Nuclear Physics B (Proc. Suppl.) 2 (1987) 191-200 19 l North-Holland, Amsterdam

CATASTROPHE THEORY

Robert GILMORE*

Department of Physics and Atmospheric Science Drexel University, Philadelphia, PA 19104 USA

Catastrophe Theory is a program. The aim of the program is to classify the qualitative properties of the solutions of equations, and to determine how these qualitative properties depend on the parameters which appear in these equations. Elementary Catastrophe Theory is the implementation of this program in the simplest class of nonlinear dynamical systems: it is the study of the equilibria of gradient dynamical systems.

1. INTRODUCTION

"What is the relation between Catastrophe Theory and chaos?" This often asked question is

misleading. Chaos and Catastrophe Theory can only be compared in the same way as apples and

oranges can be compared.

Catastrophe Theory 1 is a program. The objective of this program is to classify the

qualitative behavior of the solutions of equations, and to determine how the qualitative behavior

depends on the parameters which occur in the equations. 1-6

Chaos is one kind of qualitative behavior exhibited by various types of dynamical systems.

The study of the routes to chaos, and the phenomena associated with each route, is an

implementation of Catastrophe Theory for those dynamical systems which can exhibit chaos.

Catastrophe Theory itself is a program that has barely begun. To date, we are unable to

begin the classification of the qualitative properties of the solutions of integro-differential

equations, partial differential equations, and differential equations involving time derivatives of

the state variables in arbitrary functional form. Some limited knowledge exists for systems of

equations in which the time derivatives occur in a simple form

d x l / d t = f / x , c ; t ) (dynamical system)

There is a larger body of known results in the event the forcing term is time independent

d x i / d t = f i ( x , c ; - ) (autonomous dynamical system)

Two-dimensional autonomous dynamical systems are fully understood, three dimensional

autonomous dynamical systems are partly understood, with the degree of understanding trailing

off as the dimension increases. A great deal of information is available for the very special class

of autonomous dynamical system in which the forcing term can be written as the gradient of a

* Work supported in part by NSF Grant PHY852-0634.

0920-5632/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

192 R. Gilmore / Catastrophe theory

potential

dxf'dt = -DxV(X;C ) (gradient dynamical system)

The behavior of such dynamical systems is organized by the invariant sets. In this case they are

extremely simple - these are the equilibria:

0 = DxV(X;C ) (equilibria of gradient system)

Elementary Catastrophe Theory is the study of the equilibria of gradient dynamical systems:

how they move about, coalesce and bifurcate, as a function of the control parameters, c.

The program of Catastrophe Theory applies to many classes of equations. In general, the

simpler the class, the more that can be said. Very little can be said until the simplicity of low

dimensional dynamical systems is reached. At the simplest level, Elementary Catastrophe

Theory, the program has been fully implemented. It is elementary - but important to realize -

that all bifurcation phenomena which can occur in one class of equations will occur in a more

complex class containing the first as a subset. In particular, all phenomena of Elementary

Catastrophe Theory will be encountered in the study of autonomous and nonautonomous dynamical

systems, as well as in partial differential equations.

2. THEOREMS OF ELEMENTARY CATASTROPHE THEORY

Elementary Catastrophe Theory exists at the intersection of two branches of mathematical

evolution. First, it sits at the bottom of the hierarchy of complexity in the implementation of

the program of Catastrophe Theory. Second, it stands at the top of the hierarchy of a series of

theorems from Calculus dealing with the reduction of functions to canonical form. These

theorems are the Implicit Function Theorem, the Morse Lemma, and the Thom Theorem.

2.1 Implicit Function Theorem

Let f ( x ) = f (x l ,x 2 . . . . . Xn) be a function with nonzero gradient at a point x° : Df(x °) 40.

Then it is possible to find a coordinate system, Yi--Yl~X) , with the property

f = Yl" (Implicit Function Theorem)

That is, in this new coordinate system the function is equal to Yl in the neighborhood of

y°=y(x°). 2.2 Morse Lemma

If Df(x °) = O, then the Implicit Function Theorem is not applicable. Nevertheless, it is

still possible to reduce f to a canonical form provided that the stability matrix, or Hessian,

~2f/gxidxj is nonsingular at x °, In this case, it is always possible to find a coordinate system

such that 7

f = -yl 2 - y 2 2 - . . . -y~ +Yi+l 2 +. . . + y n 2 = M n (Morse Lemma)

This canonical form is called a Morse /-saddle.

