Applications of catastrophe theory to the physical sciences

Physica 2D (1981) 245-305 North-Holland Publishing Company

A P P L I C A T I O N S O F C A T A S T R O P H E T H E O R Y T O T H E P H Y S I C A L S C I E N C E S

I a n S T E W A R T Mathematics Institute, University of Warwick, Coventry CV4 7AL, England

Received 11 November 1980

Catastrophe Theory was introduced by Rent Thorn in the late 1960's, as an attempt to model morphogentic changes in nature using ideas from topological dynamics and the theory of singularities of mappings. Thorn envisaged a very general approach to topological changes in the solutions to parametrized systems of equations (such as differential and difference equations), and in particular discussed the special case of 'elementary' catastrophe theory: singularities of smooth real-valued functions. Popular expositions have tended to overemphasize this special case, but it remains the major source of ideas and methods.

Here we survey the applications of Catastrophe Theory to the physical sciences (physics, chemistry, engineering, fluid mechanics, etc.). For brevity we confine attention to an area lying between 'elementary' and 'general' Catastrophe Theory, usually known as Singularity Theory. This is the theory of singularities of smooth vector-valued functions, which mathematically is a straightforward (though non-trivial) generalization of the real-valued case. In the last few years it has developed into a powerful and useful technique in several areas of theoretical physics, notably optics and bifurcation theory. Equivariant Catastrophe Theory, taking account of symmetry, is likely to prove especially interesting.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 2. Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

2.1. Four basic problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 2.2. Equivalence of germs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 2.3. The orbit picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 2.4. Codimension and the Jacobian ideal . . . . . . . . . . . . . . . . . . . . . . . . 250 2.5. Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 2.6. Unfoldings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 2.7. Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 2.8. Bifurcation geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 2.9. Structural stability and density . . . . . . . . . . . . . . . . . . . . . . . . . . 255 2.10. Modality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 2.11. The non-gradient case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 2.12. The equivariant case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 2.13. Other generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 2.14. Globalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

3. Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 3.1. Geometrical optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 3.2. Asymptotic solutions to wave equations . . . . . . . . . . . . . . . . . . . . . . . 262 3.3. Lagrangian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 3.4. Local asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 3.5. Crystal scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 3.6. Refraction by liquid droplets . . . . . . . . . . . . . . . . . . . . . . . . . . 266 3.7. Junctions in caustic networks . . . . . . . . . . . . . . . . . . . . . . . . . . 267 3.8. Random media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 3.9. Other applications to optics and related fields . . . . . . . . . . . . . . . . . . . . 271

4. Elastic buckling and bifurcation theory . . . . . . . . . . . . . . . . . . . . . . . . 272 4.1. The Thompson-Hunt theory . . . . . . . . . . . . . . . . . . . . . . . . . . 273

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246 I. Stewart/Physical applications of catastrophe theory

4.2. P e r t u r b e d b i f u r c a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3. T h e g e o m e t r y o f p e r t u r b e d b i f u r c a t i o n . . . . . . . . . . . . . . . . . . . . . . .

4.4. P a r t i a l c l a s s i f i c a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5. I m p e r f e c t i o n s e n s i t i v i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.6. E q u i v a r i a n t b i f u r c a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.7. M o d e - j u m p i n g in the r e c t a n g u l a r p l a t e . . . . . . . . . . . . . . . . . . . . . . .

4.8. Inf ini te d i m e n s i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5. O t h e r a p p l i c a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1. T h e s t i r r ed t a n k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. T h e w i n g e d c u s p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3. T w o r e a c t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4. T h e t h e r m a l - c h a i n b r a n c h i n g m o d e l . . . . . . . . . . . . . . . . . . . . . . . .

5.5. T h e B r u s s e l a t o r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.6. P h a s e t r a n s i t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7. F lu id f low . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.8. D e g e n e r a t e H o p f b i f u r c a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . .

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I. Introduction

A coarse system of differential equations (Andronov and Pontryagin) is one whose phase portrait retains its topological structure after any small perturbation of the system. This notion was generalized, as structural stability, to dynamical systems on manifolds (Smale [219]) and lies at the heart of modern dynamical systems theory (Smale [220]). Thom [242, 243] proposed it as a framework for modelling mor- phogenetic change. Fundamental to Thorn's approach is the concept of a catastrophe: a point (in some parameter space) at which a system loses structural stability, so that its solutions change their topological type. Zeeman [287] coined the term catastrophe theory to embrace both the mathematics inspired by this approach, and its applications to science.

This phrase has been interpreted in differing ways by different authors. Its broadest sense (Thorn [242-246]) is (roughly) "topological bifurcation theory"-where the bifurcating system may be of very general type (including inter alia partial or ordinary differential and difference equations) and where "bifurcation" means any change in topological type, as proposed first by Smale [221], rather than the classical "branching of solutions". Its narrowest interpretation is the theory of singularities of

smooth real-valued functions. More properly, this is elementary catastrophe theory (Thom [244], Zeeman [290]); but the qualifying adjective is often dropped for ease of exposition, especially in less technical articles. Thom's pro- gramme was first developed for this case, which is still its most accessible part: it has therefore been emphasized in the expository literature.

The mathematical ideas behind elementary catastrophe theory were conjectured by Thom and proved by Arnol'd, Malgrange, Mather, Thom and others during the 1960's and 70's. The early restriction to scalar-valued functions has been relaxed to include vector-valued functions: this body of work has acquired the name singularity theory. It is a direct descendant of elementary catastrophe theory and its central concepts are virtually identical. Of course some of its results are not.

Thus elementary catastrophe theory is a part of singularity theory, which in turn is a part of catastrophe theory. The latter also includes other generalizations of elementary catastrophe theory, notably to infinite dimensions or taking symmetries into account. At the level of singularity theory, catastrophe theory is essentially of a quasi-static nature, dealing with dynamic behaviour only peripherally. Non-elementary catastrophes have a more dynamic nature. The simplest example is the Hopf [118] bifurcation

I. Stewart~Physical applications of catastrophe theory 247

to a limit cycle: and an extremely important class of examples includes "chaotic" bifurcations to strange attractors.

Early attempts to develop applications of catastrophe theory included a number of tentative models in the biological and social sciences, which have proved controversial (Sussmann and Zahler [233-235], Zahler and Sussmann [286], Smale [222]). While many of these criticisms can be met (Pearson and MacLaren [172], Poston [178], Stewart and Woodcock [230]) or are felt to be misplaced (Guckenheimer [102]) there is no dispute as regards the tentative nature of these models. However, the controversy has tended to obs- cure the contributions of catastrophe theory to the physical sciences, even though these have consistently formed the majority of the literature on applications. These contributions are largely ignored in the critical literature, and on occasion their existence has been denied (Sussmann [232]) or minimized (Zahler and Sussmann [286]).

The object of this paper is to survey some of the current developments in applications of catastrophe theory to the physical sciences. The coverage is not exhaustive, and many topics have been omitted or given brief mention to conserve space: those included are a represen- tative selection. We largely restrict attention to elementary catastrophe theory and its generalization to singularity theory. This is not in- tended to deny the importance of non-elementary catastrophes, but to confine the exposition to acceptable lengths. In addition, much work on bifurcation to limit cycles or chaos has been done in other fields, and we have no wish to attempt to draw questionable lines of demar- cation. Catastrophe theory has close connec- tions with many other schools of thought: bifurcation theory, topological dynamics, non- equilibrium thermodynamics, spontaneous symmetry-breaking, synergetics- and these also overlap each other. This is hardly surprising given the rapid growth of research on nonlinear phenomena and its currently fluid state. Con-

ceivably in the future a greater degree of unification may be achieved.

2. Mathematics

First we sketch the mathematical ideas involved in elementary catastrophe theory, and show how these generalize to singularity theory and beyond. They involve many branches of modern pure mathematics, including differential topology and algebraic geometry, and rigorous proofs are accessible only to specialists (see Mather [148-154], Arnol'd [11-22], Levine [132], Siersma [216], Wassermann [278], Zeeman [291], Br6cker and Lander [45], Golubitsky and Guil- lemin [87], Martinet [145, 146]). Simplified expositions, omitting some or all proofs, are given in Gibson [78], Lu [133], and Poston and Stewart [182, 183].

Here we aim only to convey the spirit of the mathematics, and state some of the main results needed for applications below. An expanded discussion should be consulted for fine points. We use current mathematical notation to avoid clumsy circumlocutions: occasionally we require ideas on differentiable manifolds (Hirsch [111]) or topological dynamics (Hirsch and Smale [112], Arnol'd [24]). Those desiring a more 'classical' language should consult Poston and Stewart [183]. An up-to-date bibliography of catastrophe theory (mathematics and applications in all subjects) is now available in preliminary form, Zeeman and Hayden [299].

2.1. Four basic problems

Let R n denote n-dimensional real Euclidean space. A smooth (that is, infinitely differentiable) function f : R n ~ R has a singularity at x E R ~ if its derivative dr(x) vanishes. In coordinate form, f(xl . . . . . Xn) is a real-valued function of n real variables, and

df(x)= (~x (X)," o f ) " " ox~ (x) = (0 . . . . . 0).


By translation of coordinates we may assume x = 0 and this is usually done. Analogous definitions may be made when the target space is not R, but R m, so that f is vector-valued, f ( x ) = (fl(x) . . . . . fro(X)); for ease of exposition we restrict to m = 1.

A first approximation to f is its linearization to dr, that is, the linear part of its Taylor series. At a singularity this approximation is poor, even qualitatively, and the local behaviour (plausibly) may be found from higher terms in the expansion.

If n = 1, then the first nonvanishing term ax' of the Taylor series is a good approximation (both qualitatively and quantitatively) close to the singularity, and the series may be truncated at this term. If n > 1 the recipe is less simple: the lowest nonzero terms may not suffice. For example xZy + y4 and x2y + y5 behave differently

near 0 (e.g. their sets of zeros are topologically inequivalent) and both differ from the truncation xZy at degree 3. More variables, or symmetries, complicate the picture further. We are led to the

for some choice of small constants % Again this fails to extend to n > 1: it is neither necessary nor sufficient to add all possible terms of lower degree. (In fact even in (2.1) the term in x t-I can be t ransformed away by a linear coordinate change: the rest cannot.) All perturbations of x5 + y5 can be brought to the form

xS+Y 5+ 2 eijxiy i i,j~<3

(by the rules of subsection 2.6 below), but one such term is x3y 3 which is of higher degree than x 5 + yS. It cannot be omitted in general.

Unfolding problems: Given a singularity, what are the essential perturbations?

When n = 1 we can even classify the singularity types. By scaling away the coefficient a we obtain the types _+x t (t even), x' (t odd). The only singularities not accounted for by these are the flat functions, such as e -1/~2, with zero Tay- lor series.

Determinacy problem: At what point is it safe to truncate a Taylor series?

Mathematical models of physical phenomena are idealizations, which should be "realist ic" as regards their purpose. One way to be unrealistic is to ignore perturbations having a nontrivial effect on the predictions of the model. For example the equation (x 2 - y2)2= 0 has the solutions x = _+y; but its perturbation to (x 2 - y2)2+ Ey6----O has only the origin as solution when

> O, and a complicated 8-branched curve when e < 0 .

For a function f : R ~ R the recipe is again simple. Truncate the Taylor series at ax' as before. Then any small perturbation by a smooth function is locally equivalent (by a "smal l" smooth invertible coordinate change) to a function

O X ! --~ EI_I X t - I -.1- . . . -~- El x -]- EO (2.1)

Classification problem: Classify (with the possible exception of "pathological" cases which should be " ra re" ) the local types of singularities.

Elementary catastrophe theory largely solves these three problems; its generalization to singularity theory solves the first two, and gives relatively complete information on the third in low-dimensional cases. Of course much more is desirable, for example:

Globalization problem: Classify all possible global configurations of singularities for smooth functions f :M ~ N where M and N are manifolds.

This is largely unanswered, and almost certainly over-ambitious, but some interesting partial

results exist. A good solution to the local classification problem seems an essential first

L Stewart[Physical applications of catastrophe theory 249

step. Further generalizations can be envisaged (let M and N be infinite-dimensional, require invariance under a group of symmetries, ask about vectorfields rather than functions) and are to a greater or lesser extent illuminated by the simplest case, the elementary catastrophes. To these we now turn.

This is the natural equivalence for studying topological properties of the gradient df (Poston and Stewart [183] p. 58). If f, rather than df, is important, the term ~/ is omitted (see section 2.6).

2.3. The orbit picture

2.2. Equivalence of germs

In its local version, elementary catastrophe theory deals with functions f : U ~ R where U is a neighbourhood of 0 in R". The cleanest way to handle such functions is to pass to germs, a germ being a class of functions which agree on suitable neighbourhoods of 0. All operations on germs are defined by performing similar operations on representatives of their classes. In the sequel we shall usually make no distinction between a germ and a representat ive function.

We let E~ be the set of all smooth germs R n ~ R , and let E~m be the set of all smooth germs R ~ ~ R m. Of course En,l = E~. These sets are vector spaces over R, of infinite dimension. We abbreviate (xl . . . . . xn) E R" to x. If f E E~,, then

f(x) = ( f l ( X ) . . . . . f ra (X ) )

and the f~ are the components of f. A diffeomorphism germ th :R n ~ R n satisfies

th(0)=0, and has an inverse ~b' such that ¢b($'(x))=x=$'(~b(x)) for x near 0. It represents a smooth invertible local coordinate

change. By the Inverse Function Theorem 4, is a diffeomorphism germ if and only if it has nonzero Jacobian, that is,

det[0$i/0xj(0)] # 0.

Two germs f, g : R ~ --* R are right equivalent if there is a diffeomorphism germ ~ and a constant ~ ~ R such that

g(x) = f ($ (x ) ) + % (2.2)

A type of germ is a right equivalence class; and the classification of germs up to right equivalence amounts to a classification of types. Each type forms a subset of E~, and the central object of study is the way these types fit together.

A precise description is complicated by the fact that most types have infinite dimension; but there is a measure of the complexity of a type, the codimension, which is generally finite. Heuristically it is the difference between the dimension of the type and that of E , (even though both are infinite). A rigorous definition is given below.

The largest types have codimension 0 and form open sets in En. Their boundaries contain types of codimension 1; the boundaries of these in turn contain types of codimension 2, and so on, with higher codimensions revealing pro- gressively more complex types. Fig. 1 represents the decomposit ion schematically. Types of infinite codimension exist, but form a very small set in a reasonable sense.

~ - - - 0 - ~ / / ~

Fig. 1. Decomposition of En (actually infinite-dimensional) into types is illustrated schematically. Numbers denote codimensions.


The decomposit ion by codimension is called the natural strati f ication of E,. It in some ways resembles the structure of the set of orbits of a Lie group acting on a manifold (Gibson [78]) and in fact the types may be viewed as orbits for the action of an infinite-dimensional group of diffeomorphisms (4~, ~') for which

( ~, "y)f(x ) = f(cb(x )) + "y.

This analogy gives useful insight to those familiar with Lie group actions, although rigorous proofs proceed less directly. For example the effect of a t ransformation (~b, ~/) can be related to that of a suitable "infinitesimal" transformation, thereby linearizing some aspects of the structure: this idea leads to a criterion for determinacy. Perturbations of a given type correspond to neighbouring types (of equal or lower codimension) and may be detected by a transverse disc whose dimension is equal to the codimension of the type: all nearby germs may be slid along their orbits by the group action until they encounter such a disc; and the same goes for entire parametrized families of perturbations. The result is the notion of a universal

unfolding of a given germ, studied more fully below. Individual types form submanifolds of E,, embedded in a nice way.

The advantage of this geometric description is

that, in modified theories, it usually suggests the correct modification of the main results: adapt- ing standard techniques usually provides the proofs once the problem is correct ly formulated.

where the right-hand side is the Taylor series, or jet, of f. Note that it exists as a f o rma l power series for all smooth f : convergence is not required in what follows. The map j is onto, linear over R, and preserves products (i.e.

i ( f g ) = f t . jg). Let m. be the set of f E E. such that f(0) = 0.

This is an ideal of E. (meaning that if f E m . and g E E. then fg E m., which we write briefly as m . E . C_ m.) . Its kth power ink. consists of all f E E. such that 0 = f(0) = df(0) = d2f(0) . . . . . dk-~f(0). In particular f is a singularity if and only if f E m . 2. The ideals m. k form a decreasing sequence

2 E , ~ _ m , ~ _ m , ~ _ m 3 ~ _ . . . .

There is a similar chain in F,. Let M, = j(m.): this is the set of formal power series with zero constant term. Then M k = j ( m k) is the set of formal power series without terms of degree ~<k-1 . The intersection of all M k is 0; the intersection of all m, k is the set m~ of flat germs, having zero Taylor series.

The Jacobian ideal of a singularity f is the set of all germs expressible in the form

af gl + " ' " + g" Ox.

for arbitrary germs gi. We denote it by A(f), or merely A when f is understood. Its image jA(f)C_ F, has an analogous definition, where the partial derivatives are defined formally.

2.4. Cod imens ion and the Jacobian ideal Since f is a singularity, A(f)C_ m.. The codimens ion of f is defined to be

Let E. be the set of germs R" ~ R , and let F. be the set of formal power series in x~ . . . . . x.. There is a map j : E. --* F. defined by

M a2f /f = f (0) + Y- o~,~ (O)x, + ~ ~, ~ (O)x,xj + . .

cod(f) = dimR m.]A(f) .

Similarly, at the formal power series level, we define

cod(if) = dimR M . / j A ( f ) .

L Stewart/Physical applications of catastrophe theory 251

These codimensions are the analogues in E~ and Fn of the "geometric" codimension defined in section 2.3. This is perhaps not obvious: it occurs roughly as follows. The codimension of an orbit is the same as that of its tangent space T. This is the same as the dimension of the quotient vector space E/T. In E, the analogue of this tangent space is the Jacobian ideal, so the codimension should be dim E,/A(f). This measures the number of independent directions in En "missing" from Aft), or equivalently missing from the orbit of f.

The computation of codff) is effected by means of the following result: i[ either codff) or cod(jr) is finite then so is the other, and they are then equal. Thus the computation may be carried out on the formal power series level, where it is a combinatorial calculation. For examples, in classical notation, see Poston and Stewart [183].

2.5. Determinacy

Let I E E~, and define the k-jet jkf to be the Taylor series of f up to and including terms of order k. For example

iS(sin(x)) = x - x 3/3 ! + x 5/5 !.

We say that f is k-determinate (or k-determined) if whenever g E E, has jkg = j~f it follows that g is right equivalent to f.

