The Topology of Chaos E-Book

Robert Gilmore

• Herausgeber: Wiley-VCH
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 1083

Veröffentlichungsjahr: 2012

Beschreibung

A highly valued resource for those who wish to move from the introductory and preliminary understandings and the measurement of chaotic behavior to a more sophisticated and precise understanding of chaotic systems. The authors provide a deep understanding of the structure of strange attractors, how they are classified, and how the information required to identify and classify a strange attractor can be extracted from experimental data. In its first edition, the Topology of Chaos has been a valuable resource for physicist and mathematicians interested in the topological analysis of dynamical systems. Since its publication in 2002, important theoretical and experimental advances have put the topological analysis program on a firmer basis. This second edition includes relevant results and connects the material to other recent developments. Following significant improvements will be included: * A gentler introduction to the topological analysis of chaotic systems for the non expert which introduces the problems and questions that one commonly encounters when observing a chaotic dynamics and which are well addressed by a topological approach: existence of unstable periodic orbits, bifurcation sequences, multistability etc. * A new chapter is devoted to bounding tori which are essential for achieving generality as well as for understanding the influence of boundary conditions. * The new edition also reflects the progress which had been made towards extending topological analysis to higher-dimensional systems by proposing a new formalism where evolving triangulations replace braids. * There has also been much progress in the understanding of what is a good representation of a chaotic system, and therefore a new chapter is devoted to embeddings. * The chapter on topological analysis program will be expanded to cover traditional measures of chaos. This will help to connect those readers who are familiar with those measures and tests to the more sophisticated methodologies discussed in detail in this book. * The addition of the Appendix with both frequently asked and open questions with answers gathers the most essential points readers should keep in mind and guides to corresponding sections in the book. This will be of great help to those who want to selectively dive into the book and its treatments rather than reading it cover to cover. What makes this book special is its attempt to classify real physical systems (e.g. lasers) using topological techniques applied to real date (e.g. time series). Hence it has become the experimenter?s guidebook to reliable and sophisticated studies of experimental data for comparison with candidate relevant theoretical models, inevitable to physicists, mathematicians, and engineers studying low-dimensional chaotic systems.