R. Gilmore / Catastrophe theory 193

2.3 Thom Theorem

If Df = 0 and det a2f/dxF~xj = 0 at a point, then neither the Implicit Function Theorem

nor the Morse Lemma can be used to reduce f to a canonical form in the neighborhood of that

point. There is a theorem (Splitting Lemma), due to Thom, which goes part of the way towards

establishing a canonical form. A second theorem (Classification Theorem), also due to Thorn,

completes the machinery required to construct canonical forms. 24

2.3.1 Splitting Lemma. If exactly keigenvalues of a2f/axl~xjvanish at a point, then

f = fNM(Yl , Y2 . . . . . Yk ;c) + Mp-k(yk+1 . . . . . Yn) (Splitting Lemma)

This reduces the problem of finding a canonical form for the function f(x) of n state variables

(x 1, x 2 . . . . . Xn) to the simpler problem of finding a canonical form for the function fNM of only

k variables.

2.3.2

function

Classification Theorem. This provides a canonical form for the non-Morse

fNM(Y l , Y2 . . . . . Yk ;c) = Cat(k,m) (Classification

Cat(k,m) = CG(k) + Pert(k,m) Theorem)

The function Cat(k,m) is the "elementary catastrophe." It is a function of k state variables and

m control parameters. Further, this function has a canonical decomposition into two functions:

1 ) CG(k), called the germ of the catastrophe. This function describes the greatest possible

degeneracy that can occur at a critical point in an m parameter family of functions.

2) Pert(k,m), called the universal perturbation of the catastrophe germ. This function

describes all possible qualitatively distinct behaviors of the catastrophe germ under arbitrary

perturbation. Further, the m control parameters enter linearly into the canonical form for the

perturbation.

3. THE ELEMENTARY CATASTROPHES

The Implicit Function Theorem, the Morse Lemma, and the Thom Theorems provide

canonical forms for a function, or more generally a family of functions, in the neighborhood of

any point.

The Implicit Function Theorem and the Morse Lemma simply provide canonical forms. The

Thom Theorem provides both a canonical form for a function in the neighborhood of a degenerate

critical point as well as a list of perturbations of that degenerate critical point. Why should the

Thom Theorem have a complexity not present in the two earlier results?

The reason is straightforward. If Df ~ Oand f is perturbed, the perturbed function, f°,

will also have a nonzero gradient at the same point: Dr' ~ O. Both will have the same canonical

form by the Implicit Function Theorem. If Df = 0 but det ~2f/axiaxj ~ 0 at a point, and if f is

perturbed, then the perturbed function f°will have a critical point near (but not necessarily at)

194 R. Gilmore / Catastrophe theory

the critical point of f. Thus, both can be reduced to the same canonical Morse form, since the

Morse index (~) is invariant under perturbation. However, if f has a degenerate critical point,

an arbitrary perturbation is likely to lift that degeneracy. The degeneracy can be lifted in a

number of ways. The role of the perturbation, P e r t ( k , m ) , is to describe all the possible ways

that the degeneracy can be lifted under an arbitrary perturbation. For example, the function

f(x) = x 3 has a doubly degenerate critical point at x ° = O . Under the perturbation ClX, the

doubly degenerate critical point is split into two nondegenerate critical points for c I < O; for c 1

> 0 there are no (real) critical points. A generic perturbation of x 3 will destroy the

degeneracy, either by splitting or by annihilating the critical points. As a result, any

perturbation of x3can be represented by the canonical perturbation ClX.

Table 1 contains a list of the Elementary Catastrophes in one or two state variables,

depending on fewer than six control parameters. Families of functions depending on six or more

control parameters may possess critical points which are not simple, in the sense that the

canonical form of the germ will contain one or more parameters (called moduli) which cannot be

given canonical values. An example of such a germ, and the reasons why moduli occur, will be

encountered in Sec. 5.

4. ELEMENTARY CATASTROPHE THEORY: WHY IT EXISTS

To illustrate how the canonical forms of Elementary Catastrophe Theory arise, we first

carry out a simple computation. This involves constructing the canonical form for a two control

parameter family of functions depending on one state variable: f = f ( x ; c l , c2 ) . This function is

expanded around an aribitrary point, x °, and the properties of the Taylor series expansion are

studied. 8,9

The general Taylor series expansion is shown in Line 1 of Table 21 to which the remainder

of this discussion is keyed. All coefficients fi in this expansion are functions of the two control

parameters: fi = f i ( x ° ; c l , c 2 ) • Where df /dx ~0the Implicit Function Theorem is applicable.

More interesting qualitative behavior is observed at the critical points, where d f /dx = O. We

therefore locate a critical point. By shifting the origin of coordinates to this point, the first two

terms in the Taylor series expansion are annihilated (Line 2, Table 2).

So far, the control parameter degrees of freedom have not been used. By varying these

parameters, it is possible to annihilate one or even two, but typically no more than two, terms

in the Taylor series expansion. Qualitative changes in the shape of the potential do not occur if

only terms higher than quadratic are eliminated from the Taylor series, since the Morse Lemma

remains applicable. The most significant qualitative changes occur when the lowest degree

terms are eliminated. We therefore assume that suitable values of the control parameters can

be chosen to annihilate f2 and f3" Then typically f4 ~ 0 (Line 3, Table 2).

