Beyond Elementary Catastrophe Theory

Ian Stewart

Mathematics Institute Uflivfmity of Warwick Coventry CV4 7AL, England

ABSTRACT

We give a brief discussion of the relations between elementary catastrophe theory, general catastrophe theory, singularity theory, bifurcation theory, and topological dynamics. This is intended to clarify the status, and potential applicability, of “catastrophe theory,” a phrase used by different authors and at different times with different meanings. Catastrophe theory has often been criticized for (supposed) applicability only to gradient systems of differential equations; but properly speaking this criticism can apply only to the elementary version of the theory (where it is in any case wrong). Roughly speaking, elementary catastrophe theory deals with the singular- ities of real-valued jkctiorts, general catastrophe theory with singularities of flows. Between these lies singularity theory, which deals with oector-valued functions. All relate strongly to bifurcation theory and topological dynamics. The issue is more subtle than it appears to he, and we describe an example where elementary catastrophe theory has heen used to solve a long-standing problem about nongradient flows: degenerate Hopf bifurcation.

This is a brief description of some recent progress in catastrophe theory, and how it relates to certain other areas, notably bifurcation theory and topological dynamics. It is often claimed that catastrophe theory is a very limited subject, but that analysis does not stand up to serious scrutiny.

Bifurcation theory, at least in its classical form, uses methods from analysis to study the way that solutions to partial differential equations can branch off each other as parameters vary. Topological dynamics uses more modem ideas, basically qualitative in nature, but rigorous and powerful, to look at systems of urdinuy differential equations, and see how the solutions to a single equation behave. This perhaps sounds more restrictive. Catastrophe theory, in the most general sense, is really a combination of the two: topological bifurcation theory for ODES (that is, dynamical systems). So it ought than topological dynamics, but less so than bifurcation

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to be more general theory (because it

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26 IAN STEWART

doesn’t handle PDEs). But what is often claimed is that in practice general catastrophe theory doesn’t say anything new; in practice catastrophe theory reduces to elemmtay catastrophe theory, which is about the topological bifurcations of real-valued functions (or, even more restrictively, germs: locally defined functions). With some extra generality one gets a theory for vector- valued functions: this is called singularity theory. So the sort of picture often presented is this:

BIFURCATION THEORY / \

TOPOLOGICAL DYNAMICS GENERAL CATASTROPHE THEORY

(which doesn’t really exist)

SINGULARITY THEORY

(ELEMENTARY) CATASTROPHE

THEORY

I suggest that this is actually a misleading way to look at how useful the ideas are.

First, general catastrophe theory is much more extensive now than it was ten years ago, and it has largely developed out of ideas first worked out on the elementary catastrophes. For example, singularity theory is clearly a part of general catastrophe theory. There have been attempts to draw a heavy line between elementary catastrophes and singularities, based on the fact that numerical-valued functions and vector-valued functions are very different- most vector-valued functions aren’t gradients. Certainly there is an important technical distinction; but the mathematics, once one looks below the surface, works out almost the same. A few changes to the definitions are needed, such as replacing “right equivalence” with “contact equivalence,” but the same main ideas and techniques are required: ideas like determinacy, unfoldings, and the Malgrange preparation theorem. Neither mathematically nor histori- cally is it possible to argue that any such hard-and-fast line really exists. However, there certainly exist many good applications of singularity theory now, whereas many critics of “catastrophe theory” have claimed they can prove it has no applications. So there is a tendency to try to pretend that some fundamental difference exists, even though it does not.

In addition, a theory should be judged, not by the generality of the problems it attempts to tackle, but by what it actually manages to achieve. Most ways to solve PDEs reduce them to various kinds of ODE: and what’s more, one can often approach a PDE as a kind of infinite-dimensional ODE,

Beyond Elementary Catastrophe Theoy 27

and most of catastrophe theory has at least potential extensions to infinite dimensions [l]. I don’t mean there’s nothing different about PDEs-quite the contrary. But in practice any new depth of insight into ODES is bound to carry over to PDEs in some important ways. (For example, not only do reaction-diffusion equations look rather like dynamical systems-there’s a very close mathematical link.)

The elementary catastrophes may appear to be very limited, but they are so fundamental that they show up in a wide variety of circumstances, even where they have no right to if one thinks unimaginatively about gradient dynamics. Rather than go into this in general, let me just give one surprising example, where the whole tower of subjects collapses together.

