F = 0.39 the orbits have sufficient energy to visit both wells (f). The corresponding trajectories are shown with thicker darker lines, while the thin lighter lines indicate the background phase space for γ = 0 = F

Figure 4.12a–d show typical trajectories at specific values of F to illustrate the above list of events. The bifurcation diagram for the Duffing oscillator is shown in Fig. 4.13 for one well (the left one). Here, the stable equilibrium points are shown as a function of the driving strength F. Finally, Fig. 4.14 reports a Poincaré map for specific initial conditions. It shows one point plotted in phase space for each period of the drive, see Sect. 3.8.1. The Poincaré SOS appears not space filling, yet there seems to be structure on all scales. This is similar to fractals, hence we speak of a strange (chaotic) attractor in this case [10]. Strange attractors represent chaotic motion in dissipative systems [5,10]. They show a sensitive dependence on the initial conditions since the contraction of volume

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Fig. 4.13 Bifurcation diagram for the left well of the Duffing oscillator as a function of the driving amplitude F. The right panel shows a zoom into the interesting parameter region F = 0.34 . . . 0.37. The other parameters are as in Fig. 4.12

Fig. 4.14 Poincaré SOS for ωt = 0 mod 2π for one trajectory of the Duffing oscillator in the chaotic regime at F = 0.385. The other parameters are as in Fig. 4.12. For this value of the drive, the trajectory explores both, left and right regions of the phase space

does not imply that length scales must shrink in all directions. Points, infinitely close, will exponentially increase their relative distance on the attractor as time evolves. The necessary conditions for the occurrence of a strange attractor are: • stretching of volume elements in at least one dimension; • shrinking of volumes in all other directions; • a finite allowed region in phase space that guarantees the necessary folding of trajectories (c.f. Sect. 3.8.6). The properties of the chaotic attractor are illustrated further by another example in the next section.

4.6

Lorenz Model

The Lorenz model is a system of ordinary differential equations first studied by Edward Lorenz in 1963 [14]. Lorenz was a mathematician and meteorologist who established the theoretical basis of weather forecasts, as well as of computer-aided

4.6 Lorenz Model

121

Fig. 4.15 (Upper panel) A sketch of heat transport in the atmosphere in the three space dimensions (x, y, z), where heat flows mainly in the Z direction from lower and hotter air layers to higher and colder layers. (Middle panel) At small temperature differences, the heat flow is in stationary equilibrium and the temperature changes linearly, i.e. with a constant gradient. (Lower panel) At larger temperature differences, vortex-like structures in the heat flow up and down the layers occur and the flow may even become turbulent

meteorology. In such contexts, he coined the famous butterfly effect, the idea that small changes in the initial conditions (small causes) can have large effects, even in a remote region of the physical space. The Lorenz system was born as a model for heat transport in the atmosphere. A detailed discussion may be found in [15,16]. As sketched in the upper panel of Fig. 4.15, the vertical flow of heat along the z-axis is investigated for small heights h compared to the extension of the horizontal plane in x and y direction. For small temperature gradients, the drop in temperature is usually linear, see the middle panel in Fig. 4.15, whilst for large temperature differences heat convection may occur as sketched in the lower panel of Fig. 4.15. The Rayleigh– Bénard experiment [5,17] realises the essential idea of the model under idealised laboratory conditions. The system of three coupled ordinary differential equations brought forward by Lorenz as minimal model for the temporal evolution of heat transport is given in the three variables X (t), Y (t), and Z (t) which describe the circular flux velocity, the temperature difference between rising and falling flux, and the amplitude of oscillations in the vertical temperature profile, respectively. The coupled system of equations reads X˙ (t) = s(Y − X ) Y˙ (t) = r X − Y − X Z

(4.6.2)

Z˙ (t) = X Y − bZ .

(4.6.3)

(4.6.1)

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Here, we have three parameters: s = ξv is the so-called Prandtl number, defined as the ratio of the viscosity of the liquid v and the heat conductivity ξ . r the Rayleigh number that is proportional to the temperature difference. b = 4/(1 + h 2 /l 2 ) characterizes the physical dimensions h and l of the convection cell, see the lower panel in Fig. 4.15. Lorenz fixed s = 10 and b = 8/3. Since the Lorenz model is described by three nonlinear ordinary differential equations, it can bear chaotic motion. Obviously, it is just a crude deterministic model for the much more complicated process of heat flow in the atmosphere. Our analysis of the Lorenz system starts with the search for fixed points of the evolution, hence points with x˙ = (0, 0, 0), where the vector x represents the three variables (X (t), Y (t), Z (t)) of (4.6.3). The fixed-point relation implies that X =Y X (r − 1 − Z ) = 0 X 2 = bZ ,

(4.6.4) (4.6.5) (4.6.6)

as can easily be verified. For the linear analysis, we compute the tangent matrix, see (4.2.3), that equals the Jacobian matrix ⎛

⎞ −s s 0 J = ⎝r − Z −1 −X ⎠ . Y X −b

(4.6.7)

The eigenvalues (λ1 , λ2 , λ3 ) of this matrix determine the stability properties of the evolution. The motion around a fixed point remains stable for Re(λ j ) < 0 for all j = 1, 2, 3. One may easily see that the stability properties depend in particular on the parameter r . So let us distinguish the following cases: • 0 < r < 1: x0 = (0, 0, 0) is a stable fixed point of the evolution given by (4.6.6). This case corresponds to stable heat conduction. • r > 1: now x0 = (0, 0, 0) becomes unstable, and two new equilibrium points appear, namely:   x± = (± b(r − 1), ± b(r − 1), r − 1).

(4.6.8)

Around the latter two stable points stable heat conductivity occurs in the form of circular rolls, see the lower panel in Fig. 4.15. • r > rc = s(s+b+3) s−b−1 : now x± become unstable, and the motion approaches a chaotic attractor, which was proved only recently by Warwick Tucker [18]. This scenario is similar to the Duffing oscillator studied in the previous section for F  F∞ . Close to the attractor, an exponential spreading of originally nearby trajectories occurs in the time evolution.

4.6 Lorenz Model

123

Fig. 4.16 Bifurcation diagram for the Lorenz model for the parameters b = 8/3 and s = 10. Plotted are the positions of the fixed points in the X direction against the parameter r . Whilst at r = 1 a normal pitchfork bifurcation occurs (the stable fixed point at the origin (0, 0, 0) turns unstable and produces two new stable ones), at the critical value r = rc ≈ 24.7 a subcritical pitchfork bifurcation is seen that will be better explained in Sect. 4.8.1. The solid lines result from numerical simulations of the true asymptotic time evolution, whilst the dashed lines are drawn in order to represent the unstable branches

First, we note that rc > 1 for choices of s  1 and b of the order 1 that correspond to the values mentioned above. Using the definition of the Lyapunov exponent from Sect. 3.9.4 with h t (x) = (X (t), Y (t), Z (t)) starting at x0 = (X (0), Y (0), Z (0)) 

1 h t (x0 + δx) − h t (x0 ) , σ = lim lim ln t→∞ δx→0 t |δx|

(4.6.9)

one can prove that indeed we obtain a spectrum of three exponents satisfying σ1 > σ2 = 0 > σ3 .

(4.6.10)

Here, σ1 is responsible for the chaotic expansion, σ3 for the convergence to the attractor, and σ2 describes the motion along the flow lines of the evolution. The mentioned chaotic attractor cannot have dimension D = 2 since this would contradict the Poincaré–Bendixson theorem. Its dimension cannot be exactly D = 3 either since then there would be stretching or folding of the trajectories along all directions, in particular along the flow itself. Hence, the dimension of the attractor must be noninteger and lie between two and three: 2 < D < 3. Non-integer dimensions are well known from fractals. We conclude that our attractor has a fractal support embedded in three dimensions. This is the reason why we may call such an attractor strange. The attractor of the Lorenz model has dimension D ≈ 2.05, as can be computed numerically [19]. Fractals will be the subject of the next section. Before that we plot the stability diagram for the Lorenz model in Fig. 4.16 and show the trajectories converging to the Lorenz attractor in configuration space in Fig. 4.17. At the time of Lorenz, computer power was still restricted, and he had the idea to reduce the complexity of the problem to a one-dimensional map. This now called Lorenz map is very useful in interpreting the results of the temporal evolution of the full system of equations. The definition of the Lorenz map is sketched in Fig. 4.18,

124

4

(a)

(b)

(c)

(d)

Dissipative Systems

Fig. 4.17 Chaotic attractor of the Lorenz equations (4.6.3) for the parameters r = 99 > rc , b = 8/3, s = 10 in the variable space (X , Y , Z ) (a) and in its two-dimensional projections (b)–(d). The initial conditions are [X (0), Y (0), Z (0)] = (−20, −15, 113)

(a)

(b)

Fig.4.18 Construction of the Lorenz map: From the time series Z (t) in (a) the maxima are extracted from the region t ∈ [0, 100] and plotted as the map Z n+1 versus Z n . b A slope in its absolute value larger than one in this Lorenz map, as compared to the dashed line with slope one, indicates a globally unstable temporal evolution

where the maxima in the evolution of one variable, typically Z (t) define the discrete set of new variables Z n . The latter can be plotted as a map of the form Z n+1 = f (Z n ) by graphical extraction of the maxima from the exact time evolution. Now, if the absolute value of the discrete derivative of the map f is larger than 1, in formula | f  (Z )| > 1, we may conclude that the Lorenz map is unstable. If this is true for all initial conditions Z (0), then the embedded map is globally unstable, c.f. Problem 2.1. The typical kink form of the Lorenz map is shown in Fig. 4.17b in the chaotic parameter regime.

4.7 Fractals

4.7

125

Fractals

Fractals are famous representatives of chaotic systems, often shown in nice color pictures, see e.g. the book by Peitgens and Richter [20], or Mandelbrot’s own book from 1982 [21]. Fractals are geometric objects that have a scale-invariant structure. What this means is that they look precisely the same (for true fractals) and practically the same (for physical fractals) on all scales one looks at them. In the previous two sections, we have seen that the dynamical evolution of chaotic systems may settle down onto strange attractors that are examples of fractal geometric structures with a non-integer dimension. This section introduces some of the simplest fractal sets to grasp their characteristic properties. Furthermore, we will introduce a practical method to actually compute the dimension of fractal sets. This method can then be applied to characterize complicated sets one might encounter when studying dynamical systems. For applications in higher-dimensional system, we explicitly refer to [19].

4.7.1

Simple Examples

4.7.1.1 Cantor Set Arguably the simplest fractal is the Cantor set that is well known from any introductory analysis course, see e.g. [22,23]. The Cantor set is embedded into the real line and is by construction self-similar. The construction of the set starts by deleting from the interval [0, 1] the open middle third interval (1/3, 2/3). We are left with the union of the two line segments [0, 1/3] and [2/3, 1]. This process of deleting the open middle third is repeated for each remaining segment ad infinitum. More precisely, after the second step we are left with the union of the following four line segments [0, 1/9], [2/9, 1/3], [2/3, 7/9], and [8/9, 1]. The Cantor set contains then by definition all points in the interval [0, 1] that are not deleted in any step in this infinite process. The construction of the set is sketched in Fig. 4.19. The Cantor set is an example of a fractal string. Obviously, one may start from a broader or smaller interval [0, L], with any 0 = L ∈ R and by rescaling all lengths we just get the very same construction. After n steps, we are left with N = 2n intervals (or objects in the set) with a length each of = 1/3n (choosing again L = 1). As shown in analysis text books [24], the Cantor set has Lebesgue measure zero but it is uncountable. Obviously, one might repeat the construction deleting other length of intervals. Therefore, Fig. 4.19 The first four iterations of Cantor’s one-third set

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Fig. 4.20 The first three steps of the construction of the Koch curve (b)–(d). a shows the starting line

sometimes we speak more precisely of the ternary Cantor set for the construction given above. We may define the ratio − ln N / ln = ln 2/ ln 3 ≈ 0.631 that describes the fact that in each step we create two copies of length 1/3. This ratio is the Hausdorff dimension of the set. Pictorially, the Hausdorff dimension counts how many balls N (R) of a certain radius R are necessary to cover a set in one or more dimensions. The limit of this number (if existing) for vanishing radii defines the Hausdorff dimension as DH = − lim R→0 lnlnN (R) R , see e.g. [24]. For smooth curves in D = 1, 2, 3 dimensions it is easy to see that N ∝ R D . For instance, take a square of length 1 and divide it into nine squares with side length 1/3, this gives DH = ln 9/ ln 3 = 2 = D. The Hausdorff dimension is, however, capable of assigning non-integer numbers to sets embedded into R D . The Cantor set is the simplest example with DH = ln 2/ ln 3. An approximation of the Hausdorff dimension can be obtained for more complicated sets numerically, e.g., with the box-counting algorithm to be discussed below.

4.7.1.2 Two Dimensional Fractals There are two-dimensional extensions of the construction used for the Cantor set. The most famous ones are the Koch curve [25] and Sierpinski’s triangle [26]. The construction of the Koch curve (or Koch star or snowflake) is sketched for the first three steps in Fig. 4.20. For this curve, one again cuts the open middle sets of each of three sides, but in addition a triangular tip is added within the cuts, the two sides of which are again cut in the middle third. This procedure goes on ad infinitum. The boundary length of the Koch curve is infinite in the limit of infinite steps since it increases by a factor of 4/3 in each step. Its effective dimension must be larger than 1 but smaller than 2. Effectively, its Hausdorff dimension is DH = ln 4/ ln 3 ≈ 1.26. Another example of a deterministically constructed fractal set is the Sierpinski triangle (or Sierpinski gasket or sieve) [26]. Its overall shape is of an equilateral triangle, subdivided recursively into smaller and smaller equilateral triangles. One shrinks the triangle to 1/2 in height and 1/2 in width, makes three copies, and positions the three shrunken triangles such that each triangle touches the two other triangles at a corner, see Fig. 4.21. The result is the emergence of the central hole, because the three shrunken triangles can cover only 3/4 of the area of the original triangle. As for the Cantor set and the Koch curve, holes are the important feature of these simple fractals. Finally, we repeat the previous step with each of the smaller triangles, see Fig. 4.21 on the very right, and so on ad infinitum. It is somewhat clear that the area remaining must be of dimension smaller than 2 since parts have been cut out. The Hausdorff dimension of the Sierpinski triangle is indeed

4.7 Fractals

127

Fig. 4.21 The first six steps of the construction of the Sierpinski triangle. The full starting triangle is shown on the left. Moving to the right, we remove first one internal triangle, second three smaller ones, then nine even smaller triangles, and so on

DH = ln 3/ ln 2 ≈ 1.59, the inverse of the dimension of the Cantor set. This is actually not an accident since both sets are mathematically related [27]. Practical information on how to actually compute fractal dimensions can be found, in e.g. [19,28,29] where various standard method are compared, in particular the box counting and its refinement in [29]. The latter we will briefly discuss in the following as the simplest and most easily accessible technique.

4.7.2

Box-Counting Dimension

The box-counting algorithm gives a practical and intuitive recipe for determining the “effective” dimension of a geometrical object in n ∈ {1, 2, 3, . . .} dimensional space. The algorithm goes as follows: • cover the object with n-dimensional cubes of length • count the number N = N ( ) of cubes which cover the object. One looks for a scaling of the form

D

0 for → 0. (4.7.1) N  N0

• The limit → 0 defines the box-counting dimension ln N ( ) .

→0 ln

DBC ≡ − lim

(4.7.2)

The scaling of N ( ) ∼ −D determines the effective dimension D of the object. We illustrate the procedure in Fig. 4.22a for a single point in R2 where always exactly one square (or box) is sufficient to cover the point, hence N = 1 and DBC = 0 as should be for a point like zero-dimensional object. For a smooth line segment in R2 , the necessary number of boxed scales inversely proportional to the length of the line, hence we have N ( ) ∼ 1/ , and DBC = 1, see Fig. 4.22b. For an area enclosed by a smooth curve in R2 , we similarly have N ( ) ∼ 1/ 2 , and DBC = 2, see Fig. 4.22c. Here, in all the three cases the box-counting dimensions equals the dimension of the underlying space as it should be for smooth curves. Our examples from the previous subsections show the usefulness of the box-counting dimension, as a good practical approximation to the mathematically more precise Hausdorff dimension, also for fragmented or fractal geometrical objects with non-integer dimension. We leave as

128

(a)

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(b)

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(c)

Fig.4.22 Sketch of box-counting for two (a) and three box-sizes (b), (c): a a single point (red/grey dot) is covered by exactly one box no matter of what size, hence DBC = 0. b to cover a smooth line (red/grey line) usually the number of boxes is inversely proportional to its length , or in other words N ( ) ∼ −1 leading to a dimension DBC = 1. c a smooth area (enclosed by the red/grey line) is covered by a number of boxes scaling with the box size as N ( ) ∼ −2 , giving a dimension DBC = 2

an easy exercise the computation of the box-counting dimension of the Cantor set to the interested reader. There exists a refinement of the box-counting algorithm, known as variational method for determining the fractal dimension of an object. This method typically gives more reliable results, as discussed e.g. in [29]. Applications of the algorithm to the quantum version of the standard map, see Sect. 5.5.2, are found e.g. in [30]. The variational method is best explained for the analysis of functions of one variable f (x) over an x-axis. The smoothness of such curves embedded in two-dimensional space, can then be analysed by dividing the full interval along the x-axis to be studied in M subintervals, and the total variation of the curve on groups of 2 adjacent subintervals is computed. The average of these quantities is called VM ( ) and the value of M which gives the best scaling of the form VM ( ) ∼ DV is then used to extract the variational fractal dimension DV . The advantage with respect to the simpler box-counting method is given by the fact that the latter uses only the number of adjacent squares of fixed width along the x-axis necessary to box all points of the curve. In the variational method the box sizes are variationally adapted which typically gives more reliable results. We finally recommend careful checks when computing fractal dimensions numerically. All numerical methods are somewhat problematic since fractal curves have, in principle, exploding derivatives. Their behaviour at any edge of a given grid (box) is therefore, notoriously hard to sample. The best recipe is certainly to apply various different techniques and compare their results.

4.7.3

Examples from Nature

There are many examples from nature whose fractal structures have been analysed in detail, see e.g. [20,21,31]. Arguably the most famous is the Romanesco broccoli (or Roman cauliflower), which shows self-similar form on many length scales.

4.8 Bifurcation Scenarios

129

Many examples stem from an analysis of pattern formation, e.g. in trees where scale-free ramification can occur. Also ice-crystals show a self-similar structure that originates from their formation. This fractal pattern is best seen in frost crystals that sometimes form on cold glass. Similar examples of dendritic crystal formation that show self-similarity can also occur in natural crystal growth, or in copper and plexiglass under stress (e.g. when applying high electric currents). Finally, we mention the self-similar structure of some coast lines. Good examples are the coasts of Great Britain or Norway. The west coast of Britain has an estimated fractal dimension of about 1.25 [32]. The interested reader may test this by using an enlarged photocopy of his atlas and applying the box-counting algorithm to it. It should be clear that anything occurring in nature has a finite maximal and a minimal length scale. Hence, self-similarity will always be found only on certain characteristic length scales in nature. This is contrary to the idealised mathematical objects discussed above, which show self similarity on infinitely small scales. Typically, one would need a self-similar power-law scaling, like the number of boxes versus box sizes, over at least two to three orders of magnitude in order to realistically speak about a fractal-like structure and to determine a reasonable fractal dimension.