R. Gilmore / Catastrophe theory 195

Finally, it is possible to exploit a nonlinear change of coordinates to transform away the

higher degree terms in the Taylor tail, and to transform the quartic term to canonical form

(+_x 4 or +_(114)x4), depending on the sign of f4 (Line 4, Table 2).

The result of this sequence of transformations is a canonical form (catastrophe germ) for

a potential with a three-fold degenerate critical point at the origin.

The effect of a perturbation on the degenerate critical point is determined by essentially

repeating these steps. First, an arbitrary perturbation, e(x) = ,Tzj~/, is added to the catastrophe

germ. Since we are interested in qualitative properties (shape), the value of the perturbation

at the degenerate critical point is unimportant, so e o = 0. The resulting perturbed function is

shown in Line 5, Table 2.

This perturbation is reduced to a finite polynomial perturbation (Line 6) by following

the same procedure used to effect the transition from Line 3 to Line 4 in the computation of the

catastrophe germ. This reduces the "infinite-dimensional" perturbation e(x)to a finite

(three)-dimensional perturbation.

Finally, a shift of the origin can be carried out to eliminate the parameter e 3 (Line 7,

Table 2). The result is the two-parameter perturbation, Pert(l,2) = e(x)= e lx + e2 x2, which

is the most general possible perturbation of the catastrophe germ +_x 4 of smallest dimension.

The procedure for computing the catastrophe germ and the universal perturbation involve

a series of transformations which exhibit an elegant symmetry. 9

Catastrophe Germ 1. Shift origin.

2. Input control parameter degrees of freedom to

annihilate leading Taylor series coefficients.

3. Nonlinear transformation to finite polynomial form.

Universal Perturbation 3'. Nonlinear transformation to finite polynomial form.

1'. Shift origin. Output is Universal perturbation.

This sequence of transformations is summarized in Figure 1.

5. AN APPLICATION

To illustrate this algorithm in a less trivial case, we compute the catastrophe germ and

universal perturbation for the family of functions f(x,y,z;cl,c2,c3,c4,c5,c6)depending on

three state variables and six control parameters. The resulting germ is not an Elementary

Catastrophe. It depends on a parameter (modulus) which cannot be given a canonical value (e.g.,

+1, 0). Such germs first occur in families of functions depending on six control parameters.

The existence of such germs, of which this is the simplest, is the reason for the truncation of

Table 1 at k = 5, with 11 Elementary Catastrophes. 8,9

196 R. Gilmore / Catastrophe theory

1 ) To apply the general algorithm, we first expand fabout an arbitrary point (x°,y°,z°):

f = Z, fijk ( x - x ° ) i ( y - y ° ) J ( z - z ° ) k

2) By searching for a critical point, and then shifting the origin to that point, we eliminate

the Taylor coefficients of the degree zero and degree one terms.

3) The six control parameter degrees of freedom can be used to eliminate up to six additional

Taylor series coefficients. The possibility exists of annihilating all six coefficients of the degree

two terms. The Taylor series then begins with terms of degree three.

4) By carrying out a nonlinear transformation, it is possible to eliminate all remaining

terms of degree greater than three. In addition, the 10 terms of degree three can be put into

canonical form. The homogeneous linear transformation which does this is represented by a real

3 x 3 matrix. This is insufficient to fix the value of all 10 cubic terms, leaving one modulus

(10 - 9 = 1) without canonical value. The resulting catastrophe germ is 4,8,9

T3,3,3 = x 3 + y3 + z 3 + axyz

Nine of the degree three coefficients can be given canonical values (+1,0), the remaining

coefficient (a) is the modulus of this unimodular catastrophe germ.

5) The universal perturbation is obtained by first adding an arbitrary perturbation to

T3,3,3.

6) A nonlinear transformation can be chosen to eliminate all terms of degree > 3. The terms

of degree three can be put into the canonical form of T3,3, 3 (a-->a3. The result is a nine

parameter perturbation (eoo o = 0).

7) Three of these terms (x 2, y2, z 2) can be eliminated by shifting the origin. The resulting

universal perturbation, Pert(3,6), is

Pert(3,6) = ClOOX + cOlOY + Co01 z + c110xy + c101xz + co11YZ

The resulting canonical form for the family of functions f(x;c) depending on three state

variables and six control parameters, in the neighborhood of the eight-fold degenerate critical

point (Of = D2f = D3f = O) is

f(x;c) = T3,3, 3 + Pert(3,6).