It has to do with the Hopf bifurcation [2, 31, where a sink turns into a source and throws off a stable limit cycle, as in Figure 1. The original theory of Hopf can be stated as follows. Consider the differential equation

$=f(o,h), (*)

FIG. 1.

28 IAN STEWART

where u E BB *, and X E IL! is a parameter. Suppose f(O,O) = 0. If further

(Hl) the Jacobian of f at (0,O) has simple eigenvalues on the imaginary axis,

(H2) the eigenvalues cross the imaginary axis transversely as A passes through 0,

then there exists a unique branch of nonconstant periodic solutions bifurcat- ing from the zero solution at X = 0.

This solution can be parametrized by amplitude x, and then

If

(H3) ~2 # 0,

then the solution branch is either super- or subcritical depending on the sign

of P2*

The first obvious point about the Hopf bifurcation is that it cannot happen in a gradient system. There can be no limit cycles in a gradient system. This

’ appears to rule out elementary catastrophe theory as a way of studying the Hopf bifurcation; indeed, the Hopf bifurcation is “dynamic” and does not look much like the “static” bifurcations of elementary catastrophe theory. Many critics of catastrophe theory have so reasoned-for example, Smale [4] considers the Hopf theorem to be deeper than catastrophe theory (meaning elementary catastrophe theory) and exhibits it as a counterexample to the claim sometimes made that catastrophe theory is the first theory of discon- tinuous change.

Recent work of Golubitsky and Langford [5] casts a new light on such arguments. They show that not only can the Hopf theorem be proved using elementary catastrophe theory, but that the same method allows a lot to be said when hypotheses (H2) and (H3) are both false. There has been a lot of research by bifurcation theorists where either (H2) or (H3) is dropped, but not both: the Golubitsky-Langford approach seems the natural way to get a full understanding of the phenomena and is definitely more powerful than the previous methods. The actual proof, catastrophe theory aside, is reasonably straightforward, using one extra standard trick and then applying existing machinery.

Thus not only is the Hopf theorem not deeper than elementary catastrophe theory, being in fact a relatively simple consequence of it; it also takes its rightful place as the first in a more general sequence of theorems (dropping H2 and H3) which together constitute a theory of degenerate Hopf bifurca-

Beyond Elementary Catastrophe Theoy 29

x

t

FIG. 2.

tion. This theory provides a full analysis of perturbations, by way of unfold- ings of the degenerate cases; it is easy to read the results directly from catastrophe theory.

This looks surprising at first; but in fact it is entirely natural. It is necessary only to take proper note of a fact which has been well known for some time: the amplitude in a Hopf bifircation looks exactly like a symmetrized CMQ cata.stmphe. (See Figure 2.)

Here’s a sketch of the argument. Seek periodic solutions of (*) by resealing time so that the period is exactly

2a, and working in the space C&, of periodic functions. Perform a Lyapunov- Schmidt reduction (essentially a use of the implicit-function theorem for a Banach space-a standard but extremely useful trick) to convert to a problem in two dimensions. Observe that phase-shifts in periodic solutions introduce a circular symmetry. Take the two-dimensional circularly symmetric problem and slice it along a diameter to get a one-dimensional problem with mirror symmetry. Apply the Z s-equivariant version of elementary catastrophe theory to this (that is, do ECT but use only even functions-a straightforward extension of the classical case). That’s it.

In other words, a degenerate Hopf bifurcation is just an elementary catastrophe, symmetrized to even functions, and spun in a circle.

ActuaUy doing this is by no means trivial: Golubitsky and Langford put in a lot of effort. But the deepest part of the whole argument is without doubt the catastrophe theory.

Finally, note that the Hopf theorem is often applied to a differential equation arising from a PDE, and that the new theory is tailor-made for the same kind of use -and watch that tower concertina together.

There’s a great deal of new work using elementary catastrophes or singularities, often with symmetry conditions, to solve problems in bifurcation

IAN STEWART

theory. [To forestall some objections, I should point out that Zs-equivariant singularity theory for real-valued functions (1) is catastrophe theory-the equivariant version has been around since Poenaru [6], and nobody has ever tried to suggest it is not catastrophe theory; and (2) is certainly no deeper than elementary catastrophe theory itself. The main deep result is the Malgrange preparation theorem, and this plays a similar role throughout all versions of the theory-unless you think that the representation theory of Z, is deeper than Malgrange.]