Leseprobe

Contents

Cover

Half Title page

Related Titles

Title page

Copyright page

Preface to Second Edition

Preface to the First Edition

Chapter 1: Introduction

1.1 Brief Review of Useful Concepts

1.2 Laser with Modulated Losses

1.3 Objectives of a New Analysis Procedure

1.4 Preview of Results

1.5 Organization of This Work

Chapter 2: Discrete Dynamical Systems: Maps

2.1 Introduction

2.2 Logistic Map

2.3 Bifurcation Diagrams

2.4 Elementary Bifurcations in the Logistic Map

2.5 Map Conjugacy

2.6 Fully Developed Chaos in the Logistic Map

2.7 One-Dimensional Symbolic Dynamics

2.8 Shift Dynamical Systems, Markov Partitions, and Entropy

2.9 Fingerprints of Periodic Orbits and Orbit Forcing

2.10 Two-Dimensional Dynamics: Smale’s Horseshoe

2.11 Hénon Map

2.12 Circle Maps

2.13 Annulus Maps

2.14 Summary

Chapter 3: Continuous Dynamical Systems: Flows

3.1 Definition of Dynamical Systems

3.2 Existence and Uniqueness Theorem

3.3 Examples of Dynamical Systems

3.4 Change of Variables

3.5 Fixed Points

3.6 Periodic Orbits

3.7 Flows Near Nonsingular Points

3.8 Volume Expansion and Contraction

3.9 Stretching and Squeezing

3.10 The Fundamental Idea

3.11 Summary

Chapter 4: Topological Invariants

4.1 Stretching and Squeezing Mechanisms

4.2 Linking Numbers

4.3 Relative Rotation Rates

4.4 Relation between Linking Numbers and Relative Rotation Rates

4.5 Additional Uses of Topological Invariants

4.6 Summary

Chapter 5: Branched Manifolds

5.1 Closed Loops

5.2 What Does This Have to Do with Dynamical Systems?

5.3 General Properties of Branched Manifolds

5.4 Birman–Williams Theorem

5.5 Relaxation of Restrictions

5.6 Examples of Branched Manifolds

5.7 Uniqueness and Nonuniqueness

5.8 Standard Form

5.9 Topological Invariants

5.10 Additional Properties

5.11 Subtemplates

5.12 Summary

Chapter 6: Topological Analysis Program

6.1 Brief Summary of the Topological Analysis Program

6.2 Overview of the Topological Analysis Program

6.3 Data

6.4 Embeddings

6.5 Periodic Orbits

6.6 Computation of Topological Invariants

6.7 Identify Template

6.8 Validate Template

6.9 Model Dynamics

6.10 Validate Model

6.11 Summary

Chapter 7: Folding Mechanisms: A2

7.1 Belousov–Zhabotinskii Chemical Reaction

7.2 Laser with Saturable Absorber

7.3 Stringed Instrument

7.4 Lasers with Low-Intensity Signals

7.5 The Lasers in Lille

7.6 The Laser in Zaragoza

7.7 Neuron with Subthreshold Oscillations

7.8 Summary

Chapter 8: Tearing Mechanisms: A3

8.1 Lorenz Equations

8.2 Optically Pumped Molecular Laser

8.3 Fluid Experiments

8.4 Why A3?

8.5 Summary

Chapter 9: Unfoldings

9.1 Catastrophe Theory as a Model

9.2 Unfolding of Branched Manifolds: Branched Manifolds as Germs

9.3 Unfolding within Branched Manifolds: Unfolding of the Horseshoe

9.4 Missing Orbits

9.5 Routes to Chaos

9.6 Orbit Forcing and Topological Entropy: Mathematical Aspects

9.7 Topological Measures of Chaos in Experiments

9.8 Summary

Chapter 10: Symmetry

10.1 Information Loss and Gain

10.2 Cover and Image Relations

10.3 Rotation Symmetry 1: Images

10.4 Rotation Symmetry 2: Covers

10.5 Peeling: a New Global Bifurcation

10.6 Inversion Symmetry: Driven Oscillators

10.7 Duffing Oscillator

10.8 Van der Pol Oscillator

10.9 Summary

Chapter 11: Bounding Tori

11.1 Stretching & Folding vs. Tearing & Squeezing

11.2 Inflation

11.3 Boundary of Inflation

11.4 Index

11.5 Projection

11.6 Nature of Singularities

11.7 Trinions

11.8 Poincaré Surface of Section

11.9 Construction of Canonical Forms

11.10 Perestroikas

11.11 Summary

Chapter 12: Representation Theory for Strange Attractors

12.1 Embeddings, Representations, Equivalence

12.2 Simplest Class of Strange Attractors

12.3 Representation Labels

12.4 Equivalence of Representations with Increasing Dimension

12.5 Genus-g Attractors

12.6 Representation Labels

12.7 Equivalence in Increasing Dimension

12.8 Summary

Chapter 13: Flows in Higher Dimensions

13.1 Review of Classification Theory in R3

13.2 General Setup

13.3 Flows in R4

13.4 Cusps in Weakly Coupled, Strongly Dissipative Chaotic Systems

13.5 Cusp Bifurcation Diagrams

13.6 Nonlocal Singularities

13.7 Global Boundary Conditions

13.8 From Braids to Triangulations: toward a Kinematics in Higher Dimensions

13.9 Summary

Chapter 14: Program for Dynamical Systems Theory

14.1 Reduction of Dimension

14.2 Equivalence

14.3 Structure Theory

14.4 Germs

14.5 Unfolding

14.6 Paths

14.7 Rank

14.8 Complex Extensions

14.9 Coxeter–Dynkin Diagrams

14.10 Real Forms

14.11 Local vs. Global Classification

14.12 Cover–Image Relations

14.13 Symmetry Breaking and Restoration

14.14 Summary

Appendix A: Determining Templates from Topological Invariants

A.1 The Fundamental Problem

A.2 From Template Matrices to Topological Invariants

A.3 Identifying Templates from Invariants

A.4 Constructing Generating Partitions

A.5 Summary

Appendix B: Embeddings

B.1 Diffeomorphisms

B.2 Mappings of Data

B.3 Tests for Embeddings

B.4 Tests of Embedding Tests

B.5 Geometric Tests for Embeddings

B.6 Dynamical Tests for Embeddings

B.7 Topological Test for Embeddings

B.8 Postmortem on Embedding Tests

B.9 Stationarity

B.10 Beyond Embeddings

B.11 Summary

Appendix C: Frequently Asked Questions

C.1 Is Template Analysis Valid for Non-Hyperbolic Systems?