6. LESSONS LEARNED

The reduction to canonical form described above has been presented in an intuitive but

fundamentally correct manner. A great deal of mathematical machinery has been developed to

provide a rigorous underpinning for these results. Since Elementary Catastrophes occur in the

simplest dynamical systems, they occur in all dynamical systems. Therefore, the. associated

mathematical underpinnings, as well as the procedures developed to describe this reduction,

must play a role in the description of all more complex dynamical systems. To be sure, the

R. Gilmore / Catastrophe theory 197

TABLE 1. The Elementary Catastrophes of codimension less than six. Each catastrophe function consists of two terms: the catasstrophe germ, which characterizes the type of degeneracy of a critical point, and the Universal perturbation, which describes the effect of an arbitrary perturbation on the degenerate critical point.

Name k m CG(k) Pert(k, m)

A 2 1 1 x 3

A+_ 3 1 2 _+_x 4

A 4 1 3 x 5

A_+ 5 1 4

A 6 1 5 x 7

o.4 2 3 x2y_ y3

o+4 2 3 x2y + y3

D 5 2 4 x2y+y 4

o_8 2 5 x2y_

o+8 2 5 x2y +

E+6 2 5 x3 +_ y 4

alx

alx + a2 x2

a lx + a2x2 + a3x3

a lx + a2 x2 + a3 x3 + a4 x4

a lx + a2 x2 + a3x3 + a4 x4 + a5x5

a lx + a2Y + a3x2

a lx + a2Y + a3 x2

a lX + a2Y + a3x2 + a4Y 2

a lx + a2Y + a3x2 + a4Y 2 + a5y3

alx + a2Y + a3 x2 + a4Y 2 + aSY 3

alx + a2Y + a3xY + a4Y 2 + a5xY 2

TABLE 2. Taylor series coefficients of a two-parameter family of functions of a single state variable, and effect of sequential transformations.

Object Line Procedure x 0 x I ~2. x3

Find Can- 1 Taylor Expansion fo f l f2 f3

onical Germ 2 Shift origin 0 0 f2 f3

3 Use control parameters 0 0 0 0

to annihilate coefficients

4 Nonlinear coordinate 0 0 0 0

transformation

Find Canonical 5 Add arbitrary perturbation 0 e I e 2 e 3

Perturbation 6 Nonlinear coordinate 0 e 1 e 2 e 3

7 Shift origin 0 e I e 2 0

x 4 ~5x6

f4 f s f 8

f4 f s f 6

f4 f s f 8

+_1 0 0

-+1+e 4 e 5 e 6

+_I 0 0

±1 0 0

198 R. Gilmore / Catastrophe theory

relative importance of the mathematics developed for the description of gradient dynamical

systems will decrease with the complexity of the dynamical system (autonomous,

nonautonomous, partial differential equation, etc.).

We have found the concepts developed for the treatment of gradient dynamical systems to

be useful for the description of nonautonomous dynamical systems. For classes of periodically

driven two dimensional oscillators, we have found that it is not only possible, but useful to

identify a germ (Poincare section); for each germ an unfolding is possible - these are

distinguished by an intertwining matrix; for each unfolding different traverses (paths through

parameter space) describe different bifurcation diagrams. These three concepts have been

developed for the description of the Elementary Catastrophes. Nevertheless, they play a

fundamental role in our understanding of simple nonautonomous dynamical systems.

7. CONCLUSIONS

Catastrophe theory is a program. 4 Its implementation has barely begun. It has been

successfully implemented only at the simplest level - the classification and description of the

equilibria of gradient dynamical systems. The resulting canonical forms are the Elementary

Catastrophes. A complete classification exists for the Elementary (zero-modal) Catastrophes,

as well as for the unimodal and bimodal Catastrophes.

The mathematics and procedures developed for construction of these canonical forms are

just now finding application in the description of more complicated dynamical systems.

ACKNOWLEDGEMENTS

I would like to thank Profs. T. Poston and I. N. Stewart for enlightening discussions.

REFERENCES

1 ) R. Thom, Structural Stability and Morphogenesis (Benjamin-Addison-Wesley, Reading, MA, 1975).

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

T. Poston and I. N. Stewart, Catastrophe Theory and its Applications (Pitman, London, 1978).

R. Gilmore, Catastrophe theory for Scientists and Engineers (Wiley, New York, 1981 ).

V. I. Arnol'd, Singularity Theory (University Press, Cambridge, 1981).

J. M. T. Thompson, Instabilities and Catastrophes in Science and Engineering (John Wiley & Sons, London, 1982).

M. Morse, Transactions Am. Math. Soc. 33 (1931) 72.

2)

3)

4)

5)

6)

7)

R. Gilmore / Catastrophe theory 199

8) V.I . Arnord, Functional Analysis and Applications 7 (1973) 230.

9) R. Gilmore, Catastrophe Theory: What it is, Why it Exists, How it Works, in: Mathemat- ical Analysis of Physical Systems, ed. R. E. Mickens (van Nostrand Reinhold, New York, 1985) pp. 299-356.