A germ is 1-determined if its linear part is nonzero, that is, its derivative does not vanish. So the non-l-determined germs are the singularities. If [ is a singularity and f(0) = 0 (as we can assume) then the second derivative gives the 2-jet of f in the form

J2f (Xl . . . . . Xn) = X n i jx iYJ , i,j

where the matrix H = (Hij) is symmetric. We call H the Hessian of f at 0. It can be shown that [ is 2-determined if and only if det(H) # 0;

and in this case f is right equivalent to

- X l : - ' ' . --- x~. (2.3)

This is a reformulation in determinacy terms of the Morse Lemma (Milnor [162]). A germ equivalent to (2.3) is said to be Morse. Morse germs are precisely those of codimension 0. The number l of negative signs in (2.3) is the index of f, and f is an l-saddle. Morse theory (Milnor [162]) describes the global properties of a function f : X - - , R where X is a smooth manifold, and f has only Morse singularities.

There exist rules for computing the determinacy of a given germ: an easy necessary condition, an easy (different) sufficient condition, and a harder necessary-and-sufficient condition. We describe these.

Let A be the Jacobian ideal of f. Then: (i) If m k C_ m,A then f is k-determined;

(ii) If f is k-determined then mk,÷JC_m,A; (iii) f is k-determined if and only if m,k÷~C_ m~A(f +g) for all gEm~ ÷1.

There is a slightly stronger form of (i), namely (i') If m~+lC_m~A then f is k-determined.

Result (iii) is due to Stefan [225]. Numerous examples in Poston and Stewart [183] and Gib- son [78] show how to compute the determinacy of a given f. For example, suppose f is in Morse form (3). Then A = ( + 2 x l . . . . . _+2xn) = m~ and

2__ m,-moA. By (i), f is 2-determined as asserted above.

A germ is finitely determined if it is k-determined for some finite k. The following are equivalent: (iv) f has finite codimension; (v) f is finitely determined;

(vi) m~CA for some t. The solution to the Determinacy Problem of

Section 2.1 is thus that it is safe (up to right equivalence) to truncate a k-determined germ at degree k of its Taylor series. For a germ such as x2y~E2, which is not finitely determined, it is not safe to truncate higher order perturbing terms (and indeed x2y+y t has a type that


depends on t). Germs that are not finitely determined either arise in a context where some symmetry is acting (and should be analysed by methods similar to those above but which take symmetry into a c c o u n t - w h i c h can be done) or must be viewed with suspicion. By (iv), we may summarize: "nice" germs have finite codimension.

Suppose that f is not 2-determinate, so that det (H)=0. Let the rank of the matrix H be r, and call n - r its corank. A useful result, called the Splitt ing L e m m a , says that f is right equivalent to a germ of the form

g(xl . . . . . Xn-r)+X2-r+l + ' " "+X 2.

For most purposes the sum of squares may be ignored. So the Splitting Lemma reduces the effective number of variables to n - r . In infinite dimensions this result is due to Gromoll and Meyer [98]. A simple proof for finite dimensions is in Poston and Stewart [183].

The determinacy calculations, and the application of the Splitting Lemma, may be carried out equally well on jf in Fn, provided the codimension of f is finite. The formal power series setting is better for computations.

2.6. Unfoldings

where

8 = ( ~ . . . . . 8m) ~ R m,

o~ : R" ~ R",

~b : Rm --, R t,

3,:R~ ~ R.

Two unfoldings are equivalent if each can be induced from the other. An /-parameter unfolding is versal if all other unfoldings can be induced from it; universal if in addition l is as small as possible.

Suppose that f has finite codimension c. Let uj . . . . . uc be a basis for m,/A( f ) . Then a uni-

versal unfolding is given by the germ

F (x, E ) = f (x )-JrEIUl(X )-~ -" " "-]-EcUc(X ). (2.4)

While different choices of the ui can be made, a universal unfolding is unique up to equivalence. The existence of universal unfoldings in finite codimension, and the method for computing them, is probably the most significant and useful result in elementary catastrophe theory. (Note that (2.4) is linear in the unfolding variables ~. This is a theorem, and is not built into the definition of an unfolding.)

For example, if f ( x , y)=x3+y 4,

An unfolding of a singularity is a "parametrized family of perturbations". The notion is useful mainly because, for finite codimension singularities, there exists a "universal unfolding" which in a sense captures all possible unfoldings.

More rigorously, let f E E , . Then an l- parameter unfolding of f is a germ FEE,+t , that is, a real-valued germ of a function F(xl . . . . . x~, el . . . . . el)=F(x, ~), such that F(x, 0)=f(x).

An unfolding P is induced from F if

F(x, ~)=F(p~(x), ~(~))+y(~),

then a basis for m2/A(f) is {x, y, xy, y2, xy2}. So a universal unfolding is given by

F(x , y, E)= X3+ y4+EIX +E2y+E3xy+E4y2+EsXy 2.

The codimension of a germ f has several interpretations:

(i) The codimension of the Jacobian ideal in m, ; (ii) The number of independent directions

"missing" from the orbit of f, (iii) The number of parameters in any universal unfolding of f.

In addition, if the codimension of f is c, it can


be shown that any small perturbation of f has at most ca-1 critical points.

In the version omitting y in (2.2), a basis for En[A is needed to obtain a universal unfolding. That is, while in the gradient version a universal unfolding of x 3 is x 3 + ax, in the 'value' version it

is x3+ax +b.

2.7. Classification

We sketch how these ideas may be used to classify germs of codimension at most 4.

Le t f E E , . If f is not a singularity then f ( x ) is right equivalent to Xl. If f is a singularity and its Hessian has nonzero determinant, then f is right equivalent to +-x2- . . .+-x 2. Otherwise, de t (H)= 0. Let k = n - r be the corank of H, and split f as

f ( x )=g(x l . . . . . xk)+-x~+l +-" " "'+'X2..

No higher term added to x 3 produces a codimension 4 result; and no higher term added to 0

does. Finally let k~>3. Then the codimension can be

proved to be at least 7, so this case does not

arise. Thus we have classified the germs of codi-

mension ~<4 into the canonical forms

X1,

.+. X 2 + . + 2 • " X n ,

x ] + ( M ) , +_x4+(M),

x~+(M),

+-x~+(M),

X~-XlX2+(N),

x]+xtx2+(N),

+-(x~x2 + x~) + (N),

It can be proved that the classification of pos- sibilities for f depends only on the similar classification for g.

The Taylor series of g begins with cubic or higher terms. First suppose that k = 1, and let the first nonzero jet of g be atx t. This is t-determined, and scales to +-x t (t even), x t (t odd). The codimension is t - 2 , so t = 3, 4, 5 or 6.

Next let k=2, and let

j3g(x, y)=ax3+bx2y+cxy2+dy 3.

By a linear change of variable this cubic may be brought to the form x3+xy 2 (one real root), x 3 - xy 2 (three distinct real roots), x2y (three real roots, one repeated), x 3 (three real roots, all coincident), or 0.

The forms x3+--xy 2 are 3-determined, and of

codimension 3. The form x2y is not 3-determined, so we

consider higher terms. A series of changes of variable bring any higher order expansion to the form x2y---y ', which is t-determined and of codimension t. Only t =4 is relevant to our problem here.

where (M)=+-x2- . . . - x 2, (N)=+-x]--- -. - - x 2.

The celebrated elementary catastrophes of Thorn are the universal unfoldings of the singularities on this list, or its extension to higher codimensions. The universal unfolding arises when we try to classify, not germs, but l- parameter families of germs. For 1~<4, "almost all" such are given by universal unfoldings of germs of codimension ~<4.

Table I summarizes the list of germs and their unfoldings up to codimension 5, together with their customary name and symbol in the systematic notation of Arnol 'd [15]. The terms (M) and (N) are omitted for clarity; x and y replace Xl and x2; and unfolding parameters are listed as (a, b, c, d, e) rather than (~t, ~2, e3, e4, es).

There are many gaps to be filled before the above sketch becomes a proof; but what it does show is how the classification problem reduces to the determinacy and unfolding problems (and is relatively easy once these are solved). In applications the main influence of the classification is an organizing one: the determinacy and unfolding theorems play a more direct role.

254 I. Stewart/Physical applications o[ catastrophe theory

Table I The elementary catastrophes of codimension ~<5. When the -+ sign occurs, germs with sign (+) are called standard, (-) are called dual.

Symbol Name Germ Universal unfolding Corank Codimension

A2 fold x 3 x~+ax 1 1

-+A3 cusp ~-x 4 ~x 4 q- ax 2 -~- bx l 2

A4 swallowtail x 5 x ~ + ax 3 + bx 2 + cx 1 3 -+ A5 butterfly +_ x6 +- x6+ax 4+bx 3+ cx2+dx 1 4

A6 wigwam x 7 x7 +axS+bx4+cx3+dx2+ex 1 5

D4 elliptic umbilic x 3 - x y 2 x 3 - x y 2 + a x 2 + b x +cy 2 3 D~ hyperbolic umbilic x3+xy 2 x 3 + x y 2 + a x 2 + b x + c y 2 3

- + D 5 parabolic umbilic ±(x2y+y 4) +-(x2y+y4)+ax2+by2+cx+dy 2 4

D6 second elliptic umbilic x S - x y 2 x S - x y 2 + a y 3 + b x 2 + c y : + d x +ey 2 5 D~ second hyperbolic umbilic xS + xy 2 xS + xy Z + a y 3 + bx2 + c y 2 + dx + ey 2 5

±E6 symbolic umbilic -+(x3+y 4) +-(x3+y4)+axy2+by2+cxy+dx+ey 2 5

2.8. Bifurcation geometry

Let F : R n × R ~ ~ R be a parametr ized family of smooth functions,

F(x, a )= F(xl . . . . . x, ; a~ . . . . . aj).

Define the catastrophe set (or equilibrium set)

M =fix, a) [dr(x, a)=0},

the singularity set

E={(x, a) I dr(x, a )=0 , det d2f(x, a)=0},

the catastrophe map

x : M ~ R ,

X(x, a)=a,

and the bifurcation set

B =X(~)={a ](x, a ) ~ E for some x}.

For "a lmos t all" F, and in particular for the e lementary catas t rophes , M is an embedded submanifold of R" x R ~, and hence is called the catastrophe manifold.

For a given a E R ~, imagine a sys tem whose states x are determined by the condition

dr(x, a )=0 . As a varies, the possible x trace out the ca tas t rophe set. As a crosses the bifurcation set B, x may cross the singularity set E, which results in a change in the topology of possible states x - h e n c e the term "bi furca t ion" in this context. In many applications the geomet ry of B, and the project ion x : M ~ B , are important. These have been investigated for the seven e lementary catas t rophes of codimension ~<4 (Poston and Woodcock [184], Thom [242, 243], Br6cker and Lander [45], Pos ton and Stewart [183]) and for -var ious higher catast rophes, notably E6, Ds, and the "double cusps" X 4 - t - y 4 - t -

hx2y 2 called X9 in Arnol 'd ' s notation (Callahan

[48, 49], Zeeman [292]). Fig. 2 shows the bifurcation sets of the e lementary catas t rophes of codimension ~<3. Fig. 3 is taken f rom Callahan [49] and pertains to the bifurcat ion set of X9 (in

A2

Fig. 2. Bifurcation sets of the elementary catastrophes of codimension ~<3.


explained by finding new parameters (which need not have physical interpretations, but usually do) that merge all the component singularities into a single, highly degenerate one: this acts as an organizing centre, and the original arrangement may be located within its universal unfolding. The technique is highly successful (see e.g. section 5.2), partly because it turns a (semi-)global problem into a local one.

2.9. Structural stability and density

Fig. 3. Bifurcation geometry occurring in an unfolding of °X9 at an E6-point, after Callahan [49] fig. 29.

a way that we shall not describe here): it illustrates the complicated form that a higher catastrophe may take.

In many physical applications, a system ad- mitting a multiplicity of states must choose a single state from those available. Changes in this state, as parameters vary, can occur: sometimes these are forced to be discontinuous (via bifurcation). The commonest conventions used are perfect delay (the state varies continuously if possible) and the Maxwell convention (adopt a global minimum of the potential). The former is usual for quasistatic bifurcation problems, the latter is mainly used in thermodynamic applications, or for other systems subject to statistical "noise". Gilmore [80] suggests using the Fokker-Planck equation to interpolate between these two extremes. In many cases (e.g. optics) all states are realised simultaneously.

The elementary catastrophes in table I have an important geometrical property: they are topological cones. This fact, which is due to the quasihomogeneity of the universal unfoldings, means that an arbitrarily small neighbourhood of the origin has the same topological singularity structure as the entire universal unfolding: the singularity at the origin captures the surround- ing structure "in microcosm". The origin is said to be an organizing centre. In many applications, a global arrangement of singularities is

A family of germs F(x ,e ) is structurally stable if any small perturbation of it is equivalent to it (as an unfolding). Every universal unfolding of a germ of finite codimension is structurally stable. Also, if 1~<5, "almost every" /-parameter family is equivalent to the universal unfolding of a germ of codimension ~<i. So table I lists "almost all" 5-parameter types.

"Almost all" is here used in the following sense. There is a topology (the Whitney topology) on En÷l in which the relevant germs form an open dense set. The precise definition of this topology may be found in Golubitsky and Guillemin [87] or Zeeman [291]; roughly, two germs are near each other if all their partial derivatives of arbitrary order are close but for technical reasons this is not quite adequate as a description.

A property satisfied by an open dense set of objects is often said to be generic. Table I thus provides a generic classification of families of germs. By abuse of language, individual germs having a given generic property are themselves called generic.

Both structural stability and density a r econ- sequences of a transversality property (Poston and Stewart [183]); and the phrase "almost all" carries the same overtones as "almost all systems of n equations in n unknowns have isolated solutions" or "almost all pairs of curves in R 3 fail to meet".

We can sketch at a geometric level how the

256 L Stewart~Physical applications of catastrophe theory

argument goes. Any /-parameter family of functions F forms an /-dimensional subset of E,; we ask how it cuts the orbits (equivalence classes of germs). Fig. 4 is a schematic representation of the typical case, with F cut- ting orbits transversely. If the codimension-I orbits fill out the region of En concerned, then F must meet at least one of them. Further, any nearby family must also be transverse to the orbits, and meet the same one. By "pushing along orbits" any two transverse families are equivalent. Hence F is structurally stable. Since any non-transverse F may be made transverse by a small perturbation, transversality is dense; hence so is structural stability.

However, from codimension 6 onwards in F, when n~>3, and from codimension 7 in E2, the picture changes, because there exist regions which are not filled out by the codimension-I orbits. We consider this in the next section.

2.10. Modality

Fig. 5 illustrates this possibility. Here one region Y breaks up as a continuous family of orbits Y(tx) parametrized by t~. If the codimension of the region Y is l, then an /-parameter family F can meet Y transversely, and it must then meet some orbit Y(~) . But this orbit has codimension l+l (assuming/~ 1-dimensional). A small perturbation of F must still meet Y, but can do so in a different orbit Y(/~0. Then F

2

2 2

Fig. 4. Transverse family of germs F meeting orbits in E,. Note how codimensions are unchanged (up to the dimension of F).

Fig. 5. Modal family of orbits Y is met transversely by F0 and Ft, but individual orbits Y(/~) are not. A small perturbation F, of a disc F0 meeting Y at Y(/z0) continues to meet Y, but now at Y(~0.

cannot be structurally stable. In fact an open set of F, namely those meeting Y transversely, lack structural stability: the set of structurally stable /-parameter families is no longer dense.

The set Y is an example of a moduli space or modal family of orbits, and the dimension of the parameter /.~ is its modality. The phenomenon of modality is less of a problem than it may seem, because a dense set of families may still be found: we have to include not only the universal unfoldings of all germs of codimension ~<l, but also all unfoldings transverse to Y of germs Y(/~), for all the modal families Y such that

cod(Y)=cod(Y (/~))-dim/~ ~<1.

Arnol'd has classified all germs of modality ~<2 (up to complex coordinate changes) which includes all orbits up to codimension 14.

Density can be recovered another way, by relaxing the equivalence relation to permit continuous coordinate changes rather than differentiable ones. There are then finitely many orbits (relative to this weaker equivalence relation) in each codimension, and each modal family collapses to a finite number of topological types. Structurally stable families are then always dense (Mather [154], Gibson et al. [77]).

An example may clarify these ideas. Among

I. Stewart/Physical applications of catastrophe theory 257

orbits of codimension 8 are a family X9 corresponding to homogeneous quartics

f ( x , y) = ax 4 -t- bx3y + cx2y 2 + dxy 3 + ey 4,

whose associated polynomial f(x, 1) has no repeated roots. These can be transformed by linear coordinate changes to the form -+x 4+ - y4+

hx2y 2 where h # ---2 for the case of equal signs. For a given choice of signs, h is a modal parameter, and distinct h give differentiably inequivalent germs; but two values of h in the same connected component of the moduli space give topologically equivalent germs. In fact h is related to the cross-ratio of the root-lines of the quartic (see e.g. Poston and Stewart [182]) which is a linear, hence differential, invariant; but clearly not a topological invariant. The topological classification yields exactly four orbits,

°X9: X 4 "t- y4 -F Ax2y 2, )t > --2, A # 2,

0X9: _X 4 _ y4 -t- Ax2y 2, A < 2, X ~ --2,

2X9: x 4_ y4 + Ax2y2, any A,

4X9.- x 4 -t- y4 -t- Ax2y 2, A < --2,

(Callahan [49]). Differentiably each orbit has codimension 8, but their union has codimension 7; topologically the orbits have codimension 7.

Topological equivalence preserves many useful properties, and is often sufficient for application (especially to bifurcation theory, see section 4.7). However, there are pitfalls. Callahan [49] has shown that the topological type of the associated catastrophe map X : M --) B may Change within a topological equivalence class of germs. In fact it does so for ---°X9 when A = 0, +--4%/2/3; and for 2X9 when A = +_4V~(c + 1)/(1 - c) where 7c 3 - 5c 2 + 21c + 9 = O.

2.11. The non-gradient case

It is often thought that catastrophe theory applies only to gradient systems. Apart from

confusing general and elementary catastrophe theory, this places undue emphasis on a technical restriction which can often be avoided. Some clarification may be in order.

A function f : R " ~ R defines a gradient function df : R ~ -~ R n and a vectorfield Xj = (x, dr(x)) on R" whose integral curves satisfy

dx d---t = d/(x). (2.5)

Singularities of f produce zeros of d[, hence singularities of X~; and higher singularities of f produce singularities of dr. The problem of extending elementary catastrophe theory differs between the "stat ic" case of dr, or the "dynamic" case Xf.

A perturbation from f to g changes df to dg and X I to X s, and the latter remains a gradient vectorfield. Often it is unnatural to consider only gradient vectorfields (or perturbations): the direct translation of elementary catastrophe theory into vectorfield language then has little significance, except when the gradient setting or a generalization to gradientlike bifurcations (only point attractors) is appropriate. The latter is an important extension, since restriction to gradient differential equations (2.5) trivializes the dynamics.