C.2 Can Template Analysis Be Applied to Weakly Dissipative Systems?

C.3 What About Higher-Dimensional Systems?

References

Index

Robert Gilmore and Marc Lefranc

The Topology of Chaos

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The Authors

Prof. Robert GilmoreDrexel UniversityDept. of PhysicsPhiladelphia, [e-mail]

Prof. Marc LefrancLaboratoire de Physique des Lasers, Atomes, MoléculesUniversité des Sciences et Technologies de Lille Villeneuve d’[e-mail]

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Preface to Second Edition

Since the appearance of the First Edition a decade ago our understanding of the relation between Topology and Chaos has grown quite a bit. This growth is reflected in the increased size of the Second Edition. For the most part the additional material is present in Chapters 11 and 12, and Appendix B of the current version. The two new chapters have been inserted between the first ten and the last two chapters of the First Edition.

We have made small changes in the first ten chapters. The principal change can be seen in Chapter One. This Chapter has been largely rewritten to make the entire work more accessible to someone first coming to the field. Chapter 2 contains two new short sections about homoclinic tangles and annulus maps. A brief discussion of embeddings has been largely expanded and relocated from Chapter 6 to Appendix B of the present work. Some recent beautiful experimental work done in Zaragoza, Spain, has explored the perestroikas that branched manifolds can undergo. This work has been included in Chapter 7. In Chapter 9, a new section summarizes the essential mathematical aspects of orbit forcing and topological entropy, including train track algorithms, and points to the relevant litterature. How knots can be used to compute entropy in real systems is also illustrated in a fluid experiment and in an optical system.

Unfortunately there has been little progress in our understanding of flows in higher dimensions (Chapter 11, First Edition) and too little development in the program for dynamical systems (Chapter 12, First Edition). These appear essentially unchanged as Chapters 13 and 14 in the present edition. Still, a new section in Chapter 13 presents an interesting proposal to generalize braids to dynamical triangulations of periodic points. This approach appears to be equivalent to the conventional one in three dimensions, and adapts naturally to phase spaces of any dimension.

Bounding Tori are introduced in Chapter 11. These two-dimensional surfaces enclose three-dimensional strange attractors. They have an elegant classification that goes back more than two centuries to the earlier great topologists (Euler). These structures serve to place yet another identifying tag on low-dimensional strange attractors.

There is a ‘Representation Theory’ for strange attractors that is similar in spirit, if not in detail, to the ‘representation theory’ for groups and algebras developed over a century ago. The representation theory for three-dimensional dynamical systems is now complete and presented in Chapter 12. For higher-dimensional dynamical systems the path has been blazed but not yet traversed. The representation theory provides a satisfying answer to the troubling question: “When you analyze an embedding of data generated by a chaotic dynamical system, what do you learn about the dynamical system and what do you learn about the embedding?”

The new Appendix B is devoted to the black magic of Embeddings. There are procedures for attempting to create embeddings and there are several different types of tests to assay whether an embedding has in fact been achieved. Some procedures are more reliable, others less so. These and other questions are explored in this Appendix.

We have taken this opportunity to correct mistakes that have crept into the First Edition. Hopefully, there are none in this version, but corrections and suggestions are welcomed and will be available online at the book’s website: http://www.thetopologyofchaos.net/. We would like to thank friends and colleagues for pointing out mistakes (with a special mention to Michel Nizette and Mihir Khadilkar), pointing to places where our writing could have/should have been clearer, and most important, for their support and encouragement during the preparation of the Second Edition.

This work was supported in part by the National Science Foundation under grant number NSF-0754081.

June 2011

Rouen, FranceLille, France

Robert Gilmore ([e-mail])Marc Lefranc ([e-mail])

Preface to the First Edition

Before the 1970s opportunities sometimes arose for physicists to study nonlinear systems. This was especially true in fields like fluid dynamics and plasma physics, where the fundamental equations are nonlinear and these nonlinearities masked (and still mask) the full spectrum of spectacularly rich behavior. When possible, we avoided being dragged into the study of abstract nonlinear systems. For we believed, to paraphrase a beautiful generalization of Tolstoy, that

All linear systems are the same.Each nonlinear system is nonlinear in its own way.