I’m not trying to say that catastrophe theory solves all of the problems in bifurcation theory-it doesn’t-and I’m not saying that it is better than the classical approach, because it isn’t. The point is that the whole tower of subjects represent different ways of looking at the same kinds of problem, and that these approaches complement each other. In the Golubitsky-Langford work, observe how the “classical” Lyapunov-Schmidt reduction fits like a glove with the equivariant singularity theory. Both approaches contribute.

Catastrophe theory, nowadays, is a much broader and better-developed area than it was a decade ago. It can handle vector-valued instead of real-valued functions; add symmetries; put in boundaries, distinguish im- portant parameters, and preserve their special nature; work in infinite-dimen- sional spaces; and add constraints such as volume preservation. Notions such as unfoldings and organizing centers [7] generalize to some dynamic bifurca- tions including chaotic ones. There is plenty left to do, and not all of it will follow the present pattern or be associated with the particular school of thought initiated by Thorn and Zeeman; but the basic ideas are absolutely fundamental and extremely important, and they apply much more widely than a superficial look at the setup might suggest. Many of the criticisms that have been leveled at “catastrophe theory” are directed at a pale shadow of the real thing, and this helps nobody and advances science not a single step.

I believe catastrophe theory has promise; but if the experience of other subjects is anything to go by, it will take a lot of work directed at specific issues peculiar to a given application to fulfill that promise. And it might not work at all. But, in a lot of areas, it certainly does [B]. It is true that the physical sciences were not the area at which some criticisms were aimed; but many of those criticisms, if correct, rule out any successful applications, with no discussion of the particular area. So successes in physics show that the critics cannot be totally correct. There are more modest successes in biology and behavioral sciences too. Catastrophe theory may never live up to its image in Newsweek-but I don’t believe scientists should judge subjects by reports in the popular press. It is certainly an important contribution to science in general; but I suspect that all of the various schools of thought that border on the area of nonlinear modeling have a lot of overlaps: some kind of unification will eventually emerge. Bifurcation theory, topological dynamics,

Beyond Elementary Catastrophe Theory 31

catastrophe theory, singularity theory, nonequilibrium thermodynamics, lattice dynamics, renormalization group, spontaneous symmetry-breaking, synerget- its: these are all part of the same big picture, and it’s no wonder proponents of the various approaches often argue, because they tend to tread on each other’s toes, and they don’t always recognize when they’re saying the same kind of thing in different languages. It is this broad picture that Thorn [9] was advocating; and it is clear from Zeeman’s early papers that he intended “catastrophe theory” to refer to an overall viewpoint on this broad picture- not a very specialized technique or theorem. As it happens, even that specialized technique has proved far more useful than many people thought; but it is the broad picture that is the really important thing, and the special results are most interesting, in the long run, because they open up paths into the broader area.

It has been suggested that it is a waste of time to try to apply catastrophe theory because it is the wrong kind of mathematics. This suggestion hints at the idea that research is only worth doing if you can guarantee results beforehand, which I think quite wrong. The point is that catastrophe theory does work, in many areas: the important thing is to use it intelligently and not just as some kind of catchall. And the mathematics is so basic, and so beautiful, that nobody should feel that effort put into learning it is wasted.

REFERENCES

1 L. Arkeryd, Thorn’s theorem for Banach spaces, .I. Lot&n Math. Sot. 19:359-370 (1979). (See also Reference [8] for other papers.)

2 E. Hopf, translated in The Hopf B’f E urcution and Its Applications (J. E. Marsclen and M. McCracken, Eds.), Springer, New York, 1976.

3 B. D. Hassard, N. D. Kazarinoff, and Y.-H. Wan, Theory and Applications of Hopf Bifurcation, LMS Lecture Notes 41, Cambridge U.P., 1981.

4 S. Smale, Book review, Bull. Amer. Math. Sot. 84:136&1368 (1978). 5 M. Golubitsky and W. F. Langford, Classification and unfoldings of degenerate

Hopf bifurcations, to appear. 6 V. P&naru, Singularitb C” en Prt%etace de Sytie, Springer, 1976.

7 E. C. Zeemau, Bifurcations, Catastrophes, and Turbulence, in New Directioru in Applied Mathematics, Case Western Reserve Univ., 1980.

8 I. N. Stewart, Applications of catastrophe theory to the physical sciences, Physica 2D:245-305 (1981).

9 R. Thorn, St&W Structurelle et Morpho&&se, Benjamin, New York, 1972.