Direct translation of results is usually less successful than generalization of methods, and here some progress is possible. Takens [240] has classified singularities of vectorfields on R 2, and their unfoldings, for codimension ~<2. At codimension 3 global bifurcations occur, making even the definition of a universal unfolding problematic. Elementary catastrophe theory and singularity theory may be applied to Hopf bifurcation to a limit cycle, and generalizations of this (Takens [239], Golubitsky and Langford [89], section 5.8 below), with some success. As regards "chaos" and strange attractors, singularity theory has yet offered little.

In the "stat ic" interpretation of df as a map R " - > R n, much more has been done. Not every

258 L Stewart/Physical applications of catastrophe theory

map h : R" ~ R" is a gradient df ; but the methods used in e lementary catastrophe theory readily extend to the case of general mappings R" ~ R m, including therefore non-gradient mappings R" R" as well as much else. This generalization is usually called Singularity Theory. We outline some sample results: for more details see Gibson [78] or Martinet [146].

Recall that Enm is the set of germs f : R" ~ R " at 0.

Two germs f and g E E , ~ are (left-right)- equivalent (or isomorphic) if there are diffeomorphism germs qb : R" ~ R" and tk : R m Rm such that

g(ck(x)) = qJ(f(x)), x E R". (2.6)

That is, we permit coordinate changes in both R" and R m. (In e lementary catastrophe theory m = 1, and only shifts of origin 7 in R m are allowed; and not even these if the value of f, rather than dr, is important.)

Unfoldings and universal unfoldings are defined in the obvious way. The Jacobian ideal Aft) is replaced by the natural " tangent space" T(f ) in E.m relative to the transformations (2.6), and the codimension cod(f) is defined to be the dimension of the vector space m.m/T([), where

m,m = {f E E,m If(0) = 0}. A germ of finite codimension has a unique universal unfolding (up to equivalence), which can be computed from a basis for mnra/T(f).

There is a technical problem with this set-up: T(f ) is not an E,-module. To overcome this the equivalence relation is changed to contact equivalence (or V-isomorphism, Martinet [146]; or K-equivalence, Gibson [78]) as originally defined by Mather [149]. This can be formulated by replacing (2.2) above with the condition (for f(O)=O=g(O))

g(x) = r(x)" f(c~(x)), (2.7)

where r (y) is a family of m×m matrices smoothly parametrized by y E R " (that is,

r :R" ~GLm(R)) . Roughly, f and g are contact equivalent if their graphs have the same contact with 0.

Now the tangent space T(f ) is modified to the natural analogue

[ o f . of }+E.m{f,. .fro} Tv(f)=E" tOXl . . . . ox . " '"

and the contact codimension of f is coda(f)= dim m,~,/T~.(f).

The interplay between the two equivalence relations is intricate, but simplifies in one useful case. Define f to be (structurally) stable if any nearby g is equivalent (in the sense (2.2)) to f. Then two stable germs f and g are equivalent if and only if they are contact equivalent.

Moreover , the stable germs are dense (that is, any germ can be approximated arbitrarily closely by a stable one) provided the dimensions (n, m) are nice in the following sense. Let q = m - n ; then (n, m) is nice if and only if: q~>4, m < 7 q + 8 ; q = 3 , 2 , 1,0, m < 7 q + 9 ; q = - l , m <8 ; q = - 2 , m <6 ; q~<-3, m < 7 (see Golubitsky and Guillemin [87] p. 163). Note that (n, m) is always nice if n ~<7 or m ~<5.

Classification theorems exist, but tend not to be written out in full. Thus Gibson [78] proves results which classify all stable singularities R" ~ R m for n, m ~< 4 (table II).

2.12. The equivariant case

A very common structurally unstable family is

X 4 X 2 F~ (x)=~- -A -~-

for which the equation dFa/dx=O gives the familiar pitchfork bifurcation diagram (fig. 6)

x3-Ax =0.

Why is it that a structurally unstable family is so


Table II Stable Singularities for n, m ~< 4. The symbol 2 ijk.. denotes the T h o m - B o a r d m a n singularity type, Gibson [78].

Spaces Stable germs Type

R ~ R x~x 2 ° X__)X 2 21o

R -') R 2 x-*(X, O) 2 °

R ~ R 3 x~(x,O,O) 2 ° R ~ R 4 x~ (x , O, O, O) 2 0 R2"-~R (x, y)--~x 2 j

(x, y)~-+x2-+y 2 250 R2-~ R 2 (x, y)~(x, y) 2 0

(x, y)-'*(x, y2) 21o (x, y)~(X, y3+xy) 2 II°

R 2 + R 3 (x, y)~(x, y, 0) 2 ° (X, y)-->(X, xy, y2) 210

R 2 ~ R 4 (x, y)~(x, y, 0, 0) 2 0 R3-~ R (x, y, z)-'~x 2 2

(X, y, Z) --~ -----X 2 --+ y2 ~ Z2 230 R 3 --, R 2 (x, y, z)~(x, y) 2 ~

(x, y, z)--,(x,-+ y2-+z2) 2 20 (X, y, Z)--->(X, +---y2+z3+XZ) 2 21°

R 3 -~ R 3 (x, y, z)-~(x, y, z) 2 0 (x, y, z)~(x, y, z 2) 2 ~° (x, y, z)-~(x, y, z% xz) 2 "0 (x, y, z)--,(x, y, z%xz+yz 2) 2 m°

R 3 ~ R 4 ( x , y , z ) ~ ( x , y , z , O ) 2 0 (X, y, Z)-"~'(X, y, Z, Z 2) 2 10

R 4 ~ R (x, y, z, t )--,x 23 (x, y, z, t)--~-+x2-+y2-+z2---t2 2 40

R 4--~ R 2 (X, y, Z, t)---~(x, y) 2 2 (X, y, Z, t)--~(X, -----y2-----Z2-----t2) 2 30 (X, y, Z, t)"~(X, -----y2--+Z2+t3+.~t) 2 31°

R4"-'~R 3 ( x , y , z , t ) - ' ( x , y , z ) 21 (X, y, Z, t)'~(X, y, - -z2+t 2) 2 20 (X, y, Z, t) --~ (X, y, +- Z 2 + t 3 + x t ) 2 21° (X, y, Z, t )-.~(X, y, +--z2+t4+xt + yt 2) 2 21'°

R'-'~R 4 ( x , y , z , t ) - ~ ( x , y , z , t ) 2 0

(x, y, z, t)-~(x, y, z, t 2) 2 ~° (x, y, z, t)-~(x, y, z, t3+xt) 2 "0 (x, y, z, t)-'+(x, y, z, t4+ xt + yt 2) 2 m° (x, y, z, t )~(x , y, z, tS+xt + yt2+zt 3) 2 m~° (X, y, Z, t)-'*(X, y, zt, z 2+t2+xz + yt) 2 50

common, if the exceptions to the classification theorem are " r a r e"? The answer seems to be symmetry. F x ( x ) is an even function in x, that is, symmetric under the transformation x ~ - x . If the class of perturbations permitted is restricted to even functions as well, then FA(x) becomes structurally s tab le within the symmetry class.

Fig. 6. The pitchfork: a Z2-equivariant bifurcation.

In general, when symmetries are present, and when the perturbations of interest may be expected to share those symmetries, it is more appropriate to use an equ ivar ian t version of catastrophe theory, which respects the symmetry. Po6naru [174] has developed such a theory, and it has been applied by Golubitsky and Schaeffer [91] as we discuss in section 4.6.

Classifications of equivariant germs for crystallographic point groups are given by Ascher, Gay and Poston [26] and Poston [180]. Was- sermann [280] considers compact abelian groups. Z2-actions are discussed by Golubitsky and Langford [89] as reductions of circle-group actions.

2.13. O t h e r genera l i za t ions

Elementary catastrophe theory, and its methods, have been extended to numerous settings, each of potential interest for applications. The non-gradient and equivariant cases above are examples. In addition, we mention here: s p a c e - t i m e catastrophes, Wassermann [279]; catastrophes on manifolds with boundaries or corners, Pitt and Poston [173], Arnol 'd [25], Siersma [217]; infinite-dimensional state spaces, Chillingworth [53], Magnus [134-137], Arkeryd [7-9], Michor [161], Quinn [186], Shearer [215]. For applications to bifurcation theory, see section 4.8. Golubitsky and Schaeffer [90-94] have developed a form of catastrophe theory especially appropriate to bifurcation theory, by the simple device of throwing in a distinguished parameter: see section 4.2.

260 L Stewart/Physical applications of catastrophe theory

The potential for generalization of the results and methods of 'e lementary catastrophe theory ' was evident early on, and this is one reason why it was at first greeted with enthusiasm. The realization of this potential makes the common assumption in the critical literature, that 'catastrophe theory ' and 'Thorn's Theorem' (i.e. table I) are essentially identical, unreasonably res- trictive.

2.14. Globalization

b c a

Fig. 7. With 1 state variable, only (a) is realizable as a bifurcation set of a global catastrophe. If there are 2 state variables, all three are realizable.

There are two global settings of interest. The analogue of elementary catastrophe theory concerns parametrized families of smooth maps

F : X x C ~ R ,

where X and C are manifolds. The analogue of singularity theory deals with maps

f : M ~ N ,

where M and N are manifolds. Notions of equivalence are obtained by replacing germs by globally defined functions, and diffeomorphism germs by globally defined diffeomorphisms. Hayden [106--108] gives a comprehensive survey and many new results, making heavy use of algebraic topology. We content ourselves with some simple examples, and concentrate on the analogue of e lementary catastrophe theory.

One series of results asserts the homological triviality (mod 2) of various singularity sets. As a low-dimensional example, suppose X and C are compact , d i m X = l , d i m C = 2 , and f : X x C ~ R is generic. Then the bifurcation set of f (defined much as in section 2.8 above) consists of finitely many closed curves, each smooth except for an even number of cusps. For instance, fig. 7a may occur, but figs 7b, c cannot. If dim X =2, both of the latter can occur (note that (b) happens in the elliptic umbilic). Thus the dimension of the state space X can

influence the possible forms for the bifurcation set.

Define f : X × C ~ R to be locally stable if its germ is stable near any point. For dim C~<5 locally stable maps are dense (Zeeman [291]) and their singularities are all of elementary catastrophe type.

Markus [142] solves the extension problem for locally stable maps. Le t P be a compact connected r-manifold whose boundary OP is embedded in an r-manifold Q, and let F : X × OP ~ R be locally stable. Then there exists a locally stable extension G : X × P ~ R. Hayden [106] shows that if G1 and G2 are any two such extensions, with compact catastrophe manifolds, they have the same number (mod2) of singularities of a given type E of codimension r.

A closely related problem is exemplified by the following result. Suppose F : X × C ~ R where d i m C = 2 . Let D be a disc in C and suppose that above the boundary OD the catastrophe manifold M has two fold points, arranged as in the cusp catastrophe (fig. 8), and is otherwise regular. Then, if F is generic, M has an odd number (hence at least one) of cusp points inside D. This theorem is proved in various versions by Whitney [282], Zeeman [298], Poston [178]. It underlies many of Zeeman's more tentative applications of catastrophe theory outside physics (Poston [178]) and appears to have applications elsewhere (Stewart [227-228]). A generalization to singularities of


Fig. 8. Boundary conditions implying a cusp point.

type Ak holds. Note that when dim X =l there must be at least one cusp on the bifurcation set that is connected to the fold points on the boundary as in fig. 9; but for dim X~>2 fig. 9b can occur.

A final example: if C is orientable, then so is the catastrophe manifold M, for any generic [ : X x C--*R.

3. Optics

3.1. Geometrical optics

In geometrical optics, and related fields such as acoustics, catastrophe theory facilitates the description and analysis of caustics and wavefront singularities. Its emphasis on structural stability renders it applicable to naturally occurring caustics, lacking any special symmetries: it is thus complementary to traditional optical theories which generally assume a high degree of symmetry, appropriate to carefully

a b

Fig. 9. Whether the cusp point must join to the boundary depends on the dimension of the state space.

designed optical instruments. A forthcoming survey of this catastrophe optics, Berry and Upstill [43], runs to some 88 pages, and we shall , be less ambitious here.

The involvement of catastrophe theory may be seen as follows. Consider light rays emitted at some fixed initial position or direction P and observed at a variable position or direction c. Let Xc be the set of all paths from P to c. By Fermat's Principle a path x~Xc represents a ray if and only if the time T(x, c) taken to traverse it is stationary.

Ignoring various technicalities, assume Xc finite-dimensional: then dxT(x,c)=O and elementary catastrophe theory applies provided (as is usual) T is smooth. In particular the caustics correspond to the bifurcation set of T.

There are various ways to make this idea precise. Often there is an obvious, though ad hoc way of restricting X~ to some finite-dimensional set, parametrized independently of c by a space X', and containing all plausible physical paths. In a medium of constant refractive index, piecewise linear paths with vertices on special surfaces (representing lenses, mirrors, etc.) usually suffice; for a simple example see Poston and Stewart [183] p. 250. The unfolding methods of section 2.6 apply, and the geometry of a caustic follows from that of the corresponding singularity.

However, from a theoretical viewpoint, an ad hoc choice for X' fits uneasily with generic classifications. (In practice, reasonable choices yield structurally stable caustics generically, and it is generic to be reasonable- but this is a little hard to formalize.) More satisfactory would be a version of elementary catastrophe theory in the infinite-dimensional space X=U~Xc of all (perhaps piecewise smooth) paths. The recent progress in developing catastrophe theory on Hilbert or Banach spaces suggests that this can indeed be done, but an explicit treatment for optics appears not to have been published. By analysing the singularities of the wave-function, a wave-optical treatment should be feasible.

262 I. Stewart~Physical applications of catastrophe theory

A third approach, by way of Lagrangian manifolds (Maslov [147]) has distinct mathematical advantages, and leads to the key idea of an oscillatory integral, where the connect ion with elementary catastrophes runs deep. Alternatives to Maslov's formalism also lead to oscillatory integrals, with the same results. The mathematical theory has been developed by Arnol 'd [17], Duistermaat [70, 71], Guillemin and Schaeffer [103], and Guckenheimer [101]. Our discussion below is based on these, together with expositions by Chazarain [50] and Poston and Stewart

[1831.

3.2. Asymptotic solutions to wave equations

Let t ~ R, x = (x~ . . . . . x , ) E R", and let v = v(t, x) be a solution to the wave equation

tor of the form

{x 1 0 ~.)=k~_o~.,, kp k {x 1 0 P \ ' i Ox" \ '-i-~x/'

where the degree of Pk is at most k and its coefficients are smooth and defined on an open set X C_ R". Define the principal symbol of P to

be

f(x, ~)=~ Pk(x, ~), k=0

where ~ E R" and pk is the homogeneous part of Pk of degree k. For example in (3.1) we have m=2, Po(x,,~l,,~2, 2 2 2 ~3)=-(~1+~z+~3), PI=0, Po=l , so

f ( x , 2 2 2 ~)= 1-~,-~2-#3.

02/) ~ 2 - - ~ - - V V

( w h e r e V2=02/0x~q - . - .--[-02/0x2). Call v stationary if

v(t,x)=e'~tu(x).

Then u satisfies

(V2+rZ)u =0. (3.1)

Hencefor th we require f to be real, and OJ(x, ~)~0 when f(x, ~)=0. This fails in example (3.2) at points where V(x)=E. Here 0 J =

(of]o~ . . . . . of]o~.). The asymptotic behaviour of solutions to

Pu=O as ~-~oo may be studied by introducing the notion of an asymptotic solution, a smooth function u(x, ,c) such that

P \[x, iox,1 0 ,1") u(x,'r)=G('r ~),

Similarly for the Schr6dinger equation

ih av+ h 2 Ot 2m V2v-V(x)v=O

a stationary solution v(t, x ) = e itE/hU(X) satisfies

( V 2 + ~ - ( E - V ( x ) ) ) u=O, (3.2)

and 1/h plays a role analogous to ~- above. In particular small h corresponds to large r.

More generally, consider a differential opera-

where the right-hand side indicates a function decreasing rapidly with r (uniformly on compact sets). A typical boundary condition is that on some hypersurface SC_X we have u= Uo(X, ~)=e~°(X)b(x). We seek a solution

u(x, .r)=ei'*~X)a(x, .r), (3.3)

where 4>:R" ~ R is a smooth phase and a :R" × R ~ R is the amplitude. Let a have an asymptotic expansion (for large r)

a(x, r)-';"[ao(x)+al(x)r-%a2(x)r-2+ . . .].

Substituting this in Pu =0 we get S

O=Pu ~q-"÷~'[f(x, d~b(x))ao(x)]+. • . , (3.4)

where f is the principal symbol of P and dtk = (O~)]OX 1 . . . . . O~)/OXn). For a solution with a0(x) not identically zero we must impose the con-

dition

f(x, d~(x) )=0 , (3.5)

called the characteristic equation. We may now use Hamil ton-Jacobi methods (subject to a mild transversality condition on f). There is a vectorfield f(x, d~b(x)) whose integral curves are the characteristics (some authors prefer bicharac- teristics) of f. Terms in r"+"÷J in (3.4) suc- cessively determine a0, al . . . . as solutions to certain linear differential transport equations.

This method is valid locally, near S, but it breaks down on caustics. For example if S is a wavefront in a medium of constant refractive index then characteristics are rays normal to S by Huyghens ' principle. If S is a parabola these rays envelop a cusped caustic (fig. 10). On a set U avoiding the caustic, initial data propagate uniquely along rays, but at a caustic rays overlap. More general methods are needed to obtain global solutions in the presence of caustics.


Fig. 10. Propagation of a parabolic wave front leads to a cusped caustic.

dimension of A, is correct ly represented. Now A6 has four simple properties:

(i) A6 is an n-dimensional submanifold of T ' X ; (ii) The symplectic 2-form to = E d~j ^dx~

vanishes on A,; (iii) f vanishes on A6; (iv) The projection (x, ~ ) ~ x from A~ to U is one-one.

Moreover , A6 determines ~b, so we may think of A~ itself as the solution to the problem. Maslov suggested dropping the awkward condition (iv), the obstacle to global extension of solutions. Define a Lagrangian manifold A C_ T * X to be an n-dimensional submanifold on which to vanishes; it is a Lagrangian solution if

3.3. Lagrangian manifolds

Maslov [147] introduced the notion of a Lagrangian manifold to circumvent this difficulty. Suppose that tk is defined on U and satisfies (3.5). The graph of d~b is

A~ = {(x, dtk(x)) I x E U) C_ T ' X ,

where T * X denotes the cotangent bundle, a topologist 's formulation of a 'global phase space'. Fig. 11 illustrates this, but with the "vert ical" dimension R n collapsed to 1 for artistic reasons, for X 2-dimensional. The Fig. l 1. Lagrangian singularities, after Chazarain [50].


further f vanishes on it. That is, A satisfies (i), (ii) and (iii) above. If 7r : A ~ X is the projection, 7r(x, ~) = x, then the set 2 of critical points of 7r is the singular set of A, and its image B = 7r(X) is the caustic. (Note the analogy with a catastrophe manifold and its bifurcation set.)