4.8

Bifurcation Scenarios

This section reviews only the most important scenarios of bifurcations. More systematic overviews over the topic can be found in the literature [3,5,12,15]. Generally, a bifurcation is a sudden qualitative change of a fixed point, or of a periodic orbit, when varying a control parameter in a given system. Most of our analysis will be based on simple one-dimensional potentials and hence on systems with a two-dimensional phase space. Their relevance lies in the fact that more complex systems may be locally reduced to them. The theory of normal forms is a method for transforming the ordinary differential equations for nonlinear dynamical systems into certain standard forms. Using particular coordinate transformations, one may thus remove the inessential part of higher-order nonlinearities, and a two-dimensional theory sometimes may suffice to characterize the local stability properties [2,9,12,33–35].

4.8.1

Examples of Pitchfork Bifurcations

We have already encountered in this book the so-called pitchfork bifurcation when discussing the period doublings in the logistic map in Chap. 2 and in the Duffing oscillator in Sect. 4.5.2. Also the splitting of stable fixed points (surrounded by stability islands) in the standard map, e.g. around K ≈ 4.25, see Fig. 3.27, is an example of a bifurcation looking like a pitchfork with three teeth. One of the simplest systems showing a pitchfork bifurcation is a particle in a one-dimensional nonlinear potential of the form V (x) = x 4 − αx 2 .

(4.8.1)

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Fig. 4.23 Potential model for a pitchfork bifurcation. a α < 0 and b α > 0, for a bifurcation of the fixed point xFP sketched in (c) with αcrit = 0. Stable fixed points are marked by green circles and unstable ones by a green cross

Fig. 4.24 Potential model for a subcritical pitchfork bifurcation. a α < 0 and b α > 0, for a bifurcation of the fixed point xFP sketched in (c) with αcrit = 0. Stable fixed points are marked by a green circle and unstable ones by green crosses

For negative α < 0, there is only one minimum, hence one stable equilibrium point at the origin x = 0, see Fig. 4.23a. For α > 0, there are two symmetric minima and the origin is unstable (this point corresponds to the pencil kept upward on a finger tip), see Fig. 4.23b. Changing the control parameter α over the zero point, we observe a bifurcation of pitchfork type, see the sketch in Fig. 4.23c. The corresponding fixed points are √ (4.8.2) x± = ± α, with

 x± = ±i |α|,

(4.8.3)

for negative α. The precise name for such a bifurcation is supercritical pitchfork bifurcation. A subcritical pitchfork bifurcation occurs for a nonlinear potential of the form V (x) = −x 4 − αx 2 ,

(4.8.4)

plotted in Fig. 4.24. Here, we have a local stable point at the origin x = 0 and two unstable points for α < 0. For α > 0, there is only one unstable point at the origin, and the particle escapes to infinity at the slightest perturbation of the initial condition

4.8 Bifurcation Scenarios

131

Fig. 4.25 Potential model for a tangent bifurcation. a α < 0 and b α > 0, for a bifurcation of the fixed point xFP sketched in (c) with αcrit = 0. Stable fixed points are marked by green circles and unstable ones by a green cross

x = 0. Having a high enough energy to overcome the barrier, the same happens in the case of negative α. The subcritical bifurcation is sketched in Fig. 4.24c. Pitchfork bifurcation are important because they occur quite often in nonlinear dynamical systems and they represent one possible route to chaotic motion by a cascate of period doublings (a sequence of pitchfork bifurcations), as in the logistic map for r → r∞ = 3.57 . . ., or in the Duffing oscillator for F → F∞ , c.f. Sect. 4.9.

4.8.2

Tangent Bifurcations

Next we treat tangent bifurcations, sometimes also known as saddle-node bifurcations. A system showing a tangent bifurcation is a particle in a one-dimensional nonlinear potential of the form V (x) = x 4 − αx.

(4.8.5)

This potential has a saddle point at the origin x = 0 for negative α < 0, see Fig. 4.25. For positive α > 0, there is one local stable minimum at positive x, and symmetrically at negative x an unstable maximum. In consequence, the bifurcation diagram looks as sketched in Fig. 4.25c.

4.8.3

Transcritical Bifurcations

Transcritical bifurcations are not generic but do occasionally occur. An example is the nonlinear potential V (x) = x 3 − αx 2 ,

(4.8.6)

plotted in Fig. 4.26 for negative and positive α. The corresponding bifurcation diagram looks as sketched in Fig. 4.26c. Here, at α = 0, a stable fixed point exchanges

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Fig. 4.26 Potential model for a transcritical bifurcation. a α < 0 and b α > 0, for a bifurcation of the fixed point xFP sketched in (c) with αcrit = 0. Stable fixed points are marked by green circles and unstable ones by a green cross

with an unstable one describing the movement of the stable equilibrium point at the origin towards positive x.

4.8.4

Higher-Order Bifurcations

We saw an example for the development of higher-order bifurcations when looking at the appearance of resonance chains within regular stable islands, see Sect. 3.8.5. The Poincaré–Birkhoff theorem predicts the creation of pairs of stable and unstable fixed points around resonance points. We show this feature again in some sketches of SOS plots in Fig. 4.27a, b. The trace of the monodromy matrix trM determines the criterion for stability in the two-dimensional SOS, c.f. Sects. 3.8.4 and 4.2.1 above. This is shown in Fig. 4.27c for the three points A, B, C indicated in Fig. 4.27b as a function of a control parameter that, in this case, is the winding number α = ω1 /ω2 from Sect. 3.8.3. Its critical value determines the intersection of stable and unstable evolutions. The minimal number is m ≥ 4 for the iterations of the SOS map in order to resolve more than two new stable fixed points. In our case of Fig. 4.27b m = 6, i.e. one torus, which is stable below the critical parameter value, transforms into 6 stable and unstable fixed points, respectively, above the critical point. The monodromy matrix above is understood after m iterations of the Poincaré map in order to study the local evolution around an island in the island chain. For the case of Fig. 4.27b after m = 6 iterations.

4.8.5

Hopf Bifurcations

The last bifurcation we look at here are Hopf bifurcations named after the GermanAmerican mathematician Eberhard Hopf. They are important and occur regularly in dynamical systems’ theory [12,36]. They generalise the concept of a pitchfork bifur-

4.8 Bifurcation Scenarios

(a)

133

(b)

(c)

Fig. 4.27 Sketch of higher-order bifurcation scenario: a integrable motion along a two-dimensional torus. b Development of an island chain at a 1/6 resonance. c The trace of the monodromy matrix trM for the points A, B, C indicated in (b)

cation to higher dimensions. The bifurcation can again be supercritical or subcritical, resulting in a stable or an unstable limit cycle, respectively. One of the simplest examples, is the motion of a particle in the following twodimensional nonlinear and non-separable potential V (x) = (x 2 + y 2 )3 − α(x 2 + y 2 ).

(4.8.7)

Figure 4.28a reports the form of the potential for negative α, while Fig. 4.28b shows it for positive α. Changing the control parameter from negative to positive values transforms the stable equilibrium point at the origin (x, y) = (0, 0) into a limit circle around the origin. This corresponds to a supercritical Hopf bifurcation. A subcritical one occurs for the modified potential of the form V (x) = −(x 2 + y 2 )3 − α(x 2 + y 2 ),

(4.8.8)

much along the same lines as discussed in Sect. 4.8.1. Another example for the occurrence of a Hopf bifurcation is the van der Pol oscillator given by the following equation of motion, see Sect. 4.3.3, q¨ + γ (q 2 − a 2 )q˙ + ω02 q = 0.

(4.8.9)

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Fig. 4.28 Sketch of a Hopf bifurcation, i.e., the transformation of a stable equilibrium point (a) into a stable limit cycle (b) Fig. 4.29 Sketch of the Hopf bifurcation occurring in the Van der Pol oscillator. The phase space motion transforms from an attractive sink (a) into a limit cycle (b)

Here, γ is our control parameter. For γ < 0, we have a stable sink at the origin of phase space, while for γ > 0 we obtain a limit circle, see Fig. 4.29. The crossover is again a bifurcation of the Hopf type.

4.9

Two Routes to Chaos

4.9.1

Landau’s Transition to Chaos

After the discussion of typical bifurcation scenarios occurring in dynamical systems, we can now briefly review an old idea by Lev Landau for the development of chaotic motion in dynamical systems or for the occurrence of turbulence in fluid (liquid or gaseous) flows, c.f. Fig. 4.15, [37]. In fluid dynamics, turbulence is a motion characterized by chaotic changes in e.g. pressure and/or flow velocity. It contrasts a laminar flow, which occurs when a fluid flows regularly in parallel layers, with no interference between those layers. Landau’s idea is only sketched in Fig. 4.30. There we see how a stable fixed point on the left may transform into a limit cycle by a Hopf bifurcation. This brings a new frequency into the motion. Further changes of a control parameter may lead to another Hopf bifurcation adding another independent frequency. Quasi-periodic motion on a two-torus, for instance, can hence transform into a motion on a three-torus etc. Quasi-periodicity means that the participating frequencies ω1 , ω2 , ω3 , . . . are all pairwise incommensurate. A whole (infinite) series of Hopf bifurcations may increase the number of participating frequencies further and further until a (quasi-)continuous power spectrum is reached. Kolmogorov’s theory of turbulent motion [38,39] actually predicts a scale-free algebraic power spectrum of modes in the turbulent limit.

4.10 Intermittency

135

Fig. 4.30 Sketch of Landau’s transition to chaos

4.9.2

Ruelle–Takens–Newhouse Route to Chaos

Today, Landau’s idea, sometimes also known as Landau–Hopf theory [40] is substituted by a more modern theory of turbulence. Such a theory was developed by Ruelle, Takens, and Newhouse (RTN) in the 1970s [41–43]. The reader is referred to the literature for deepening this rather difficult topic, see, e.g., the readable review on the different roads to chaos by Eckmann [43], or the collection of Ruelle’s reprints in the book [44]. In the language used above, we may just say here that the RTN theory predicts that a three-dimensional torus may become structurally unstable when changing some system parameters, and that it may eventually decay into a chaotic attractor. Hence, regular motion becomes unstable already after the third Hopf bifurcation. This scenario dramatically shortens Landau’s route of an infinite series of bifurcations. No chaos is possible on the two-torus before the third bifurcation due to the Poincaré–Bendixson theorem. RTN showed that for a motion in dimensions larger or equal to three every regular vector field on the torus can be perturbed by an arbitrarily small amount towards a new vector field with a chaotic attractor [42]. It may sometimes even happen that the two-torus gets directly destroyed without showing a third frequency (with a motion on a quasi-regular three-torus). Real as well as numerical experiments with turbulent flows, see e.g. [45–50] and Chap. 6.5 in [51], similar to the Rayleigh–Bénard heat convection problem mentioned in Sect. 4.6, seem to confirm that already after two or three Hopf bifurcations a (quasi-)continuous power spectrum can set in. This practically excludes Landau’s route to chaos and favours the other scenarios based on period doubling (Sect. 4.8.1), the RTN route, and intermittency (see next section).

4.10

Intermittency

Next to the two major routes to chaos, by period doubling (Feigenbaum scenario) and by the occurrence of strange attractors (Ruelle–Takens–Newhouse), there is another standard route [43] going back to works of Yves Pomeau and Paul Manneville, also in the 1970s [52,53]. The Pomeau–Manneville scenario is the transition to chaos (or turbulence in fluids) due to tangent bifurcations leading to a phenomenon that is known as type-I intermittency. In the following, we try to give a brief introduction into the phenomenon of intermittency.

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Fig. 4.31 Sketch of a tangent bifurcation at a critical control parameter rc . Below r < rc the motion is quasi-regular with chaotic bursts, as described in the main text and shown in the following figures

We start out with a concrete example, namely the logistic map, introduced in Sect. 2.2.1. There, we saw that the logistic map shows one of the possible routes to chaos due to the cascate of period doublings approaching r → r∞ ≈ 3.57 from below. Above r > r∞ , stable √ p-periodic cycles exist, in particular a stable 3-cycle that disappears at rc = 1 + 8 ≈ 3.8284 [5]. This 3-cycle has a stable branch for r > rc (regular motion), but the motion is unstable at values of r just below r < rc (chaotic motion). This crossover at r = rc is an example of a tangent bifurcation [3,5], see Fig. 4.31. The quasi-regular behaviour—observed for r < rc but very close to the critical parameter—is interrupted by irregular chaotic bursts. This temporal change between quasi-regular and chaotic motion is called intermittent chaos, or simply intermittency. Figure 4.32 shows the value of the logistic map after three iterations f 3 (x), as a function of the variable x, above (in a and c) and below (in b and d) the critical parameter. Just below, r < rc , the motion is trapped for a long time within a small boundary, but it can eventually escape and change more drastically again, see Fig. 4.32d. We want to estimate the temporal length of the quasi-regular (non-chaotic or laminar) phases. We do this following closely the treatise in Ott’s book [5], see Chap. 8 therein. Let us define Fr (x) ≡ f 3 (x), i.e. the function of three-step iterations of the logistic map. r stands for its control parameter. At r = rc , we define the threeperiod fixed point of the map Fr (xc ) = xc . Looking at the behaviour of Fr (x) close to this fixed point, see Fig. 4.33, we observe that the first and second derivatives of the function with respect to x are Fr (xc ) = 1 Fr (xc ) = 2a > 0.

(4.10.1) (4.10.2)

The derivative with respect to r is  ∂ Fr (x)  = b < 0. ∂r r =rc

(4.10.3)

Now we quadratically expand the function in a region x1 < x < x2 , where x1 and x2 are points close to the critical point xc , see Fig. 4.33, for r ≈ rc : Fr (x) ≈ xc + (x − xc ) + a(x − xc )2 + b(r − rc ) . . . .

(4.10.4)

137

1

1

0.8

0.8

0.6

0.6

f 3 (x)

f 3 (x)

4.10 Intermittency

0.4

0.4

0.2 0

0.2 0

0.2

0.4

x

0.6

0.8

0

1

0

0.2

0.4

(a)

0.6

0.8

1

(b) 0.54

0.52

0.52

f 3 (x)

f 3 (x)

0.54

0.5 0.48 0.48

x

0.5

0.5

x

(c)

0.52

0.54

0.48 0.48

0.5

x

0.52

0.54

(d)

Fig. 4.32 Three-step iterations of the logistic map f 3 (x) above and below the critical parameter r = 3.855 > rc (a) and r = 3.815 < rc (b), respectively. There is a stable fixed point at x ≈ 0.49 (circle) and an unstable one at x ≈ 0.54 (cross). Zoom into the region around these fixed points for r = 3.835 > rc (c) and r = 3.827 < rc (d). While c shows the convergence to the fixed point (accumulation point) on the left (red dots connected by solid lines), in d we observe the trapping and escaping of an exemplary trajectory (red dots connected by solid lines). The different r parameters are chosen for better visualisation Fig. 4.33 Behaviour of Fr (x) close to this critical point xc

The second term on the right hand side is close to zero, so we may neglect it in the following. Substituting again x3n for x and x3(n+1) for the function, we have the approximation x3(n+1) ≈ x3n + a(x3n − xc )2 + b(r − rc ) . . . .

(4.10.5)

The difference x3(n+1) − x3n per step n is just a discrete derivative for x3(n+1) ≈ x3n [54]. By doing so, we substitute discrete differences by continuous changes in order to simplify the following calculations. From dx ≈ x3(n+1) − x3n ≈ a(x − xc )2 + b(r − rc ), dn

(4.10.6)

138 Fig. 4.34 (Upper panel) Scan into the bifurcation diagram for the logistic map of Fig. 2.1 around the critical value rc ≈ 3.828427. The period-3 attractor can be seen for r > rc . In the time evolution of the logistic map close to this critical value, typical bursts of irregular motion occur as a manifestation of intermittency in this parameter region, for r = 3.828424 (middle panel) and for r = 3.82841 (lower panel)

Fig. 4.35 Bifurcation diagram for the Duffing oscillator between chaotic and regular dynamics with intermittent motion around the critical force value of F  0.5421

4

Dissipative Systems

4.10 Intermittency

139

Fig. 4.36 Stroboscopic map of the Duffing oscillator taken at periods nT with T = 2π/ω for γ = 1/2, β = 1, ω = 1. From F = 0.54215 to F = 0.54, a transition from quite regular to more and more irregular dynamics is observed. The chaotic bursts occur for F = 0.54215 and F = 0.54211 at distances of ca. 400 and 100 periods, respectively

1

F = 0.54215

q

0 −1 −2

0

1

200

400

n

600

800

1000

600

800

1000

600

800

1000

F = 0.54211

q

0 −1 −2

0

1

200

400

n

F = 0.54

q

0 −1 −2

0

200

400

n

we obtain 

x2

dx 2 + b(r − r ) a(x − x ) c c x1  x2 a 1 r 1, a unique inverse cannot be defined any more [57,58]. Expanding the recurrence relation (4.11.3), one obtains   n K sin(2π θi ) mod1. (4.11.4) θn = θ0 + αn − 2π i=0

Suppose now that θn = θ0 mod 1, so they are periodic fixed points of period n. Then the number of oscillations of the sine per cycle will be (θn − θ0 ) · 1, thus characterizing a phase locking. For K = 0, the winding number is not α any more, but we may use a definition going back to Poincaré for the new winding or rotation number W of torus maps: θn − θ0 f n (θ0 ) − θ0 = lim . n→∞ n→∞ n n

W (K , α) = lim

(4.11.5)

Please note that in this definition it is reasonable not to rescale to the unit interval, i.e. not to apply the mod operation in the iterations of the map (4.11.3).