At that time we believed that one could spend a whole lifetime studying the non-linearities of the van der Pol oscillator [1, 2] and wind up knowing next to nothing about the behavior of the Duffing oscillator.

Nevertheless, other intrepid researchers had been making an assault on the complexities of nonlinear systems. Smale [3] described a mechanism responsible for generating a great deal of the chaotic behavior that has been studied up to the present time. Lorenz, studying a drastic truncation of the Navier–Stokes equation, discovered and described “sensitive dependence on initial conditions” (1963). The rigid order in which periodic orbits are created in the bifurcation set of the logistic map, and in fact any unimodal map of an interval to itself, was described by May [4] and by Metropolis, Stein, and Stein [5].

Still, there was a reluctance on the part of most scientists to indulge in the study of nonlinear systems. This all changed with Feigenbaum’s discoveries (1978). He showed that scaling invariance in period-doubling cascades leads to quantitative (later, qualitative) predictions. These are the scaling ratios:

that are eigenvalues of a renormalization transformation. The transformation in the attitude of scientists is summarized by Gleick’s [6] statement:

“It was a very happy and shocking discovery that there were structures in nonlinear systems that are always the same if you looked at them the right way.”

This discovery launched an avalanche of work on nonlinear dynamical systems. Old experiments, buried and forgotten because of instabilities or unrepeatability due to incompetent graduate students (in their advisors’ opinions) were resurrected and pushed as groundbreaking experiments exhibiting “first observations” of chaotic behavior (by these same advisors). And many new experiments were carried out, at first to test Feigenbaum’s scaling predictions, then to test other quantitative predictions, then just to see what would happen.

Some of the earliest experiments were done on fluids, since the fundamental equations were known and are nonlinear. However, these experiments often suffered from the long time scales (days, weeks, or months) required to record a decent data set. Oscillating chemical reactions (e.g., the Belousov–Zhabotinskii reaction) yielded a wide spectrum of periodic and chaotic behavior that was relatively easy to control and to tune. These data sets could be generated in hours or days. Nonlinear electric circuits were also extensively studied, although there was (and still is) a prejudice to regard them with a jaundiced eye as little more than analog computers. Such data sets could be generated very quickly (seconds to minutes) – almost as fast as numerical simulations. Finally, laser laboratories contributed in a substantial way to very quickly (milliseconds to minutes) building up extensive and widely varying banks of chaotic data.

It was at this time (1988), about 10 years into the “nonlinear science” revolution, that one of the authors (R.G.) was approached by his colleague (J.R. Tredicce, then at Drexel, now at the Institut Non Linéaire de Nice) with the proposition: “Bob, can you help me explain my data?” (Chapter 1). So we swept the accumulated clutter off my desk and deposited his data. We looked, pushed, probed, discussed, studied, etc. for quite a while. Finally, I replied: “No.” Tredicce left with his data. But he is very smart (he is an experimentalist!) and returned the following day with the same pile of stuff. The conversation was short and effective: “Bob,” (still my name), “I’ll bet that you can’t explain my data.” (Bob sees red!) We sat down and discussed further. At the time two tools were available for studying chaotic data. These involved estimating Lyapunov exponents (dynamical stability) and estimating fractal dimensions (geometry). Both required lots of very clean data and long calculations. They provided real number(s) with no convincing error bars, no underlying statistical theory, and no independent way to verify these guesses. And at the end of the day neither provided any information on “how to model the dynamics.”

Even worse: Before doing an analysis I would like to know what I am looking for, or at least know what the spectrum of possible results looks like. For example, when we analyze chemical elements or radionuclides, there is a periodic table of the chemical elements and another for the atomic nuclei that accommodate any such analyses. At that time, no classification theory existed for strange attractors.

In response to Tredicce’s dare, I promised to (try to) analyze his data. But I pointed out that a serious analysis couldn’t be done until we first had some handle on the classification of strange attractors. This could take a long time. Tredicce promised to be patient. And he was.

Our first step was to consider the wisdom of Poincaré, who had suggested about a century earlier that one could learn a great deal about the behavior of nonlinear systems by studying their unstable periodic orbits, which

“… yield us the solutions so precious, that is to say, they are the only breach through which we can penetrate into a place which up to now has been reputed to be inaccessible.”