To construct Lagrangian solutions of (5) we seek integral curves in T * X of the Hamiltonian vectorfield

Hs(x, ~i) = ( ad (x, ~), - o f f ( x , ~)),

called (bi)characteristic strips by Duistermaat [71]. These are drawn in fig. l l as curves on A: their projections into X are the characteristics, and even when these meet, the characteristic strips above them do not. If a characteristic strip meets a Lagrangian solution then it lies entirely inside one, so the solution A, may be extended globally to A by flowing along the characteristic strips.

This is the geometric view. For asymptotic computations, Maslov suggested using oscillatory integrals

u,o(x, = f ei**(x'")a(x, 1-, a) da, R m

(3.6)

where a C R " is a parameter. Intuitively (3.6) represents an infinite superposition of stationary waves eiT*a. The amplitude a is smooth, and zero outside some compact set (to ensure convergence); it has an asymptotic expansion

a(x, r, ~) ~ z~[a0(x, a) + a,(x, ~)~- '

+ a2(x, ot)'r -2 + • " "].

An oscillatory function associated with a Lagrangian manifold A is a locally finite (in x) sum of terms u+,, where ~b runs over the phases of A. If OA denotes the class of all such functions, it may be proved (Duistermaat [71]) that a global asymptotic solution to the wave equation

exists in OA. The proof makes use of the principal symbol and extends some results of H6r- mander on Fourier integral operators.

The local classification of singularities of the project ion map ~r : A---> X for a Lagrangian manifold A is identical with that for the catastrophe map x : M ~ C of an elementary catastrophe; and locally every catastrophe manifold can be viewed as a Lagrangian manifold in T*C (Weinstein [281]). However , not every Lagran- gian manifold can be obtained globally by this method: the Maslov index (Arnol'd [10]) is an obstruction. In fact, there exist manifolds which occur as Lagrangian submanifolds but cannot be catastrophe manifolds at all. Hayden and Zeeman [109] construct a Lagrangian Klein bottle in T*R 2. This cannot be a catastrophe manifold because it is not orientable, whereas R 2 is (see section 2.14).

3.4. Local asymptotics

The next step is to study the asymptotic behaviour of oscillatory integrals u,o near a point x0, as ~-~oo. By repeated partial in- tegration (as in the method of stationary phase) it follows that the only contributions not decreasing rapidly with ~- occur on the critical set

c , = {(x, '~)l 4,'(x, ~) = o}

and depend only on the Taylor expansions of ~b and a" and C,. There are three cases: (i) The shadow-zone: xo~ ,r(A). Then C6 is empty and u,o is rapidly decreasing; (ii) The illuminated zone: x0E1r(A), Xo~-B. Then (x0, a0) E C, for some a0, and the Hessian H~b is nonsingular at (x0, a0). That is, (x0, a0) is a Morse point of ~b. Here the classical method of stationary phase provides the estimate

u,a(x, ~) z~(2w/~-) "/2

det H4~ [a0(x0, a0)

+ ~(,r-1)] ei~+(xo.~o)+#,m), (3.7)


where Htk is evaluated at (x0, a0) and tr is its signature as .a quadratic form. The phase factor ~r~/4 is constant on any sheet of C~, and is significant only when the contributions from separate sheets are summed; (iii) The caustic: Xo E B. Then det H~b = 0 and (2.7) blows up at the degenerate singularity of ~b. Finding a replacement is a local problem, solved by catastrophe theory. The main objective is to compute the order of the caustic, defined as the least upper bound K of the numbers v such that u(x, T) = ~ ( ~ ) near x0. Suppose we use a right equivalence to change the variables in u~, say

variables and the original variables x are con- trols. Associated to each canonical form is a unique "special funct ion"

F(x) = f e it°(~)+~'°'(~)] da. (3.10) R m

For the fold catastrophe this is the classical Airy [1] function Ai(x): for the cusp it was investigated by Pearcey [171].

The factor z can be scaled out of the exponential in (3.6) provided that (3.9) is quasihomogeneous, that is, there exist numbers r i such that

dP(X (x), A(x, a)) = ~b(x, a) + T(x).

Then

U(X, ~) = f ei~*(x'A)b(X, A, ~) dA

becomes

U (X (x), ¢) = e i~v(~) f ei'~'tx'~)a(x, a, "r) da, J

(3.8) O(t"ot l . . . . . t '~otm) = tO(a t . . . . . Otto)

identically for t > 0. It is a curious consequence of the classifcat ion theorem that this is true up to high codimensions. We may assume that 0 < rj ~<~: they are then uniquely defined rational numbers. We can also arrange for each 01 to be a monomial, say

Ol(a) = af" . . , a~'.

where Set

a(x, a, I") = b (X (x), A(x, a), "0 " det d~A(x, a).

Apart from the oscillatory factor e ~(x) the asymptotic behaviour is the same. Thus we may replace $ by any right equivalent function: in particular, put it into any convenient canonical form. If the codimension of $(x, ao) at x0 is finite (and not too large) and if a versally un- folds it around a0 (which is generically true) then the classification theorem provides a list of the required cases; and we seek a single asymptotic estimate for each type of catastrophe.

We then have a canonical form

O ( a ) + x , O , ( a ) + . • • + x . O . ( a ) (3.9)

for ok. Note that the parameters a define state

st = ~. slirj > O, I

Then (Duistermaat [71]) up to oscillatory factors e i~*(x) there is an asymptotic formula

U , a ( X , "r) "r ~ - g r l - v v r ( x ) F ( ' r l - s t g t . . . . . "rl-s"xn) r=0

+ ~ vt(x)r-" W-"x,

(3.11)

for r ~ ~, uniformly for x in a fixed neighbourhood of Xo in R". Here the functions v' and vt are determined by the asymptotic development of a, that is, by a0, at . . . . . in a computable way.

If the function F is bounded for all x (as may


be shown for the simple singularities, hence in particular for codimension ~<5, essentially by induction on the codimension of the singularity, see Duistermaat [71]) it follows that the order of the caustic is given by

m ri.

J

This leads to Table III below, which also includes information on the earliest detailed in- vestigation of the corresponding special function F. Further work on these special functions is desirable in view of their universality. Some more extensive computations of • are given by Arnol'd [17, 19] as singularity indices.

3.5. Crystal scattering

Berry [34] applies an optical rippling mirror model (Garibaldi et al. [74]) to the scattering of atoms off a crystal surface, deducing the general form of the caustic by a topological analysis of the critical point structure of a Wigner-Seitz lattice cell, and a brief appeal to Thorn's classification theorem. The simplest generic caustic consists of two curves; the outer one smooth, the inner with four cusps. For non- rectangular lattices, six cusps will occur. This calculation provides the first theoretical evidence for the occurrence of cusps in rainbow

Table IIl Orders of caust ics for singularities of codimens ion ~<5, and their associa ted special func t ions

Singularity Order Special function

A2 fold 1/6 +-A3 cusp 1/4

A4 swallowtail 3/10 -+ A5 butterfly 1/3

A6 wigwam 5/14 Di elliptic umbilic I/3 D~ hyperbolic umbilic 1/3

-+ D5 parabolic umbilic 3/8 D6 2nd elliptic umbilic 2/5 D~" 2nd hyperbolic umbilic 2/5

-+ E6 symbolic umbilic 5/12

Airy ll] Pearcey [171] Wright [283]

Berry, Nye and Wright [42] Wright (unpublished, partial)

lines for classical scattering of particles from surfaces: the numerical work of McClure [157, 158] suggests the presence of folds, but is too coarse to show cusps.

The classical caustic is blurred by three effects: diffraction, which decorates the caustic with Airy and Pearcey fringes (see also Poston and Stewart [183] pp. 267-271) with intensity - h -1/3 on fold lines, - h 1/2 at cusps, via the singularity index; Bragg diffraction; and thermal noise. All three are discussed in detail.

3.6. Refraction by liquid droplets

Light refracted through an irregular droplet produces only folds and cusps in the far field (Berry [35]). By varying the screen position A4 and D4 can also be stably observed. They are loci associated with umbilic points on the droplet surface (or more accurately the wavefront, which for thin drops approximates the droplet surface) (Nye [163]), that is, points at which the principal curvatures are equal.

In the absence of gravitational distortions, D~ and A4 occur but D~ does not (Nye [163]). Further, Poisson's equation implies that all D4 points occur at (approximately) the same height above the droplet, so that it has a well-defined plane of focus. Experiments (Nye [163] plate 5g) confirm this. Changes in the form of the caustic as the plane section under observation varies may be classified, and observed experimentally.

Nye [164] considers further the effects of gravity, and of tilting the droplet, varying its boundary, or altering the quantity of liquid. Gravity alone permits the stable occurrence of DL Two sequences of interactions between umbilic points are commonly observed (fig. 12) and these are partially explained by considering sections of the higher catastrophes D5 and E6, which appear to act as "organizing centres". These in turn may be observed in full by permitting further physical variables as parameters (shape of drop boundary, quantity of liquid).


a

b

Fig. 12. Typical sequences of caustics observed in light refracted by a liquid droplet, after Nye [164].

The global topology has extra features which suggest that still higher catastrophes are involved as organizing centres that compactify (Thom [242]) the singularities involved. A likely candidate is the double-cusp Xg. Apart from its organization role, catastrophe theory is used here to justify the Taylor series truncations used (by appeal to classification and structural stability, although it could be done directly via the Mather determinacy criteria of section 2.6).

The nature of umbilic points in general has been studied by Berry and Hannay [39] and Porteous [177] who combine three different classifications: elementary catastrophe (elliptic or hyperbolic), topological (index ½ or -½), and differential-geometric (pattern of lines of cur- vature of "lemon", "star" or "monstar" type).

A detailed study of the D~ oscillatory integral and its diffraction pattern is given in Berry, Nye and Wright [42]. The results are compared with experimental observations of triangular droplet diffraction. The theoretical results "reproduce virtually every detail of the experimental pic- tures" (Nye [165]), and a typical comparison is shown in fig. 13. The integral is studied in the form

lff E(x, y, z) = ~ exp[i(~ 3 - 3~7~ 2 -- Z (~ 2

-oo

- x~ - yrl)] d~ d ~ ,

-I- 1~ 2)

a function also studied by Trinkaus and Drepper [268]. At the level z = 0 of the umbilic focus,

there is a triple-rayed diffraction star, many experimental examples of which are given in Nye [163, 164, 165]. As z moves away from 0 a three-cusped caustic, with the expected pattern of Pearcey functions at the cusps and Airy folds, appears. In its interior a complicated pattern of distorted hexagonal cells of varying brightness appears. These are caused by a structure of diffraction maxima stacked like the atoms of a distorted crystal whose space group is of type R3m. The point maxima are separated by lines on which IE[ vanishes, forming wavefront dislocations (Nye and Berry [167]), discussed elsewhere under the name quantized vortices (Riess [191-193]; Hirschfelder and Christoph [ll3]; Hirschfelder, Goebel and Bruch [114]; Hirschfelder and Tang [115, 116]). These are also explained in terms of a four-wave theory. The dislocation lines form distorted hexagons near the z-axis, but change to "hair- pins" near the cusp lines. This is explained quantitatively. Dislocations outside the caustic are also discussed. A sample of the com- putational results is given in fig. 14.

3.7. Junctions in caustic networks

Berry and Nye [41] discuss the apparent triple-junctions in networks of caustics formed by water-waves. Triple junctions occur stably in a variety of circumstances (cracks in mud, the markings on giraffes), but catastrophe theory shows that they cannot occur stably in caustics in R 2, so some more delicate mechanism must be responsible. Berry and Nye suggest that the apparent network of caustic lines is a poorly resolved image of a caustic with finer structure. Their experiments revealed the fine structure of fig. 15, with each apparent line being doubled, and a cusped curve sited at the junction. The diagram is similar to suitable sections of the double-cusp °X9, and this was conjectured to be an organizing centre. Since it is the first compact catastrophe beyond the butterfly, and optical caustics ultimately must be embedded in

Fig. 13(a)

Fig. 13(b)


theoretical and experimental results on the caustic patterns produced by superimposed sinusoidal wavetrains, where the unresolved network of fig. 16a has, for example, the fine structure of fig. 16b, an array of overlapping "giant lips". The relation of these ideas to °)(9, and considerable information on that singularity, is given in Upstill [271].

3.8. Random media

If a wave encounters a random structure S then its wave function and intensity acquire randomness. Examples in practice include twinkling starlight and fluctuations in radio waves caused by atmospheric randomness, the scintillations of sunlight reflected off water, and sound or radio waves reflected off a landscape.

Berry and Hannay [39] not only classify umbilics: they also find their statistical dis- tribution on a Gaussian random surface. The proportion of umbilic points of a given type is computed as follows (in the isotropic case):

Fig. 13(c)

Fig. 13. Theory (top) and experiment (bottom) compared for the elliptic umbilic diffraction catastrophe. After Berry, Nye and Wright [42].

compact catastrophes, its occurrence is likely; and has since been confirmed in a variety of situations. The theory of caustic networks develops at various levels. First, there is the poorly resolved network of lines. At a finer level, each line becomes a pair of folds, and the main feature becomes the junction between pairs: here catastrophe theory provides a strong grip on the geometry, mostly by way of °X9. At a further level, the organization of clusters of junctions is presumably described by sections of some still higher compact catastrophe: the mathematical complexity is probably too great for more than exploratory results.

Upstill [271, 272] has obtained detailed

elliptic 0.268.

hyperbolic 0.732.

index ½ 0.500.

index -½ 0.500.

lemon 0.447.

star 0.500.

monstar 0.053.

The non-isotropic case is also considered. The low number for the monstar may help to explain why it has gone largely unnoticed in the literature.

There is in general no simple relation between the statistics of the random structure S and of the wave function ~ or its intensity. Fluc- tuations in the intensity are described by the moments

=


6 B D B D B D B B D B B D B B B D B B B B D I B B B B B I D I B B B B B B B B B B o 81 1"~ ~ -~- .-~--~-~ -- ~ -~"~ , -rv~-~- ' -r~-~-~----~-_.r--~. ~ ¢ - - - ~ $ ~ . . . . . . .

• I . / ' / ~ . . i - ~ . . ~ dP ~ ,---, ~ , , . , . . . -- . C ; b ~ ~ ,~--.(C'$"-,,D-..I_.'~ ~ . ~ - s . . , . , ,~ :~-~ . . d , ~ r - - " - --

" - 0 .-.,, . . . . . . . . . . . . . . ~ ~ . . ~ < ~ . . . . . . . . , . ; ; r ' ,£ ' --~ , _ ~ - " ~ l ' ~ . z - ~ > _ ~ o ~ o . £ " - ~ o ~-.-,,, ~ _ ~ ~ - ~ . - r . ~ - - , . < J-

/ L > < / . . < ; ) . - . . . / I - ' ) ~ . ~ / - ~ . ' , , , , S - - ~ . . . ."~,. , ._/--~ - , . . . . . . . . . . . . , . . . X ,

v • / / ,, ~ ',, ~ • \ \ \ \ ~

• ; • -~r, '~ \ , , ' \ \

/ / " I \ "~ \ \ \ \ ,, , , ,

/ / / , i z ' , \ \ \ \ \ . \ ~ . . . . , 2

o I , i I t , , 1 1 , , , -10 0 10 20 30

X

Fig. 14. Plane section of the elliptic umbilic diffraction catastrophe, showing the caustic (chain line), distorted lattice planes (broken lines), and dislocation lines (thick). B, D, I signify bright, dark and intermediate hexagonal cells. After Berry, Nye and Wright [42] fig. 8.

averaged over the ensemble of possible S. If

has Gaussian statistics then

V n I. = n . I i .

This arises at great distances behind finite slabs

of media (Mercier [160]) but numerical cal-

culation of I2 (Mercier [160] Bramley and Young [44]) and experimental data (Jakeman et al.

[122], Pusey and Jakeman [185]) show that it

fails even if the medium itself has Gaussian statistics•

In this case, Berry [36, 37] has computed the short-wave asymptotics of I, for (in principle)

Fig. 15. Fine structure in a caustic triple-junction, after Berry and Nye [41].

all n when the space in which the light pro-

pagates has dimension d = 2 , and for n ~< 13

(subject to some plausible assumptions) when d = 3. The method makes extensive use of the

classif ication of elementary catastrophes. Define critical exponents v, so that I, - )t ~. as

the wavelength A-->0. The ensemble of possible S is defined by a torus of random phases

0, . . . . , ON, and ensemble averages are computed

Fig. 16. Caustic from two interfering wave-trains, (a) unresolved, (b) resolved to show the fine structure of overlapping "giant lips". After Upstill [271,272].

L Stewart[Physical applications of catastrophe theory 271

by integrating over this torus. For a Gaussian random medium we make N --> oo.

For large enough N, any given catastrophe ~b contributes to v, provided its corank is less than or equal to d - 1. High codimension catastrophes occur rarely, but give rise to large intensity fluctuations and hence tend to dominate higher moments. The computations support this heuristic belief, and show that generally only one or two catastrophes dominate. The problem is resolved by finding how much each catastrophe contributes, and selecting the dominant ones.

For a catastrophe ~b, let K(~k) be its order as defined in section 3.4, and let V(tk) be a measure of the size of its brightest region: the hyper- volume of the diffraction maximum is asymptotically h ~*) as A ~ 0 . Thus for the fold catastrophe, K = 16, V = ~. The rational number y is called the Berry index or fringe index, and it is invariant under diffeomorphism, Romero [195]. The analysis shows that

v. = max,(2nK(~b)- 7(~b)).

For d = 2 the only catastrophes that need be considered are those of type Aj. Any particular vn may be computed explicitly, although there is no simple analytic formula. The approximation

Here X denotes one of the family of catastrophes X4-Fx2y2W ay j (j I> 4) Wl2 = X4"~ y5 ~_ ax2y3, W13 = x 4+ xy 4q- ay 6. There are some sources of

uncertainty about this table related to the present lack of knowledge of the full classification. It is not known which members of the X family dominate for n = 6-10. Catastro- phes not included in the existing classification might alter this table, but the trend so far visible makes this unlikely. Even an approximate general form for vn is not known.

These predictions are capable of experimental testing, and this is currently in progress. Their failure, however , would tend to demonstrate that the medium concerned is non-Gaussian, rather than to cast doubt upon the theory.

Berry [37, 38] discusses also the case when the random structure S is "rough on all scales", that is, fractal in the sense of Mandelbrot [141]: he calls the resulting waveforms diffractals. They pose new problems for the computat ion of short-wave asymptotics. Berry and Upstill [43] explain how catastrophe optics may be viewed within this more general context , where wave motion is viewed in terms of the interplay between three regimes, depending on the scale of structure relative to wavelength: the main features being caustics, diffractals and dislocations.

