Jeff Rubard
2010-02-06 21:47:15 UTC
: http://www.jstor.org/stable/20115070
Catastrophe Theory
Author(s): Hector J. Sussman
Source: Synthese, Vol. 31, No. 2, Mathematical Methods of the Social
Sciences (Aug., 1975), pp.
229-270
Published by: Springer
Accessed: 06/02/2010 16:44
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HECTOR J. SUSSMANN
CATASTROPHE THEORY*
1. Introduction
The purpose of this paper is to give an account of the basic
mathematical
facts pertaining to 'catastrophe theory'. No applications to concrete
problems will be discussed here. However, some consideration will be
given in Section 2 to a number of questions which can be Regarded as
'applications' in a broad sense, for they are of a philosophical
nature. In
particular, we will attempt to show how some of the basic ideas of
cata
strophe theory and systems theory are related.
In deciding what to include in this work, we took the view that 'cata
strophe theory', as mathematics, is a not very well defined part of
the
theory of singularities of smooth maps. It is not clear to us what
specific
problems, or points of view, are the ones which give catastrophe
theory
its own identity. In Section 2 we attempt to isolate one possible
charac
terization, but the reader is advised not to concentrate on this
point.
Indeed, we believe that the main fact about the theory of
singularities of
smooth maps, which the reader may find relevant, is that it leads
naturally
to a number of concepts that are not frequently encountered in Applied
Mathematics: classification of objects according to their qualitative
properties, genericity, stability (in the 'second sense', of Section
2). This
belief has lead us to include a somewhat larger amount of information
than might be considered to be of direct relevance to the theory. On
the
other hand, we give no proofs. (For references to where those proofs
can
be found, see the Appendix.)
Finally, a disclaimer. This author takes no position on the hotly
debated
question as to whether catastrophe theory really has important
applications.
We believe, however, that it is important for a mathematically
oriented
social scientist to be acquainted with the theory, so as to be able to
use it
if he can, and to criticize it if it cannot be used. In the meanwhile,
it must
be remembered that catastrophe theory resembles the classical theory
of
differential equations in that it only claims to provide a general
frame
Synthese 31 (1975) 229-270. All Rights Reserved
Copyright ? 1975 by D. Reidel Publishing Company, Dordrecht-Holland
230 HECTOR J. SUSSMANN
work for modelling certain types of phenomena. Good models for parti
cular phenomena are as unlikely to be derived from the general theory,
as
it is for Newtonian mechanics to be derivable from the existence
theorem
of ordinary differential equations. But, in both cases, the fact that
such
models cannot be derived from the general theory does not qualify as a
valid objection against it.
2. What is catastrophe theory?
The framework which has been most widely used for the mathematical
modeling of processes evolving in time is, undoubtedly, that of the
theory
of ordinary differential equations (O.D.E.'s). In this approach, one
selects a number n of 'state variables' xl9..., xn9 and thinks of each
'state'
of the system as characterized by the values of the state variables
for the
given state. The time evolution of the system is then described by a
system of O.D.E.'s
(1)
?'
=fi(xi9..., x?), i =
1,..., n.
(A number of situations lead to states that are represented by points
in
infinite dimensional spaces. This is the case, for instance, of
Maxwell's
equations and of Quantum Mechanics. In this way, there appear new and
extremely interesting mathematical problems. However, the basic
heuristics for such infinite dimensional systems resembles the finite
dimensional case.)
A system (1) functions very much like a set of traffic signs
consisting of
arrows which, at each point, indicate which way to go (and how fast).
Clearly, such signs will determine the trajectory of any motorist who
follows them, provided only that we know the point from which he
departed. Similarly, the system (1) tells us, for any given state
xl9...9 xn9
what the state will be after an 'infinitesimal' time dt has elapsed.
