CATASTROPHE THEORY AND ITS APPLICATIONS

CATASTROPHE THEORY

AND ITS APPLICATIONS

Tim Poston

INSTITUTE OF SYSTEMS SCIENCE (R & D)

NATIONAL UNIVERSITY OF SINGAPORE

Ian Stewart

MATHEMATICS INSTITUTE

UNIVERSITY OF WARWICK

COVENTRY, ENGLAND

Dover Publications, Inc.

Mineola, New York

Copyright

Copyright © 1978 by T Poston and I N Stewart.

All rights reserved.

Bibliographical Note

This work, first published by Dover Publications, Inc., in 1996, is an unabridged, unaltered republication of the work first published by Pitman Publishing Limited, London, 1978, in the series Surveys and Reference Works in Mathematics.

Library of Congress Cataloging-in-Publication Data

Poston, T.

Catastrophe theory and its applications / Tim Poston, Ian Stewart,

        p.          cm.

Originally published: London ; San Francisco : Pitman, 1978. (Surveys and reference works in mathematics ; 2).

Includes bibliographical references and index.

eISBN 13: 978-0-486-14378-1

1. Catastrophes (Mathematics) I. Stewart, Ian, 1945-II. Title.

QA614.58.P66 1996

514’.74—dc20

96-21795

CIP

Manufactured in the United States by Courier Corporation

69271X03

www.doverpublications.com

to Christopher Zeeman

at whose feet we sit

on whose shoulders we stand

Preface

Since the first rumours in the mid-1960’s of René Thom’s Stabilité Structurelle et Morphogénèse, which finally appeared in 1972, there has been a rapid growth of interest in the subject now known as catastrophe theory. Thom suggested using the topological theory of dynamical systems, originated by Poincaré, to model discontinuous changes in natural phenomena, with special emphasis on biology; and he pointed out the importance in this context of structural stability, or insensitivity to small perturbations. He further remarked that in one context this requirement implied that the system could be described, locally, by one of seven standard forms, the elementary catastrophes.

As well as great interest, Thom’s ideas have generated great confusion and, more recently, controversy. Early claims for the theory’s universality (partly misstatements based on confusion between elementary and non-elementary catastrophes, and partly overstatements attributable to the ‘youthful enthusiasm’ for a new subject) have been repeated too often without proper qualification. In some circles, too, the belief has arisen that catastrophe theory is ‘purely qualitative’, with a split between those who think this is a good thing and those who do not. The wide range of precursors to the theory in many fields (itself a product of the typicality that we shall account for in Chapter 7) has led some to infer that the theory contains no new material at all. Speculative extensions of the theory to realms where its applicability is not guaranteed by the appropriate mathematical formalism have been misinterpreted as definitive applications; and disputes arising in these areas have contaminated other fields where the problems are quite different. These misunderstandings may largely be traced to the unfamiliar mathematical language in which the theory is couched, and the tendency of mathematicians to emphasize aspects of the theory that are not always in sympathy with the practical requirements of the working scientist. Thus Turing, charged that computers only operated deterministically, replied that was how he was asked to design them. The same goes for qualitatively with topologists and catastrophe descriptions, except that they asked themselves. You want numbers, we have numbers; but, simply, most topologists don't want numbers, they want qualities – though these sometimes acquire a fearsomely algebraic, even numerical, expression. The problems have been exacerbated by the lack of suitable source material between the extremes of hard-core topology and soft-centre popularization.

Our first aim in this book is to explain the mathematical ideas involved, in terms which will be accessible to the practising scientist who is familiar with calculus in several variables and a little linear algebra. This places catastrophe theory in its rightful position as an extension of, or a development within, the calculus (rather than a radical new departure, or a replacement for current methods, as is sometimes thought to be the case). It also makes clear the limitations of the theory. Unless one understands in some detail the precise mathematical hypotheses involved, and the way in which they lead to the conclusions, one will fail to have an adequate feel for what the theory can or cannot be made to do. It has been said more than once that it is possible to apply Thom’s theorem without understanding the mathematics behind it: we disagree. In fact we disagree with the implication that it is Thom’s theorem that should be applied: analysis of the most solid and successful applications shows that the methods and concepts that lie behind the theorem are often of greater importance than the result itself.

Our second aim is to explode the myth that catastrophe theory is purely qualitative. We achieve this by the direct method of surveying some of its quantitative applications. We concentrate on the physical sciences, where the existing mathematical theory leads naturally to problems that fall within the domain of catastrophe theoretic methods, and where these methods may be used as mathematical tools to provide quantitative information that may be tested by experiment. We place considerable emphasis on the computational aspects of the subject and the explicit calculations that may be performed using it, illustrating these both by mathematical examples and in applications. Catastrophe theory methods have a clearly defined, though not universal role to play in the physical sciences, and it is important that controversy over the less well established applications should not be permitted to obscure this fact.