3.9. Other applications to optics and related fields

is very accurate. Results for n ~< 13 are: Applications of catastrophe theory to optics

n 2 3 4 5 6 7 dominant catastrophe A2 A2 A3 A3 A4 A4 Vn 0 1/3 3/4 5/4 915 12/5 V, (approx) 0.051 0.338 0.758 1.257 1.810 2.401

8 9

A4, As A~ 3 1113 3.023 3.669

10 11 12 13

As As, A6 A6 A6 13/3 5 40/7 45/7 4.336 5.019 5.718 6.429

For d = 3 the results are different:

n 2 3 dominant catastrophe A2 A2, D4 v, 0 1/3

4 5 6 7 8 9 10 11 12 13

D4 D4, E6 E6, X X X X X, W12 Wn Wi2 W13 1 5/3 5/2 7/2 9/2 11/2 13/2 38/5 87/10 157/16


are not confined to caustics. Wavefront singularities and their pattern of evolution have been classified by Arnol'd [23] up to codimension 6 (and applied to a simple model of Zeldo- vi~ for the shape of galaxies). The stability of wavefront singularities is studied by Rand [187]. Arnol'd calls them Legendre singularities to distinguish them from Lagrange singularities (i.e. singularities of the projection map of a Lagrangian manifold). For technical reasons their classifications diverge at high codimension.

Mirages may be described by elementary catastrophes (Poston and Stewart [183])but this idea remains to be developed.

Caustics and oscillatory integrals have applications outside optics: for example to acoustics, hydrodynamics, and quantum mechanics. Some of these are sketched in Poston and Ste- wart [183] pp. 275-283. An extensive application to the development of uniform asymptotic semiclassical approximations to Feynman path integrals is given by Dangelmayr, Gfittinger and Veit [67] who give among other examples an application of the double-cusp to electron optics (Veit and Dangelmayr [273]).

Wright [283, 284] applies catastrophe theory to wavefront dislocations, relating it to complex wave theory.

Oscillatory integrals in molecular collison theory are studied by catastrophe-theoretic methods by Connor [61, 62].

Catastrophe optics also sheds light on some traditional analyses, where symmetry plays a greater role. Seidel aberrations from a perfect focus are discussed by Berry and Upstill [43], reporting unpublished work of Hannay, as a partial unfolding of the infinite-codimension degenerate double-cusp x 4+2x2y 2+y4. Their analysis lends weight to the conjecture (Poston and Stewart [183] p. 257) that classical aberrations represent some of the more important terms in the unfolding.

It appears likely that application of equivariant methods may shed further light, since this singularity has S~-symmetry: one is seeking cer-

tain symmetry-breaking imperfections. Similarly the glory (Berry and Upstill [43], Khare and Nussenzveig [127, 128]) has circular symmetry. Berry and Upstill [43] point out that the technique of uniform approximation is more general than catastrophe optics, since it appl ies- for example - to the glory: a suitable equivariant version of Duistermaat [71] might lead to a unified theory of equivariant catastrophe optics, bringing all these phenomena together. Current techniques should be able to decide this.

4. Elastic buckling and bifurcation theory

Elastic deformation of structures is governed by extremization of a potential. As for optics, the variational problem is infinite-dimensional; but may often be reduced to finite dimensions. Infinite-dimensional versions of catastrophe theory are also available for many problems.

The engineering literature contains numerous examples of elementary catastrophes (Sewell [206, 213], Thompson and Hunt [259, 260, 263]). This abundance illustrates not only their ubiquity, but also the apparent ability of the engineering community to handle such problems without explicit use of catastrophe theory. But the latter becomes essential in more recent work, involving higher singularities, and resolv- ing some problems concerning the rather ad hoc treatments found in the literature.

In particular the relation of catastrophe theory to bifurcation problems has been clarified by the work of Golubitsky and Schaeffer [90], who develop a direct generalization of elementary catastrophe theory to perturbed bifurcations. This technique applies very generally, unlike traditional methods of bifurcation theory which require solutions to branch from a trivial zero solution: an equivariant version [91] is also available. Golubitsky [86] considers these methods to be a part of catastrophe theory, since the mathematics is identical in spirit. According to Golubitsky and Schaeffer

L S t e w a r t ~ P h y s i c a l a p p l i c a t i o n s o f c a t a s t r o p h e t h e o r y 273

[90] a key step towards their method was the paper of Zeeman [293] relating catastrophe theory to buckling phenomena.

4.1. The Thompson-Hunt theory

Independently of catastrophe theory, Thom- pson and Hunt [259] developed a classification of certain buckling phenomena as part of a general perturbation-theoretic analysis. This applies (Thompson, Tan and Lim [266]) to a potential energy reducible to the form

F(x, A, •) = G(x) + AQ(x) + • l X l "3 L " " " "3 L E n X n ,

where x, • ~ R n and )t E R. A perturbation scheme eliminates passive coordinates (corresponding to zero eigenvalues of the Hessian) to leave only active coordinates xi relative to which the Taylor series of G starts with cubic or higher terms. Q is a negative definite quadratic form since A usually represents load. The ¢~ are called principal imperfections. Thompson and Hunt list the main types of buckling problem when n ~ 2, assuming F to be even in xz when n = 2 (semisymmetric buckling).

It was noticed almost immediately that the Thompson-Hunt list correlates closely with Thorn's list of elementary catastrophes via the singularity type of F(x, 0, 0). This is shown in table IV. For simplicity we have set (Xl, x2)=

(x, y) and scaled certain variables before writing down the formula for F.

In some cases several distinct buckling problems define the same elementary catastrophe (Thompson and Hunt [260]), an observation that in the case of Az goes back to Huseyin [119]). The reason (Thompson [256], Thompson and Hunt [260]) is that the parameter X plays a special role in bifurcation problems. Fixing the • i and varying A corresponds in general to a path (parametrized by A) in the control space of the relevant catastrophe; and in some cases several distinct paths must be considered. Thomspon, Tan and Lim [266] show the choices of path appropriate to table IV.

One consequence of the correlation with catastrophe theory is that rigorous proofs become available for many results previously obtained by perturbation-theoretic methods (which tend to lack proofs that the perturbation series actually converge). The symmetry restrictions imposed by Thompson and Hunt can presumably be relaxed without any essential changes, but there are difficulties in making this precise in a variational setting.

A number of explicit applications of these ideas exist. Thompson and Shorrock [264, 265] discuss the stress response of a crystal lattice, finding both a dual cusp and a hyperbolic umbilic. Thompson and Gaspar [258]show that all umbilic catastrophes up to codimension 5

Table IV Correlation between Thompson-Hunt list and Thom list. (Note: the non- standard germ x3+ 3xy z for D~ is equivalent to the usual x3+ y3, but in addition has even symmetry in y.)

Catastrophe Thompson-Hunt classification Name

A2 x 3 + ~x limit point x 3 - Ax e + Ex asymmetric

A3 x 4 - ~x e + ~x stable-symmetric x 4 - ~x cut-off point

- A3 - x 4 _ ~.x e + ex unstable-symmetric D~ x 3 - 3xy: - ~(ax 2 + by z) + ~lx + eey anticlinal D~ x 3 + 3xy 2 - ~ ( a x e + by e) + etx + azy monoclinal (2b > a)

homeoclinal (2b < a) D s xyZ + y4 _ ~(axe + bye) + Emx + e2y paraclinal


occur in a variant of the Augusti model, and Gaspar [75] develops this. Thompson, Tan and Lim [266] describe a general topological classification of postbuckling phenomena (of a somewhat heuristic nature). Thompson [257] discusses the prediction of changes in stability through a succession of fold catastrophes (in a multidimensional state space). The two basic 'theorems', then hypotheses, of Thompson and Hunt [259] on the branching of equilibrium paths have been proved (subject to clarified hypotheses) by Kuiper using cohomology theory (see Chillingworth [52]). Chillingworth, Marsden and Wan have obtained a rigorous solution [55] of the Stoppelli traction problem, which sheds new light on, for example, unique- ness of solutions. Buzano et al. [46] describe the buckling of a triangular beam. Engineering texts 'prove' that this will be similar to that of a circular beam (Poston and Stewart [183] p. 316) but a rigorous analysis shows that this is false (though the practical implications may not be crucial).

4.2. Perturbed bifurcation

Care is required in interpreting the structural stability of the elementary catastrophes in connection with bifurcation problems. We can illustrate this on a model of the Euler arch (Zeeman [293]). Here the height x of the arch under sideways compression/3 and zero loading is found by extremizing a potential

/.L 4 F ~ ( x ) = ~ x - Xx 2,

where h =/3 - 2ix and/z is the modulus of elasticity of the hinge. This gives the bifurcation diagram

x 3 - hx = 0, (4.1) 3

the familiar pitchfork (fig. 6) or symmetric point

of bifurcation. The family Ex is structurally unstable, and a universal unfolding

G~A(X) = F~(x) + ax (4.2)

may be realised physically by introducing a vertical load a. That this gives a cusp catastrophe is made explicit in Zeeman [293]. Structural stability now implies that no further unfolding parameters are required-provided we treat (4.2) as a two-parameter bifurcation problem. However, this does not imply that every one- parameter bifurcation problem obtained by perturbing (4.1) is of the form (4.2) for fixed h.

To see why, think of the bifurcation problem (4.1) as a path through the cusp catastrophe, parametrized by h. For fixed a, (4.2) describes a family of parallel paths. However, the original path can also be perturbed by tilting. This corresponds to a bifurcation parameter obtained as a mixture of contributions from both h and a, say h = h()~) and a = a()~) for a new parameter ;~. In fact, as is well known, a perturbed problem of the form

~ x 3 - (,X +a~)x +a2x =0 3

can display hysteresis loops for arbitrarily small a~ (e.g. Golubitsky and Schaeffer [90, 93]), not obtainable from (4.2) for fixed a: see fig. 17.

Fig. 17. Full unfolding of the pitchfork bifurcation diagram, after Golubitsky and Schaeffer [93].


The coordinate changes used in elementary catastrophe theory, which define structural stability, allow diffeomorphisms which “mix together” all control parameters. But in a bifurcation problem with a distinguished parameter A, it is more useful to keep A distinguished: this requires less general coordinate changes and changes structural stability properties.

This equivalence relation preserves the differentiaI topology of the bifurcation diagram G(x, A) = 0 relative to the A-coordinate.

An E-parameter unfolding of G is a germ

F:R” xR xR’-+R”,

Similarly the fact that Thompson and Hunt’s anticlinal point of bifurcation is a universal unfolding of the elliptic umbilic catastrophe renders it structurally stable as a 3-parameter bifurcation problem, but not as a family of perturbations by (ty,, ~2) of a l-parameter problem with distinguished parameter A. Varying (aI, CIZ) produces paths through the elliptic umbilic parallel to its axis: tilted paths lead to other types of bifurcation diagram. Of course, changes such as tilts may be less important than those described by (or, a*), and indeed often are; but it must not be thought that elementary catastrophe theory rules them out altogether.

such that F(x, A, 0) = G(x, A). An unfolding is universal if every unfolding Hs can be induced from it in the sense that

Hdx, A) = Z-(x, A, W%sk A), As(A), Jlt@),

where T is a matrix-valued function and the germs involved have appropriate domains of definition. The universal unfolding, which is essentially unique if it exists, captures all families of perturbations, up to contact equivalence respecting the special role of A.

Golubitsky and Schaeffer [90] found that a simple modification of elementary catastrophe theory, to take account of the distinguished parameter, is available. The relevant equivalence relation, a generalization of contact equivalence, works equally well in the non- variational case, permitting a general theory of quasistatic bifurcation on R”. We summarize some of their results.

Put G(x, A) in component form as

(gt(x, A), . . . , gn(x, A)) for real-valued gi. The ideal (G) generated by the components gi of G is, by definition, the set of all germs of the form

uig1+ * * * + U”&,

A bifurcation problem is an equation

where Ui(x, A) E En+l, and then (G)” is the set of all germs (&(x, A), . . . , &(x, A)) where 4 E ((3).

Next, let aG/axi denote the germ

(%l/axj, * * * 9 @“/axi) E E,+1,,. Denote by E,+~{~G/8x} the set of all germs of the form

G(x, A) = 0,

where G : R” x R + R” is a smooth germ. Two problems G and HI are contact equivalent if there are smooth germs X :R” x R +R” and A: R +R, and nonsingular n x n matrices T(x, A), such that

8G v’axl+

. ..+v E fl ax,'

where Oi E .&+I. This is the E,+t-module generated by {aGlaxi : j = 1,. . . , n}.

Finally, let E,{aG/aA) be the set of germs

H(x, A) = T(x, A)GtX(x, A), MA)). 40) $

We require det d,x(O, 0) > 0 and an/ah(O) > 0. for d, E E,.


Define two subsets of E.+t.. as follows: there is a corresponding determinacy result. If

~'G = (G)" + E.÷,{aG/ax},

TG = (G)" + E.+,{aG/ax} + E~{aG/aA}.

Intuitively, TG is the tangent space to the orbit of G under the group of contact equivalences; TG is the tangent space to the orbit under a slightly smaller group, which is technically more manageable.

Golubitsky and Schaeffer [90] prove that an

unfolding F, of G is universal if and only if

E . + l . = TG + aF . + aF , ~ (0) +'" ~ (0),

as vector spaces over R. In particular, if we choose germs p~(x, )t) . . . . . p~(x, ;t) that afford a basis for E,+t,,/TG, then

F(x, ;~, E) = G(x, X) + E~p~(x, ;t ) +. • • + E~p~(x, X)

is a universal unfolding. This result holds only under a finiteness

hypothesis. Golubitsky and Schaeffer assume that dim E,+L,/TG is finite. The occurrence of TG here, rather than TG, is inelegant: it was forced by the current state of knowledge. Recently, Damon and Mather [64] have replaced it by the condition that dim E,+~,,/TG is finite. This number is the codimension cod(G) and equals ! above. An advantage of the Damon- Mather improvement is that the theory now becomes useful when 2t E R k, for k > l, and not just for ~ 1-dimensional. (The hypothesis on TG used by Golubitsky and Schaeffer holds only in trivial cases, when k > 1.) The universal unfolding is built from the quotient by TG, and is not changed by the Damon-Mather improvement. The availability of these results for multidimensional )t is obviously of interest for applications, though little work has yet been done since the result is so new.

As well as an unfolding theorem strictly analogous to that for elementary catastrophes,

mkE.+l,. C_ mTG

then G is k-determined (its k-jet determines its contact equivalence class). Any problem of finite codimension is k-determined for some k, hence both it and its universal unfolding are contact equivalent to polynomials. The proofs are routine but non-trivial generalizations of those used for the elementary catastrophes.

Despite the abstract appearance of the above results, they translate into efficient com- putational tests, closely analogous to those for the elementary catastrophes: they may be described in "classical" terms, resembling the examples in Poston and Stewart [183] chapter 8. To save space we shall not attempt to do this here: Golubitsky and Schaeffer [90] give enough examples for the method to be clear. Chilling- worth [54] gives a careful exposition of the meaning of the mathematics.

4.3. The geometry of perturbed bifurcation

The universal unfolding of a bifurcation problem gives a global picture of all small perturbations. We illustrate this first on the pitchfork, and then describe some general features.

Consider the universal unfolding of the pitchfork, in the form

0 = G(x, ,~, El, E2) = X 3 + AX + EA + E2 (4.3)

(which is a slight variation on that used by Golubitsky and Schaeffer [90], explaining the difference between our picture above, and theirs). The topological form of this diagram varies with (el, E2) as follows. There are two curves in (El, EE)-space, with equations

B : E2 = E~,

H: E2=0,

dividing the space into four regions. In each


region, and on the dividing curves, the form of the bifurcation diagram is as shown in fig. 17. These are the only topological types of small perturbations on the pitchfork. Note that, near the origin in (E,, e&space, the regions in which hysteresis loops can occur become very thin- explaining the infrequent observation of them in the literature.

Note that eq. (4.3) may be interpreted as a section of the cusp catastrophe manifold

x’+ax+b=O

above a curve (a, b) = (A, EJ + Q) parametrized by A (and varying with tl and ~2). This reiates the unfolding to Thompson and Hunt’s ‘path’ idea. Such paths have considerable heuristic value, but do not appear to be the best way to formalize perturbed bifurcation as far as computations are concerned: we mention this only to exhibit the connection with the Golubitsky- Schaeffer method on the geometric level.

Of the diagrams in fig. 17, those not lying on the curves B and H are (sr~~cf~ru~~y) stable in the sense that their contact equivalence class is unchanged by small perturbations; those on B or H are (s~rucr~ra~~y) unstable.

Next we explain how fig. 17 fits into a more general picture. Consider an arbitrary bifurcation problem G with universal unfolding F = F(x, A, E). Then again e-space is divided up into regions, one corresponding to each contact equivalence class of perturbed diagrams. Tran- sition between regions corresponds to loss of structural stability (exactly as in Thorn’s general viewpoint on morphological changes).

In general, there are three “accidents” that can cause a diagram to be unstable: b~~~~c~f~o~, hysteresis and double limit points. These (and their instability) are shown in fig. 18. Their occurrence defines three sets in e-space, as follows:

B ={E [3x, A such that F(x, A, E) =O and rank d,F < n};

If = (E 13x, h such that Ffx, h, E) = 0,

det d,F = 0, d:F(v, u) E range d,F},

where 0 # D E ker(d,F);

D={E I3x,y,h with xfy, F(x,h,~)=0= WY, A, ~1, and

det d,F(x, A, E) = 0 = det d,F(y, A, E)}.

By equation-counting, each of these sets in general has dimension I - 1 where E E R’. Be- tween them, they divide e-space into finitely many connected components (assuming F polynomial) within each of which the contact equivalence class is unchanged, and the diagram is stable. On B, Ei or D, it is unstable (either with a bifurcation, hysteresis, or double limit point - or something more degenerate, still lying in B, H or D.) Hence the analysis of perturbed diagrams reduces to the geometry of B, H and D. This can be very complicated (see section 5.4) and other methods from singularity theory are sometimes used (Golubitsky, Keyfitz and Schaeffer [SS]).

Examples given in Golubitsky and Schaeffer [90] include a rigorous analysis, from the point of view of Sobolev spaces, of the Euler beam. Further applications will be discussed below: without doubt these methods are among the most important yet to emerge as far as physical applications of catastrophe theory are concerned.

I< 2

27>

X --\ 3:

x \ ‘C B H D

Fig. 18. Three types of structural instability in a bifurcation diagram: bifurcation (B), hysteresis (H), and double limit points (II). Each sequence runs vertically, with the central diagram being the unstabte one: those above and below are perturbations.