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Dissipative Systems

For rational W = p/q, for p, q ∈ Z relatively prime, mode coupling occurs as in the simple example above of a fixed point of period n. For such rational winding numbers, the iteration of the map, (4.11.3), starting from any initial angle will converge to a limit circle (regular motion). For irrational W , we either see quasi-periodic motion or chaotic behaviour due to the nonlinearity now. The type of motion observed will depend on the nonlinear control parameter K . Here, we just review the phase diagram of the circle map, see Fig. 4.38: • K = 0: without perturbation (arising in a real system e.g. from an interaction between different oscillators) the oscillator proceeds with his unchanged frequency. For irrational α, the motion is quasi-periodic. This quasi-periodic behaviour dominates the line K = 0 in Fig. 4.38, since irrational numbers are much more frequent than rational ones. • 0 < K < 1: the phase-locked regions (blue in Fig. 4.38) in the parameter plane (K , α) form so-called Arnold tongues. Here, V-shaped regions appear with phase locking, touching down to a specific rational value α = p/q in the limit of K → 0. The values of (K , α) in one of these regions will result in a motion with the rotation number W = p/q. One reason the term “locking” is used is that the values θn can be perturbed by a rather large random disturbance, without disturbing the limiting rotation number. That is, the sequence stays “locked on” the signal, despite the addition of significant noise to the series θn . The notion to “lock on” in the presence of disturbance or noise is central, e.g., to the utility of phase-locked electronic circuits. The discussed deformation of the tongues is due to the effect of frequency pulling, already touched open in Sect. 3.7.2. • K = 1: here the Arnold tongues move toward each other. The consequence is that the regions in W without mode coupling form a self-similar (fractal) Cantor set with measure zero. • K > 1: chaotic behaviour is possible once the Arnold tongues overlap. Here, two or even more tongues can share the same regions of parameter space, giving the possibility that multiple stable limit cycles coexist. Chaotic and non-chaotic regions (green and red in Fig. 4.38) are then strongly entangled in the parameter plane (K , α). Please note the resemblance of Chirikov’s resonance overlap argument reported for Hamiltonian systems in Sect. 3.9.6. The interested reader may consult [5,15,56–58] for more details on the phase diagram and the behaviour of the sine-cycle map. In the following, we will concentrate on the mode locking that is interesting for quite a wide class of dynamical systems. A condition for model locking is obtained by fixing 0 < K < 1 and searching for a rational α = p/q such that a stable q-cycle is obtained with f q (θi∗ ) = p + θi∗ ,

(4.11.6)

4.11 Coupled Oscillators

143

Fig. 4.38 Arnold tongues for the sine-circle map (4.11.3) in the parameter plane of coupling K and unperturbed winding number Ω = α. The blue regions are the ones in which the oscillator is locked and the corresponding tongues are labeled with their rational values α = p/q. The white regions represent intermixed quasi-periodic and periodic behaviour. The green and the red parts also show intermixed behaviour with chaotic motion above where the tongues start to overlap. The dashed horizontal line at K = 1 indicates the onset of multi-stability and chaos, as described in the main text. This figure, adapted from [58,59], is obtained from M. H. Jensen and used with his kind permission

for certain angles θi∗ i ∈ {1, 2, . . . , q}. The stability criterion for one-dimensional maps is given by the derivative as follows  q   q      q ∗      ∗  ∗   f (θ ) =  1 − K cos(2π θi )  < 1. f (θi ) =  i     i=1

(4.11.7)

i=1

For arbitrary rational α = p/q [15,57] show that, for 0 < K < 1, there is a final interval of parameters α(K ) in which mode coupling occurs and the motion is stable. In Problem 4.6, the reader is invited to find the stability window for the first step at α = 0 that is |α ≤ K /(2π )|. One can show that the measure of the union of all intervals Δα where regular mode locking occurs is strictly positive [57]. A particular case occurs for K = 1 for which the regular intervals have full measure one and they form a so-called devil’s staircase. The latter is a monotonously increasing function on [0, 1] that shows a step of finite length for any rational p/q. The size decreases with increasing denominator q. For two rational winding numbers W = p/q and W  = p  /q  , the largest step in-between the two has a winding number of W  = ( p + p  )/(q + q  ). This rule that from two rationals one obtains the next number by adding the numerators and denominators, respectively, forms the so-called Farey tree of rational numbers. The non-locked regions on the devil staircase show a fractal structure at the critical parameter value K = 1 [57]. The Arnold tongues and the corresponding Farey tree-structure occur in many dissipative dynamical systems, see e.g. [5,15,60,61]. The quasi-classical approxi-

144

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Dissipative Systems

mation to the so-called accelerator modes of the quantum kicked rotor shows their relevance even for quantum systems [62].

4.11.3 Synchronization Linear (harmonic) mechanical oscillators are well known to support common oscillatory, so-called normal modes when linearly coupled [13,63]. For two such oscillators, these modes correspond, for instance, to swinging in phase or in opposite phase. For small excursions, arbitrary couplings may always be linearised to show exactly such a behaviour, see any advanced book on analytical mechanics, e.g. [63,64]. The more surprising result is that stable synchronization and mode locking of oscillators are ubiquitously found, even in highly nonlinear systems. The last subsection was devoted to such a simple nonlinear model showing mode locking. This means that regular motion converges to a limit cycle or periodic orbit due to the nonlinear coupling, that was represented by the sinusoidal term in (4.11.3). This behaviour is characteristic of many periodic and also chaotic systems that can move in synchronous lockstep under certain conditions [65–67]. The first one to notice such a synchronization in coupled pendulum clocks was Christian Huygens in the 17th century [68]. Conservative oscillators cannot synchronise, apart from what was said above on linear oscillators. Their dynamics is determined by linear or nonlinear resonances, see our previous chapter on Hamiltonians systems. True stable synchronization requires a certain level of irreversibility that occurs naturally in dissipative oscillatory systems. As we have seen above, their oscillations converge to certain forms, so-called attractors, depending on the initial conditions and system parameters. The oscillations are stabilized by a balance between energy pumped into the system, e.g. by an external drive, and energy lost by dissipation. Examples for synchronising systems towards a regular attractor are the Van der Pol oscillator and the nonlinear circle map from Sects. 4.3.3 and 4.11.2, respectively. The simplest model for synchronization is given by the Adler equation that describes the evolution of the relative phase of two nonlinearly coupled oscillators. Each self-excited periodic oscillation can be described by an amplitude and a phase. Now, the important point is that a weak coupling changes in first order only the phase, keeping the amplitudes approximately constant. Under such conditions, we may only describe the phase evolution of the oscillators, simplifying the problem quite a bit. Then the two oscillators (1 and 2) can be reduced to study only their relative phase Δφ = φ2 − φ1 . The Adler equation [69] for the phase difference is dΔφ = Δω − K sin(Δφ), dt

(4.11.8)

where Δω = ω2 − ω1 is the frequency mismatch of the free oscillations and K is the nonlinear coupling parameter. The equation is a simple model that describes the synchronization of a dynamical system with an external driving force. The transition to synchronization occurs for a coupling strength |K | > |Δω| that guarantees a stable

4.11 Coupled Oscillators

145

Fig. 4.39 The phase diagram for the Adler equation. Phase synchronization happens within the shaded area at which |K | > |Δω| is satisfied

K

Δω

equilibrium state. This is easily seen by setting the time derivative dΔφ = 0, dt

(4.11.9)

from (4.11.8). Another explanation is that the motion can be interpreted as the one of a particle in the nonlinear potential V (Δφ) = −ΔωΔφ − K cos(Δφ).

(4.11.10)

This potential shows no minima for |K | < |Δω| and the phase difference increases monotonously in time. But for |K | > |Δω| local minima appear that correspond to the synchronised equilibrium solution. The stability diagram is plotted in Fig. 4.39. The wedge form in the parameter plane (K , Δω) in which |K | > |Δω| holds, is just an example of an Arnold tongue known from the previous subsection. We conclude that the oscillation frequency of a periodically driven oscillator adapts itself perfectly to the excitation frequency (the one of the second oscillator) whenever the amplitude of the drive (or the coupling) increases beyond a critical value K c . The latter threshold depends on the frequency difference Δω. Synchronization can occur in higher order as well at which the oscillators lock at rational winding numbers W = p/q, with p, q = p/q, p, q relatively prime integers. This leads to plateaus in the graph W (Δω, K = const.), e.g. for the Van der Pol oscillator, see [67,70], similar to the devil’s staircase from the previous subsection.

4.11.4 Kuramoto Model Just as two oscillators can adapt their oscillation frequency also a larger number of coupled oscillators may synchronise their motion. Yoshiki Kuramoto introduced in 1975 a mathematical model that can be solved [71]. This by now standard model for the synchronization of oscillator ensembles is reviewed nicely in [66,67,72,73]. The model describes an ensemble of N oscillators whose frequencies are drawn from a certain distribution g(ω). All oscillators are long-range coupled with all other oscillators. This leads to the following system of ordinary differential equations for the phases of the oscillators N K dφi − sin(φ j − φi ). = ωi dt N i=1

(4.11.11)

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Dissipative Systems

Again, K is the coupling strength usually taken identical for all oscillators. The condition of equal and long-range couplings may sometimes be relaxed in practical application (and a synchronization transition may still occur), but it simplifies the mathematical treatment quite a lot. In order to quantify the transition from the nonsynchronised to the synchronised state, one introduces a real order parameter R ReiΦ =

N 1 φi e . N

(4.11.12)

i=1

The global phase Φ is usually not important. If all oscillators show different independently evolving phases, the order parameter vanishes in the limit N → ∞. If there is phase correlation present, at least between a subset of oscillators, then R increases from zero to finite positive values. It goes to one, R → 1, at complete synchronization of all the oscillators. This shows that the Kuramoto transition is a phase transition, described by a smooth change of the order parameter R from zero to one. The role of “temperature” in this transition is played by the width of the frequency distribution g(ω) that is usually supposed to be unimodal (otherwise the occurrence of synchronization and its form depend on the precise frequency distributions). The critical coupling parameter is essentially proportional to this width [67,72,73]. ReiΦ can also be interpreted as the complex mean field that is driven by the terms dφi = ωi − K R sin(Φ − φi ). dt

(4.11.13)

The connection between (4.11.11) and (4.11.13) requires the use of simple trigonometrical formulae [67,72]. The oscillators’ equations (4.11.13) are no longer explicitly coupled. The order parameters R and Φ govern their evolution now. A mean-field theory can then be applied to arrive at a formal solution of the problem [67,72]. Just as in the case of Adler’s equation (4.11.8), synchronisation typically occurs for K R > |ωi | in (4.11.13). Many nonlinear systems can generate aperiodic, chaotic oscillations. We have seen some examples in the previous sections. Coupling now two chaotic oscillators can also lead to synchronization in the same sense as above, that the two systems align their frequencies with increasing coupling strength. Such transitions from individual chaotic motion to synchronised motion are useful for controlling chaotic systems [74,75] and were studied, in particular, by Pecora and Carroll [76,77]. The effects of synchronization in various dynamical systems form a bridge between the properties found in low-dimensional dynamical systems and the statistical methods used to describe more complex systems. This includes also synchronization induced by random noise rather than by deterministic motion, synchronization in networks of lasers, see for instance [78–80], or in neurobiological and other complex networks [66,67,81–83]. A recent review that nicely brings together limit cycles, Arnold tongues and synchronization, and the emergence of chaos with references to a series of biological applications is found in [59]. Second-order Kuramoto models [73] include the phase velocities (angular momenta) in the evolution and are used, e.g., to

4.12 Increasing Complexity

147

investigate power grids [84,85]. Synchronization of quantum mechanical oscillators is also well established in the presence of dissipation, see e.g. [86–94]. Remarkably, even in non-dissipative Hamiltonian classical [95] and quantum systems [96] phase synchronization of local but coupled modes was found. In the quantum case, this is made possible—according to Heisenberg’s uncertainty relation—by a fast increase in amplitude fluctuations otherwise neglected as in the simplified description above. For instance, a strong phase synchronization of localised quantum mechanical modes may lead to strong fluctuations of the conjugate (amplitude or population) variables, implying, for instance, the creation of strong quantum entanglement between these modes, see [96].

4.12

Increasing Complexity

Real complex systems in nature are much more complicated than the abstract mathematical models we have discussed so far, e.g., in the present or the two previous chapters. Nevertheless, the following models are examples that show and pin down phenomena occurring in nature: • the Hénon–Heils H (x, y) problem from Sect. 3.8.7.1 models a reduction of the motion of celestial bodies to three dimensional phase space, leaving, however, the possibility of chaotic evolution actually present in the three-body planet problem that originally lives in a much higher-dimensional space. • the Lorenz system models heat convection in the atmosphere. • simple fractal models reflect the structure of natural phenomena such as the Romansco broccoli. • the logistic map is a minimal model for population evolution in biology. In population biology, for instance, there are many more models, e.g. a class of predator-prey models in two-dimensions going back to Kolmogorov in 1936 [97]. These models are described mathematically by the following coupled system of ordinary differential equations x(t) ˙ = r (x)x + f (x)y y˙ (t) = g(x)y.

(4.12.1) (4.12.2)

The models are based on the following reasoning. The number of prey is given by x(t) and the one of predators by y(t). Moreover, it is reasonable to assume that 1. if there are no predators, the birth rate of preys r (x) decreases with increasing population becoming eventually negative. The reason is the lack of food once the population becomes too large; 2. the function g(x) should increase with growing prey population going from negative (food shortage) to positive (food abundance);

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3. the function f (x) is expected to have the following behaviour: f (0) = 0 f (x) > 0 for x > 0,

(4.12.3) (4.12.4)

since no loss occurs if there are no preys, and loss starts as soon as there are preys available. Since autonomous models in two-dimensions cannot sustain chaos by the Poincaré–Bendixson theorem, see Sect. 4.4, we need at least three species to generate a chaotic population evolution. Extensions are systems of equations of the form x˙i (t) = xi Mi (x1 , x2 , . . . , x N ),

(4.12.5)

for i = 1, 2, . . . , N (N ≥ 3) and Mi are appropriate coupling functions with certain properties that guarantee a reasonable evolution, as described by the points 1.–3. mentioned above. Mathematical models for such systems were studied, for instance, by May [98] or by Smale in the 1970s [99]. Recent experiments in the laboratory with plankton food webs actually show chaotic behaviour similar to the one predicted by simplifying models, see [100]. Of course, to get closer to real complex problems, one would have to study much higher-dimensional systems. In the continuum limit, coupled discrete elements (mass points) will have to be substituted by a field description. The governing equations are then much more complicated partial differential equations. This actually links us to quantum mechanics which is governed by such partial differential equations, as soon as more than one dimension and/or more than one point particle is involved. A classification of higher-dimensional models is found in the book [51], where space, time and the underlying state space can be all either discrete or continuous. If all are continuous the models are coupled partial differential equations. If only space is discrete, we deal with simpler coupled ordinary differential equations. If only the state space is continuous we have coupled lattice models, while cellular automata models are used for discrete space, time and state space. A deeper analysis of higherdimensional or field-theoretical models is beyond the scope of this book.

Problems 4.1 Prove that the circle map introduced in Sect. 4.11.1 is ergodic for any irrational parameter α. 4.2 Determine the fractal (Hausdorff) dimensions of a modified Cantor set that is constructed by deleting the middle open half (not the middle third) of each interval of the previous construction step. Start out, as in Sect. 4.7.1.1, from the interval of unit length [0, 1].

4.12 Increasing Complexity

149

4.3 This exercise makes you construct the Mandelbrot set that is obtained by iterating the complex valued map z n+1 = z n2 + c,

(4.12.6)

where z n , c ∈ C and the starting point is z 0 = (0, 0). The Mandelbrot set is now the set of all complex constants c for which the orbits (z n )n∈N do not leave the cycle of radius 2 in the complex plan, i.e. for which |z n | ≤ 2 for all n ∈ N,

(4.12.7)

remains satisfied. You may stop after M = 100 iteration each. Plot the set in the complex plane as a black pixel. If the value of c is not in the set, plot it as white pixel. Check what happens if you stop after e.g. M = 200 iterations (Refinements of this procedure show the famous color plots of the Mandelbrot set that are obtained by assigning a further color code to values of c depending on after precisely how many iterations the corresponding orbit leaves the confined region of radius 2.). 4.4 Study the following nonlinear dynamical map xn+1 = (xn2 + α)xn ,

(4.12.8)

with real parameter α. (a) Find a simple (!) fixed point of the evolution determined by the map (4.12.8). (b) Study the stability of this fixed point. Verify that it is stable for the interval of parameters α1 < α < α2 . Determine α1 and α2 . (c) Characterize the bifurcations occurring at α = α1 and α = α2 . 4.5 We are given the following system of two ordinary differential equations x(t) ˙ = −y + x(α − x 2 − y 2 ) y˙ (t) =

x + y(α − x − y ), 2

2

(4.12.9) (4.12.10)

with real parameter α. (a) Find a simple (!) fixed point of the evolution determined by (4.12.10). (b) Study the stability of the motion around this fixed point. Check, in particular, how the eigenvalues of the stability matrix evolve when changing α. (c) Transform the system (4.12.10) to polar coordinates (r , φ) defined by x(t) = r (t) cos(φ(t)) y(t) = r (t) sin(φ(t)),

(4.12.11) (4.12.12)

and try to solve it. How does the solution behave in the asymptotic limit t → ∞?

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4.6 We study the first step of the devil’s staircase of the nonlinear map, (4.11.3), i.e. the one with winding number W = q0 = 01 = 0. The solution corresponds to a fixed point f (θ ∗ ) = θ ∗ that becomes unstable for | f (θ ∗ )| ≤ 1. From the latter condition, find the window of the parameter α for which the motion is mode locked.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

28. 29. 30. 31.

Forster, O.: Analysis 2. Vieweg+Teubner, Wiesbaden (2010) Crawford, J.D.: Rev. Mod. Phys. 63, 991 (1991) Strogatz, S.H.: Nonlinear Dynamics and Chaos. Westview Press, Boston (1994) Van der Pol, B.: Radio Rev. 1, 701 (1920) Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (2002) Bendixson, I.: Acta Math. 24, 1 (1901) Perko, L.: Differential Equations and Dynamical Systems. Texts in Applied Mathematics, vol. 7, 3rd edn. Springer, New York (2001) Teschl, G.: Ordinary Differential Equations and Dynamical Systems (Graduate Studies in Mathematics). American Mathematical Society, Providence (2012) Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (2003) Ott, E.: Rev. Mod. Phys. 53, 655 (1981) Duffing, G.: Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz. F. Vieweg u. Sohn, Braunschweig (1918) Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, Berlin (1983) Kittel, C.: Introduction to Solid State Physics. Wiley, New York (2005) Lorenz, E.N.: J. Atmos. Sci. 20(2), 130 (1963) Lichtenberg, A.J., Lieberman, M.A.: Regular and Chaotic Dynamics. Springer, Berlin (1992) Sparrow, C.: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Springer, New York (1982) Ahlers, G., Grossmann, S., Lohse, D.: Rev. Mod. Phys. 81, 503 (2009) Tucker, W.: C. R. Acad. Sci. Paris 328, 1197 (1999) Grassberger, P., Procaccia, I.: Phys. D: Nonlinear Phenom. 9(1), 189 (1983) Peitgen, H.O., Richter, P.H.: The Beauty of Fractals. Springer, Berlin (1986) Mandelbrot, B.B.: The Fractal Geometry of Nature. W.H. Freeman and Company, San Francisco (1982) Rudin, W.: Principles of Mathematical Analysis. McGraw Hill, New York (1964) Walter, W.: Analysis 1. Springer, Berlin (2004) Walter, W.: Analysis 2. Springer, Berlin (2002) von Koch, H.: Ark. Mat. Astron. Fys. 1, 681 (1904) Sierpinski, W.: Comptes rendus hebdomadaires des séances de l’Academie des sciences 160, 302 (1915) Aleksandrov, P.S.: Einführung in die Mengenlehre und in die allgemeine Topologie. Hochschulbücher für Mathematik, Band 85. Deutscher Verlag der Wissenschaften, Berlin (1984) Farmer, J., Ott, E., Yorke, J.A.: Phys. D: Nonlinear Phenom. 7(1), 153 (1983) Dubuc, B., Quiniou, J.F., Roques-Carmes, C., Tricot, C., Zucker, S.W.: Phys. Rev. A 39, 1500 (1989) Tomadin, A., Mannella, R., Wimberger, S.: J. Phys. A: Math. Gen. 39(10), 2477 (2006) Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester (1990)