The mathematical chapters which form the first half of the book do no more, in principle, than to expound the theory as it now stands; but the novelty of our approach is to use the mathematics of the working scientist to motivate the style of thinking involved and the results obtained. We do not give the rigorous proofs of the more powerful theorems (where the deepest and newest mathematics lies) but we do give in a new way the geometric heart of these arguments, which explains (better than a strictly formal treatment) why the results are true. Independently of any question of applications, the mathematical theorems of catastrophe theory form an essential contribution to an important and natural problem: the study of singularities of families of smooth functions. Our treatment of them here may also prove useful as introductory motivation for those who wish to study the mathematics in its full depth.

One result that we do prove explicitly (using only elementary calculus) is the important Splitting Lemma whereby the number of variables in a problem may be (often drastically) reduced. To some scientists this result has appeared as the most surprising of the theory: its essentially classical nature deserves to be better known, as does the result itself.

The second half of the book is on applications. We not only discuss the more established and familiar of these, but include some very recent work that is less well known, and a certain amount of material that has not seen print before. In the latter category are the stability analysis of an idealized oil-rig in Chapter 10; the sections on mirages and sonic booms in Chapter 11; the quantitative exploitation of the non-local bifurcation set of the elliptic umbilic catastrophe in Chapter 12 in connection with fluid flow; parts of Chapter 13 on elasticity, especially the treatment of a double-cusp catastrophe in a buckling plate; much of Chapter 14 on thermodynamics; the bee theory and the new constraint catastrophes of Chapter 16, along with other material on the formation of biochemical and ecological frontiers. Chapter 15 is also new, and is due entirely to Bob Gilmore and Lorenzo Narducci. The reader who has had previous contact with the catastrophe literature will find much that he has not seen before, as well as some of the old favourites. Indeed, the current explosive growth of catastrophe theory is well illustrated by the fact that if this book had gone to press in Summer 1976, roughly half of our material on applications could not have been included. The time pressures on the writing (which we hope are not too visible via consequent errors) were only partly exerted by the publisher: the subject itself was breathing down our necks.

A detailed list of the applications discussed will be found on the contents pages and will not be given here; but two points should be made. The first is that we have attempted, as far as possible, to consult research workers in the fields of application, to check that our exposition is in line with current thinking in those subjects. This of course does not imply that the experts agree with our theorizing. But it helps to avoid the effect, too apparent in some of the literature, whereby the physics encountered in applications is that of the nineteenth century. When arguing that physics relies too heavily on the mathematics of the nineteenth century – as in some areas we feel it does-it seems best to avoid committing the same error in reverse. Not being physicists, naval architects, biologists, engineers ourselves, it is too much to hope that we have entirely succeeded in this attempt, but we have at least tried.

The second point is that many of the applications, notably in Chapters 11, 12 and 13, are to systems governed by partial differential equations. This is so despite the fact that the obvious application is only to a special class of ordinary differential equations, a fact that is often advanced as an objection. It happens this way because mathematics does not always respect the tidy categories into which it is habitually placed. The bifurcations describable by (elementary) catastrophe theory occur commonly in a much wider class of equations than in the special class (gradient ordinary differential equations) in which it is most obvious that they are the only ones that can stably occur. Rigorous studies of partial differential equations can often-though certainly not always-lead to elementary catastrophes. Any study of a mathematical problem may transfer it to a different area from that in which it was posed.

The penultimate chapter gives a brief survey of some of the uses that have been made of catastrophe-inspired models in the social and behavioural sciences. By relating these to our earlier work we are able to offer reasoned and (we hope) constructive criticism of these types of model. The chapter also serves to exhibit the wide spectrum across which attempts to apply catastrophe theory have been made. We hope that a non-partisan discussion of this controversial area may help to set it in perspective. We have therefore concluded with one simple example where both elementary catastrophe modelling, and the implicit equilibrium hypotheses of traditional verbal economics (‘the invisible hand of the market’) fail in spectacular fashion.

We do not consider the resolution of this particular controversy to be of essential importance to the development of catastrophe theory in general; any more than the arguments about astrology or biorhythms or general relativity can affect the status of spherical trigonometry, Fourier analysis, or differential geometry. Our own views on the likely future development of catastrophe theory are given briefly in the final chapter, and may be summarized as follows. In the immediate future only the physical sciences will see solid benefits, because of their selection of ‘simple’ systems, and more recently of those with disorganized complexity, which can be ‘statistically simple’. The organized complexity of biology offers the best hope for the medium term, but here it is the whole of dynamical systems theory that will be required (with catastrophe theory a small but essential component). The organized complexity of social systems is unlikely to be well understood until after we have come to grips with biological systems. The only important question to be resolved at the moment is whether catastrophe theory is worth pursuing at all. We feel that, if only for the immediate returns to be gained in physics, the answer must be ‘yes’: whether we may thereby be taking a small step towards understanding the more spectacular problems of human existence is a question that may reasonably be left to future generations.