The bifurcations to which the methods apply are essentially quasistatic (though limit cycle behaviour can be handled, see section 5.8). More "classical" methods are also available for such problems, and it is worth remarking here on a technical advantage of the Golubitsky- Schaeffer method. Because it works with germs on R" x R x R 1, it provides descriptions valid in a neighbourhood of the origin in R" x R, that is, (x, / )-space, that depends uniformly on ~. The classical tools of bifurcation theory tend to provide non-uniform estimates, so that the size of the neighbourhood shrinks to zero as the singularity is approached. The difference may be traced to the greater power of the Malgrange Preparation Theorem compared to the Implicit Function Theorem. As a result, while some papers in bifurcation theory contain the same diagrams as those found by singularity theory, their results are valid in regions that disappear as the origin is approached, and are thus weaker than those obtained by singularity theory methods. The advantages of uniform estimates should not be underrated.

winged cusp quintic hysteresis

cusp pitchfork quartic Fold

isola bifurcation hysteresis

~ f ! l d

F regular

Fig. 19. Subordination diagram for bifurcation problems in 1 variable.

where p and q are homogeneous quadratic forms, L is a linear map on R 2, and c ~ R 2. These are classified and described in Golubitsky and Schaeffer [90]. Under the assumptions that L is invertible and that H = 0 has a "trivial" solution depending smoothly on h (the traditional hypothesis of classical bifurcation theory) this can be reduced to

G(x, y, x) = (p(x, y), q(x, y ) ) - (Xx, Xy) = 0.

There is a second reduction to one of five types:

4.4. Partial classification

Classifications of bifurcation diagrams com- parable in detail to those for elementary catastrophes have not yet been obtained, but some partial results exist.

When n = 1, Golubitsky and Keyfi tz [88] list normal forms for bifurcation diagrams up to (local) contact equivalence, which with a few additions is complete up to codimension 2. Table V presents this in flowchart form. Fig. 19 is the corresponding subordination diagram, where a bifurcation problem is subordinate to another if it occurs locally in its universal unfolding.

When n =2 , the simplest problems are quadratic:

(i) (x 2 - y2, -2xy ) + (ax + by - h)(x, y);

(ii) (x 2 + y2, -2xy ) + (ax + by - h)(x, y);

(iii) (x 2, - 2xy ) + (ax + by - X)(x, y);

(iv) (0, x 2) + (ax + by - h)(x, y);

(v) (ax + by - h)(x, y).

Of these, (i) and (ii) are the only nonde-

generate problems (McLeod and Sattinger [156]), that is, p and q have no common factor, and the surfaces p(x, y) = hx and q(x, y) = hy are nowhere tangent other than at the origin (or equivalently rank dx.yaG/>2 for nonzero (x, y, A) on the intersection of the two surfaces). For any nondegenerate problem we must have

(a + 1)[(a - 2)2~ 3b ~] # 0,

H ( x , y , h ) = ( p ( x , y ) , q ( x , y ) ) - h L ( x , y ) + h 2 c = O , respectively in cases (i), (ii). It is proved that


Table V Identification and partial classification of bifurcation problems.

Condition Formula Codimension Name

G = O

1 G~ = 0?

yes

G~x = 0?

yes

G~ = 0?

Gx~ = 0?

no ~ x 0 regular

. O , G x = 0 ? .o~ xZ_+X 0

yes [

det d2G . o x2_+A2 1

= 0?

yes

d3G(v, v, v)

=0 , where ,o, xZ +_A3 2

d2G(v, v) = 0?

yes

> ? />3

.o)Gxxx=O? .o, x3__A 1

yes

G . . . . = 0 7 " ° , x 4 +- X 2

yes [

G ..... = 0? .o ~ x 5 -+ A 3

yes ' ? ~>4

.O~GAx=0? .o> x3+_Ax 2

yes

GAA =07 .o, x3___ A2 3

yes I , ? / > 4

, ? I>3

fold

bifurcation

exchange of stability

isola

cusp

hysteresis

quartic fold

quintic hysteresis

Note: x m +- A if finite codimension m - 2

pitchfork

stable-symmetric

unstable-symmetric

winged cusp

280 I. Stewart/Physical applications o/catastrophe theory

any nondegenerate problem of type (i) or (ii) is 2-determined (so that it is safe to truncate at the order written above) and of codimension 7, with universal unfolding

G ( X , y , A ) + (ri + s jy + s3x + t lx 2 + t2xy, r2 + s2x

- s 3 y + t~xy + t2y2).

Imperfect ion analysis of this is complicated by the need to discuss seven imperfections, and the occurrence within this universal unfolding of problems of types x4+_Ax and x3+_A2x, occurring in table V only under the heading '?'. (They appear, of course, if the table is extended slightly.)

Two of the 7 parameters are modal (namely a and b) and a full description of how the un- perturbed diagram varies with a and b is given in Golubitsky and Schaeffer [91].

4.5. Imperfection sensitivity

Consider a buckling path that loses stability by snap-buckling (at a limit point or fold point), an effect normally equated with mechanical failure. Imperfect ions e~ can alter the load at which failure occurs, sometimes drastically. The graph of maximum sustainable load against imperfect ion values is the imperfection sensitivity diagram. For dual-cusp unstable-symmetric buckling

F(x,A,E) = - x 4 - A x 2 + e x

the maximum sustainable load is that at which the path through control space of the dual cusp corresponding to imperfection of size ~ meets the bifurcation set of the dual cusp in (E,A)- space (fig. 20), that is, the imperfection sensitivity diagram is the bifurcation set, viewed as a graph of A against E. The cusp in this curve shows an imperfection sensitivity varying with E 2/3 (a result due to Koiter [129] in a functional- analytic setting). The exponent 2 in this asymp-

L~

~o

Fig. 20. Bifurcation set of the cusp catastrophe interpreted as an imperfection sensitivity diagram. The maximum sustainable load )to at E = 0 drops rapidly for nonzero imperfections.

totic relation is the imperfection sensitivity exponent.

In a similar way the bifurcation sets of the elliptic and hyperbolic umbilic catastrophes in figs. 21a and 21b may be interpreted as imperfection-sensitivity graphs, and the corresponding exponents computed. For the anticlinal, monoclinal and homeoclinal bifurcations, the exponent is ½ indicating much greater sensitivity to imperfections.

The standard bifurcation sets of elementary catastrophes appear here because ~ and e between them provide a universal unfolding. A different choice of imperfections can change the diagram. Use of the Golubi tsky-Schaeffer methods can illuminate this point, by means of a simultaneous analysis of all imperfections. As a bifurcation problem, the unstable-symmetric buckling above requires not one, but two unfolding parameters (imperfections)

- -X 4 _ }~X 2 ~ EiX --[- E2,~X.

This form is derived by unfolding the derivative as a bifurcation problem and integrating the result, a method that works for variational problems with a single x-variable. To the principal imperfection E l we adjoin another, •2,

which in an arch-buckling model may be real-


Fig. 21. Bifurcation sets of the elliptic (a) and hyperbolic (b) umbilics, as imperfections sensitivity diagrams for (a) the anticlinal point of bifurcation, (bf the homeoclinal point of bifurcation. After Thompson and Hunt (2621.

ised by tilting the arch away from the horizontal.

For given load ho let IC(&) be the set of point in (E,, e&pace whose corresponding bifurcation diagrams do not branch for A < ho. Then the

boundary of I(&,) is a curve delineating those imperfections for which buckling at ho first occurs. In this example the curves are as shown in fig. 22. As ho approaches the ‘perfect’ buckling point ho = 0 from below, the changing effects of e1 and l 2 are clear. There is a drastic sensitivity to el. asymptotically 1~~1~‘~; the sensitivity to 62 is only le212. This confirms that cl is the principal imperfection in the sense that it is the most severe. Note that the exponents f and 2 here are diffeomorphism-invariant.

To carry out such calculations for two or more state variables is problematic. Consider the anticlinal point of bifurcation

x3 - 3xy2- A(x* + y*) + el.x + ~2y.

A (non-variational) unfolding as a bifurcation problem requires not 2, but 7 imperfection parameters (Golubitsky and Schaeffer [90]), which is rather large. A variational unfolding is not known (and may not even exist) and on heuristic grounds will require 5 or 6 parameters. What is really required is a hybrid theory: equivalence up to contact transformation of variational perturbations. For further discussion see Golubitsky and Schaeffer [90] section 8.

Fig. 22. Imperfection sensitivity of the pitchfork to simultaneous imperfections. Within the shaded region, the system buckles at loads <ho < 0.

282 1. Stewart/Physical applications of catastrophe theory

Notice that, despite the tendency of expositors to identify catastrophe theory with variational problems, here it is the non-variational case that is most easily handled.

Golubitsky and Schaeffer 1911 generalize their method to the equivariant case, where a symmetry group I’ is acting on the germs. The proofs are analogous, and we sketch the main changes needed in the formulation.

Let G : R” x R -P R” be a bifurcation problem. Let r be a compact Lie group acting orthogon- ally on R” (possibly with different actions in the two copies of R” involved). This in particular includes the case of a finite group. Call G r-equiuariant if

G(yx, A) = rG(x, A)

for all y E r. Let Ei+r,, be the set of r-equivariant germs. Then a bifurcation problem G E EC,,,, will be called a bifurcation problem with symmetry group r. Let G and hl be two such: they are r-equivalent if

H(x, A) = T(x, A)G(X(x, A), A(A))

as before, but with the symmetry conditions

X(7x, A) = rX(% A),

y-lT(yx, A>r = T(x, A),

where in the latter we think of y as an ortho- gonal matrix.

An l-parameter f-unfolding of G is a germ

F(x, A, E) E E;+,+r,n (with trivial action on the E parameters) such that F(x, A, 0) = G(x, A). Uni- versality is defined as before, with obvious symmetry conditions.

Instead of ‘i’G and TG, in the earlier notation, we introduce the analogous IY-equivariant tangent spaces

I-G = @#)(X(x, A)) + T(x, A)@,

TG = i+G + E,{aG/aA}.

Here X and T satisfy the symmetry conditions: apart from this the computation is as for i;G and TG.

Suppose now that p,(x, A), . . . , p,(x, A)E

Ef;+i,, project to give a spanning set for Ef .+,,JTG. Then a universal unfolding for G is

F(x, A, E) = G(x, A) + E,P,(x, A) + * - . + EI~I(X, A).

Again there is a finiteness hypothesis: Golubit- sky and Schaeffer require dim Ef;+,,,/fG to be finite, but the Damon-Mather improvement holds, so that Ef;,,,, ITG finite is sufficient. The dimension of the latter is the I’-codimension of G.

It has long been recognized that symmetry tends to lead to multiple eigenvalues in bifurcation problems, so that Lyapunov-Schmidt reduction produces problems with a large number of parameters. Sattinger 11971 observes that symmetry also places strong restrictions on the Taylor expansions of the functions concerned, to some extent compensating for this. Golubit- sky and Schaeffer [91] note that a similar effect occurs with the codimension: symmetry forces high codimension (within the class of all perturbations, including those that break the symmetry) but the equivariant codimension can remain low. Their method combines two powerful mathematical machines: group representation theory and singularity theory.

Illustrations in Golubitsky and Schaeffer [91] include the buckling of an annular plate, with symmetry group O(2) x Z2, see also Majumdar [140]; quadratic bifurcation problems in two variables; and the symmetrized double cusp. Further applications are discussed below in section 4.7 (mode-jumping in the rectangular plate), section 5.7 (the spherical BCnard problem), and section 5.5 (the Brusselator reaction). The most complicated group action so far occurring in a computable problem is the 5- dimensional irreducible representation of O(3).


4.7. Mode-jumping in the rectangular plate The linearized Von Khrmfin equation

Golubitsky and Schaeffer [201] apply their methods to the buckling of a rectangular plate whose aspect ratio is l : l . In 1959 Stein [223, 224] showed experimentally that when a plate with l = 5.38 buckles under longitudinal compression it first buckles into a mode with wave- number 5; then at a load approximately 1.7 times this initial loading, it suddenly and violently jumps to wave-number 6. This phenomenon, called mode-jumping, has hitherto eluded theoretical explanation. Bauer, Keller, and Riess [30] suggest treating it as a secondary bifurcation caused by imperfections away from a double eigenvalue: the analysis of Golubitsky and Schaeffer show this to be correct, but not under the assumptions made by Bauer, Keller and Riess.

The Von K~mfin equations for the rectangular plate are

02W V4w = [~, w ] - - 0 ~ x '

V4~ = -½[w, w],

where w is the vertical deflection in the z- direction, ~ the Airy stress function, V 4 is the biharmonic operator (~72) 2, and

02U 02t~ 02U 02U 02U 02/3 [u, v ] = ~-~ ~ - 20xOy OxO---y + ~-~-fx "

For t#, Golubitsky and Schaeffer impose the boundary conditions ~N=(V2&)N=0 on the boundary of the rectangle. This is not the usual choice (t# =V2& =0). For w there are two common choices of boundary condition:

Clamped: w = w N = 0 on ends; w = V 2 w = 0 on sides

Simply supported: w = V2w = 0 on boundary. Here N indicates differentiation normal to the boundary.

02W V4w +-~-~x = 0

has a double eigenvalue for a discrete set of values of l, namely

! = X/k(k +2) (clamped),

l = V~-(k + 1) (simply supported).

Wave-numbers 5 and 6 occur together when k = 5; this occurs for I = 5.92 (clamped) and 5.48 (simply supported). Stein's actual value of 5.38 is slightly smaller than either of these.

Lyapunov-Schmidt reduction (see section 4.8) implies the existence of a smooth function

G : R2 x R -~ R2; G(x, y, A ) = (gl(x, y, A), g2(x, y, A)),

whose zeros parametrize the solution to the nonlinear von Kfirmfin equation, when A is near a double eigenvalue. Here x~b5 + Y~6 parametrizes the eigenspace to the linearized equation, where tb~ is the eigenfunction with wave-number k for the chosen boundary conditions.

The problem has several natural symmetries: Z2 x Z2 for the rectangle itself, and Z2 for up- down symmetry in the buckling. Ostensibly this gives a symmetry group Z2 x Z2 x Z:, but some of the action on the eigenfunctions is trivial and it reduces to a Z2×Z2-action which implies (since the reduced equation shares these symmetries) that

g~ is odd in x and even in y,

g2 is even in x and odd in y.

To lowest order, G therefore has the form

G(x, y, A) = (ax 3 + bxy e - x, cxEy

+ dy 3 - Ay)+ ~'(5),


where (~(5) denotes higher order terms. For the von Kfirmfin equations ad > 0, and this can be scaled to make a = d = 1. The determinacy criteria imply that the ~(5) term can be transformed away by a contact equivalence, provided the problem is nondegenerate: the cubic parts h a v e no common factor, and the curves g~ = 0, g2 = 0 are transverse in the (x, y)-plane. Such degeneracies occur (assuming a = d = 1) for b = 1, c = 1 and bc = 1. These curves divide (b, c)-space into seven regions as in fig. 23.

The parameters (b, c) are modal: they cannot be changed by a smooth contact equivalence but they can (within a region) by a continuous but non-differentiable analogue of contact equivalence (see section 2.10). There is thus one topological type of bifurcation diagram for each region.

Moreover , a universal unfolding within the symmetry class, in this nondegenerate case, takes the form

F(x , y, A, b, c, L ) = (x 3 + bxy 2 - x,

cxZy + y 3 (A - L)y)

with one extra parameter L. Some computations using Lyapunov-Schmid t show that; (i) For clamped boundary conditions, (b, c) lies in region 4 of fig. 23;

Q

bc =1

Q

b=l

Fig. 23. Modal plane for the buckling plate, after Schaeffer and Golubitsky [201].

(ii) For simply supported boundary conditions, (b, c) lies in region 1.

To model Stein's experiments, L must be taken posi t ive (since it represents the deviation of the aspect ratio from the ideal case). The bifurcation diagrams from regions 1 and 4 are shown in fig. 24 for positive L. In region 1 no secondary bifurcation occurs f rom the primary branch, so this cannot explain mode-jumping. But in region 4 we do get secondary bifurcation and mode-jumping. So secondary bifurcations can explain m o d e - j u m p i n g - b u t only under c lamped boundary conditions. It has long been customary to use simply supported conditions for this p r o b l e m - p r e s u m a b l y for mathematical conven i ence -a l t hough Stein [223, 224] states that clamped conditions seem more appropriate.

These conclusions should be compared with a recent paper by Matkowsky, Putnick and Riess

u

S UIII t; U U f / S

s ~ 5 ~

b i I u~ l u

Fig. 24. Bifurcation diagrams for the buckling plate. (a) when the modal parameter is in region I (simply supported boundary conditions) secondary bifurcations are on the wrong branch to explain mode-jumping from wave-number 5 to 6 (marked on the corresponding branches). (b) for modal parameters in region 4 (clamped boundary conditions) secondary bifurcations (arrowed) and stability assignments predict mode-jumping. After Schaeffer and Golubitsky [201]. (s = stable, u = unstable.)

I. Stewart]Physical applications of catastrophe theory 285

[155] which establishes the occurrence of secondary bifurcations (and suggests this may explain mode-jumping) but fails to notice the importance of the modal parameter, which determines whether the secondary bifurcations are of a suitable nature for this. The authors do not mention the results of Golubitsky and Schaeffer [201], a surprising omission.

One "philosophical" point should be noted here. Since singularity theory works in terms of germs, the results obtained by Golubitsky and Schaeffer here are only known to be correct in some (possibly small) neighbourhood of the double eigenvalue case. Without making delicate estimates, it is not clear that they hold when, for example, the aspect ratio is 5.38 as in Stein's experiment. However, this is within 10 percent of the double-eigenvalue ratio 5.92 for the clamped version of the problem (and closer still to the simply-supported). It is assumed, on grounds of common practice in the subject, that these differences are probably small enough for the local analysis to retain validity. For that matter, the Von Kfirmfin equations themselves are a small-deflection approximation. The criticism [235] that catastrophe theory methods, being "local" in nature, cannot have any practical implications, relies on an interpretation of mathematical results that is so strict that it effectively rules out all local methods-perturbation theory, asymptotic expansions, in fact most of the tools in the mathematical physicist's armoury. It is circumvented by the standard device: assume that the local analysis actually holds good in a wider range than the general theory can guarantee (which is commonly true) and compare the results with experiment, or numerical simulations. The criticism is in any case overstated. While "local" to a mathematician is defined as meaning "within an arbitrarily small neighbourhood" this does not imply that a local result can only hold in a neighbourhood so small as to be invisible. It implies that it holds in some neighbourhood, whose size is immaterial to the mathematician. The physicist, to whom

the size is more important, may well find that the neighbourhood is quite large. For example, the theory of the elliptic umbilic diffraction catastrophe is based on a purely local approximation to the optics. According to [235] it cannot therefore carry implications for experiment. If this is correct, the agreement between theory and experiment in fig. 16 is mere coin- cidence.