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32. Mandelbrot, B.: Science 156(3775), 636 (1967) 33. Chow, S.N., Hale, J.R.: Methods of Bifurcation Theory. Springer, New York (1982) 34. Chicone, C.: Ordinary Differential Equations with Applications. Texts in Applied Mathematics, vol. 34. Springer, New York (1999) 35. Murdock, J.: Normal Forms and Unfoldings for Local Dynamical Systems. Springer, New York (2003) 36. Marsden, J.E., McCracken, M.: Hopf Bifurcation and Its Applications. Springer, New York (1976) 37. Landau, L.D.: Dokl. Akad. Nauk SSSR 44, 339–342 (1944) 38. Kolmogorov, A.N.: Proc. USSR Acad. Sci. 30, 299 (1941) 39. Kolmogorov, A.N.: Proc. USSR Acad. Sci. 32, 16 (1941) 40. Hopf, E.: Commun. Pure Appl. Math. 1(4), 303 (1948) 41. Ruelle, D., Takens, F.: Commun. Math. Phys. 20(3), 167 (1971) 42. Newhouse, S., Ruelle, D., Takens, F.: Commun. Math. Phys. 64(3), 35 (1978) 43. Eckmann, J.P.: Rev. Mod. Phys. 53, 643 (1981) 44. Ruelle, D.: Turbulence, Strange Attractors and Chaos. WORLD SCIENTIFIC, Singapore (1995) 45. Giglio, M., Musazzi, S., Perini, U.: Phys. Rev. Lett. 47, 243 (1981) 46. Guzman, A.M., Amon, C.H.: Phys. Fluids 6(6), 1994 (1994) 47. Budroni, M.A., Masia, M., Rustici, M., Marchettini, N., Volpert, V., Cresto, P.C.: J. Chem. Phys. 128(11), 111102 (2008) 48. Kühnen, J., Braunshier, P., Schwegel, M., Kuhlmann, H.C., Hof, B.: J. Fluid Mech. 738, 463 (2014) 49. Kühnen, J., Holzner, M., Hof, B., Kuhlmann, H.C.: J. Fluid Mech. 770, R3 (2015) 50. Kühnen, J.: Experimental investigation of transition to turbulence in a torus. Ph.D. thesis, TU Wien, Vienna (2012) 51. Cencini, M., Cecconi, F., Vulpiani, A.: Chaos: from Simple Models to Complex Systems. Cambridge University Press, Cambridge (2010) 52. Pomeau, Y., Manneville, P.: Commun. Math. Phys. 74(2), 189 (1980) 53. Manneville, P., Pomeau, Y.: Phys. D: Nonlinear Phenom. 1(2), 219 (1980) 54. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes 3rd Edition: the Art of Scientific Computing, 3rd edn. Cambridge University Press, New York (2007) 55. Cornfeld, I.P., Fomin, S.V., Sinai, Y.G.: Ergodic Theory. Grundlehren der mathematischen Wissenschaften, vol. 245. Springer, Berlin (1982) 56. Leven, R.W., Koch, B.P., Pompe, B.: Chaos in Dissipativen Systemen, 2nd edn. Akademie Verlag, Berlin (1994) 57. Jensen, M.H., Bak, P., Bohr, T.: Phys. Rev. Lett. 50, 1637 (1983) 58. Jensen, M.H., Bak, P., Bohr, T.: Phys. Rev. A 30, 1960 (1984) 59. Heltberg, M.L., Krishna, S., Kadanoff, L.P., Jensen, M.H.: Cell Syst. 12(4), 291 (2021) 60. Schuster, H.G.: Deterministic Chaos. VCH, Weinheim (1988) 61. Belykh, V.N., Pedersen, N.F., Soerensen, O.H.: Phys. Rev. B 16, 4860 (1977) 62. Buchleitner, A., d’Arcy, M.B., Fishman, S., Gardiner, S.A., Guarneri, I., Ma, Z.Y., Rebuzzini, L., Summy, G.S.: Phys. Rev. Lett. 96, 164101 (2006) 63. Goldstein, H., Poole, C.P., Safko, J.L.: Klassische Mechanik. Wiley-VCH-Verlag, Weinheim (2006) 64. Fließbach, T.: Mechanik. Spektrum Akademischer Verlag, Heidelberg (2009) 65. Strogatz, S., Stewart, I.: Sci. Am. 269, 68 (1993) 66. Strogatz, S.H.: Sync: the Emerging Science of Spontaneous Order. Hyperion, New York (2003) 67. Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: a Universal Concept in Nonlinear Dynamics. Cambridge University Press, Cambridge (2001) 68. Huygens, C.: Horologium Oscillatorium: sive de motu pendulorum ad horologia aptato demonstrationes geometricae. F. Muguet, Paris (1673) 69. Adler, R.: Proc. IRE 34(6), 351 (1946) 70. Parlitz, U., Pikovsky, A., Rosenblum, M., Kurths, J.: Phys. J. 5(10), 33 (2006)

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5

Aspects of Quantum Chaos

Abstract

This chapter discusses two important ways of defining quantum chaoticity. One access to characterize dynamical quantum systems is offered by the powerful approach of semiclassics. Here we compare the properties of the quantum system with its classical analogue, and we combine classical intuition with quantum evolution. The second approach starts from the very heart of quantum mechanics, from the quantum spectrum and its properties. Classical and quantum localization mechanisms are presented, again originating either from the classical dynamics of the corresponding problem and semiclassical explanations, or from the quantum spectra and the superposition principle. The essential ideas of the theory of random matrices are introduced. This second way of characterizing a quantum system by its spectrum is reconciled with the first approach by the conjectures of Berry and Tabor and Bohigas, Giannoni, and Schmit, respectively.

5.1

Introductory Remarks on Quantum Mechanics

The origins of quantum mechanics lie in the attempt to explain the behaviour and the dynamics of systems at atomic scales. One important motivation to search for a new theory in the 1920ties was the stability of the hydrogen atom. While classical electrodynamics predicts that an electron encircling the nucleus loses energy emitting electromagnetic radiation and quickly falls into the nucleus, experiments tell us that the hydrogen atom has certainly a stable ground state. In analogy to the wavelike nature of light, it was assumed that atomic and elementary particles exhibit a wave behaviour. Famous experiments that support this theory are the diffraction of electrons on crystalline lattices (Davisson–Germer experiment [1]) as well as Young’s double slit experiment conducted with electrons [1]. On the other hand it was already clear from the onset of the twentieth century that light not only exhibits the characters of a wave, but interacts as if it was composed of particles. This is for example illustrated by the photo-electric effect explained by Einstein in 1905 [2].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. Wimberger, Nonlinear Dynamics and Quantum Chaos, Graduate Texts in Physics, https://doi.org/10.1007/978-3-031-01249-5_5

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Starting from this wave-particle duality, Louis de Broglie postulated the wave nature of particles. One fundamental mathematical description of the quantum theory lies in Schrödinger’s wave mechanics. The wave equation of a particle in an external field is given by the Schrödinger equation: i

∂ψ(r, t) = Hˆ ψ(r, t). ∂t

(5.1.1)

Equation (5.1.1) is a partial differential equation which describes the time evolution of the given physical system. Moreover, it contains information about the energy spectrum. The function ψ(r, t) is called wave function and completely determines the state of a physical system of N particles for r ∈ R3N at all times (as far as this is possible within the statistical interpretation of quantum mechanics). Schrödinger’s wave mechanics is hence a fully deterministic theory. From ψ(r, t), which is also interpreted as a probability amplitude, we can draw information about distributions in momentum and position space as well as the expectation value of quantities such as the energy [1,3,4]. The Hamiltonian Hˆ is a linear operator acting on the wave function ψ(r, t). Its linearity is the origin of the superposition principle in quantum mechanics. It is responsible for the wave-like character of the solutions of Schrödinger’s equation. Consequently, it may be difficult to define chaos in quantum mechanics. As we have seen in Chap. 3, chaos in classical systems relies on the nonlinear nature of the corresponding evolution equations. Therefore, the quantum evolution cannot be chaotic in the sense of the Lyapunov instability discussed in Sect. 3.9 and Fig. 3.30. We even have that the overlap between two different wave functions remains identical for all times     ˆ† ˆ ψ(r + δr, t)|ψ(r, t) = ψ(r + δr, 0)| U  U |ψ(r, 0) = ψ(r + δr, 0)|ψ(r, 0) , =1

(5.1.2) ˆ with the quantum mechanical evolution operator1 Uˆ = e−i H t/ . Instead of the initial conditions, a parameter of the system may be varied. This idea goes back to Asher Peres [6] who introduced the so-called fidelity as a measure of stability in quantum mechanics. It is defined by the overlap of two initially identical wave functions which have a slightly different time evolution. Let us assume a small perturbation δλ of some parameter λ of the Hamiltonian operator. Then the following overlap is time dependent and its evolution may teach us something about the stability of the quantum system with respect to the corresponding change of the parameter λ:   | ψ(r, t; λ)|ψ(r, t; λ + δλ) |2 ≡ F(t, δλ) .   

(5.1.3)

“fidelity”

relation between the hermitian operator Hˆ and the unitary Uˆ is due to a theorem of Stone [5]. We assume Hˆ was time-independent, otherwise time-ordering must be used. 1 This

5.2 Semiclassical Quantization of Integrable Systems

155

Since there are many, partly contradicting results established for the fidelity, we will not elaborate on it here. We rather refer the interested reader to the specialized literature [7–9]. We may thus ask ourselves: What is quantum chaos? The original idea is to study the relations between a quantum system and its corresponding classical analogue. Starting from this idea, we present some methods of the phenomenology of quantum chaos in this chapter. On the one hand, we focus on semiclassical quantization techniques of integrable systems (Sect. 5.2). On the other hand, we describe semiclassical tools for fully chaotic systems in Sect. 5.3. Having these tools at hand, we study the evolution of quantum mechanical wave packets projected into phase space, see Sect. 5.4, and classical as well as quantum mechanical localization of the wave-packet evolution, see Sects. 5.4.3.3 and 5.5.1, respectively. Doing so we can characterize also generic systems whose classical analogues show a mixed regularchaotic phase space, c.f. Sect. 3.8.7. Quantum chaology, a term tossed by Michael Berry [10,11], can arguably be put onto firm grounds by analyzing the spectral properties of quantum systems. These properties can be compared with predictions from the theory of random matrices. In this way, all quantum systems can be formally analyzed without referring to a classical analogue. The introduction to this vast field of research, presented in Sect. 5.6, should help to appreciate the concepts and allow the reader its practical application.

5.2

Semiclassical Quantization of Integrable Systems

5.2.1

Bohr–Sommerfeld Quantization

In 1885 Balmer investigated the spectral properties of light emitted by a hydrogen lamp and discovered the following empirical law describing the energy levels of the hydrogen atom. The frequency derived from the wave length of the corresponding spectral lines are given by  1 1 (5.2.1) − 2 , ν=R n2 k with positive integers n and k. Here, R denotes the Rydberg constant and k = 2 holds in case of the Balmer lines. For the energy we have the relation: hν = ω = E k − E n , 4

where E n = − me 22 e2

1 n2

(5.2.2)

= − 2n1 2 mc2 α 2 for the hydrogen atom. m is the atomic mass

and α ≡ c denotes Sommerfeld’s fine-structure constant, with the speed of light c, electron charge e and Planck constant . This experimental law can also be derived from the action integral along a closed trajectory q(t), i.e. along a quantized torus, see Sect. 3.4, which is described by the atomic model after Bohr and Sommerfeld

S(E) = pdq = hn = 2π n, n ∈ N. (5.2.3)

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Fig. 5.1 Simple model of the helium atom as an example for a non-integrable three-body problem with core and two electrons denoted by e. For generic initial conditions the classical dynamics can be chaotic [13]

It is possible to derive the Bohr–Sommerfeld quantization from the WKB approximation which is considered in detail in the next Sect. 5.2.2. The mentioned quantization condition was postulated in 1915 and belongs to the so-called “old quantum theory”. This old quantum theory does not really constitute a consistent theory but was instead a collection of heuristic results from early twentieth century. These results, achieved before the advent of a full quantum theory, were able to describe the phenomenon of discrete energy levels for systems as simple as the hydrogen atom. Today, the old quantum theory can be interpreted as giving low-order quantum corrections to classical mechanics and is thus known as the simplest version of a semiclassical theory. Already from a look at the quantization condition (5.2.3), one can see that the formulation relies on assumptions that will necessarily break down in more complex (in the sense of more degrees of freedom) systems: • This approach does not work for non-integrable systems. The formulation relies on the existence of tori that satisfy the quantization condition. Einstein noticed this already in his pioneering paper from 1917 [12]. • This means that many-body problems, e.g. the three-body helium atom (c.f. Fig. 5.1), cannot be analyzed with this method, see also our discussion in Sect. 1.2. Separable many-body systems may be quantized using a proper extension of WKB presented in Sect. 5.2.3.

5.2.2

Wentzel–Kramer–Brillouin–Jeffreys Approximation

In this section we present a famous and powerful semiclassical approximation method which was formulated by Wentzel, Kramers, and Brillouin in 1926. Jeffreys had already found a general method for the approximation of linear differential equations of second order in 1923. WKB(J) treats the solution of the Schrödinger equation in the semiclassical limit, i.e. for particles with large momenta and consequently short de Broglie wave lengths. In other words, the action of the particle must be large compared to . As mentioned in Sect. 1.3, the semiclassical limit  → 0 intends exactly this. It stands in analogy to the transition from wave-optical to geometric optics which results from λ  L (in short “λ → 0”), where λ denotes the wave length of the light and L is the characteristic length of the optical system. The development

5.2 Semiclassical Quantization of Integrable Systems

157

E 2λ

λ

0

δλ V (x) x

δx

Fig. 5.2 The WKB approximation is valid in the range of short wavelengths, i.e. the potential varies slowly with respect to the particle wavelength: δx  δλ

of the method roots also in the work of Liouville, Green, Stokes, and Rayleigh in connection with geometric and wave optics. Further historical information may be found in Appendix 11 of [14]. In the case of one-dimensional problems, the approximation is valid if the de Broglie wave length varies slowly over distances of its order. In other words the change of the potential is small against the particle wave length: δλ  λ  δx. δx is the characteristic distance for which there is no significant change of the potential. This is illustrated in Fig. 5.2. The most important applications of the WKB method are the calculation of semiclassical approximations of eigenenergies of bound states and of tunneling rates through potential barriers. Examples will be shown below.

5.2.2.1 Derivation of WKB Wave Functions We start with the stationary Schrödinger equation describing the one-dimensional motion of a particle in an external time-independent potential  2 d 2 − + V (x) ψ(x) = Eψ(x). 2m d x 2   

(5.2.4)

= Hˆ

The WKB ansatz leads to an analytical solution for the approximation in the limit “ → 0”, i.e.   S for typical classical actions S of the system. In order to find an approximate solution of (5.2.4), we start with the following ansatz for the wave function  i ψ(x) = exp g(x) (5.2.5) 

  i d 2 g(x) 1 dg(x) 2 i g(x) d 2 ψ(x) = − 2 (5.2.6) e ⇒ dx2  dx2  dx  1 dg(x) 2 i d 2 g(x) + + V (x) = E. (5.2.7) ⇒ − 2m d x 2 2m dx

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Here g(x) is assumed to be a complex valued function for x ∈ R. The first term in (5.2.7) is a correction term proportional to , whilst the -independent part of the equation can be compared to the classical Hamilton-Jacobi equation (3.3.25) from Sect. 3.3.4. From (5.2.7) we may write  d 2 g(x) dg(x) , = ± p 2 (x) + i dx dx2

(5.2.8)

√ where p(x) ≡ 2m(E − V (x)) denotes the classical momentum. The solution in lowest order of  is given by dg(x) = ± p(x) + O() dx dp(x) m d V (x) d 2 g(x) =± + O() = ∓ √ + O(). ⇒ dx2 dx p(x) d x

(5.2.9) (5.2.10)

An improved solution can be derived by putting (5.2.10) into (5.2.8) to obtain   dg(x) d 2 g(x) 1 dp(x) 2 + O(2 ). ≈ p(x) ±1 + i 2 = ± p (x) + i dx dx2 2 p (x) d x (5.2.11) Here, we expanded the square root in the approximation    1 dp(x)  1    p2 d x    .

(5.2.12)

The latter condition is equivalent to    d λ¯     1, dx 

(5.2.13)

with the de Broglie wave length of the particle λ¯ ≡

 . p

(5.2.14)

This condition is the semiclassical condition, and it gives the range when our approximation is valid. It can be rewritten as follows, with m dV dp =− , dx p dx        d λ¯     1 ⇒ m  d V   1.  dx  3 p dx 

(5.2.15)

(5.2.16)

5.2 Semiclassical Quantization of Integrable Systems

(a)

159

(b)

Fig. 5.3 The naive WKB approximation diverges close to the turning point x0 of the classical trajectory as seen in (a). This problem can be solved by approximating the solution close to these turning points by the Airy functions sketched in (b)

This summarizes what we said earlier. Consequently, the semiclassical approximation will not be applicable if the momentum is too small. Particularly, it is not directly applicable at the turning points of the classical dynamics. Going on with our solution, it follows that 

x

i p(x )dx + ln( p(x)) + const. + O(2 ) 2 x0   A i x ⇒ ψ(x) = √ exp ± p(x )dx .  x0 p(x) g(x) = ±

(5.2.17) (5.2.18)

The following relation corresponds to the classically allowed region, see Fig. 5.3: p(x ) > 0 ∀ x ∈ [x0 , x] ⇔ V (x ) < E ∀ x ∈ [x0 , x]. Consequently, the general WKB solution in this region is given by   x   i i x



A exp . p(x )dx + B exp − p(x )dx  x0  x0 (5.2.19) The general WKB solution in the classically forbidden region, where E < V (x ) ∀ x ∈ [x0 , x] yields 1 ψ(x) = √ p(x)



    x  1 x 1



C exp − . ψ(x) =  p(x ˜ )dx + D exp p(x ˜ )dx  x0  x0 p(x) ˜ (5.2.20) √ Here, we define p(x) ˜ ≡ 2m(V (x) − E). The wave function in this region is of purely quantum mechanical origin and can be used to describe, e.g., tunneling processes. An obvious problem occurs near the classical turning points, c.f. Fig. 5.3. In the limit E → V (x), the momentum vanishes, that means that the de Broglie wave length diverges. As p, p˜ → 0, the wave function from our ansatz becomes singular. The probability |ψ(x)|2 dx that the particle is located at x ∈ [x, x + dx] is proportional to p(x)−1 , meaning that the particle rests only a short time at a certain place where p(x) is large, i.e. where it moves fast. On the other hand, in accordance with the 1

160

5

Aspects of Quantum Chaos

classical intuition, the particle can most likely be found close to the turning points where it is the slowest and spends most of the time. There are several ways to solve the turning point problem. An excellent way is presented in [3], where the singularity is encircled in the complex plane. Here, we follow the simpler and more standard approach and consider a linear approximation of the potential close to the turning points. Naturally, this approximation will produce good results only for smooth potentials. We expand x ≈ x0 :

V (x) ≈ V (x0 ) +  α (x − x0 ) + O[(x − x0 )2 ].   