A book of this type, cutting across traditional subject lines, would not have been possible without the generous assistance of experts in the fields involved, which we acknowledge with gratitude. Christopher Zeeman alone is almost too numerous to mention; it is questionable whether, without his pioneering efforts and enthusiasm, the subject would have advanced to the stage at which sufficiently many people were even aware of its existence, let alone embarking on criticism of it. Both of the authors of this book made their initial contacts with catastrophe theory by way of his lectures; and we hope he will consider it a compliment if we say that his teaching was so successful that we not only began to understand a tiny part of Thom’s theories, and Zeeman’s efforts to develop them, but on occasions found ourselves in disagreement with both! To him this book is respectfully dedicated.

Every contact with René Thom shed a new and often mysterious light on the beauties of mathematics and the sciences. Ken Ashton supplied us with his ecological data. Ruth Bellairs told us of her biological experiments and corrected our use of biological language. Michael Berry allowed us to borrow his extensive knowledge of optics, along with many of his photographs; he and Malcolm Mackley were similarly generous regarding fluid dynamics, and Malcolm Mackley went to considerable lengths to supply us with prepublication copies of the beautiful photographs of fluid flow that adorn Chapter 11, each of which represents many hours of work on his part. As experimental evidence for a solid, technological application of the theory, these photographs are essential to the message of the book. Bill Supple, Giles Hunt, Michael Thompson and Michael Sewell kept us informed of developments in engineering and admonished our ignorance. Edgar Ascher and Bob Gilmore instructed us in thermodynamics. Colin Renfrew, Alan Wilson, Robert Magnus and David Pitt allowed us to quote from unpublished work; Bob Gilmore essentially wrote Chapter 15 for us. Our original plan to collaborate with Ted Woodcock, who was to have provided several chapters on biology, was prevented by the pressures of time: however, he is with us in spirit and, more physically, represented by some of his computer graphics. The exigencies of space forbid the further listing of several dozen people who made important contributions, either to the content of the book or to the morale of its authors.

I. N. S.&T. P.

March 1977

Coventry and Geneva

The work of Tim Poston was supported by the Fonds National Suisse de la Recherche Scientifique (grant no. 2.461-0.75) with additional support by Battelle Institute, Ohio (grant no. 333-207).