This observation was compatible with what we learned from experimental data: the most important features that governed the behavior of a system, and especially that governed the perestroikas of such systems (i.e., changes as control parameters are changed) are the features that you can’t see – the unstable periodic orbits.

Accordingly, my colleagues and I studied the invariants of periodic orbits, their (Gauss) linking numbers. We also introduced a refined topological invariant based on periodic orbits – the relative rotation rates (Chapter 4). Finally, we used these invariants to identify topological structures (branched manifolds or templates, Chapter 5), which we used to classify strange attractors “in the large.” The result was that “low-dimensional” strange attractors (i.e., those that could be embedded in three-dimensional spaces) could be classified. This classification depends on the periodic orbits “in” the strange attractor, in particular, on their organization as elicited by their invariants. The classification is topological. That is, it is given by a set of integers (also by very informative pictures). Not only that, these integers can be extracted from experimental data. The data sets do not have to be particularly long or particularly clean – especially by fractal dimension calculation standards. Further, there are built-in internal self-consistency checks. That is, the topological analysis algorithm (Chapter 6) comes with reject/fail to reject test criteria. This is the first – and remains the only – chaotic data analysis procedure with rejection criteria.

Ultimately we discovered, through analysis of experimental data, that there is a secondary, more refined classification for strange attractors. This depends on a “basis set of orbits” that describes the spectrum of all the unstable periodic orbits “in” a strange attractor (Chapter 9).

The ultimate result is a doubly discrete classification of strange attractors. Both parts of this doubly discrete classification depend on unstable periodic orbits. The classification depends on identifying:

A branched manifold – which describes the stretching and squeezing mechanisms that operate repetitively on a flow in phase space to build up a (hyperbolic) strange attractor and to organize all the unstable periodic orbits in the strange attractor in a unique way. The branched manifold is identified by the spectrum of the invariants of the periodic orbits that it supports.A basis set of orbits – which describes the spectrum of unstable periodic orbits in a (nonhyperbolic) strange attractor.

The perestroikas of branched manifolds and of basis sets of orbits in this doubly discrete classification obey well-defined topological constraints. These constraints provide both a rigidity and a flexibility for the evolution of strange attractors as control parameters are varied.

Along the way we discovered that dynamical systems with symmetry could be related to dynamical systems without symmetry in very specific ways (Chapter 10). As usual, these relations involve both a rigidity and a flexibility that are as surprising as they are delightful.

Many of these insights are described in the paper [7], which forms the basis for part of this book. We thank the editors of this journal for their policy of encouraging the transformation of research articles into a longer book format.

The encounter (falling in love?) of the other author (M.L.) with topological analysis dates back to 1991, when he was a Ph.D. student at the University of Lille, struggling to extract information from the very same type of chaotic laser that Tredicce was using. At that time, Marc was computing estimates of fractal dimensions for his laser. But the estimates depended very much on the coordinate system used and gave no insight into the mechanisms responsible for chaotic behavior, even less into the succession of the different behaviors observed. This was very frustrating. There had been this very intringuing paper in Physical Review Letters about a “characterization of strange attractors by integers,” with appealing ideas and nice pictures. But as with many short papers, it was difficult to understand how you should proceed when faced with a real experimental system. Topological analysis struck back when Pierre Glorieux, then Marc’s advisor, came back to Lille from a stay in Philadephia and handed him a preprint from the Drexel team, saying, “You should have a look at this stuff.” The preprint was about topological analysis of the Belousov–Zhabotinskii reaction, a real-life system. It was the Rosetta Stone that helped put pieces together. Soon after, pictures of braids constructed from laser signals were piling up on his desk. They were absolutely identical to those extracted from the Belousov–Zhabotinskii data and described in the preprint. There was universality in chaos if you looked at it with the right tools. Eventually, the system that had motivated topological analysis in Philadephia, the CO2 laser with modulated losses, was characterized in Lille and shown to be described by a horseshoe template. Indeed, Tredicce’s laser could not be characterized by topological analysis because of long periods of zero output intensity that prevented invariants from being reliably estimated. The high signal-to-noise ratio of the laser in Lille allowed us to use a logarithmic amplifier and to resolve the structure of trajectories in the zero intensity region.