(5.2.21)

d V (x ) = dx 0

=E

The Schrödinger equation for the linearized problem reads −

2 d 2 ψ(x) + α(x − x0 )ψ(x) = 0. 2m d x 2

(5.2.22)

With a change of variables,  z=

3

2mα (x − x0 ), 2

(5.2.23)

the equation can rewritten in the following form d 2 ψ(z) = zψ(z). dz 2

(5.2.24)

The last equation is known as the Airy equation, its solutions are the Airy functions Ai(z) and Bi(z) [15]. This yields

 ψ(x) = a Ai

3



  2mα 3 2mα (x − x0 ) + b Bi (x − x0 ) , 2 2

(5.2.25)

with constant coefficients a and b. In the following, we discuss the asymptotic behaviour of the Airy functions Ai(z) and Bi(z): ⎧ 3 ⎨ 1 e− 23 z 2 , 1 2   z → +∞ Ai(z) =  √ 3 2 π ⎩ π |z| cos 3 |z| 2 − 4 , z → −∞ ⎧ 3 (5.2.26) ⎨e 23 z 2 , 1 z → +∞   Bi(z) =  √ 3 π |z| ⎩− sin 2 |z| 2 − π , z → −∞. 3

4

Let us now consider what this means for the wave function ψ(x) close to a classical turning point. We want to look at the limit  → 0, where the argument z of the Airy

5.2 Semiclassical Quantization of Integrable Systems

161

functions will tend to ±∞. We can thus replace the Airy functions in (5.2.25) by their asymptotic form given in (5.2.26). The wave function ψ(x) close to the turning point thus must have the form ⎧ 3 2 a − 2 (k0 (x−x0 )) 23 ⎪ ⎪ + be 3 (k0 (x−x0 )) 2 , x > x0 e 3 ⎪ ⎪2  ⎪ ⎨ 3 2 π →0 1 2 − a cos (k0 (x0 − x)) − ψ(x) ≈  √ 4 ⎪ 3 π k0 |x − x0 | ⎪ ⎪ 3 ⎪ 2 π ⎪ ⎩ b sin , x < x0 , (k0 (x0 − x)) 2 − 3 4 (5.2.27)  3 2mα where the new constant k0 ≡ denotes the wave number. We see already that 2 we obtain oscillating, wave-like solutions in the classically allowed region, and exponential behaviour inside the classically forbidden region. We can now compare this wave function to the results of the WKB ansatz in the classically allowed, (5.2.19), as well  as classically forbidden region, (5.2.20). Close to the turning point, the value of p(x)dx takes the form: Case x < x0 : p(x) =

 x≈x0  2m(E − V (x)) ≈ 2mα(x0 − x)  x0  2 ⇒ p(x )dx ≈ 2mα(x0 − x)3 3 x

(5.2.28) (5.2.29)

Case x > x0 : p(x) ˜ =

x≈x0  2m(V (x) − E) ≈ 2mα(x − x0 )  x  2 ⇒ p(x ˜ )dx ≈ 2mα(x − x0 )3 . 3 x0



(5.2.30) (5.2.31)

This leads to an improved WKB ansatz that replaces (5.2.19) and (5.2.20)    ⎧  1  x0 1 ⎪

)dx − π + B sin 1 x 0 p(x )dx − π ⎪ , x < x0 A cos p(x ⎨√ x x p(x)  4   4  ψ(x) = 1 1 x 1 x ⎪ ⎪ ˜ )dx + D exp p(x ˜ )dx , x > x0 . C exp − x0 p(x ⎩   x0 p(x) ˜

(5.2.32) By comparison with the solution (5.2.27) at the right turning point, it follows immediately that C = A2 and D = −B. Analogously, one derives the solutions at the left turning point, see Fig. 5.4, from the matching conditions there by just replacing  x0  x p dx → p dx (5.2.33) x

x0

in the discussion above. To better illustrate the procedure we now look at some explicit examples.

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Aspects of Quantum Chaos

Fig. 5.4 Left turning point of a classical trajectory at V (x0 ) = E in analogy to Fig. 5.3a

Fig. 5.5 A one-dimensional bounded potential with two turning points x1 and x2 for an energy E

5.2.2.2 Example 1: Bound States in a Potential V (x) A bound particle in one-dimensional must have two classical turning points. According to the WKB method from above one has to consider a left turning point x1 as well as a right turning point x2 , see Fig. 5.5. We already know the form of the WKB ansatz for one turning point, (5.2.32), such that our ansatz for the system with two turning points takes the following form (at fixed energy E):  ⎧ 1  √ C1 e−σ˜ 1 (x)/ + D1 eσ˜ 1 (x)/ , x < x1 ⎪ ⎪ ⎪ p(x) ˜ ⎪      ⎪ 1 π π ⎪ ⎨√ A1 cos σ1 (x)/ − 4 + B1 sin σ1 (x)/ − 4 p(x) ψ(x) = !      ⎪ A2 cos σ2 (x)/ − π4 + B2 sin σ2 (x)/ − π4 , x1 < x < x2 = √1 ⎪ ⎪ p(x) ⎪   ⎪ ⎪ ⎩√ 1 C2 e−σ˜ 2 (x)/ + D2 eσ˜ 2 (x)/ , x > x2 , p(x) ˜

(5.2.34) where the σ ’s are defined as xx  2

σ˜ 2 (x) ≡ x2

x1 0. This theorem implies, in practice for large L, that the matrix T (L) has L two eigenvalues e± λ since its determinant must be one. T (L) is so to speak an areapreserving map similar to the ones studied in Sect. 3.8 and 1/λ corresponds to the Lyapunov exponent. We may collect all these insights in the following theorem: Theorem (Anderson localization) In our one dimensional lattice model described by Hˆ lattice , (5.5.4), all eigenstates ψλ(E) are exponentially localized in configura-

tion space for any disorder strength Wt > 0, i.e. any eigenstate with eigenenergy E behaves asymptotically for large |i| as |ψλ(E) (i)| ∼ e−

|i− j(E)| λ(E)

.

(5.5.11)

The truly exponential scaling with localization length λ(E) is sometimes understood as a manifestation of strong localization implying Anderson localization, see our sketch in Fig. 5.26, in contrast to weak localization effects which manifest in a slower than exponential decay of the wave functions and other transport properties in solid-state samples [76]. In a sample of infinite length all eigenstates (for all eigenenergies, even much higher than the actual random potential peaks) are indeed localized in our model. For finite length L, some states may be practically extended if their localization length becomes comparable with L. This explains that diffusive transport may be possible in all finite-length disordered samples, provided that the energy-dependent localization length of the participating states is sufficiently large.

5.5 Anderson and Dynamical Localization

(a)

211

(b)

j(E)

i

j(E)

i

Fig. 5.26 Sketches of an exponentially decaying eigenfunction |ψλ(E) (i)| centred around j(E) on a linear (a) and a semilogarithmic (b) scale

5.5.2

Dynamical Localization in Periodically Driven Quantum Systems

The above Anderson model describes a static disordered system. There exists a dynamical version of strong localization which we are going to present in the following. The dynamical systems are periodically driven ones, i.e. there is a well defined driving frequency (which can be the common frequency of several commensurate ones as well [77–79]). An example of such systems is the driven Rydberg problem of (3.7.58), which was presented in the previous section for initial conditions attached to regular phase-space structures. In the present context of dynamical localization we assume that the initial wave packet starts in the chaotic part of phase space, i.e. typically at large energies or principal quantum numbers. Intuitively, the drive in the Rydberg atom couples the initial state with principal quantum number n 0 to the next state with n 0 + 1 quasi-resonantly. This implies that there will always be a state within reach by a two-photon transition even closer to the continuum threshold and so forth, see the sketch in Fig. 5.27. Since the multiples of frequencies never hit exactly the electronic states there will always be essentially random detunings. Altogether, this semiclassical picture describes a ladder of quasi-resonantly coupled states in energy space, similar to the ladder of irregularly coupled lattice sites due to the disorder in our Anderson model. Further information on this analogy can be found in the original references [71,80–84]. Another example for a periodically driven system is the quantized version of the standard map introduced in Sect. 3.8.3. In what follows we present this quantum kicked-rotor model since it is reasonably simple and its classical version was extensively studied Chap. 3. The classical Hamiltonian of the kicked rotor from (3.8.14) can be recast for L = I /T in a quantum Hamiltonian ∞ ' ˆ2 ˆ t) = L T + k cos(θˆ ) ˆ L, δ(t − j). Hˆ (θ, 2 j=−∞

(5.5.12)

212

5

Fig. 5.27 The energy spectrum of a hydrogen Rydberg ladder with E(n) = − 2n12 in atomic units. The initial state with principal quantum number n 0 is coupled by the “photons” of the external drive with frequency ω to higher excited states. The detunings δi in the couplings of the states towards the electronic continuum (of free states) are essentially random. This forms a ladder of pseudo-randomly coupled states, similarly to the situation of Anderson localization

Aspects of Quantum Chaos

continuum

E=0

...

δ3

n0 + 4 n0 + 3

δ2

n0 + 2

δ1

n0 + 1

hω n 0 > 20

...

n0 − 1

In this units the angular momentum eigenvalues are integers, which is rather convenient for practical purposes. Since δ(t) =

∞ ' n=0

 2π cos n t , T

(5.5.13)

we get back in zeroth-order (n = 0) approximation a quantum pendulum Hamiltonian. The quantum kicked rotor has the great advantage that its evolution operator from right after one kick until right after the next one factorizes into a kick and a free rotation part: ˆ Uˆ (nT , (n + 1)T ) = Uˆ (0, T ) = e−ik cos(θ ) e−iT

Lˆ 2 2

.

(5.5.14)

This makes the analytical and numerical treatment of this model much simpler than for a system, like a driven Rydberg atom, for which the temporal dependence is continuous. The wave function after n kicks is then just given by applying n-times the kick-to-kick evolution operator: ψ(t = n) = Uˆ (0, nT )ψ(t = 0) = Uˆ (0, T )n ψ(t = 0).

(5.5.15)

From the form in which we have written Hˆ and Uˆ , we also see that the two parameters T and k are quantum mechanically independent, and we cannot just reduce them to a single parameter K = kT as done in (3.8.14). The classical limit is obtained for K = const. (constant classical phase-space structure) while simultaneously T → 0 and k → ∞. In the scaling of variables used above, T is then the effective Planck ˆ = iT. ˆ L] constant, and we have for the commutator [θ,

5.5 Anderson and Dynamical Localization

213

We can write the kick-to-kick evolution operator in matrix form expressing it in the basis of angular momenta |mm∈Z in our units. Then the matrix elements are 

 m2 1 m  Uˆ (0, T ) |m = e−i 2 T 2π



2π

dθ e−ik cos(θ ) ei(m −m)θ = e−i

0

m2 2

T

Jm −m (k),

(5.5.16) with the ordinary Bessel functions of the first kind of integer order m − m. For k → 0, the matrix is essentially diagonal meaning that the kick hardly couples the different angular momentum states. For fixed kick strength, the off-diagonal elements sooner or later will quickly decay to zero since the Bessel functions are approaching zero rapidly for (m − m)  k [15]10 1 Jm −m (k) ∼ √ 2π |m − m|



k e 2 |m − m|

|m −m|

.

(5.5.17)

Here e = 2.71828 . . . is Euler’s number. For periodically time-dependent Hamiltonians, we can still arrive at stationary equations from the Schrödinger equation, but not directly for the Hamiltonian, yet for the time-evolution operator. One can try to diagonalize the kick-to-kick operator and compute its eigenphases φ = −εT :     Uˆ (0, T ) ψφ = eiφ ψφ .

(5.5.18)

ε is the so-called quasi-energy corresponding to the eigenphase φ. This name indicates that the above ansatz is indeed the same as for a periodic dependence of the Hamiltonian in real space, for which we can use the Bloch ansatz from (5.5.2). Equation (5.5.18) describes translations in time induce by the so-called Floquet operator Uˆ (0, T ), as similarly true for translations in space used to show that the solutions for transitionary invariant problems are indeed of Bloch type, see e.g. [4]. Now the eigenvalue problem for the quantum kicked rotor can be cast into a form very similar to (5.5.7): 

   m  Uˆ (0, T ) |m

−e I iφ

m ,m

   m ψφ m = 0.

(5.5.19)

This system of equations is related to the Anderson model studied above. The only difference is that the disorder is not built in by random onsite energies but by the dynamics itself. The quantum kicked rotor can indeed be written in exactly the same form as (5.5.6) with diagonal elements which give, for typical choices of the parameter T , a pseudo-random sequence. This is explicitly shown by Fishman et al. in [70,85], cf. also Chap. 7 in [22]. The result is that the eigenstates of the kickto-kick operator Uˆ (0, T ) are exponentially localized but now in angular-momentum

10 Please

see the asymptotic expansion with number [9.3.1] in Chap. 9 of [15].

214

5

Aspects of Quantum Chaos

Fig. 5.28 Sketch of an exponentially localized eigenfunction |ψφ (m)| ∼ e−|m− j(φ)|/λ(φ) of the Floquet operator (5.5.18) centred around the angular momentum j(φ) with localization length λ(φ)

j(φ)

m

space, c.f. Fig. 5.28. If the parameters of the kicked rotor are such that the classical phase space is chaotic, the quantum evolution nevertheless will localize after a finite number of kicks whose number is proportional to k [the effective bandwidths of the Floquet matrix in angular-momentum space, see (5.5.17)] [86,87]. This type of localization is of pure quantum origin just like Anderson localization. It arises from many-path destructive interference in the semiclassical picture of the time evolution, see Sect. 5.3.2. Because of its dynamical origin this type of quantum localization is known as dynamical localization.11 The effect of dynamical localization is visualized in Fig. 5.29. This figure shows the second moment of the angular-momentum distribution divided by two, i.e. the kinetic energy of the system, as a function of the number of applied kicks. While initially the kinetic energy grows just like in the classical kicked-rotor system (dashed line), after about ten kicks the growth stops, and finally turns into a saturation. This is quite fascinating since the system is continuously kicked and, naively, one might expect that it would absorb more and more energy, yet it does not. The underlying classical model has a completely chaotic phase space, imagine Fig. 3.27d for still larger kick strength K = 7.5, hence without any visible regular islands. For such a case the classical trajectories lose memory quickly, allowing to approximate the classical energy by the diagonal terms of the following sums obtained by iterations of the standard map In = I0 + K

n '

sin(θ j )

(5.5.20)

j=1

E class (n) =

n I2 I2 In2 K2 ' 2 K2 sin (θ j ) ≈ 0 + ≈ 0 + n. 2 2 2 2 4

(5.5.21)

j=1

The classical diffusion in the chaotic phase space is drastically stopped in the quantum system as soon as it realizes its coarse-grained nature [92]. Hence, the quantum

11 The

notion dynamic or dynamical localization is unfortunately also used in different contexts, one example being the suppression of tunneling by a periodic driving force [88], for experimental realizations of this latter effect see, e.g. [89,90]. The classical description of dynamical localisation in [91] is nevertheless based on quantum concepts.

5.6 Universal Level Statistics

215

Fig. 5.29 The energy E(n) as a function of the number of kicks n for the classical (black dashed line) and quantum (red symbols) kicked-rotor problem. The blue arrow marks the quantum break time when both evolutions start to deviate from each other significantly. Both curves originate from averaging over different initial conditions to smooth the strong fluctuations characteristic of localized systems [72,83,94]

system shows much more regular behaviour than its classical version due to dynamical localization. This regularity of the quantum system is seen also in the spectral fluctuations in the quasi-energies [93]. These spectral properties of regular and chaotic quantum systems are studied in Sect. 5.6.

5.5.3

Experiments

Dynamical localization was experimentally observed in the two systems mentioned above. Independently, Bayfield [18,95] and Koch [96,97] and their respective collaborators measured the rise of the ionization threshold of driven alkali Rydberg states with increasing frequency. This counterintuitive effect had been predicted before as a direct consequence of dynamical localization [71,98,99]. The experimental implementation of the kicked-rotor system was pioneered by Raizen’s group using the centre-of-mass degree-of-freedom of cold atoms. Dynamical localization and the exponential form of the momentum distribution of the atoms is largely discussed in the references [100,101]. It took about ten years more until Anderson localization could be measured in situ with light, see [102] and references therein, or dilute Bose-Einstein condensates [103] propagating or expanding in a disordered spatial potential, respectively.

5.6

Universal Level Statistics

Up to now we have not yet defined what chaos really means for a quantum system in general. We discussed the properties of quantum systems with a well-defined classical counterpart, which can be chaotic, regular or mixed, as we introduced in Chap. 3. In

216

5

Aspects of Quantum Chaos

the previous section, we have seen that there may be a time for which the quantum system follows the classical motion, the so-called quantum break time, but after which the dynamics are different from the classical one. Chaos on the quantum level may, however, be introduced without a priori relation to any classical correspondence. This can be achieved by studying the properties of the very characteristics of quantum systems, their eigenspectra. The properties of the quantum mechanical spectrum are, of course, intimately related to the temporal evolution. This is a consequence of the spectral theorem [3,5], which states that the time-evolution operator can always be expanded in the eigen(energy)basis of the Hamiltonian. Hence, irregular and complex dynamics go hand in hand with the quantum chaotic properties of the spectra. This section gives a brief introduction to the theory of random matrices of finite size, modelling real physical systems with discrete spectra.12 All of our sketches of proofs will be given for the simplest type of matrices, namely 2 × 2 ones. The shown ideas carry over to matrices of arbitrary but finite sizes, yet the proofs and computations are then rather difficult and call for more mathematical language [104]. We refer to the specialized literature for further details, see e.g. Haake’s book [22] or Mehta’s standard reference for random matrices [105,106].

5.6.1

Level Repulsion: Avoided Crossings

The essence in the study of the dynamical properties of quantum systems lies in the coupling of different states. The sensitivity of energy eigenlevels can be tested by varying, for instance, a control parameter in the Hamiltonian. If the states are coupled the levels will repel each other as predicted by perturbation theory [4]. Let us study the eigenvalues of an arbitrary hermitian 2 × 2 matrix H = H † : 

H11 H12 . ∗ H H12 22

(5.6.1)

The eigenvalues are easily computed 

√ 1 1 (H11 − H22 )2 + |H12 |2 ≡ (H11 + H22 ) ± D. 4 2 (5.6.2) We can now distinguish the following possible cases:

1 E ± = (H11 + H22 ) ± 2

(a) No interaction between the levels, i.e. H12 = 0. Then the eigenvalues are simply E + = H11 and E − = H22 . Both eigenvalues can be forced into a crossing by a single real parameter, let’s say an appropriate parameter λ which enters the element H22 , such that H11 − H22 (λ) = 0.

12 A short overview of possible spectra of a quantum system is found in the appendix of this chapter.

5.6 Universal Level Statistics

E m5

217

E (a)

(b)

λ

λ

m4 m3 m2 m1

Fig. 5.30 Sketch of the energy levels for the same cases (a, b) as discussed in the text: an integrable systems (a) and of a typical system (b) versus a single control parameter λ. The good quantum numbers are labeled by m i in (a). The avoided crossings in (b) make a global assignment of quantum numbers along the λ-axis practically impossible

(b) More generally, H12 = 0, and we have a determinant D = 41 (H11 − H22 )2 + Re (H12 )2 + Im (H12 )2 . For real matrices, Im (H12 ) = 0, implying that we need to change two matrix elements simultaneously to obtain a level crossing, i.e. a degeneracy of the two eigenvalues. This can be formalized as 1 (H11 − H22 (λ1 ))2 + H12 (λ2 )2 = 0 4 ⇔ H11 − H22 (λ1 ) = ±2iH12 (λ2 ),

(5.6.3) (5.6.4)

for an appropriate pair of real parameters (λ1 , λ2 ). If also Im (H12 ) = 0, we can easily see that three independent real parameters are, in general, necessary to enforce a level crossing. The typical behaviour of the energy levels while changing just one control parameter is sketched in Fig. 5.30. For randomly chosen parameters, the crossings will most likely be avoided ones! For classically regular systems of arbitrary dimension, which can be quantized by the EBK method introduced in Sect. 5.2.3, we have a similar situation as in case (a) above. The quantum mechanical levels are approximated by the semiclassical EBK torus quantization as  μ ! ,λ , E m (λ) ≈ H I =  m + 2

(5.6.5)

where the dimension of the Hilbert space dim (m) = d. The levels are well characterized by their respective quantum numbers m j . It is thus quite intuitive that locally one real parameter, effecting just one level E m j (λ), may suffice to enforce a level crossing between two of the eigenvalues. From this heuristic argument it is expected that a classically regular system does show level crossings with relatively large probability. We will come back to the case of regular classical systems more quantitatively in Sect. 5.6.5.