Contents

Preface

1Smooth and sudden changes

1Catastrophes

2The Zeeman catastrophe machine

3Gravitational catastrophe machines

4Catastrophe theory

2Multidimensional geometry

1Set-theoretic notation

2Euclidean space

3Linear transformations

4Matrices

5Quadratic forms

6Two-variable cubic forms

7Polynomial geometry

3Multidimensional calculus

1Distance in Euclidean space

2The derivative as tangent

3Contours

4Partial derivatives

5Higher derivatives

6Taylor series

7Truncated algebra

8The Inverse Function Theorem

9The Implicit Function Theorem

4Critical points and transversality

1Critical points

2The Morse Lemma

3Functions of a single variable

4Functions of several variables

5The Splitting Lemma

6Structural stability

7Manifolds

8Transversality

9Transversality and stability

10Transversality for mappings

11Codimension

5Machines revisited

1The Zeeman machine

2The canonical cusp catastrophe

3Dynamics of the Zeeman machine

4The gravitational machines

5Formulation of a general problem

6Structural stability

1Equivalence of families

2Structural stability of families

3Physical intercontenttations of structural stability

4The Morse and Splitting Lemmas for families

5Catastrophe geometry

7Thom’s classification theorem

1Functions and families of functions

2One-parameter families

3Non-transversality and symmetry

4Two-parameter families

5Three-, four- and five-parameter families

6Higher catastrophes

7Thom’s theorem

8Determinacy and unfoldings

1Determinacy and strong determinacy

2One-variable jet spaces

3Infinitesimal changes of variable

4Weaker determinacy conditions

5Transformations that move the origin

6Tangency and transversality

7Codimension and unfoldings

8Transversality and universality

9Strong equivalence of unfoldings

10Numbers associated with singularities

11Inequalities

12Summary of results and calculation methods

13Examples and calculations

14Compulsory remarks on terminology

9The first seven catastrophe geometries

1The objects of study

2The fold catastrophe

3The cusp catastrophe

4The swallowtail catastrophe

5The butterfly catastrophe

6The elliptic umbilic

7The hyperbolic umbilic

8The parabolic umbilic

9Ruled surfaces

10Stability of ships

Static equilibrium

1Buoyancy

2Equilibrium

3Stability

4The vertical-sided ship

5Geometry of the buoyancy locus

6Metacentres

Ship shapes

7The elliptical ship

8The rectangular ship

9Three dimensions

10Oil-rigs

11Comparison with current methods

11The geometry of fluids

Background on fluid mechanics

1What we are describing

2Stream functions

3Examples of flows

4Rotation

5Complex variable methods

Stability and experiment

6Changes of variable

7Heuristic programme

8Experimental realization

Combing polymer molecules

9Non-Newtonian behaviour

10Extensional flows

Degenerate flows

11The six-roll mill

12The non-local bifurcation set of the elliptic umbilic

13The six-roll mill with polymer solution

14The 2n-roll mill

12Optics and scattering theory

Ray optics

1Caustics

2The rainbow

3Variational principles

4Scattering

Wave optics

5Asymptotic solutions of wave equations

6Oscillatory integrals

7Universal unfoldings

8Orders of caustics

Applications

9Scattering from a crystal lattice

10Other caustics

11Mirages

12Sonic booms

13Giant ocean waves

13Elastic structures

General theory

1Objects under stress

2Elastic equilibria

3Infinite-dimensional peculiarities

Euler struts

4Finite element version

5Classical (1744) variational version

6Perturbation analysis

7Modern functional analysis

8The buckling of a spring

9The pinned strut

The geometry of collapse

10Imperfection sensitivity

11(r, s)-Stability

12Optimization

13Symmetry: rods and shells

Buckling plates

14The von Karman equations

15Unfolding a double eigenvalue

Dynamics

16Soft modes

17Stiffness

14Thermodynamics and phase transitions

Equations of state

1van der Waals’ equation

2Ferromagnetism

Thermodynamic potentials

3Entropy

4Transforming the maximum entropy principle

5Legendre transformations

6Explicit potentials

7The Landau theory

Fluctuations and critical exponents

8Classical exponents

9Topological tinkering

10The rôle of fluctuations

11Spatial variation

12Partition functions

13Renormalization group

14Structural stability of renormalization

The rôle of symmetry

15Even functions

16The shapes of rotating stars

17Symmetry breaking

18Tricritical points

19Crystal symmetries

20Spectrum singularities

15Laser physics

Preliminaries

1Atoms

2Field

3Interaction

4Measurement

The laser catastrophe

5Unfolded Hamiltonian

6Equations of motion

7Mean field approximation

8Boundary conditions

9Non-equilibrium stationary manifold

Experiments

10Laser transition

11Optical bistability

12Photocount distributions

Analytic correspondence

13Equilibrium boundary conditions

14Equilibrium manifold

15Thermodynamic phase transition

16Critical behaviour

17Analytic correspondence of experiments

18Future prospects

16Biology and ecology

The size of bee societies

1Bee economics

2The advantages of aggregation

3Catastrophe geometry

4Variation in space

5Complications

Constraint catastrophes

6Boundary effects

7Classification

Travelling waves in ecology

8Choice of convention and model

9Frontiers

10Numerical tests

11How frontiers stabilize

12How differentiation begins

Embryology

13Cell differentiation

14Switching catastrophes

17The problems of social modelling

1Identification of variables

2The archaeology of sudden change

3Catastrophes as models

4Prison riots

5Bistability of perception

6Alcohol and introverts

7Beyond elementary catastrophe theory

18Catastrophe theory: whither away?

1The contentsent state

2The future

Appendix 1

Computer program for determinacy and unfoldings

By D. R. Olsen, S. R. Carter and A. Rockwood

Appendix 2

Catastrophes in numerical analysis

Guide to the literature

Bibliography of catastrophe theory

References

Index

1Smooth and Sudden Changes

Classical physics (from Newton to General Relativity) is essentially the theory of various kinds of smooth behaviour; above all the awe-inspiring fall of the planets around the sun: unresting, unhasting and utterly regular. Even the wobbles that have dethroned Earth’s rotation as the standard clock happen smoothly. No coherent and mathematical description of celestial mechanics can allow, say, a huge comet falling into the solar system, parting the Red Sea as it passes Earth, and then losing most of its kinetic energy and settling down into an almost perfectly circular orbit as the planet Venus (a widely held pseudoscientific theory). Planets interact much too evenly for that.

1 Catastrophes

Other things, however, jump. Water suddenly boils. Ice melts. Earths and moons quake. Buildings fall. The back of a camel is stable, we are told, under a load of N straws, but breaks suddenly under a load of N+l. Stock markets collapse.

These are sudden changes caused by smooth alterations in the situation: an analogous astronomical event would be the Sun’s steady motion around the galaxy causing the Earth to switch (instantly or in a matter of days) to an orbit ten million miles wider, when some critical position was reached. Such changes are far more awkward for prediction and analysis than the stars in their courses, and the sciences (from physics to economics) are still gathering together the analytical techniques to handle such jumping behaviour.