Therefore,
it seems intuitively clear that a system such as (1) can be solved, in
the
sense that, given any 'initial state' (*?,...,*?), (1) determines
unique
functions of time xt (t)9 ...9xn(t) such that xt (0)=x?l9..., xn(0)=x?
n. This
intuition is amply justified by the existence and uniqueness theorem
of
ordinary differential equations, provided the functions f satisfy some
reasonable technical conditions.
CATASTROPHE THEORY 231
The state-space-O.D.E. approach has certainly had such an impressive
number of successes, that it is both impossible and unnecessary to
list
them. On the other hand, there are limitations as well, and we wish to
discuss two of them which are of significance for catastrophe theory.
First, the method does not lend itself very easily to the study of
situa
tions which essentially involve objects which remain unchanged for
long
intervals of time and then experience abrupt changes. The state of a
system described by O.D.E.'s is either always changing or, if it is
constant
for a very short time, it remains constant forever, unless it is
disturbed by
an agent extraneous to the system. It is true that some aspects of
this
'permanence and sudden change' can be captured by the O.D.E. approach,
but it would be desirable to have a framework in which it appears
natur
rally.
A second and much more serious difficulty arises from the inadequacy
of the state-space O.D.E. approach for the study of complex systems.
This inadequacy is seen whenever one has to deal with a very large
number of variables which cannot easily be 'aggregated' into a small
number of 'macroscopic' ones. Examples of this situation appear when
ever the variables cannot be grouped into a small number of classes of
variables of the same type. Moreover, they also appear in cases in
which
such grouping is possible, but there is no reasonable way of 'adding'
or
'averaging' the variables within each class (e.g., the well known
difficulties
that arise in any attempt to aggregate individual utility functions
into a
'social welfare function').
Mathematically, the reason why the state-space-O.D.E. approach is
inadequate to handle complex systems is the bad behavior of O.D.E.
systems under the operation of 'pushing forward'. Indeed, suppose we
have a system (1), with n very large, and suppose that there is a
small
number m of 'important' variables yl9..., ym which are functions of
the
xi9 namely,
(2) Ji =
0i(*i,...,*?), i=l,...9m.
Suppose that the yt are the only variables that really matter to us.
It
would then be desirable to use (1) to get a similar system of O.D.E.
for
the yt. Now, it is easy to see that, in general, such a system will
not exist.
For instance, (a) the future behavior of the y? will not necessarily
be
232 HECTOR J. SUSSMANN
determined by their values y? at a particular time t09 as would be the
case
if thej^ evolved according to O.D.E.'s.
Now, systems exhibiting feature (a) are far from a mathematical
novelty,
For instance, if the yt are the coordinates of the position of a
moving
particle, then (a) holds. The traditional way out of this difficulty
is approxi
mately three centuries old, and it consists of adding new state
variables,
such as the components of the moving particle's velocity vector.
However,
in recent years we have witnessed developments for which this
traditional
way does not work. In the social sciences, in biology, in geology, the
number of variables becomes too large and the models almost impossible
to handle.
We are now in a position to describe the aim of catastrophe theory. //
is
an attempt to study systems of the type (1), together with functions
(2), with
the aim of saying as much as possible about what happens in the space
of the
y?. The space of the x variables, and the properties of the O.D.E.'s
(1) are
regarded as being of lesser interest, except for those aspects which
have
'observable' manifestations in y-space. In particular, it is desirable
to
isolate statements about the dynamics in j>-space which are
independent
of particular assumptions on what happens in x-space.
We now restate the above considerations in a more modern language.
The x-space referred to above was, presumably, the set of all ?-tuples
of
values (#!,..., xn)9 i.e., the space Un. Applications make it
necessary to
consider more general spaces, such as open domains of IRno r even sets
of
points in Un which are defined by equality constraints. This leads
naturally
to the use of arbitrary smooth manifolds as state spaces. If iVis an ?-
dimen
sional smooth manifold, a system of ordinary diflerential equations on
N is,
by definition, the same as a smooth vector field V on N. A dynamical
system is a pair (N, V)9 where N is a smooth manifold and F is a
smooth
vector field on N. Catastrophe theory deals with the study of
situations in
which we are given :
(i) a dynamical system (N, V)
(ii) a smooth manifold P9 and
(iii) a smooth map FrN-* P.