218

5

Aspects of Quantum Chaos

Fig. 5.31 A classical model for many-body scattering of a neutron on a heavy nucleus consisting of many constituents. Reprinted by permission from Macmillan Publishers Ltd. [108], copyright 1936

5.6.2

Level Statistics

For classically integrable systems we have a set of “good” quantum numbers, e.g. denoted m in (5.6.5) for the semiclassical approximation of the levels. There is no such simple rule for classically non-integrable systems, whether mixed or chaotic. There may exist symmetries which allow to separate at least some degrees of freedom. Nevertheless, we typically lack a full set of good quantum numbers corresponding to the number of equations or to the size of the Hilbert space. How can we characterize the eigenspectra in these cases? Imagine that we were given spectra of a complex quantum system either from an experiment in the laboratory or from numerical computations. How can we decide that the spectra look “irregular”? The idea is to study the statistics of the levels, in particular the fluctuations of the level distances, which tells us something about the probability of crossing or anti-crossing of levels. On the other hand, there are systems much too complex for even knowing the precise form of the Hamiltonian. Heavy nuclei studied experimentally by scattering experiments are a good example. Assuming just minimal knowledge, like the existence of some symmetries, such systems can be modelled by matrices with essentially random entries. The random elements describe random couplings between the various degrees of freedom. This idea goes back to Wigner [107] and a simplified picture of Bohr who thought of the scattering process between a hadron and a heavy nucleus as a classical billiard-ball-like game, see Fig. 5.31. Now it is the theory of random matrices which brings both aspects together in the question of how good the spectra of physical systems reflect the statistics of randommatrix ensembles. Before we can introduce the two main hypotheses, which link both semiclassical and purely quantum aspects, we have to speak about symmetries, which must be considered as they can mix irregular spectra in an uncorrelated way such that they may look more regular than they actually are.

5.6 Universal Level Statistics

5.6.3

219

Symmetries and Constants of Motion

For practical purposes, an important step in analyzing complex quantum spectra is to separate them into subsystems which do not couple because of symmetries. Only after such a separation it makes sense to test the level statistics for each of the symmetryreduced subsystems independently. The concept of symmetries is well-known from quantum mechanics, see e.g. [4]. We review it nevertheless since symmetries are essential for the spectral analysis.

5.6.3.1 Examples for Symmetries • Continuous symmetry The symmetry is described by a generating hermitian operator Aˆ acting on the Hilbert space. Let λ be a real continuous parameter. If the Hamiltonian of the ˆ a symmetry may be given by translations of the form system commutes with A, ˆ ˆ ˆ iλ A ψ → e ψ, which follows from Hˆ = e−iλ A Hˆ eiλ A . An example for such a symmetry is a rotational invariance with respect to an axis, e.g. Aˆ = Lˆ z /, corresponding to a rotation around the z axis by an angle λ. • Discrete symmetry The Z2 inversion symmetry defined by the parity operator Aˆ P with Aˆ P ψ(x) = ψ(−x) = λψ(x) for λ ∈ {1, −1}. 5.6.3.2 Irreducible Representation of the Hamiltonian Matrix ˆ Hˆ ] = 0. An operator Aˆ defines a symmetry if it commutes with the Hamiltonian [ A, This means that we can find a common eigenbasis for both operators. Let us assume that we found an eigenbasis |m, n of Aˆ Aˆ |m, n = αm |m, n ,

(5.6.6)

with αm ∈ R and αm = α j for j = m, where the second index n stands for the part that remains to be diagonalised. We then can express the Hamiltonian in this basis representation   (m) (5.6.7) m , n  Hˆ |m, n = δm,m Hn,n

, which leads to the following block-diagonal form of the Hamiltonian matrix

220

5

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

.. 0 .. .

.

...

0 (m−1)

Hn,n ..

.

..

.

(m) Hn,n

(m+1) Hn,n

0

...

0

Aspects of Quantum Chaos

⎞ 0 ⎟ ⎟ ⎟ .. ⎟ . ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ ⎠ .. .

(5.6.8)

ˆ while the Here, m is the index of the block corresponding to the eigenvalue αm of A, ˆ Hˆ ] = 0 second indices n, n run from one to the size of the respective blocks. [ A, implies a degeneracy of eigenenergies, and typically different blocks contain states with the same energy. Separating the blocks effectively lifts those degeneracies what was exactly our goal. If no further symmetries are present, the above form is said to be an irreducible representation of the Hamiltonian. The analysis of the spectrum must now be restricted to a single block of the full matrix. Of course, all of the blocks can be analyzed separately, which can be useful in order to improve the statistical value of such a spectral characterization. If more than one symmetry is present, an eigenbasis can be found which commutes with all three operators, the Hamiltonian and the other two corresponding to the symmetries. Then the Hamiltonian is represented in this common eigenbasis being of block-diagonal form within a cube. In practice, any of the blocks indexed with m may then be further split into subblocks whose number is determined by the additional symmetry. To state an example, one symmetry could be the translation invariance, e.g. in a tight-binding model, see Sect. 5.5.1, and a simultaneously present symmetry could be the parity for lattices with an odd number of sites and periodic boundary conditions. The procedure is extendable to any dimension if even more symmetries are simultaneously present. The important message is that the such obtained blocks should be analyzed independently. This is necessary to avoid the mixing of various blocks, which can induce degeneracies in the spectrum which may not be present within any of the single blocks. It is clear that any operator commuting with the Hamiltonian defines a constant of the motion. This means that an initially complex system can be reduced to a smaller space which turns out to be useful for practical computations. Hence, finding all symmetries helps to reduce the complexity, too.

5.6 Universal Level Statistics

221

Fig. 5.32 Sketch of a level staircase N (E) for the ordered spectrum of eigenenergies E j . Each time a level pops up, the curve jumps by one. Differences in the local level density make the overall curve have a slope different from one

5.6.4

Density of States and Unfolding of Spectra

Yet another important step in the practical application of any spectral analysis has to do with the fact that we are interested in the statistics of local eigenvalue fluctuations and not in global trends which are system specific. The former correspond to the fluctuating part of the level density, while the latter part is determined by the smooth part in Eq. (5.3.67). In this section, we show how one can reduce the information contained in quantum spectra to the information relevant for us on local fluctuations. The aim is to renormalize the eigenenergies of one symmetry-reduced subspace, see the previous section, in such a way that the mean density of the levels is uniform over the entire spectrum of the subspace. This is known as unfolding of the spectra. We assume a discrete quantum spectrum of finite size. The algorithm for spectral unfolding is as follows. 1. Sort the energies in increasing order, i.e. such that E 0 ≤ E 1 ≤ E 2 ≤ . . . ≤ E N , where N is the dimension of the symmetry-reduced subspace. 2. / Compute the integrated density of states, the so-called staircase function N (E) = ∞ n=0 Θ(E − E n ), giving all eigenvalues up to the energy E. Here, Θ is the Heaviside step function. The construction of the level staircase function / is sketched in Fig. 5.32. The level density is formally given by ρ(E) = dd NE = ∞ n=0 δ(E − E n ). 3. Approximate N (E) by a smooth function N (E) interpolating between the jumps of the curve in Fig. 5.32. 4. With the help of the smooth function N (E) we finally can rescale the spectrum, such that the average level density is equal to one. Reading off the values of the points xn in Fig. 5.33, which are defined by xn = N (E n ), we arrive at the new density of states ρ(x) =

∞ ' n=0

δ(x − xn ), with ρ(x) ≡

1 Δx



x0 +Δx/2 x0 −Δx/2

d x ρ(x) ≈ 1. (5.6.9)

222

5

Aspects of Quantum Chaos

Fig. 5.33 N (E) from the previous figure with a smoothed version N (E). The crossing points between the two functions determine the “new” levels, which are read off at the y axis. These values xi have on average a level spacing of one by construction

The practical problem one faces is the question how to obtain a good estimate for the smoothed level density N (E). Sometimes a semiclassical estimate, e.g. based on Weyl’s law, see Sect. 5.3.6, may be possible [109,110]. In general, one will have to measure or to compute the spectrum and then to extract the local average of the level density numerically. With this average density we can finally rescale all energies to obtain approximately an average density of one. For the latter, there are many methods proposed in the literature [111], but most often a brute-force local averaging over an appropriate window of levels may suffice [112,113]. This window must be small enough in order not to average too much the fluctuations in which we are interested, yet sufficiently wide in order to smooth the local density and to obtain a local mean of one in good approximation.

5.6.5

Nearest-Neighbour Statistics for Integrable Systems

Let us focus on a subspace which is already symmetry reduced, in the sense as shown in Sect. 5.6.3. For an integrable system, we additionally have sufficient constants of motion, or quantum mechanically speaking, enough good quantum numbers. In this case, using EBK semiclassics, see (5.6.5), we can obtain corresponding estimates of the quantum levels at sufficiently high energies. Just varying a single parameter can then easily induce level crossings locally in the spectrum. This is possible since the individual levels do “not feel” each other and can be tampered individually. Assuming a random distribution of ordered numbers on the real line, −∞ ≤ x0 ≤ x1 ≤ . . . ≤ x N < ∞, the probability density of having a level within a certain distance s from another one, corresponds to the distribution of independent random numbers whose average “waiting time” of occurrence is constant. Such distances between In other words, we expect to obtain P(s) = / N −1 numbers are Poisson distributed. −s in the limit of a level continuum. The most likely δ(s − (x − x )) → e i+1 i i=0 case is then s = 0, which corresponds to a level crossing. The unfolding, described in the previous section, is responsible for the average rate of occurrence (the mean waiting time) being exactly one, i.e. s = 1. Another way of seeing this, is to ask

5.6 Universal Level Statistics

223

for the probability P(s)ds to have one level within the distance ds of a given one. This probability is proportional to the probability g(s)ds of having exactly one level in the interval [x + s, x + s + ds]. g(s) denotes the distribution of random numbers on the real positive line. It is also proportional to the probability of having no level ]x, x +s[ of length s. The latter is identical to the complement of ins the interval

), which is ∞ ds P(s ). Hence we obtain ds P(s 0 s 

∞

P(s)ds = g(s)ds ×

ds P(s ),

(5.6.10)

s

The solution of this relation is P(s) = g(s) e−

s 0

ds g(s )

.

(5.6.11)

Assuming a uniform distribution of random numbers with g(s) = 1, the conditions of normalization and average level spacing one, 

∞ 0



∞

ds P(s) = 1 and

ds s P(s) = 1,

(5.6.12)

0

immediately imply P(s) = e−s . 2 Generally, for classical integrable nonlinear systems, i.e. det( ∂∂Ii ∂HI j ) = 0, with respect to the action variables Ii , with at least two independent degrees of freedom, Berry and Tabor convincingly argued in favour of the following conjecture [114]: Berry–Tabor conjecture: In the limit of large energies (semiclassical limit), the statistical properties of the quantum spectra of classically integrable systems correspond to the prediction for randomly distributed energy levels. We met an example for a quantum system with Poisson distributed level distances in Sect. 5.5.2, the quantum kicked rotor when showing dynamical localization. This is no contradiction to the above hypothesis for integrable systems. Anderson (or dynamical) localization give eigenstates which have exponentially small overlap in real (or momentum) space, and hence essentially independently distributed energy levels which do not “feel each” other. This implies regular behaviour and Poisson distributed level spacings (despite the classical chaoticity of the kicked rotor) [93]. A better example, where we find Poisson spectral statistics, is the rectangular billiard, introduced in Sect. 5.2.3.1, if the two side-lengths of the billiard are incommensurate. Then the mixing of the levels, corresponding to the two different quantum numbers for each dimension, leads to randomly distributed level spacings. This case was analyzed also in the original paper by Berry and Tabor [114]. A similar result from [115] is shown in Fig. 5.34 and we invite the reader to test this for him/herself, see Problem 5.13. As usual for any real physical system, slight deviations are found and discussed in [115] from the exact predictions from random-matrix theory (RMT), in particular when analysing spectral correlation functions, which we briefly introduce later on in Sect. 5.6.8.

224

5

Aspects of Quantum Chaos

Fig. 5.34 Spectral analysis of a two-dimensional billiard with incommensurate side lengths (histogram) and Poisson distribution P(s) = e−s (dashed line). Adapted reprint with kind permission by Giulio Casati, copyright 1985 by the American Physical Society [115]

5.6.6

Nearest-Neighbour Statistics for Chaotic Systems

We have seen in Sect. 5.6.1 that the more degrees of freedom we have, the harder it is to enforce a level crossing. For complex hermitian 2 × 2 matrices, we needed already three real parameters to enforce a degeneracy. For larger matrices the situation becomes even worse and level crossings are, in general, unlikely. Figure 5.30 sketches the spectra for an integrable system, for which we may follow the individual levels with the control parameter λ, and for a chaotic spectrum, for which many irregular avoided crossings occur. In the language of 2 × 2 matrices, the former case may be described by the following matrix 

E 1 (λ) 0 . 0 E 2 (λ)

(5.6.13)

The latter case may be represented by additional couplings V : 

E0 − Δ V , V ∗ E0 + Δ

(5.6.14)

where we introduce  the parameter Δ for convenience. The level splitting for (5.6.14) is given by D = Δ2 + |V |2 , with V = VR + i VI , see Fig. 5.35 and Problem 5.14. We may then estimate the probability for small spacings, i.e. ΔE → 0, by    d VI dΔ P(VR , VI , Δ) δ(ΔE − D). P(ΔE) ∼ d VR (5.6.15) Here, P(VR , VI , Δ) is the probability distribution of the mentioned three independent parameters. We may assume that P(VR , VI , Δ) ≈ const. > 0 for VR , VI , Δ → 0.

5.6 Universal Level Statistics

225

Fig. 5.35 Energy eigenvalues of a 2 × 2 matrix versus the control parameter E 0 . The splitting is determined by D defined in the text

E+

D

E− E0 Then the three-dimensional volume integral over (VR , VI , Δ) can be computed as  P(ΔE) ∼ 4π

d V V 2 δ(ΔE − V ) ∝ ΔE 2 .

(5.6.16)

Our much simplified argument shows that the levels repel each other when approaching small distances. For a real matrix with just two independent parameters (V , Δ), both real, we obtain in a similar manner:    P(ΔE) ∼ d V dΔ P(V , Δ)δ(ΔE − D) ∼ 2π d V V δ(ΔE − V ) ∝ ΔE, (5.6.17) for small Δ → 0 to allow for the approximation D ≈ |V | in the argument of the δ-function. After a long experimental experience with spectral data from heavy nuclei, it was found that their excitation spectra correspond quite well to the predictions made for certain ensembles of random matrices [107,116,117]. It was Casati, Valz-Gris, and Guarneri [118], and later in more precise form Bohigas, Giannoni, and Schmit [109], after some ideas by McDonald and Kaufman [119] and Michael Berry [120], who formulated the following hypothesis, and thus generalized the statement to arbitrary quantum systems: Bohigas–Giannoni–Schmit conjecture: The eigenvalues of a quantum system, whose classical analogue is fully chaotic, obey the same universal statistics of level spacings as predicted for the eigenvalues of Gaussian random matrices. The theory of random matrices (RMT) has a long history, going back to Wigner [107]. Dyson was studying independently unitary ensembles in the context of scattering problems (of unitary S matrices) [107]. The different symmetry conditions give different predictions for the level spacings, depending on whether the Hamiltonian is real, corresponding to the so-called Gaussian orthogonal ensemble (GOE) for timereversal invariant systems with integer spin, or complex, giving the Gaussian unitary ensemble (GUE) for systems with broken time-reversal symmetry. There is another case for half-integer spin systems with time-reversal invariance which implies Gaussian symplectic ensembles (GSE). We will come back to these ensembles in the next

226

5

Aspects of Quantum Chaos

Fig. 5.36 Spectral analysis of a variant of the Sinai billiard (histogram) and GOE distribution (solid line). The billiard is shown in the inset, with a defocussing boundary in the lower left corner. Reproduced from [109], copyright 1984 by the American Physical Society

section in more detail. For now, let us just state the prediction for the level spacing distributions for the respective cases: ⎧ ⎪ e−s ⎪ ⎪ ⎪ ⎨s π e−s 2 π4 2 P(s) = 24 ⎪s 2 322 e−s π ⎪ π ⎪ ⎪ 64 ⎩s 4 218 e−s 2 9π

integrable GOE GUE

.

(5.6.18)

GSE

36 π 3

All cases obey the conditions of normalization and mean-level spacing one:  0

∞



∞

ds P(s) = 1 and

ds s P(s) = 1.

(5.6.19)

0

Unfolding automatically guarantees the latter condition. Then s is dimensionless such that all spectra can be directly compared without another scaling factor. The major difference in the distributions lies in (i) the slope at s = 0, which is negative for the Poissonian case, constant for the GOE case, and linear or cubic for the GUE and GSE cases, respectively. The second (ii) important difference between the Poisson and the other cases is the weight in the tail, which in the Poissonian case decays less fast. These most robust characteristics are usually checked first when analyzing data from an experiment or from a numerical diagonalisation. The predictions from (5.6.18) can be exactly derived for ensembles of random 2 × 2 matrices [22,106]. This will be exemplified for the GOE case in the next section. In the general case of N × N matrices, one expects the results to hold as well, which is the content of the so-called Wigner surmise. Analytic closed-form expressions can then be rigorously proven only for s → 0 and s → ∞ because of complicated integrals over matrix ensembles appearing in the derivation [106]. Figure 5.36, from the original paper by Bohigas, Giannoni and Schmit, highlights the good correspondence of the spectra of a quantum Sinai billiard (see also Sect. 3.9.1) with the GOE prediction.

5.6 Universal Level Statistics

5.6.7

227

Gaussian Ensembles of Random Matrices

As mentioned above, random-matrix ensembles were introduced as models for manybody quantum systems of heavy nuclei where the couplings are basically unknown. The idea was to assume as little as possible about the specific systems and considering mainly their symmetries. We can build a statistical ensemble of matrices with reasonable conditions for their entries. The conditions used are • Statistical independence of the matrix elements, i.e. the entries are independent random numbers. This means that the distribution of a whole matrix factorizes into its independent parts:  >  ( = pi, j (Hi, j ). (5.6.20) P H = Hi, j i, j = i, j

• The statistical properties should not depend on the representation (or basis) chosen to built the matrix. This implies that the result is stable with respect to basis transformations obeying the following symmetries: 1. The system is invariant under time-reversal, i.e. the Hamiltonian commutes with the time reversal operator Tˆ . In this case, H can, without loss of generality, be chosen real with H = H † = H t , hence H is a symmetric matrix with real entries. Our probability distribution should be independent of the basis choice, implying that it is invariant with respect to any arbitrary orthogonal transformation O −1 = O t ∈ N R N . In formulae: [ Hˆ , Tˆ ] = 0, with Tˆ 2 = 1, and P(H ) = P(O H O t ).