Now there are many kinds of jump phenomena. There are forces that build up until friction can no longer hold them: the roar of an earthquake, and the rustle of rhubarb growing, are made by the movements when friction gives way. There is a critical population density below which certain creatures grow up as grasshoppers, above which as locusts: this is why locusts, when they do occur, do so in a huge swarm. A cell suddenly changes its reproductive rhythm and doubles and redoubles, cancerously. A man has a vision on the road to Tarsus.

Many of these still defy analysis: many have been analysed, with a tremendous variety of mathematical methods. We shall be concerned in this book with one particular mathematical context which covers a broad range of such phenomena in a coherent manner. The techniques involved were developed by the French mathematician René Thom and became widely known through his book Stabilité Structurelle et Morphogénèse [1] in which he proposed them as a foundation for biology. The sudden changes involved were christened by Thom catastrophes, to convey the feeling of abrupt or dramatic change: the word’s overtones of disaster are, for most applications, misleading. The subject has since become known as catastrophe theory, a phrase which is open to a variety of interpretations depending on the scope accorded it.

These techniques apply most directly (but far from exclusively) to systems that through varying situations seek at each moment to minimize some function (e.g. energy) or maximize one (e.g. entropy). We shall clarify in Chapter 3 what this means mathematically. For the present a good picture is that of a ball rolling around a landscape and ‘seeking’ through the agency of gravitation to settle in some position which, if not the lowest possible, is at least lower than any other nearby. (But meanwhile the landscape itself is changing.) The particular geometrical forms that arise in this setting have become known, following Thom, as elementary catastrophes, in the sense of fundamental entities (like chemical elements) and their use as expounded in this book is thus ‘elementary catastrophe theory’ (a phrase misinterpreted by Sussman and Zahler [la] to resemble' ‘elementary arithmetic’), though it is deep both mathematically and scientifically. For some systems more complicated phenomena can occur (we give an easily explained example in Chapter 17 Section 7), whose onset Thom [1] classes collectively as generalized catastrophes. Their theory is by no means so complete.

Physical intuition is important for the understanding of catastrophe theory. In this chapter we shall describe three simple physical systems exhibiting typical catastrophic behaviour, having the advantage that (unlike earthquakes or stock markets) they are simple enough to build, and small enough to carry around. In addition they may be used for elementary experiments. They are well adapted to analysis, although we shall not analyse them at this stage, and will be used repeatedly as examples. The reader will find his intuition very much assisted if he actually makes them (for which reason we give some practical indications as to their construction) and plays with them. No description can compete with direct experience. But it must be emphasized that these machines bear a similar relation to catastrophe theory as do the toys known as ‘Newton’s cradle’ and ‘the simple pendulum’ to Newtonian mechanics.

2 The Zeeman Catastrophe Machine

We begin with the first machine invented. E. C. Zeeman, of the University of Warwick, devised it in 1969: after three weeks of experimentation with rubber bands and paperclips he refined it to the version we describe. The first appearance in print was Zeeman [2]: other references include Poston and Woodcock [3] and Dubois and Dufour [4].

Fig. 1.1

It consists of a wheel (Fig. 1.1) mounted flat against a board, able to turn freely, and not too heavy: too much friction resisting the movement or inertia prolonging it obscure the behaviour we wish to study. To one point (B) on its edge are attached two lengths of elastic. One of these has its other end fixed to the board at point (A), far enough from the hub (O) of the wheel to keep the elastic BA always tight. The second has its other end (C) attached to a pointer, to be held in the hand. (The position of C can thus be controlled from a little distance without obscuring it.) Dimensions which work well in practice are a wheel of radius 3 cm, OA of length 12 cm, and each piece of elastic of unstretched length 6 cm.

Regardless of the radius r of the wheel, the unstretched lengths a and b of the elastic BA and BC, and the distance OA (as long as this is more than a + r) the machine will show qualitatively the behaviour to be described below. This is a part of the property of ‘structural stability’ which we discuss later: changes in the parameters make no essential qualitative difference, in a sense to be made precise at the relevant time. However, we shall be analysing the machine for one particular set of numbers, with the aid of computer graphics; so we now give detailed instructions for a machine whose behaviour has exactly the geometry drawn by the computer.

Fig. 1.2

Photocopy Fig. 1.2. An enlarged or reduced photograph is perfectly acceptable, since scale does not affect the behaviour, or its subsequent analysis. Mount the result on board or heavy card, and attach a wheel at point O. Attach a stiff wire to the wheel, perpendicular to the plane of the board, at radius r from O. (It may be convenient to make the wheel itself a little larger, since only the position of the wire matters, and combine the mounting of the wire with that of the wheel as in Fig. 1.3.) Fix another stiff wire perpendicular to the board at A. File a groove round each wire, both at the same distance above the board, and higher than any central raised point of the wheel. Take a piece of good quality rubber cord (not a cut rubber band or sewing elastic: better the square section cord sold for catapults† or model aeroplanes), some-what longer than four times the diameter 2r shown for the wheel in your copy of Fig. 1.2. Attach the middle to the wire at A, binding it with cotton to form a loop round the groove (Fig. 1.4(a)). Mark the point whose distance along the unstretched elastic is 2r, holding the elastic doubled and straight: bind on each side of it to form a «tight loop around the groove in wire B (Fig. 1.4(b)). Bind the point whose distance is 2r further along to a pointer (Fig. 1.4(c)). Care will be needed in making AB exactly 2r long: it may help to bind the doubled end to a loop, attach point B, and only then slip the loop over wire A.