4.8. Infinite dimensions

In traditional bifurcation theory the main interest is in the branching of solutions to an equation G(x,X)= 0 defined on some infinite- dimensional Banach space, when the trivial solution x = 0 loses stability. The singularity theory approach uses a broader concept of "bifurcation" which has the advantage that a trivial solution need not exist; but as expounded above assumes x E R n, an apparently severe restriction.

However, most bifurcation problems reduce, by the Lyapunov-Schmidt procedure or some variant, to a finite-dimensional system of equations. Indeed, much effort goes into such reductions, which are often computationally difficult. But conceptually, the Lyapunov-Schmidt procedure is an easy application of the Implicit Function Theorem (Marsden [143]). Golubitsky and Schaeffer [90, 91] assume such a reduction has been performed before their methods are applied.

There are certain advantages in using a Banach space setting. Here Chow, Hale and Mallet-Paret [57, 58] apply the Thorn Trans- versality Theorem to generic bifurcation. Mag- nus [134--136] develops a version of elementary catastrophe theory on reflexive Banach spaces, extended to certain non-reflexive cases in Mag- nus [137]. A condition of Fredholm type is assumed. Magnus and Poston [138] apply these results to the buckling of a rectangular plate at a double eigenvalue (aspect ratio ~/2: 1) and find a double cusp catastrophe. They unfold this fully


to an 8-parameter family, which is partially analysed; and find physical parameters realising the full unfolding.

Arkeryd [7, 8, 9] has developed Banach space versions of Thorn's theorem and the unfolding theorem, and of the Golubitsky-Schaeffer generalization with a distinguished )~-parameter.

Dangelmayr [66] relaxes the Fredholm restriction in a way suited to bifurcational applications. Let X, Y be separable reflexive Banach spaces with duals X*, Y*. Let X C_ Y C_ H be dense embeddings, with H a Hilbert space. Assume that Y and Y* are dual with respect to the inner product on H. In practice H is usually L2, Y is Lp (p >i 2) and X is some suitably chosen Sobolev space. Applications include the buckling of columns at both simple and double eigenvalues, and non-equilibrium thermodynamics.

It should not be imagined that the Fredholm restriction, even as relaxed by Dangelmayr, necessarily holds in "most" cases. Magnus and Poston [139] discuss a case in which fa i lure of the Fredholm condition is generic, but generically the bifurcations occurring have a certain type of structural stability (and indeed resemble the fold catastrophe without technically being

one). They give a slightly artificial physical problem, a nonlinear elastic string, to illustrate these features.

5. Other applications

Even as restricted to physics, catastrophe theory has been applied to a wide range of subjects. We now look in less detail than above, at some more of these. Others, equally deserv- ing of attention, have been omitted for reasons of space: they include the study of chemical bonding in terms of molecular charge densities, and of chemical reactions at the molecular level as catastrophes, by Bader et al. [28, 29]; molecular collisions, Connor [59-62]; laser action, Gilmore [79], Poston and Stewart [183]; acous-

tics, Poston and Stewart [183]; magnetic phenomena, e.g. Gimblett and Peckover [81]; shockwaves, Golubitsky and Schaeffer [84], Guckenheimer [99, 100], Schaeffer [199]; nonlinear vibrations, Holmes and Rand [117]; aircraft stability, Mehra et al. [159]; crystal spec- trum singularities, Poston [179]; plate tectonics, Thorn [247, 248]; defects and dislocations, Thorn [249], Wright [284]; and ship stability, Zeeman [294, 295]. Not all of these involve in- vestigations as extensive or well-developed as those mentioned above, but all have significant features and suggest further work.

The topics discussed in this section are chemical kinetics, thermodynamic phase transitions, fluid dynamics and degenerate Hopf bifurcation (an example of how catastrophe theory can be used to do d y n a m i c s as well as s ta t ics ) .

5.1. The stirred tank

Chemical reactions exhibit numerous "bifurcational" phenomena, Ray [190]. Even the simplest example has interesting features. It concerns the continuous flow stirred tank reactor, or CSTR, Aris [3-6]. This comprises a reaction vessel, fed by some reactant X at concentration cf and temperature Tf. The reactant is stirred while in the vessel, and forms a product Y at a rate kc. A cooling jacket at temperature To, or a coil within the vessel, remove excess heat. If T is the temperature at which the reaction proceeds, the usual equations for c and T take the form (Aris [3, 4])

d c = c f - c _ k ( T ) c , (5.1) dt 0

d T = Tf - - T + J k ( T ) c - L ( T - Tc), (5.2) dt 0

where c is the concentration of X in the vessel, J and L are certain constants, and 0 is the residence t ime, that is, the average time a given


molecule of X remains within the reactor. The rate c o n s t a n t k depends on T. Numerous empirically derived forms for k ( T ) have been studied, but the commonest is the Arrhenius form

k ( T ) = A e x p ( - E / R T ) ,

where R is the gas constant, E the activation energy and A is a constant.

For steady-state solutions of (5.1) and (5.2) set dc /d t = d T / d t = 0, solve (5.1) for c and sub- stitute in (5.2), to get

Ok(T) (1 + L O ) T - (LOTc + Tf) = Jcf 1 + Ok(T)" (5.3)

Graphical methods show that there are in general 1 or 3 solutions for T. The graph of the right-hand side has a unique inflexion point, and when the graph of the left-hand side is tangent there, the three solutions coalesce. Thinking of (5.3) as an equation of state, this represents a cusp point: provided suitable unfolding parameters are chosen, cusp catastrophe bifurcation of the steady states is expected.

For example, Aris [5] takes as parameters Tf, and the relative cool ing

rf Fig. 25. Cusp catastrophe in the stirred tank, after Aris [5].

It is difficult to vary D experimentally without affecting other parameters, and Uppal, Ray and Poore [270] reworked their numerical results for variation of the residence time 0, obtaining sur- prisingly different bifurcation diagrams (fig. 26). Their analysis usually assumes ~/= E / R T f - - ~ ,

although numerical plots give similar behaviour for finite V. Note the occurrence of an isola in (c) and (d).

Uppal, Ray and Poore [270] studied also the d y n a m i c bifurcations finding Hopf bifurcation to limit cycles for certain parameter values.

Aris [5] attempted to explain the changed diagrams by adding a "wing" to the cusp catastrophe.

cr = LO(1 + LO) -1. 5.2. The winged cusp

Assuming that Tc = Tf and keeping all other parameters constant, the graph of T against Tf and cr is a cusp catastrophe (fig. 25). The catas- trophic jumps are conventionally interpreted as ignition and extinction of the reaction (though in practice it is not this straightforward).

Similar results have been obtained by Blum and Mehra (see [190]) using catastrophe theory; and numerically by Uppal, Ray and Poore [269], and Poore [176] as bifurcation diagrams of T relative to variation of the Damk6hler number

Starting from the idea of Aris [5], Golubitsky and Keyfitz [88] applied the methods of Golu- bitsky and Schaeffer [90] to the eqs. (5.1) and (5.2), put into dimensionless form

dx d--t- = -~x + D(1 - x)A(y), (5.4)

dy = - ( 1 ~ - - ~)y + BD(1 - x)A(y) + ~/, (5.5)

where

D = Ok(Tf). • = 110, "


T---~ !

fl

(a)

n~ff,) ~ ( e ) • m,(r+ L m,(r3, ' malt')

I m2(rz)

T " ml(r,

C 7,Xzc FIXED

fl

. (b)

G F

t

mr{r,)

m,(r 2)

rl T i,_

Oao " rn2 ( r ~ ) ~ - - - - ~ (C) t

T I

T -

Fig. 26. Bifurcation diagrams for the stirred tank reaction rate, with residence time as bifurcation parameter. After Uppal, Ray and Poore [270].

x = ( c f - c ) / q ,

y = ( T - T f ) / T t >i - 1,

"O = ( T o - T f ) / T f ,

and time t is scaled suitably. Again D is the Damk6hler number; B is proportional to the exothermicity. The Arrhenius-type function A is given by

A(y) = e x p [ - Vy/(1 + y)].

Seeking steady states solutions, one finds an equation of state

G(y, e; B, 8, n) = "0 - (1 + e)y + Be

1 + eSA(y) = 0 ,

(5.6)

where 8 = l ID. This is treated as an unfolding by (B, 8, ~1) of a bifurcation problem in y with e as bifurcation parameter. It is proved analytic-

ally that for 3,/> 8/3 there is a unique point (Y0, e0, B0, 80, ~i0) at which G is equivalent to the winged cusp

W ( x , A) = x 3 + h 2.

Further, the parameters (B, 8, ~) unfold G universally everywhere , and every diagram G = 0, for arbitrary (B, 8, ~!), is locally equivalent to one occurring in the universal unfolding of W, which perforce is an organizing centre. It is likely that the global geometry of G is the same as W, but this has not been proved.

A universal unfolding of W is

F ( x , A) = W ( x , A) + (a2 + a3X)x + al.

The hysteresis and bifurcation varieties in (al, a2, aa)-space are a Whitney umbrella and a cylinder on a cusp, arranged as in fig. 27. There are seven components of the complement, and the corresponding stable diagrams are shown in


H H

a b c

Fig. 27. Sections of the bifurcation and hysteresis varieties for the winged cusp, showing the seven stable regions. After Golubitsky and Keyfitz [88].

fig. 28. Five of these are those found by Uppal, Ray and Poore [270]: the remaining two occur for all y >i-83, but are not detected by the y = oo approximation.

To correlate these results with the cusp catastrophe, note that the problem F(x, A)= 0 may be viewed as a path parametrized by A,

a ~ 0[ 2 -J" 013A,

b -- 011 + A 2,

through the universal unfolding of the cusp

xa+ax +b =0 .

Such paths, giving rise to the seven stable diagrams, are shown in fig. 29. We see that the

0

1 2

3 4 5

6 7

Fig. 28. Bifurcation diagrams corresponding to the seven regions in fig. 27. After Golubitsky and Keyfitz [88].

use of • (or equivalently O) as bifurcation parameter introduces families of paths through the cusp catastrophe, which may relate sin- gularly to the unfolding parameters and give rise to diagrams not obtainable by straight-line paths. While considerations of this type do not appear helpful in computing unfoldings of bifurcation problems, they have considerable heuristic value in interpreting the results.

The main advantages of the Golubi tsky-Key- ritz analysis are completeness, rigour and the

¸Xj / /

Fig. 29. The diagrams in fig. 28 interpreted as parabolic paths through the cusp catastrophe, whose bifurcation set is denoted by B.


absence of restrictions on y, by which two extra bifurcation diagrams were found. (They have since been found numerically: one by Varma, the other by Russian workers.)

5.3. T w o r e a c t i o n s

More complex systems of reactions give rise to higher singularities. Two independent reactions A ~ B, C--* D, in the same reactor, yield a butterfly catastrophe, Stewart [228]. The proof is based on a general result, Stewart [227], on equations of state of the form

ax + b = cI)(x) + c~(dx),

where qb and • are s i g m o i d in a precise sense. The proof uses the Whitney cusp theorem of section 2.15. Reactions in sequence A-~B-~C and in parallel,

/ , B

A ~ C

appear also to give butterflies. Keyfitz has applied the Golubitsky-Schaeffer methods to these reactions (unpublished). Schaeffer and Shearer find that a s ingle reaction in a catalyst pellet yields a butterfly (unpublished).

%• EXPLOSION

NO EXPLOSION

T

Fig. 30. Explosition peninsula forms a protrusion of the 'explosion' region of a pressure-temperature rate diagram for a chemical reaction. Pressure (P) at which an explosion occurs becomes multivalued over temperature (T).

Experimentalists have detected at least 20 different reactions in the combustion of a hydrogen-oxygen mixture, and one proposed model is the thermal-chainbranching model, Dainton [63], Glassman [82], Semenov [204]. Four types of reactions are considered: in- itiation, branching, termination at the wall of the container, and termination in the interior. Here an active radical X undergoes, respectively, reactions of the form A + . . . ~ X + . . . ; X+ • - . ~ n X + . - - ; X + - - - - * B + . . - ; X + . . . ~ C + . - . ; where A, B, C . . . . are other reactants. If x is the concentration of X and T is the temperature, then a model for the evolution of x and T is given by the equations (Gray and Yang [97])

5.4. T h e t h e r m a l - c h a i n b r a n c h i n g m o d e l

Still more complex sequences of reactions occur in an analysis of the thermal-chainbranching model for the H~-O2 reaction, Golu- bitsky, Keyfitz and Schaeffer [95]. One feature of this reaction is the occurrence of an e x p l o s i o n p e n i n s u l a , the non-convexity of the explosion line on a pressure-temperature diagram (fig. 30), which implies that the reaction is more complicated than the simple

2H2 + 02 ~ 2H20.

dx d-T = ki + ((n - 1)kb -- kw - kg)x,

dT d---t- = hiki + (hbkb + hwk~ + h~ks)x - ( T - To).

This model has a large number of parameters, inasmuch as the reaction rate "constants" k are functions of T with extra hidden parameters. Reasonable approximations reduce the number to 18. Scaling and non-dimensionalizing reduces further to 9. Golubitsky, Keyfitz and Schaeffer argue that two of these may be fixed in the H2-O2 reaction: two more are assumed small.

I. Stewart/Physical applications ol catastrophe theory 291

The resulting five-parameter system has steady states given by an equation

G ( T , v, a, To, Z, ~2, '~3) = EsO s -- 3E2v 2 + Z E 1 v - a l ( T - To) = O,

where

Ej = exp(~/j/T), j = 1, 2, 3.

It is proved that for any value of ~/s between 0 and 1.07 there is a unique point (T °, v °, a ° T °, Z 0, ~/0, ,/~ near which G is contact equivalent (as a bifurcation problem in T, v) to

H ( x , A) = x 3 - 3mA2x +-2A 3,

where m is a modal parameter and Jm[ ~< 1. The other four parameters (along with m) unfold this universally.

The large number of parameters make it awkward to compute the bifurcation, hysteresis

and double-limit varieties (though Stewart [229] has now obtained these, at least for negative modal parameter). Nevertheless, Golubitsky, Keyfi tz and Schaeffer prove that there are precisely seventeen possible stable bifurcation diagrams in the universal unfolding of H, hence in G: these are shown in fig. 31. The proof first locates certain unstable diagrams, then uhfolds these.

As for the winged cusp, all of these diagrams may be realised as paths through a cusp catastrophe x 3 + ax + b = 0 with a(A) = ~(A 2 + 8A + E), b(A) = A 3 +/3A + a. The sequence of Ls and Rs in fig. 31 describes the order in which such curves can cross the left and right nappes of the cusp curve.

Several of the perturbed diagrams give rise to explosion peninsulas, for example L R R L L R and R L L L . It follows that the thermal chainbranching model is sufficiently complex to permit this effect: prior to the above analysis, however, the evidence for this assertion was entirely heuristic.

O RR LL

RLLR Lg'RR

LR LRRL

LLLR LRRLLR

LRRiI~RL RLLLLR LRLR LRLRRR

LLLRLR RL RRRL RLLL

Fig. 3 I. The 17 stable bifurcation diagrams occurring in a special unfolding of the organizing centre for the thermal-chainbranching model. Diagrams LRRLLR or RLLL can model the explosion peninsula.

292 L Stewart~Physical applications o/catastrophe theory

5.5. The Brusselator

The tri-molecular diffusion model of Lefever and Prigogine [131], or "Brusselator", is governed by the equations

OX ~ 0 2 X -- l . ) l -~r+ X 2 y - ( B + 1)X +A,

at

OY - 32Y Ot = L)2 - ~ - - x E y + BX,

with Dirichlet boundary conditions X(0)= X(rr) = A, Y(O)= Y(rr)= B/A. Here X and Y are chemical concentrations, A and B are externally controlled parameters also representing chemical concentrations, D1 and DE a re

diffusion coefficients, and ~ E [0, 7r] is a spatial variable. With B as bifurcation parameter the first bifurcation from the trivial solution X = A, Y = B/A can be either to a steady or periodic solution, Auchmuty and Nicolis [27]; if (say) DE/Dj > 3, the steady case holds, as is hence- forth assumed. First bifurcation is from a simple eigenvalue, unless A2=OlDEkE(k + l ) 2 for integer k, when the eigenvalue is double. The case of simple eigenvalue has been widely studied (Auchmuty and Nicolis [27], Hershkowitz- Kaufman [110]). The double eigenvalue case is discussed by Keener [123]: a rigorous and more extensive analysis has been given by Schaeffer and Golubitsky [200] using their imperfect bifurcation theory. We describe some of their results.

In general either one or three solutions bifurcate from the trivial solution, with various stability assignments: there are five cases in all, corresponding to specified regions of (A,D~,DE)-space. For each case the effect of two different perturbations is studied. The first is variation of (A, DbD2) so as to split the double eigenvalue. The second is to make A spatially variable, of the form

A c o s h X/-~(~ - Ir/2) A(~)= o c o ~ E - ~ / 2 '

which corresponds to the more realistic assumption, that experimental concentrations can be fixed only on the boundary.

The problem is analysed by equivariant methods, using the Z2 symmetry ~ ~ I r - ~. This symmetry, apparently not exploited hitherto in any systematic way, is crucial to the effects observed. In particular the occurrence of secondary bifurcation depends upon it.

Each of the two perturbations considered gives a 1-parameter family of bifurcation diagrams around one of the five "perfect" cases, giving rise to a total of 25 diagrams (some of which are "unphysical" for this problem). Per- haps the most interesting diagram for the first perturbation is fig. 32, where the first bifurcation branch exists only for a short period. Note the advantage here of treating two nearly coincident eigenvalues as perturbations away from the "organizing centre" of a double eigenvalue. Without doing this, local bifurcation analysis might locate the two bifurcation branches, and a subsequent secondary bifurcation where two branches disappear, but cannot guarantee that the branches so obtained join up. This is a general phenomenon, and a major strength of organizing centres in imperfect bifurcation methods: for a near-degeneracy, a perturbed version of the actual degeneracy has a wider range of validity than a direct local analysis of the non-degenerate problem itself.

The second perturbation gives somewhat different results. One interesting diagram is fig. 33, where tracking down stability assignments on branches leads to two branches of steady-

+ S • - + U

+ S

Fig. 32. The Brusselator at a double eigenvalue: a branch may not persist for long. Here x and y are amplitudes of eigenfunctions, and ,~ is a bifurcation parameter. After Schaeffer and Golubitsky [200].

L Stewart~Physical applications of catastrophe theory 293

÷ U ( 4 U

x\ / " ÷11 v

'( - ? 'l

Fig. 33. The Brusselator: second perturbation. Changing stability assignments along branches indicate Hopf bifurcation. Here s = stable, u = unstable, and -+ is the Leray- Schauder degree. After Schaeffer and Golubitsky [211111.

state solutions that undergo a change in stability without encountering other branches. These are Hopf bifurcations (and it can be proved that only one occurs on each branch) to periodic solutions. These will have large periods, (7(e-'/2). They can occur in the imperfect problem for much lower values of B than that which leads to a first bifurcation to periodic solutions, of the "perfect" problem, so that the "perfect" analysis alone is here somewhat misleading.