But a classification is only useful if there exist different classes. Thus, one of the early goals was to find experimental evidence of a topological organization that would differ from the standard Smale horseshoe. At that time, some regimes of the modulated CO2 laser could not be analyzed for lack of a suitable symbolic encoding. The corresponding Poincaré sections had peculiar structures that, depending on the observer’s mood, suggested a doubly iterated horsehoe or an underlying three-branch manifold. Since the complete analysis could not be carried out, much time was spent on trying to find at least one orbit that could not fit the horseshoe template. The result was extremely disappointing: For every orbit detected, there was at least one horseshoe orbit with identical invariants. One of the most important lessons of Judo is that if you experience resistance when pushing, you should pull (and vice versa). Similarly, this failed attempt to find a nonhorseshoe template turned into techniques to determine underlying templates when no symbolic coding is available and to construct such codings using the information extracted from topological invariants.

But the search for different templates was not over. Two of Marc’s colleagues, Dominique Derozier and Serge Bielawski, proposed that he study a fiber-optic laser they had in their laboratory (that was the perfect system for studying knots). This system exhibits chaotic tongues when the modulation frequency is near a subharmonic of its relaxation frequency: It was tempting to check whether the topological structures in each tongue differed. That was indeed the case: The corresponding templates were basically horseshoe templates but with a global torsion increasing systematically from one tongue to the other. A Nd:YAG laser was also investigated. It showed similar behavior, until the day when Guillaume Boulant, the Ph.D. student working on the laser, came to Marc’s office and said, “I have a weird data set.” Chaotic attractors were absolutely normal, return maps resembled the logistic map very much, but the invariants were simply not what we were used to. This was the first evidence of a reverse horseshoe attractor. How topological organizations are modified as a control parameter is varied was the subject of many discussions in Lille in the following months; a rather accurate picture finally emerged, and papers began to be written. In the last stages, Marc did a bibliographic search just to clear his mind, and … a recent 22-page Physical Review paper, by McCallum and Gilmore, turned up. Even though it was devoted to the Duffing attractor, it described with great detail what was happening in our lasers as control parameters were modified. Every occurrence of “we conjecture that” in the papers was hastily replaced by “our experiments confirm the theoretical prediction …,” and papers were sent to Physical Review. They were accepted 15 days later, with a very positive review. Soon after, the referee contacted us and proposed a joint effort on extensions of topological analysis. The referee was Bob, and this was the start of what we hope will be a long-lasting collaboration.

It would indeed be very nice if these techniques could be extended to the analysis of strange attractors in higher (than three) dimensions. Such an extension, if it is possible, cannot rely on the most powerful tools available in three dimensions. These are the topological invariants used to tease out information on how periodic orbits are organized in a strange attractor. We cannot use these tools (linking numbers, relative rotation rates) because knots “fall apart” in higher dimensions. We explore (Chapter 11) an inviting possibility for studying an important class of strange attractors in four dimensions. If a classification procedure based on these methods is successful, the door is opened to classifying strange attractors in Rn, n > 3. A number of ideas that may be useful in this effort have already proved useful in two closely related fields (Chapter 12): lie group theory and singularity theory.

Some of the highly technical details involved in extracting templates from data have been archived in the appendix. Other technical matters are archived at our web sites.1)

Much of the early work in this field was done in response to the challenge by J.R. Tredicce and carried out with my colleagues and close friends: H.G. Solari, G.B. Mindlin, N.B. Tufillaro, F. Papoff, and R. Lopez-Ruiz. Work on symmetries was done with C. Letellier. Part of the work carried out in this program has been supported by the National Science Foundation under grants NSF 8843235 and NSF 9987468. Similarly, Marc would like to thank colleagues and students with whom he enjoyed working and exchanging ideas about topological analysis: Pierre Glorieux, Ennio Arimondo, Francesco Papoff, Serge Bielawski, Dominique Derozier, Guillaume Boulant, and Jérôme Plumecoq. Bob’s stays in Lille were partially funded by the University of Lille, the Centre National de la Recherche Scientifique, Drexel University under sabbatical leave, and by the NSF.

Last and most important, we thank our wives Claire and Catherine for their warm encouragement while physics danced in our heads, and our children, Marc and Keith, Clara and Martin, who competed with our research, demanded our attention, and, in doing so, kept us human.

Lille, France, January 2002

Robert Gilmore and Marc Lefranc

1) http://einstein.drexel.edu/directory/faculty/Gilmore.html and http://www.phlam.univ-lille1.fr/perso/lefranc.html. Last access: 1 January 2002.