Fig. 1.3

only one position can be occupied by the wheel under the influence of the elastic alone. If you push it to some other position and release it, it jumps back again. This one position will depend on that of C, but a smooth change in C will lead to a smooth change in the position of the wheel.

Fig. 1.4

smoothly from one side, the wheel moves smoothly to one of them; entering on the other side and taking C to the same point carries the wheel to the other.

(four possibilities altogether) do you not need to make this machine to understand it properly.

If time is of the essence, a few minutes will make a qualitatively accurate version of the machine, using stiff card for the board, drawing pinswill change a little, but can be found experimentally. (How?)

Fig. 1.5

3 Gravitational catastrophe machines

Photocopy Fig. 1.6 (again scale does not matter) and back it with light card, about postcard thickness. Cut round the figure accurately (a knife or razor blade is best) and cut another piece of card into a ring a few centimetres wide whose outer edge is identical to that of the first. Make six triangular beams of equal length, about one quarter that of the axis of the parabola, as in Fig. 1.7. Glue them to points near the edge of the parabola, evenly spaced, with one at each corner; and to the corresponding positions on the ring, so that when laid on its face, the solid card has its boundary directly below that of the ring. A small heavy magnet behind the solid card will grip a light piece of metal in front (Fig. 1.8) and can be slid to any desired position while retaining a good grip.

Fig. 1.6

Since most of the mass of the assembled device is in the magnet, we may take the centre of gravity of the whole to be the position of the magnet. When the machine balances steadily on edge, the centre of gravity must be vertically above the point of contact. If the machine rests on a level plane, the plane must be a tangent to the edge, so the centre of gravity lies on the corresponding normal (the line through the point of contact perpendicular to the tangent). The straight lines in Fig. 1.6 are some of these normals.

Experiments with the machine, or geometric thought along the above lines, will answer the following questions.

Fig. 1.7

(a)What, if any, positions of the magnet give the machine N possible angles at which it can balance (where N = 0,1, 2, 3,... and there is a new question for each choice of N)?

(b)Putting the magnet anywhere on the normal at a point P places the centre of gravity vertically above P, so the machine can balance at P. However, for some positions of the magnet on this normal the machine will return to a point of balance after a small wobble (that is, the equilibrium is stable), for others, it will topple over like an egg that has been stood on end (the equilibrium is unstable). What distinguishes the two?

(c)When does a small change in the position of the magnet leave the machine sitting in a position which is also slightly different, and when does it make the machine roll right over (a dramatic ‘catastrophic change’ which is unmistakable in a practical experiment)?

Fig. 1.8

Now repeat the construction, with the parabola replaced by an ellipse. Fig. 1.6 is replaced by Fig. 1.9. Answer the same three questions in this case.

Fig. 1.9

These machines are not as artificial as they may seem. Both of them turn out to correspond closely (Chapter 10) to larger scale phenomena in the behaviour of ships.

4 Catastrophe Theory

The complicated behaviour of the above machines shows that even simple problems in classical statics conceal many subtleties. A deeper analysis reveals that there are some underlying regularities in the mathematical structure which permit routine calculations of how such systems behave, based on the traditional applied mathematician’s use of Taylor series approximations. But these techniques also conceal many subtleties. The main mathematical thrust of this book is to develop a proper understanding of the geometric and algebraic methods used to handle Taylor series properly. Once developed, the methods provide powerful tools for tackling a wide range of problems, going far beyond simple statics of artificial machines, and opening up perspectives to which the traditional use of Taylor expansions as a source of approximations, to be justified post hoc by experiment, is blind.

Catastrophe theory is not a single thread of ideas; it resembles more closely a web, with innumerable interconnected strands; these include physical intuition and experiment, geometry, algebra, calculus, topology, singularity theory and many others. This web is itself connected to and embedded in a larger web: the theory of dynamical systems. A proper perspective on the theory involves some appreciation of all of these strands and the way they combine. The elementary catastrophes of René Thom are but one strand, though an important one. That they only come in seven basically different shapes is an intriguing fact, but it is not the only significant feature to be dealt with. It is not Thom’s theorem, but Thom’s theory, that is the important thing: the assemblage of mathematical and physical ideas that lie behind the list of elementary catastrophes and make it work.