(5.6.21)

Now it becomes clear why the ensemble is called orthogonal ensemble (GOE) in this case. 2. The system is not time-reversal invariant, and H is, in general, hermitian with complex entries. The requirement is then invariance under any unitary basis transformation with a unitary matrix U −1 = U † ∈ N C N . This gives the Gaussian unitary ensemble (GUE), characterized by [ Hˆ , Tˆ ] = 0 and P(H ) = P(U HU † ).

(5.6.22)

3. If the system is time-reversal invariant, yet with Tˆ 2 = −1 (Tˆ being then a so-called antiunitary operator [121]), any level is twofold degenerate (the socalled Kramers degeneracy). In this case the basis transformations are given by unitary symplectic matrices S, c.f. also Sect. 3.2: [ Hˆ , Tˆ ] = 0, with Tˆ 2 = −1, and P(H ) = P(S H S † ).

(5.6.23)

This defines the Gaussian symplectic ensemble (GSE). Such a situation occurs for systems with half-integer spin that are time-reversal invariant. More information on this more exotic ensemble can be found in [22].

228

5

Aspects of Quantum Chaos

For all the three universality classes, we obtain under the above conditions of independence and invariance P(H ) = Ce−A tr



H2



,

(5.6.24)

with positive constants A and C chosen to satisfy (5.6.19). Whilst the latter result is valid for all ensembles (GOE, GUE, GSE), the following equalities follow from explicitly evaluating the trace of the product of the two identical GOE matrices H : P(H ) = Ce−A tr



H2



= Ce−A

/N i, j=1

Hi,2 j

≡

N (

p(Hi, j ).

(5.6.25)

i, j=1

Similar relations hold also for the other two ensembles, where one must introduce the real and imaginary parts of the matrices separately [22]. Finally, we can appreciate why the random-matrix ensembles are called Gaussian ensembles. The distributions of the single matrix elements must be Gaussian in order to satisfy the above criteria. It should not come as a surprise to find the matrix trace in the universal distributions (5.6.24) since the trace is naturally invariant with respect to the required transformations [104]. We conclude this section by proving for 2 × 2 matrix ensembles that (i) P(H ) is indeed as given in (5.6.24), and that (ii) we obtain P(s) as stated in (5.6.18). For simplicity, we restrict to the GOE case in what follows. A similar demonstration for the GUE case is found, e.g., in Chap. 22 of [122]. (i) Let us start with a real symmetric matrix with the three independent entries H11 , H22 , H12 :  H11 H12 . (5.6.26) H12 H22 The condition of independence of the random variables immediately gives for the probability distribution of the matrix: P (H ) = p11 (H11 ) p22 (H22 ) p12 (H12 ).

(5.6.27)

Additionally, we have a normalization condition and the requested invariance with respect to orthogonal transformations:  d H11 d H22 d H12 P(H ) = 1, (5.6.28)   P(H ) = P O H O t ≡ P(H ).

(5.6.29)

The general form of an orthogonal transformation in two dimensions is   cos(θ ) − sin(θ ) 1 −θ O(θ ) = → , (5.6.30) sin(θ ) cos(θ ) θ 1

5.6 Universal Level Statistics

229

where the latter step is valid for small θ → 0. In this limit, we obtain for the transformed matrix

= H11 − 2θ H12 H11

H22 = H22 + 2θ H12

H12 = H12 + θ (H11 − H22 ) .

(5.6.31) (5.6.32) (5.6.33)

Using the statistical independence (5.6.27) and the invariance condition (5.6.29), we get dp22 dp12 dp11 ΔH11 p22 p12 + ΔH22 p11 p12 + ΔH12 p11 p22 . d H11 d H22 d H12 (5.6.34) Remembering ΔP =

d ln p11 1 dp11 = , d H11 p11 d H11

(5.6.35)

and similarly for the other two variables, we arrive at the condition   d ln p11 d ln p22 d ln p12 . P(H ) = P(H ) 1 − θ 2H12 − 2H12 − (H11 − H22 ) d H11 d H22 d H12

(5.6.36) This implies that the correction term, the second term in the brackets, must vanish for all θ , i.e.  d ln p11 1 d ln p12 d ln p22 1 − − = 0. (5.6.37) 2(H11 − H22 ) d H11 d H22 H12 d H12 One may easily check that the following uncoupled functions solve this differential equation p11 (H11 ) = e−AH11 −B H11

(5.6.38)

p22 (H22 ) = e

2 −B H −AH22 22

(5.6.39)

p12 (H12 ) = e

2 −2 AH12

(5.6.40)

2

.

Here A and B are two positive constants. With (5.6.27), we finally arrive at P(H ) = Ce−A tr(H ) , 2

(5.6.41)

after an appropriate shift of the energy offset such that H11 + H22 = 0. This is exactly what was predicted in (5.6.24) for the general case. Hence any individual distribution does not depend on its index (i j). All of them are indeed identical. For the GOE case, as shown here, we must only be careful at counting the contributions

230

5

Aspects of Quantum Chaos

from the off-diagonal elements, i.e.13 p12 (H12 ) ≡ p(H12 )2 = p(H12 ) p(H21 ), which implies the correspondence with (5.6.25). (ii) Secondly, we want to show that the actual distribution of level spacings is given by the GOE prediction from (5.6.18). Hence we need the eigenvalues of our 2 × 2 matrix, which are  1 1 2 . (H11 − H22 )2 + 4H12 (5.6.42) E ± = (H11 + H22 ) ± 2 2 The matrix is diagonalized by an appropriate orthogonal transformation which we call O(φ) to give the diagonal matrix D± = O H O t . The entries of both matrices are then connected by the following equations H11 = E + cos2 (φ) + E − sin2 (φ)

(5.6.43)

H22 = E + sin (φ) + E − cos (φ) H12 = (E + − E − ) cos(φ) sin(φ).

(5.6.44) (5.6.45)

2

2

The corresponding probability densities in the old and new variables are connected by P(H )d H = p(E + , E − , φ)d E + d E − dφ,

(5.6.46)

or     ∂(H11 , H22 , H12 )   p(E + , E − , φ) = P(H ) det  = P(H ) |E + − E − | , ∂(E + , E − , φ) (5.6.47) which is independent of the transformation angle φ. With (5.6.41) and tr[H 2 ] = 2 + E 2 , we finally obtain E+ − p(E + , E − , φ) = p(E + , E − ) = C |E + − E − | e−A



2 +E 2 E+ −



.

(5.6.48)

Changing variables to introduce the level spacing s = E + − E − and z = (E + + E − )/2, this gives p(s, z) = C s e

−A



s2 2 2 +2z



.

(5.6.49)

The Gaussian integral over z can be performed: 

∞

−∞

dz e

−2 Az 2

 =

π . 2A

(5.6.50)

Since we fixed the energy scale H11 + H22 = 0, we may as well have directly set z = 0 in our simple example of 2 × 2 matrices. This leads to to the wanted result

13 To

make the statement as clear as possible we abuse the notation here a bit.

5.6 Universal Level Statistics

231

after computing the constants A = C = π/2, which are fixed by the conditions from (5.6.19). In conclusion, we have indeed proven that the level spacing distribution is the one of (5.6.18) for GOE. As shown above, the level distance comes into the game because of the change of variables and the corresponding functional determinant, see (5.6.47). The same mechanism carries over to the case of larger matrices, see e.g. the detailed presentation of random-matrix theory in [106].

5.6.8

More Sophisticated Methods

Up to now we have discussed just one property of the predictions for the universal random-matrix ensembles, the level spacing distributions. The latter characterize local correlations in the spectrum, dealing with the nearest-neighbour distances of levels. Much more is known from the theory of random matrices. For instance, there are predictions for two-point spectral correlation functions, which include products of spectral functions, see (5.3.124) for an example. Here we present two of such functions characterizing also the spectral correlations more globally in the spectrum. In the practical analysis of quantum spectra, the computation of these functions gives an additional test of whether and how well the analyzed quantum system follows the predictions of random-matrix theory. We state the definition of the spectral functions and the random-matrix results, while details on the derivations are found in the literature [106].

5.6.8.1 Number Variance The number variance compares the spectral cumulative density to a straight line of given length. It is defined as D$ Σ (L) ≡ 2

˜ E+L/2 ˜ E−L/2

%2 E d E ρ(E) − L

.

(5.6.51)

E˜

The densities ρ(E) must be unfolded and L is then a dimensionless positive real ˜ The number variance is a number. The average is taken over all spectral positions E. correlation function, of order two because of the square. For random-matrix ensembles in the sense of the previous two subsections, we have the following predictions [106,117] ⎧ L ⎪ ⎪ ! ⎪ ⎪2 π2 ⎨ ln(2π L) + γ + 1 − + O( L1 ) 2  8 1 Σ 2 (L) = π1  ⎪ ln(2π L) + γ + 1 + O(!L ) ⎪ π2 ⎪ ⎪ ⎩ 1 ln(4π L) + γ + 1 + π 2 + O( 1 ) 8 L 2π 2

Poisson (integrable) GOE GUE

.

GSE

(5.6.52) The Poissonian case is fairly easy to understand, when thinking about the relative √ fluctuations around the “mean” L in a Poissonian random process scaling with L

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Fig. 5.37 Analysis of the quantum spectra of a many-body system whose spectral properties can be controlled by one parameter. For a regular limit: (a) and red squares in (b); and for a chaotic  s limit: (b) and blue circles in (c). a–b show the cumulative level spacing distributions C S D(s) = 0 ds P(s ) in the main panels, and P(s) in the insets. The solid line represents the GOE prediction, while the dashed line shows the Poisson distribution. In (c) the number variance is shown for the two cases of (a) and (b), respectively. The many-body quantum systems obeys nicely the predictions of RMT in the two limiting cases. Adapted from [124]

[117]. The parameter γ ≈ 0.57722 is the Euler constant. The results for the Gaussian ensembles are valid asymptotically for not too small L. If the analyzed physical system has a typical maximal energy scale (like for the shortest periodic orbit in a billiard, for instance), its number variance will follow the above predictions maximally up to this scale. Beyond this scale, non-universal (system specific) properties will typically induce a different scaling. As an example, we show the number variance and the level spacing distributions for a quantum many-body system in Fig. 5.37. The system is a tight-binding model for interacting bosons in a one-dimensional lattice, described by the so-called Bose– Hubbard Hamiltonian [123], with a parameter controlling the spectral properties [124].

5.6.8.2 Spectral Rigidity A quantity very similar to the number variance is the so-called spectral rigidity function. It allows for an additional optimization in the comparison with a straight line with respect to (5.6.51). The spectral rigidity is defined as 1 Δ3 (L) ≡ L

D

 min{a,b}



˜ E+L/2 ˜ E−L/2

E

dE 0



2 E

d E ρ(E ) − (a + bE)

, E˜

(5.6.53)

5.6 Universal Level Statistics

233

Fig. 5.38 Spectral analysis of the Sinai billiard from Fig. 5.36, c.f. also Fig. 3.29, (symbols) and GOE prediction for the spectral rigidity Δ(L) (solid line) as compared with the Poissonian case (dashed line). Copyright 1984 by the American Physical Society [109]

with appropriate real constants a and b. The random-matrix predictions for it are [106,117]

Δ3 (L) =

⎧L ⎪ ⎪ 15 ⎪ ⎪ ⎪ ⎨ 12 ln(2π L) + γ − π 1 ⎪ ⎪ 2π 2 ⎪ ⎪ ⎪ ⎩ 1 4π 2

5 4

ln(2π L) + γ − ln(4π L) + γ −

− !

5 4 5 4

π2 8

!

Poisson (integrable) +

O( L1 )

+ O( L1 ) ! 2 + π8 + O( L1 )

GOE GUE

.

GSE

(5.6.54) γ ≈ 0.57722 is again the Euler constant. Figure 5.38 presents the spectral rigidity for the same Sinai billiard as shown in Fig. 5.36, nicely supporting the conjecture of Bohigas, Giannoni and Schmit.

5.6.8.3 Level Statistics without Unfolding We have seen above, in particular in Sect. 5.6.4, that the eigenenergies must be renormalized prior to analysing the statistics of their level spacings. Reference [125] introduced a method of analysing, instead of the level distances directly, appropriately chosen ratios of the distances. By taking these ratios the unfolding is done automatically. We will in the following briefly explain and apply this method to a many-body quantum problem. The new measure is based on the ratio of consecutive nearest-neighbour spacings sn = E n+1 − E n and sn−1 = E n − E n−1 , defined by  min(sn , sn−1 ) 1 < 1, (5.6.55) 0 < rn ≡ = min r˜n , max(sn , sn−1 ) r˜n

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Aspects of Quantum Chaos

sn with r˜n = sn−1 . Since each value is normalised by its predecessor unfolding the levels is not necessary here. This has the advantage that statistical fluctuations in the data are usually better controlled. The analytically predicted distributions for these ratios are derived in [125] for Poisson spectra

PPois (r ) =

2 . (1 − r )2

(5.6.56)

Using a 3 × 3 minimal matrix model necessary for the ratio of two spacings, Atas et al. [126] derived the corresponding results for the Gaussian ensembles Pβ (r ) =

(r + r 2 )β 1 , Z β (1 + r + r 2 )1+ 23 β

(5.6.57)

with the Dyson index β = 1, 2, and 4 for GOE, GUE, and GSE, respectively. The corresponding normalisation factors Z β given in [126] are

Zβ =

⎧ 8 ⎪ ⎪ ⎨ 27

π 4 √ 81 3 ⎪ ⎪ ⎩ 4 √π 729 3

GOE, β = 1 GUE, β = 2 . GSE, β = 4

(5.6.58)

As mentioned in [126], the measure Pβ (r ) is particularly suited to investigate many-body quantum systems, for which the local energy densities are often hard to estimate or to compute owing to the growing size of the Hilbert space with the particle number or the system size in general. We also present results of a numerical analysis of energy levels of a many-body Bose–Hubbard model, similar to the one investigated in Fig. 5.37, but now in two spatial dimensions. The systems investigated are lattices of 2 × 2, 2 × 3, and 3 × 3 sites, all sketched in Fig. 5.39, with a filling of the order unity, i.e. one particle per lattice site on average. Figure 5.40 compares the predictions from (5.6.58) with the values obtained from the many-body quantum spectra for parameters in the regular and in the chaotic regime in the left panels. The right panels report corresponding comparisons with the predictions from (5.6.18), here all for the Poisson and the GOE (β = 1) case. We see that the measure based on the ratios defined in (5.6.55) indeed works quite well. Whenever possible it may be compared with the other tests reported in (5.6.18) or Sects. 5.6.8.1 and 5.6.8.2.

5.6.8.4 More Random-Matrix Theory With the nearest-neighbour level spacings and the spectral correlation functions introduced in the previous sections, one can well characterize the properties of a given quantum system. In the two limits of being integrable or fully chaotic in the sense of the two conjectures by Berry–Tabor and Bohigas–Giannoni–Schmit, respectively. For mixed classical systems, however, the quantum spectra will typically not follow the predictions of random-matrix theory for these two limits. For such cases, it is argued that they follow distributions which interpolate in some way or

5.6 Universal Level Statistics

. (a) 2x2

235

. (b) 2x3

(c) 3x3

Fig. 5.39 Sketch of the lattice structures of the Bose–Hubbard models studied in Fig. 5.40: 2 × 2 (a), 2 × 3 (b), and 3 × 3 (c) lattice. The black thin solid lines mark direct couplings between sites that are always present. In order to make the many-body system transit from regular to chaotic, one can allow for different boundary conditions, drawn by the blue dash-dotted lines, and more or less links between the sites over which the particles can hop, see green dotted lines and red dashed lines. Figure taken from [127], Copyright 2016 by the American Physical Society

Fig. 5.40 Spectral analysis of the systems sketched in Fig. 5.39 for the 2 × 2 (a, b), 2 × 3 (c, d), 3 × 3 and (e, f) lattice. In all cases a more or less regular case (full blue squares) and a chaotic case (open red diamonds) is compared with the random-matrix theory predictions from (5.6.56), (5.6.57) (a, c, e) and (5.6.18) (b, d, f). Apart from the systematic deviations for small r and s arising from the small system sizes in the regular cases shown in (a, b, c, d), the correspondence between the data and the predictions is quite good. Figure adapted from [127]

236

5

Aspects of Quantum Chaos

another between the two limits, see e.g. [116,117,128]. While very plausible these arguments lack mathematical rigor up to now, and we do not discuss them here. Random-matrix theory offers predictions not only for the eigenvalues but also for the eigenvectors of the corresponding matrix ensembles. Properties of the eigenvectors are relevant if one wants to make statements about the behaviour of eigenfunctions of chaotic systems. Choosing a typical basis for the problem (the eigenbasis is very untypical in this sense!), the coefficients of the eigenstates of random matrices in this basis obey certain distributions. Since for a chaotic system, all coefficients can be expected to behave equally on average, one can choose one of the coefficients (the ci , for fixed i, and define η ≡ |ci |2 N , where N is the number of eigenvectors /N |ci |2 = 1, system dimension). If the coefficients are correctly normalized, i.e. i=1 random-matrix theory predicts for the distribution of η values [22]: ⎧ 1 −η ⎪ ⎨ 2πη e 2 GOE (5.6.59) P(η) = e−η GUE . ⎪ ⎩ −η ηe GSE Hence also eigenfunctions of a given system can be checked with respect to the expectations of random-matrix theory. The practical access to eigenfunction is, however, much harder than measuring the eigenvalues. The same is true for numerical diagonalizations. This is the reason why, most of the time, the eigenvalues are preferably analyzed. Practically, it may turn out to be better to study the integrated or cumulative level s spacing distribution C S D(s) ≡ 0 ds P(s ) since this quantity is statistically more robust if not too many eigenvalues are available. Integrated distributions are shown in the main panels of Fig. 5.37a, b, for instance. The predictions for Gaussian ensembles, which were presented here, carry over to ensembles of unitary random matrices with the corresponding symmetry properties. The ensembles are then called circular orthogonal, unitary, and symplectic ensembles (COE, CUE, CSE), respectively. Unitary matrices occur naturally in scattering processes (the S matrix is unitary), and for periodically time-dependent systems. In the latter case, the evolution over one period is given by the Floquet operator Uˆ (nT , (n + 1)T ), just like in Sect. 5.5.2, where T is the period and n an integer. Unitary operators have real eigenphases φ j     Uˆ (0, T ) φ j = eiφ j φ j ,

(5.6.60)

whose spectral statistics are the same as given in (5.6.18) for instance. Since these phases are understood modulo 2π , often one finds the statement that unfolding was not necessary for Floquet spectra [22]. This is true only if the quasi-energies ε j = φ − Tj uniformly fill the so-called Floquet zone of eigenvalues, i.e. if ε j ∈ [− πT , πT ] is uniformly distributed. Sometimes this may not be the case, as, for instance, for the many-body system similar as the one shown in Fig. 5.37 and analyzed in [129, 130]. In other words, unfolding, see Sect. 5.6.4, cannot hurt, if it is carefully done. Alternatively, one may use the method of Sect. 5.6.8.3.