5.6. Phase transitions

Thermodynamic phase transitions appear an attractive area for the application of catastrophe theory, inasmuch as the feature of discontinuous change with parameters, and structural instability, are clearly evident in the physical phenomena. Making use of this idea has proved hard; and a simplistic approach, either by devo- tee or critic, is liable to founder.

One of the earliest illustrations of the cusp catastrophe is as a transformation of the Van der Waals equation, Fowler [73]. Many other standard models of phase transitions may be reformulated similarly: for example the Weiss model of ferromagnetism (Poston and Stewart [183]) and the Tizza theory (Rao and Rao [188]). More generally, the Landau theory and similar mean field theories translate directly into the language of elementary catastrophe theory. Let • (x ,a) be a potential function in an order parameter x, with macroscopic variables a as parameters: assume the system attempts to

minimize ¢. A simple discussion of this idea is given by Poston and Stewart [183]: more detailed treatments are found in Dubois [69], O'Shea [170], and Vendrik [274, 275]. We describe the latter in more detail.

Vendrik assumes a model equation involving a partition function of the form

Z(/~) = (N/2) n f exp[-/3N~(x,/~)] dx,

where/~ is a vector of intensive thermodynamic variables and x is a dummy variable which integrates out of the formula. This leads to a Landau-like model in which /~ globally mini- mizes • . In addition to certain other properties,

is assumed smooth. Vendrik derives a classification of the local structure of phase diagrams, based on this model, by a version of elementary catastrophe theory, modified to place emphasis on the stability of the phase diagram itself (GM-stability). Physical examples discussed include metamagnets, He3-He 4 mixtures, ferroelectrics, and tricritical points. It is proved that the model predicts "universal" values of the critical exponents /3 and 8, as in the Landau theory, in the cases of second order phase transition and tricritical points. The Gibbs phase rule is discussed. Dukek [72] adapts a butterfly catastrophe to study the 180 ° rule at a triple point in a phase diagram. Keller et al. [126] show how various cuspoid catastrophes organize the morphology of phase transitions, including laser action, binary, ternary, quater- nary, or 5-component mixtures, ferroelectrics, ferromagnetics, superfluid helium, tricritical and tetracritical points; and with two order parameters, various coupled systems, such as ferromagnetic-ferroelectric systems (Imry et al. [121]).

Landau-type theories are common in many areas of science, and the organization produced by a catastrophe-theoretic formulation, as well as the more rigorous analysis of Taylor series truncations and perturbations that this affords, have proved useful.


The "universality" of the predicted critical exponents is a potential embarrassment, since these exponents often disagree with experiment. This has led some authors to conclude that "the" catastrophe model of phase transition is " w r o n g " - a n unfortunate oversimplification which overlooks the possibility of using a mathematical tool in more than one way. (Analogously, in optics "the" catastrophe model is ray optics, which is "wrong" on caustics. But semiclassical wave optics, which also makes good use of catastrophe theory, works well.)

The experience of the Landau theory suggests that to each critical point in a phase transition there is associated a singularity that "organizes" it, and describes the behaviour adequately everywhere e x c e p t near the critical point. The optimistic view is that this singularity is also crucial to the understanding of the critical point itself, but by some less simplistic approximation than that used in a mean field theory. The pes- simistic view is that it is some kind of artefact. This disagreement is not yet resolved, but there are some partial results in this direction.

Benguigi and Schulman [31], Schulman [202], Dangelmayr [65], and Keller [124] have all pro-

posed making a non-diffeomorphic change of variables x = m °, a = t A, in the cusp catastrophe model x 3 + ax + b = 0 derived from a Landau- type theory. Here m is some order parameter, and t = ( T - T¢)/Tc is a normalized temperature. Poston and Stewart [183] object to this, on the grounds that it is ad hoc, permits too wide a range of possible models to have good predic- tive value, and destroys the basis of transversality arguments, which lead one to expect a cusp in the first place. They note that a der iva-

t ion of sugh a form from other assumptions might overcome these objections. A possible derivation is given by a lattice model of Keller et al. [125, 126] who show that all the standard phase transition models reduce to a scaled cusp of this type. The critical exponents may be computed in terms of the two parameters 0 and A. Results are shown in tables VI and VII. Here the "cusp scaling" and "butterfly scaling" models are obtained by fitting two of the critical exponents, and predicting the remainder. The fit appears good, although the anomalous lattice dimension 3.09 requires comment (Keller et al. [126]).

A model involving only macroscopic variables

Table VI Numerica l values of critical exponents in models of second order phase transit ion, after Keller, Dangelmayr and Eikemeier [126]

Model A 0 a, a ' /3 8 3', 3"

General cusp 2(1 - A ) A/20 40 - 1 Landau 1 1 0 0.5 3 2 dim. Ising 1 4 0 0.12 15 3 dim. Ising 0.94 1.5 0.12 0.31 5 3 dim. Heisenberg (n = ~) 1 1.5 0 0.33 5 3 dim. Heisenberg

(s = +-~) 1.1 1.43 - 0 . 2 0.38 4.72 Cusp scaling 0.98 1.36 0.04 0.36 4.44 Exper iment 0.03 0.34 4.25

+-0.14 -+0.04 +-0.13

A ( 2 0 - 1)/0 1

1.75

1.25

1.33

1.42

1.24 1.20

-+0.12


Table VII Numerical values of critical exponents in models of tricritical point, after Keller, Dangelmayr and Eikemeier [126]

Model A 0 a a ' /3 8 y, y'

General butterfly 0 2 - 3A/2 A/40 60 - l Landau 1 l 0 0.5 0.25 5 Butterfly scaling 0.96 1.14 0 0.56 0.21 5.84 1.02 Experiment 0 0.56 0.21 ? 1.01

-+0.06 -+0.05 -+0.03

~(30 - 1)/20 1

is unlikely to resolve the difficulty of critical exponents (Poston and Stewart [183], Grandy [96]) since it can be traced to the failure of a finite-dimensional approximation to the partition function on the infinite-dimensional space of all states. (For a partially dissenting view see Vendrik [274].)

A standard technique that produces excellent values for critical exponents is the renor- realization group approach. Rasetti [189] notes that both Landau and renormalization group techniques may be viewed as catastrophe- theoretic methods. He proposes a model in which a thermodynamic state is realised as a cross-section of a certain fibre bundle, from which is constructed an elliptic operator T. Phase transitions correspond to changes in the (integer-valued) Atiyah-Singer index of T, a topological invariant. As an example, Rasetti rederives the critical temperature for phase transition in the 2-dimensional Ising model, originally due to Onsager [169].

5.7. Fluid flow

Berry and Mackley [40] discuss the motion of fluid in a six-roll mill, assumed two-dimensional and governed (locally) by a stream function x3-3xy 2, the elliptic umbilic singularity. Per- turbations of this "ideal" flow are given by the universal unfolding

x 3 _ 3xy2 + a(x 2 + y2) + bx + cy

whose additional terms are recognised as superimposed rotational and linear flow. By universality, no other perturbations need be considered in order to obtain a complete catalogue of the local behaviour of smooth perturbations to the stream-function. There is a rich variety of flow-patterns, illustrated in fig. 34 (see Poston and Stewart [183] for full details). Berry and Mackley use these results as an experimental technique to study the properties

@ i

@ @ @ @ @ Fig. 34. Fluid flow in the 6-roll mill: classification of flow- patterns in a cross-section of the universal unfolding. After Poston and Stewart [183].

296 I. Stewart[Physical applications of catastrophe theory

of polymer solution, obtaining good agreement between theory and experiment.

Thorndike, Cooley and Nye [267] modified this idea to derive a general way of characteriz- ing the basic structure of a flow field by way of certain singularities, namely, points at which the velocity gradient matrix becomes singular. They obtain the generic singularities of this type (those which can occur in a structurally stable fashion) and their typical evolution-patterns. Applications to the geostrophic wind and the motion of Arctic ice are discussed. Here the idea is to use the singularity structure to compare theoretical and observed data. Nye and Thorndike [168] extend their methods to three- dimensional flow fields. Theire methods are partly heuristic, but can probably be made rigorous by a systematic use of methods similar to those of Golubitsky and Schaeffer [90].

Benjamin [32, 33] describes, theoretically and experimentally, the formation of Taylor cells in the flow of a viscous fluid inside a rotating cyl inder -a famous problem (Taylor [241]) in fluid dynamics. His work takes account of end effects. Specifically, taking two parameters a (Reynolds number) and b (length of cylinder) he observes the equilibrium states after first bifurcation into Taylor cells. Depending on the values of a and b, flows with 0, 2 or 4 cells were found. These equilibria form a cusp catastrophe surface, which (together with delay convention) adequately predicts the morphology for arbitrary quasistatic variation of a and b. In particular there is a hysteresis effect.

Here catastrophe theory is used to model the steady states of a system governed by the Navier-Stokes equations (whose detailed dynamics appear insoluble): its grip on what phenomena are generically possible leads to a unique simplest "natural" model. In an attempt to strengthen the link between this model and the Navier-Stokes equations themselves, Schaeffer [198] applied the methods of Golubit- sky-Schaeffer to the problem, treating it as a perturbation of bifurcation at a multiple eigen-

value (rather as for the buckling plate, section 4.7).

Golubitsky and Schaeffer [92] have applied their methods to the B6nard problem in spherical geometry (modelling convection in the Earth's mantle) in the Boussinesq approximation. This leads to a series of O(3)-invariant bifurcation problems, of which they analyse that in which the group is represented irreducibly on R 5 (via spherical harmonics y2,,). The bifurcation problem can be reduced to an S3-equivariant problem in R 2 (where it turns out to be identical with an elasticity problem currently being studied by Buzano, Geymonat and Poston [46]). The results are compared with theoretical work of Chossat [56] and numerical results of Young [285]. The new contributions (aside from cor- rection of a minor slip in Chossat's paper) are the proof that in the self-adjoint case two branches of axisymmetric solutions bifurcate, one being stable; that the stability does not depend, as was thought, on the sign of a certain parameter; and the analysis of all 0(3)- equivariant perturbations of the problem without having to start the calculations over again.

Drazin [68] has used catastrophe theory in a rather different way, to study gravity waves. He finds a complex swallowtail singularity which may illuminate an early, naive account of wave- breaking by Zeeman [288].

5.8. Degenerate Hopf bifurcation

The simplest non-elementary catastrophe is the Hopf bifurcation to a limit cycle (Hopf [118]). The dynamic nature of this bifurcation appears quite different from the static nature of a singularity. Smale [222], for examples, suggests that 'the Hopf bifurcation lies deeper than CT'. In a recent, important paper, Golubitsky and Langford [89] show that on the contrary, the Hopf bifurcation theorem may be proved by catastrophe theory methods, and further that the same methods classify and unfold


degenerate Hopf bifurcations not accessible to classical techniques.

The setting is a differential equation

dl) m ~ dt f(v, 2t),

where f :R" x R ~ R n . As always, A is a bifurcation parameter. The hypotheses for the classical Hopf theorem are: (HI) dd has simple eigenvalue +i (after res- caling t if necessary) at (0,0); and (non- resonance condition) no other eigenvalues of the form ki for integer k (H2) As A varies through 0, the eigenvalues cross the imaginary axis at ---i transversely.

Given these, Hopf proved that there is a unique branch of periodic solutions, of period near 2¢r, bifurcating from (0,0). Along the branch, A is an even function of x, with power series expansion

~. = /./,2 x 2 "t- ~I,4X 4 + • • ".

If further the nondegeneracy condition

(H3) p2 ~ 0

holds, then the bifurcation is supercritical or subcritical as/~2 > 0 or <0.

Golubitsky and Langford [89] retain (H 1), but dispense with (H2) and (H3). Their method is conceptually simple, and its component parts more or less well known, but the use of singularity theory extracts much more from them. They seek periodic solutions of period near 2~r, and rescale time to make the period exactly 2~r. Because of (HI) the Lyapunov-Schmidt reduction applies (see references cited in [89]) and the problem can be transformed to one in equivariant singularity theory, with a circle-group action (corresponding to periodicity of the solution). This in turn reduces to a Z2-action, and a very straightforward problem in singularity theory. The degenerate Hopf bifurcations (in reduced

Z2-invariant form) are given in table VIII, for codimension ~<3; and also in the case /-~2 = 0, /~4#0 for arbitrary codimension, with the exception of certain cases of codimension i>5. Some of the resulting bifurcation diagrams are shown in fig. 35: most of these are just sample interesting cases from a multiparameter unfolding, and [89] should be consulted for further information.

Sample applications given in [89] involve glycolitic oscillations and the Fitzhugh nerve impulse equations. In the first case a bifurcation of the form shown in fig. 35a occurs: classical methods locate the two sub- and supercritical branches, but do not prove they join. The method is obviously widely applicable: for example, the results of Rinzel and Miller [194] indicate bifurcations like fig. 35b, and further work may locate a suitable organizing centre. There is thus the prospect of applying catastrophe theory to periodic solution of the Hodgkin- Huxley nerve impulse equations - a quite different use from the nerve impulse model of Zeeman [289], heavily (and largely wrongly, see [230]) criticized in [235].

Perhaps the main implication of this work is that the apparently dynamic nature of periodic motion is illusory: it is "really" a static

Table VIII Classification of degenerate Hopf bifurcations, after Golubitsky and Langford [89]

Codimension Normal form

0 x 3+-AX (classical Hopf) 1 x ~ -+ A2x 1 x 5 - + •x 2 x3+-X3x 2 x 7+ - 2tx 3 x 3 +- X4x 3* xS+2bAx3+eX2x, e=_+ l , b 2 ¢ e 3 x ~ ± 2(it -+ X2)x J + XZx 3 xS+_2Ax3±it3x 3 x T ± A x 3 ± A e x 3 x 9 +- ,~x

Note that the case * has a modal parameter b. The topological codimension is 2, not 3.


S

a b

S

c d

Fig. 35. Degenerate Hopf bifurcation: sample diagrams from various unfoldings. Here x denotes the amplitude of the periodic motion and A is a bifurcation parameter. After Golubitsky and Langford [89].

p h e n o m e n o n , d r i ven b y a c i r c l e -g roup ac t ion .

F u r t h e r d e v e l o p m e n t s of this i dea are l ike ly ,

and the re is s o m e c h a n c e tha t i nva r i an t tor i (at

l eas t , 2 -d imens iona l tor i ) m a y be a c c e s s i b l e to

s imi lar m e t h o d s - t h o u g h the e x p e r t s r e m a i n

skep t i ca l .

N o t e tha t once the c l a s s i ca l H o p f t h e o r e m is

u n d e r s t o o d in c a t a s t r o p h e - t h e o r e t i c t e rms , the

e x t e n s i o n to d e g e n e r a t e H o p f b i f u r c a t i o n is

i m m e d i a t e . S ingu la r i t y t h e o r y is a p o w e r f u l

m a t h e m a t i c a l m a c h i n e , r e a d y to run as soon as

the c o n d i t i o n s are r ight ; and the r e q u i r e d con-

d i t ions can o f t e n be u n d e r s t o o d b y e x a m i n i n g a

s ingle, p r e f e r a b l y s imple , e x a m p l e ( avo id ing the

m o r e a w k w a r d d e g e n e r a c i e s ) . I n f o r m a t i o n on

the more d e g e n e r a t e ca ses e m e r g e s a u t o m a t i c -

a l ly f rom this p r o c e s s . S imi l a r ly the en t i re

G o l u b i t s k y - S c h a e f f e r m e t h o d s t ems f rom a

p r o p e r u n d e r s t a n d i n g of the p i t c h f o r k b i fu r -

c a t i on of an Eu le r a r ch ; m u c h of the op t i c s

d e v e l o p s a u t o m a t i c a l l y f rom fo ld and cusp

caus t i c s ; and the c h e m i c a l a p p l i c a t i o n s t r ace

b a c k to the s t a n d a r d s t i r red t ank mode l .

R e f o r m u l a t i n g s t a n d a r d m o d e l s as c a t a s t r o p h e s

is no t m e r e l y an idle and a c a d e m i c e x e r c i s e in

o b f u s c a t i o n : i t is a n e c e s s a r y , and i l lumina t ing ,

p r e l i m i n a r y to the a p p l i c a t i o n of a p o w e r f u l and

gene ra l m e t h o d p r o v i d i n g genu ine ly new resu l t s .

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[289] E.C. Zeeman, Differential Equations for the Heartbeat and Nerve Impulse, in: Towards a Theoretical Biology, C.H. Waddington, ed., vol. 4 (Edinburgh Univ. Press, Edinburgh, 1972) pp. 8-67.

[290] E.C. Zeeman, Catastrophe Theory- a Reply to Thorn, Manifold 15 (1974) 4-15, also in Dynamical Systems- Warwick 1974, A. Manning, ed., Lecture Notes in Math. 468 (Springer, New York, 1975) pp. 373-383.

[291] E.C. Zeeman, Classification of Elementary Catastro- phes of Codimension ~<5, in Structural Stability, the Theory of Catastrophes, and Applications in the Sciences, P.J. Hilton, ed. (Springer, New York, 1976) pp. 373-395.

[292] E.C. Zeeman, The Umbilic Bracelet and the Double- Cusp Catastrophe, in: Structural Stability, the Theory of Catastrophes, and Applications in the Sciences, P.J. Hilton, ed. (Springer, New York, 1976) pp. 328-366.

[293] E.C. Zeeman, Euler Buckling, in: Structural Stability, the Theory of Catastrophes, and Applications' to the Sciences, P.J. Hilton, ed. (Springer, New York, 1976) pp. 373-395.

[294] E.C. Zeeman, Catastrophe Theory: Selected Papers (1972-1977) (Addison-Wesley, Reading, MA, 1977).

[295] E.C. Zeeman, A Catastrophe Model for the Stability of Ships, Geometry and Topology, Lecture Notes in Math. 597 (Springer, New York, 1977) pp. 775-827.

[296] E.C. Zeeman, Catastrophe Theory, in: Structural Stability in Physics, W. Giittinger and H. Eikemeier, eds. (Springer, New York, 1979) pp. 12-22.

[297] E.C. Zeeman, Bifurcations, Catastrophes, and Tur- bulence, New Directions in Applied Math., Case Western Reserve Univ. (1980).

[298] E.C. Zeeman, A Boundary Value Problem Involving Cusps (to appear).

[299] E.C. Zeeman and J. Hayden, 1980 Bibliography on Catastrophe Theory, preprint, Univ. of Warwick (1980).

Documents

Applications of catastrophe theory to the physical sciences