† In the USA: slingshots.

2Multidimensional Geometry

A proper understanding of catastrophe theory involves a feeling for the geometry of space of many dimensions, backed up by suitable algebraic and analytic techniques. This permits a geometric approach to the calculus of several variables: an important viewpoint which can motivate and simplify calculations by relating them to geometric insights.

The first few sections of this chapter review essential linear algebra, presenting the geometric view that is sometimes missed in treatments of ‘matrix theory’. (For a more detailed geometric account, with proofs and many more pictures, see Dodson and Poston [5], which also develops the rigorous geometry of calculus in several variables.) We then take our first, classical steps in catastrophe theory. The most widely publicized feature of the theory has been the classification theorem mentioned above and discussed in Chapter 7: up to suitable changes of coordinates, a small number of standard forms are ‘typical’ for many phenomena. Coordinate changes thus play a key role in the theory. Here we show linear coordinate changes in action, reducing polynomial functions to a few standard expressions. This is both a key example of the kind of ‘classification’ the theory achieves in a far more glorious context, and a vital ingredient in what we do later. Much of the subsequent material aims to reduce other problems to those we solve in this chapter.

1 Set-theoretic Notation

It is convenient to make use of some elementary notions from set theory. A set is a collection of objects (of arbitrary nature) and these objects are called the elements, members or points of the set. The notation

means that x is a member of the set X. Usually the sets we shall consider will be sets of points in a multidimensional space. A set with no members, we call empty.

A set X is a subset of another set Y if every element of X is an element of Y. We write

for this, and say also that X is contained in Y, or that Y contains X.

The set of all elements x for which a particular property or condition P(x) holds is denoted

The union of two sets X and Y is defined to be

and the intersection is

For arbitrary elements x, y we may introduce the ordered pair (x, y) with the property that (x, y) = (u, v) if and only if x = u and y = v. The Cartesian product of two sets X and Y is then defined to be

to denote the set of real numbers.

and

then X × Z may be thought of as a set of points (x, y, z³ looking like the surface of a cylinder, as in Fig. 2.1.

There is a more general notion of an ordered n-tuple

which may be thought of as belonging to a repeated Cartesian product

Fig. 2.1

One of the most important concepts for our purposes is that of a function. If X and Y are sets, then a function f with domain X and codomain Y is a rule† which associates to each x X a unique element f(xY. Functions are also called maps or mappings. We write

and read this as ‘f is a function from X to Y’. We say that f maps X to Y, and x to f(x). When discussing the effect of f on elements we use a different type of arrow, thus x→f(x). The image f(x) of f is the subset

of Y. The image f(A) of A X under f is

For example the function

maps x to sin x. The function log which maps x to log x cannot be defined unless x is positive. So

is a function whose domain is the set of positive real numbers.

There is a certain freedom of choice as regards the codomain. For example, sin may also be thought of as having codomain

or indeed any set containing this. It is customary to choose any codomain which is convenient.

Traditional texts usually define functions by phrases like:

‘The function f(x) = x²’.

In our context this will be interpreted as ‘the function fwhich maps x to x’. This allows us to use the slightly imprecise traditional language whenever it is clear what the precise meaning is, while retaining the option of being more pedantic if we can thereby avoid confusion in ambiguous cases. When using the traditional language x is often called the (independent) variable, a term which we retain for convenience. (The value y = f(x) is traditionally called the dependent variable, which term we avoid, though we reserve the right to use the word ‘variable’ with reference to y.)

Functions of several variables come under the same heading if we think of functions whose domain is a Cartesian product. For instance, the function of two variables

may be viewed as a function f , mapping (x, yto x²+y². A function of n variables is just a function

which maps (x1 ..., xX1 × …× Xn to f(x1,..., xY.

If f: A→B and g : C→D are functions, and if f(aC for all a A, we define the composition g°f of f and g by

Then g°f is a function A→D. In particular we can make this definition when B = C.

If f : A → B and g : B → A are such that

g(f(a)) = a

f(g(b)) = b

for all a A, b B, then we say that g is the inverse function to f and write

g=f -1.

(Note that many of the traditional ‘inverse functions’ such as sin-1 are either not functions in our strict sense, being ‘multivalued’, or must be defined on carefully chosen domains.) Even when f has no inverse we use the notation

f -¹(Y)

to denote the set of all a A such that f(aY, for a subset Y of B., we define the restriction of f to X to be the function

for which

It differs from f only in being defined on a smaller domain X. (For further treatment of these concepts and especially the nontraditional notation and terminology, see Stewart and Tall [6].)

2 Euclidean Space

High dimensional spaces are studied by using a generalized kind of coordinate geometry. For any integer n> 0 we define n-dimensional Euclidean space to be

Fig. 2.2

It is convenient to use the abbreviated notation

and refer to the xi as the (ith) components or coordinates of x.