5.7 Concluding Remarks

5.7

237

Concluding Remarks

Let us end this chapter with the important statement that random-matrix theory and the comparison of the spectral properties of physical systems with its predictions offer a way to define quantum chaos. This approach goes back to Wigner, who analyzed data of nuclear spectra, and it is obviously independent of any connection with its classical analogue and semiclassics. The conjectures by Berry–Tabor and Bohigas–Giannoni–Schmit connect then such predictions with the classical analogues, which reconciles both “definitions” of quantum chaoticity as mentioned in Sect. 1.3. Semiclassics has the advantage that we gain quite a lot of intuition for the evolution of a quantum system, simply because we are intuitively more used to classical motion. Yet, constructing semiclassical propagators for a specific problem in full detail is at least as hard as directly solving the Schrödinger equation (e.g. by numerical diagonalization). Over the last decade, in particular many-body quantum systems have entered into the focus of research. New techniques are being developed to describe such systems semiclassically, see e.g. [131–135], with methods borrowed from random-matrix theory, see e.g. [124,130,136–141], or from quantum information [142]. Hence, the field of quantum chaos is important as ever! Modern research on quantum chaos is concerned with many aspects which are not covered in this book. Random-matrix theory, for instance, is applied in many different fields of physics like solid-state theory [143,144], but also in econophysics [145]. In particular, people try to understand whether (not only full random matrices but) sparse matrices with many zero entries and special structures can model physical systems more realistically, see e.g. [143,144,146–148]. Higher-dimensional problems, for which phase-space methods like Poincaré sections are not very useful, are also in the focus of current research projects. Classically, then even the presence of KAM tori does not preclude transport in phase space since the tori do not necessarily divide the phase space into unconnected parts (as they do in a three-dimensional phase space). This transport mechanism is however extremely slow, and known as Arnold diffusion [153]. Semiclassics in higher-dimensional systems is therefore quite challenging. One branch of investigation is, for instance, connected with tunneling phenomena out of classically bound regions (like stable resonance islands) [154,155]. A recent review on this topic of dynamical tunneling is found in [55], containing also discussions of higher-dimensional tunneling. Chaosassisted tunneling is strongly influenced by the coupling of two regular states that is mediated by a quantum state lying in a chaotic region in phase space [55,155]. Experimentally, Arnold diffusion and dynamical or chaos-assisted tunneling are notoriously hard to test, but signatures of tunneling across phase-space barriers have been found recently in different experimental setups [60,156–162]. One of the paradigms of quantum chaos are driven Rydberg atoms, which we discussed in Sects. 3.7.6, 5.4.3 and 5.5.2 in various contexts. Most of the experiments were performed with hydrogen or alkali Rydberg atoms showing in good approximation only single-electron motion [18,56–58,95–97,163]. Today well controlled experiments with singly or doubly excited electronic states of helium are in reach, see e.g. [164,165]. Even if helium represents a system of very few bodies (three!)

238

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Aspects of Quantum Chaos

the excitation spectra are hard to compute numerically. This is true even more if an additional drive is added, see e.g. [165–167]. Classical helium bears similarities to the three-body Kepler problem, see Sects. 3.5 and 5.2.1, and it shows interesting classical stability properties [168]. In reality it is a quantum problem, nicely connecting Poincaré’s classical studies and classical chaos with aspects of quantum chaos [166–168]. Open quantum systems would be another topic of great interest, see e.g. [149– 152]. Some material on dissipative quantum chaotic systems is found in Haake’s book [22]. Otherwise this topic seems too vast and still not structured enough in order to include it into an introductory book.

Appendix Quantum Mechanical Spectra Spectra of quantum mechanical Hamiltonians can be categorized as follows [5]: • Point spectrum: The spectrum of the Hamiltonian is discrete. • Absolutely continuous spectrum: In physics it is usually known just as the continuous spectrum, i.e. the spectrum of a hydrogen atom above the ionisation threshold E ≥ 0. • Singular-continuous spectrum: A singular-continuous spectrum is not a point spectrum and not a continuous spectrum (it has a fractal structure, for the Hausdorff dimension D H , c.f. Sect. 4.7.1.1, of the spectrum one finds 0 < D H < 1, see [169]). Structures like this occur as well in power spectra of classical observables in the chaotic regime [170]. Additionally, a singular continuous spectrum leads to dynamics [171] that can be brought in relation to chaos. Examples of Hamiltonian operators with such exotic spectra can be found in [172]. The special structure of singular spectra makes them, in principle, a candidate to define true chaos in a quantum system with such a spectrum. In the case of a discrete quantum spectrum, the correspondence between the classical and the quantum mechanical system is certainly less good than for an absolutely continuous spectrum, simply because classically also a continuum of energies is allowed. For a quantum mechanical point spectrum the correspondence can be better if the energy levels lie closer (higher density of states), which naturally happens in the semiclassical limit. There are indeed intriguing similarities in the rigorous treatment of spectra of classical and quantum systems when discussing e.g. the spectra of classical evolution operators (known as Perron-Frobenius or Koopman operators) [170].

5.7 Concluding Remarks

239

Problems 5.1 WKB approximation — energy levels. Given the one-dimensional potentials for a particle of mass m V1 (x) = ) V2 (x) =

)1

2 mω

2 x 2,

∞, x < 0

x >0

(5.7.1)

mgx, x > 0 (g > 0) , ∞, x < 0

(5.7.2)

(a) Show that for the outer turning point a 

a 0

3 p(x)d(x) = π (n + ) (n ∈ N0 ) 4

(5.7.3)

implicitly defines the WKB-quantized energy levels. (b) Compute the two series of WKB energy levels E(n) and extract the scaling of the density of states (the inverse level distance) with energy E. (c) Imagine a point mass with m = 1 kg (with g being the gravity constant on earth). In which quantum state n would it be if you let it fall from a height of h = 1 m above the floor? How large would h be in the quantum mechanical ground state with n = 0? 5.2 WKB approximation II — tunneling. A particle is trapped in the 1D potential ) V (x) =

−U0 , |x| < a A/|x|, |x| > a,

(5.7.4)

with U0 , A > 0. What is the maximal value of E such that the particle is still trapped? (a) Calculate the probability of the particle escaping if it bounces on a wall once after starting at x = 0 with momentum p. (b) Using the above result, calculate the half-life of the particle escaping from the potential as a function of its energy E. (Hint: Compute the number of bounces for which the particle has the chance of 1/2 for being still inside) 5.3 EBK quantization. For V (r ) = −e2 /r we get in spherical coordinates (r , θ, φ), with r ∈ [0, ∞), θ [0, π ], φ ∈ [0, 2π ] the following Hamiltonian p2 e2 1 H= − = 2m r 2m

$

pφ2 pθ2 2 pr + 2 + 2 2 r r sin θ

% −

e2 . r

(5.7.5)

240

5

Aspects of Quantum Chaos

(a) Show that pr = m r˙ ,

pφ = mr 2 sin2 (θ )φ˙ (= L z )

pθ = mr 2 θ˙ ,

(5.7.6)

(b) Show that the system is integrable by finding three constants of the motion. (c) Using the integrals

1 me2 pr dr = √ −L 2π −2m E

1 pθ dθ = L − |L z | Iθ = 2π

1 pφ dφ = |L z |, Iφ = 2π Ir =

(5.7.7)

show that one can obtain the quantized energies in the form E =−

me4 1 . 2 (Ir + Iθ + Iφ )2

(5.7.8)

Use now the quantization conditions of EBK to arrive at the usual well-known quantum mechanical eigenenergies of the Coulomb problem. (d) Show that the above formulae for the integrals are correct. Hint 1: Use the identities L2 = L 2 = pθ2 +

pφ2 sin2 θ

= pθ2 +

L 2z sin2 θ

(L ≥ 0, −L ≤ L z ≤ L)(5.7.9)

Hint 2: Determine first the turning points for the motion corresponding to the tori actions Ir ,θ,φ , i.e. for Veff (r ) = −

e2 L2 + r 2mr 2

(5.7.10)

from sin2 θ1,2 = (L z /L)2 ≤ 1, and for the cyclic coordinate φ. Then show that one can write the integrals as an integral in the complex plane such that one uses a contour which encircles the real axis between the endpoints, for Ir  √

2m 1 L 2 dr Ir = −Er 2 − e2 r + (5.7.11) 2π i 2m r and for Iθ 1 Iθ = 2π



 L 2z

L L2 − dθ = − 2πi sin2 θ

√ 2 a − x2 dx . √ 1 − x2 x

(5.7.12)

5.7 Concluding Remarks

241

Study the singularities of the integrand and use Cauchy’s theorem of residues to calculate this integral. (e) Compare the semiclassical and quantum mechanical quantization of L and L z . (f) What are the multiplicities of the energy levels, i.e. how many different orbits are possible for some energy level satisfying the EBK quantization conditions? 5.4 Poisson summation formula. (a) Show for x ∈ R ∞ '

∞ '

e2πimx =

m=−∞

δ(x − n).

(5.7.13)

n=−∞

(b) Show that for α ∈ R ∞ ' n=−∞

√  ∞ 2π ' 2π m , f (αn) = F α m=−∞ α

(5.7.14)

for a periodic function f (x) and its Fourier transformation 1 F(k) = √ 2π

∞ f (x)eikx d x.

(5.7.15)

−∞

5.5 Mean Density of States — the Weyl formula. A semiclassical expression for the mean density of states in a system with N degrees of freedom is given by the Thomas–Fermi formula of (5.3.68)   1 N d ρ¯ = p d N r δ(E − H (p, r)), (5.7.16) (2π )n where r = (r1 , . . . , r N ) are the space coordinates, p = ( p1 , . . . , p N ) are the corresponding momenta and H (p, r) is the classical Hamiltonian of the system. The formula is based on the consideration that there is, on average, exactly one quantum state per Planck cell within the phase-space volume (2π )n . (1) Show for N = 3 and H = p2 /2m + V (r):   m 3 d ρ¯ = r 2m(E − V (r))Θ(E − V (r )), 2π 2 3 ) 1, x > 0 where Θ(x) = is the Heaviside step function. 0, x < 1 (2) Calculate the mean density of states for:

(5.7.17)

242

5

Aspects of Quantum Chaos

(a) the three-dimensional infinite box potential with edge lengths a, b and c, (b) the three-dimensional isotropic harmonic oscillator with the frequency ω, (c) the hydrogen atom. Compare each case with the exact density of states in the quantum mechanical system. Hint: 1

  x 1 − x 2d x = x 2 1

2

0



1 π − 1d x = x 16

(5.7.18)

0

5.6 Propagator I. From (5.3.26), we deduce the propagator of a free particle:  m  1 im(x−x0 )2 2 e 2t . 2πit

(5.7.19)

0 for x > 0, ∞ for x < 0,

(5.7.20)

0 for 0 < x < a, ∞ for x < 0 or x > a.

(5.7.21)

K f (x, t|x0 , 0) = Consider now the potentials ) V1 (x) = and

) V2 (x) =

(a) Show that the free propagator satisfies the rule  ∞ dz K f (x, t|z, t )K f (z, t |y, 0) K f (x, t|y, 0) = −∞

(5.7.22)

with 0 < t < t. To which property of the paths in the path integral does this correspond? (b) Show that the propagator K 1 (x, t|x0 , 0) of a particle moving in the potential V1 (x) is K 1 (x, t|x0 ) = K f (x, t|x0 , 0) − K f (x, t| − x0 , 0)

(5.7.23)

by noticing that each non-allowed path (i.e. which crosses the wall) is equivalent to a path of a free particle from −x0 to x. Show that K 1 also satisfies the rule given in (a). (c) Consider a particle moving in the potential V2 (x). Show that its propagator is K 2 (x, t|x0 ) =

∞ ' j=−∞

K f (x, t|2 ja + x0 , 0) − K f (x, t|2 ja − x0 , 0) (5.7.24)

5.7 Concluding Remarks

243

(d) Show that both K 1 (x, t|x0 , 0) and K 2 (x, t|x0 , 0) satisfy K 1 (0, t|x0 , 0) = 0

(5.7.25)

K 2 (0, t|x0 , 0) = K 2 (a, t|x0 , 0) = 0, i.e. the propagator is zero if the endpoint is on the wall. (e) Particle on a ring. Consider a free particle moving on a ring with radius R. Its position can be described using an angle θ . Calculate the propagator of the particle as a sum of all the free paths which go around the ring n ∈ Z times. Using Poisson’s summation formula from Problem 5.4, show that we have K ring (θ, t|θ0 , 0) =

1 ' ik(θ−θ0 ) −ik 22t e e 2m R . 2π R

(5.7.26)

k

5.7 Propagator II — the harmonic oscillator. The propagator of a particle in a harmonic potential is given by the path integral formula 

x

K (x, T |x0 , 0) =

i

D[x]e  S[x(t)] ,

(5.7.27)

x0

where the action of the particle is given by 

T

S[x(t)] = 0



1 2 1 2 2 m x˙ − mω x dt. 2 2

(5.7.28)

(a) Show that the propagator can be written as i

K (x, T |x0 , 0) = e  Scl (x0 ,x,t) K (0, T |0, 0),

(5.7.29)

where Scl (x0 , x, T ) is the action of the classical path between x0 at the t = 0 and x at t = T . Hint: Write x(t) = xcl (t) + y(t) where xcl is the classical path. (b) Show that the classical action defined above satisfies: Scl (x0 , x, T ) =

 mω  2 (x + x02 ) cos ωT − 2x x0 2 sin ωT

(5.7.30)

and argue that this means imω

K (x, T |x0 , 0) = F(T ) e 2 sin ωT



(x 2 +x02 ) cos(ωT )−2x x0



.

(5.7.31)

(c) To calculate F(T ) consider the path integral 



0

K (0, T |0, 0) =

D[y] 0

0

T



1 2 1 m y˙ − mω2 y 2 dt, 2 2

(5.7.32)

244

5

write y(t) as a Fouier series y(t) =

∞ /

Aspects of Quantum Chaos

nπt T . Replace the integral over

an sin

n=1

y(t)

with an integral over the Fourier coefficients. After performing the integral show that  K (0, T |0, 0) = C

ωT sin ωT

1/2 (5.7.33)

with some ω-independent factor C. By requiring that the ω → 0 limit gives the known result for the free particle, show that the propagator of a particle in a harmonic potential is given by  K (x, T |x0 , 0) =

mω 2πi sin(ωT )

1/2 e

(

imω (x 2 +x02 ) cos(ωT )−2x x0 2 sin(ωT )

)

. (5.7.34)

Hint: use the following formula due to Euler: sin(π z) = π z

∞  (

1−

n=1

z2 n2

.

(5.7.35)

5.8 Method of stationary phase. The Bessel function of the first kind Jn (x) (for n ∈ Z) can be expressed by the integral representation 1 Jn (x) = 2π

π

dθ ei[x sin(θ )−nθ ] .

(5.7.36)

−π

Use the method of stationary phase to derive the following asymptotic expansion:  Jn (x) ∼

2 cos(x − nπ/2 − π/4) + O(x −3/2 ). πx

(5.7.37)

5.9 Density operator. Given the density operator of a quantum system ρˆ =

'

cn |ψn ψn | with cn ∈ [0, 1] ,

'

n

cn = 1,

(5.7.38)

n

answer the following question: Have the states |ψn  to be orthogonal in general? and prove the following statements: (a) the eigenvalues λi of ρˆ are real; (b) λi ∈ [0, 1]; (c)

/ i

λi = 1; (d)

/ i

λi2 ≤ 1.

5.7 Concluding Remarks

245

5.10 Wigner and Husimi functions. (a) Prove for a 1D system that both of the following definitions give the same Wigner function W|ψ (x, p):  s 1 s ˆ p − )ei xs/ ds ψˆ ∗ ( p + )ψ( (5.7.39) 2π  2 2  1 s s ds ψ ∗ (x + )ψ(x − )e−i ps/ . 2π  2 2 (b) Prove the trace-product rule for two density matrices ρˆ1 and ρˆ2 :     tr ρˆ1 ρˆ2 = 2π  d x dp Wρˆ1 (x, p)Wρˆ2 (x, p).

(5.7.40)

(5.7.41)

(c) A quantum state |ψ may be projected onto phase space by the following definition: 2 1 Q |ψ (x, p) ≡ Φ p,q |ψ , (5.7.42) π with 2

Φ p,q (x) ≡ N e

i − (x−q) 2 −  px 2σq

,

(5.7.43)

1/2 with appropriate normalization constant N =(π 1/4 σq )−1 . Prove that Q |ψ (x,

p) coincides with the definition of the Husimi function H|ψ (x, p) defined in (5.4.20).

5.11 Quantum kicked rotor. The QKR is described by the following kick-to-kick time evolution operator ˆ with Kˆ = e−ik cos(θˆ ) and Fˆ = e−iT nˆ 2 /2 . Uˆ = Kˆ F,

(5.7.44)

k and T are two independent parameters characterizing the kick strength and period, respectively. For a rotor, periodic boundary conditions are assumed, i.e. in angle representation we have for all times ψ(θ = 0) = ψ(θ = 2π ). (a) Compute the matrix elements of Kˆ in the angle representation for a finite basis of length N and a grid θ j = 2π N j, for j = 1, 2, . . . , N . (b) Compute the matrix elements of Kˆ in the angular momentum representation for a finite basis of length N and a grid n = −N /2, −N /2 + 1, . . . , N /2. (c) For k = 5 and T = 1 and an initial state in momentum representation of the form ψ(n) = δ0,n , compute numerically the temporal evolution induced by Uˆ , i.e. compute ψ(t) = Uˆ t ψ(0), for t = 1, 2, . . . , 200.

(5.7.45)

246

5

Aspects of Quantum Chaos

(d) Plot the angular momentum distribution |ψ(n, t)|2 for t =1, 5, 10, 50, 100, 200 sure that your wave function is properly normalized, i.e. that / kicks, making 2 n |ψ(n, t)| = 1 for all times t. (e) Compute the second moment of the angular momentum distribution, i.e. the 2 expectation value of the energy E(t) = ψ(n, t)| n2 |ψ(n, t) for all t = 1, 2, . . . , 200, and plot it versus t. (f) For a better interpretation of your results, plot also the/ time-averaged quantities corresponding to (d) and (e), i.e. the quantities X (t) = tt =1 Y (t )/t, for Y (t) = |ψ(n, t)|2 or Y (t) = E(t), again as a function of t. Hints for the time evolution in (c): ˆ (1) First compute Fψ(0) in (angular) momentum representation, then use a Fast Fourier Transform (FFT), see e.g. [173], into angle representation ˆ before computing Kˆ Fψ(0). Then use the inverse FFT to get back to ˆ momentum representation to compute Fˆ Kˆ Fψ(0), and continue now this procedure iteratively. Choose a grid which is adequate for applying FFT, i.e. N = 2x for some integer x. (2) Make sure that the matrix dimension N is large enough to guarantee a numerically exact evolution. You can directly check this by looking at |ψ(n, t)|2 versus n for various values of N = 2x and x=7, 8, 9, 10, 11, . . .. 5.12 Poisson distribution. Given n real random numbers x1 , ..., xn , uniformly distributed in the interval xi ∈ [0, n]. Compute numerically for n = 105 the probability density P(s) such that the distance on the x-axis of a given xi to its nearest (larger) neighbour x j equals s. Hints: – Compute 105 real random numbers in [0, 105 ] using a standard random number generator (cf. e.g. [173]) – Sort the x j in such a way that xi < x j ∀i < j with i, j ∈ {1, 2, . . . , 105 }. – Count how many distances s = x j+1 − x j are in the intervals [0, 0.1], [0.1, 0.2], . . . , [9.9, 10], and plot the normalized numbers (relative frequencies) in a histogram. – What is the functional form of the obtained normalized distribution P(s), obeying the conditions from (5.6.19)? 5.13 Rectangular billard. (a) Show that for

) V (x, y) =

0, ∞,

0