If x, y we define addition and multiplication by a scalar λ by

n the structure of a real vector space. When we have this structure in mind, its points are called vectors.

³. The addition rule corresponds to the ‘parallelogram law’ n: this gives a vivid language but needs algebraic verification that expected properties carry over. In particular when x≠0, the set of all λx ) is called the straight line through 0 and x; and 0 is called the origin.

A subspace n is a subset W with the properties

³. Condition (2.3) says that if any x W then so does the line through x and 0; then condition (2.2) says that given any two points in W, the vertex of the corresponding parallelogram lies in W. We have several cases to consider.

(a)W = {0}. This certainly is a subspace.

If W≠{0} we can find x W with x ≠ 0. Then the line through x and 0 also lies in W. This line may be the whole of W:

(b)W = }, a line through 0.

If not, there exists y W not lying on the line {λx}. Then the line through y and 0 is in W, and also the vertices of all parallelograms whose sides lie along these two lines, namely, points λx + μy for λ, μ . Clearly (Fig. 2.3) these are the points in the plane through 0, x and y. If this is the whole of W,

(c)W = {λx + μy | λ, μ, }, a plane through 0.

Finally, W may contain another point z not on this plane. A picture like Fig. 2.3, but with three lines and using parallelepipeds, shows that W = {λx + μy + vz | λ, μ, v ³. Hence the last possibility is:

(d)W ³.

Fig. 2.3

³ ³). To generalize these ideas and make them precise we introduce some algebra.

A set of points {v¹ v²,..., vn is linearly dependent if there exist scalars λ1 λ2,..., λr, not all zero, such that

If no such equation holds (or in other words if such an equation implies that all scalars λi = 0) then the set is linearly independent.

Geometrically, two points are linearly independent if neither lies on the line through 0 to the other; three points are linearly independent if none lies on the plane through 0 and the other two; and so on.

A set of points v¹,..., vr is said to span a subspace W if every element of W can be written as a linear combination

and if all such linear combinations lie in W (or equivalently each vi does). A basis for a subspace W is a linearly independent set of elements which spans W. The dimension of W is the number of elements in a basis: an important theorem states that this is independent of the basis chosen. As a convention, {0} has dimension 0. We write

dim W

to denote the dimension of W.

In some contexts we have to refer to infinite-dimensional spaces, namely, those for which no finite set is a basis. (For instance, the vector space of polynomials in x: any finite list has a highest degree, say k, so that xk+1 cannot be a linear combination of polynomials in the list.) But we will usually be able to avoid most of the technical complications that can result from this.

It can be proved that a subspace W n must have dimension in the range

and that any dimension in this range occurs for suitably chosen W. Usually the choice of W is not unique, but if dim W = 0 then W = {0}, and if dim W= n then W n. The difference n — dim W is called the codimension of W n. A cobasis for W n is a set of vectors v¹,..., vr which, together with a basis for Wn. Necessarily r then equals the codimension of W. Note that a cobasis is not uniquely determined by W, but involves arbitrary choices.

3 Linear Transformations

A linear transformation or linear map n m is a function f : n→ m with the properties

for all x, y . To find the general form of a linear transformation we take bases u¹,..., un and v¹ ,..., vm.Then for each i, f(uim, so there must exist scalars λji, for which

n is uniquely expressible in the form

so that

Thus every linear transformation is of this form. It is easy to verify the converse: everything of this form, is a linear transformation, no matter what values the scalars λJi take. (Physicists please note: we never sum over repeated indices without a Σ to say so.)

n the standard basis

with a similar choice of v j, so that

Then the μi are the coordinates of x, which we usually write xi Making this notation change, we have

What, geometrically, is a linear map? To see this we take the easiest case, maps f². Suppose that (Fig. 2.4)

Fig. 2.4

Then, for example f(l, 1) = (α + γ, β + δ). The effect of f is to distort the plane in a manner which preserves straight lines through the origin, sending squares into parallelograms (Fig. 2.5(a)). This, at least, is the case when (α, β) and (γ, δ) are linearly independent. If they are linearly dependent (but not both 0) then f ² to a line, squashing it flat (Fig. 2.5(b)). Should α = β = γ = δ = 0, then f ² to the origin (Fig. 2.5(c)), squashing still further.

²; and so we view any linear transformation (even in higher dimensions) in geometric terms as a distortion which preserves straight lines through 0 and maps (multidimensional) cubes to (multidimensional) parallelepipeds.

² is mapped to a line, every point on that line is the image of a whole line of points; when f maps to 0, this is the image of the whole plane; but when f ² to itself, each point is the image of a unique point. Roughly, the greater the amount of squashing required (in terms of dimension), the more things get squashed to a point (in the same sense). To make this observation respectable, define the rank of f to