As was explained above, the aim is to describe what happens in the
space
P of the 'observed' variables.
It would now be desirable to give a sketch of the important theorems
of
the theory, and examples of applications. As was explained in the
intro
CATASTROPHE THEORY 233
duction, this paper will not discuss applications. As for the
theorems, it is
a feature of the theory that there are very few whose statement can be
explained without the use of sophisticated mathematical machinery. On
the other hand, in most applications what is used is not catastrophe
theory as a set of results, but catastrophe theory as a conceptual
frame
work (for an example, see Zeeman's beautiful paper [23] on heartbeat
and nerve impulse). Therefore, we shall devote the remainder of this
sec
tion to a discussion of this way of thinking. In subsequent sections
we will
give a more detailed presentation of the mathematics.
The main aspects which we will concentrate on here are the ideas of
stability and catastrophes. The concept of stability has emerged in
modern
science as one of central importance, especially where a 'systems' ap
proach is utilized. Roughly, a stable object, or system, is one that
reacts to
small changes by returning to its original state. This idea of
stability can
be modelled mathematically within the context of dynamical systems
theory by means of the concepts of 'stable equilibrium' and
'attractor'.
If (N9 V) is a dynamical system, a closed subset C of N is said to be
an
attractor if every trajectory of the system which begins sufficiently
close
to C converges to C. A stable equilibrium is an attractor which
consists of a
a single point. This idea of stability exhibits some features in
common with
the 'real life' concept. But it has the drawback it does not allow for
'jumps'
from one attractor to another to be incorporated into the dynamics of
the
system. As we will see below, when this idea is combined with the
frame
work of catastrophe theory, the treatment of such jumps becomes
possible.
In addition to what was explained above, there is a second sense of
the
word 'stability' which is of paramount importance in contemporary
mathematics. Suppose that a class X of objects is given, and that we
have
a notion of 'nearness' and a concept of what it means for two objects
of
Zto be 'qualitatively the same'. We then call an x of X stable if
every y in
X which is sufficiently near x is qualitatively the same as x (cf.
Section 3
for a precise definition). Stable objects in this sense have appeared
in
mathematics in several contexts. Here we shall mention one of them:
consider the class of all smooth maps from N to P9 where N and P are
manifolds. There is a notion of nearness and one of 'being
qualitatively
alike' (cf. Sections 4 and 6). When one attempts to classify the maps,
by
giving a description of the various kinds of qualitative behavior
which
they might exhibit, it is found that such a description is practically
im
234 HECTOR J. SUSSMANN
possible, because some maps may be highly 'pathological'. On the other
hand, the behavior of stable maps is much easier to describe.
Therefore,
stability appears naturally as a technical assumption which can be
made on
a map and which makes it possible to prove interesting theorems.
Moreover,
the point has been made by some, and especially by Thorn, that the no
tion of stability has philosophical significance, and that it is
essential for
the understanding of what we mean by an 'object'. A real object is
charac
terized by the values of a large number of variables, and these values
are
constantly experiencing 'random' changes. Yet we perceive the object
as
remaining the same. This presumably means that the object is stable in
our more technical'sense.
Having explained what shall be meant by stability, we can describe the
technical meaning of 'catastrophes'. An object x will be said to
belong to
the catastrophe set if it is not stable. The following trivial example
will
illustrate the above notions and, in particular, how a catastrophe can
be
stable or, more precisely, how it can occur in a stable way. (In view
of our
definitions, the reader should find this mysterious). The answer is
indeed
simple. Say, for instance, that X is the set of all points in the
U.S.A. and
Canada. Qualitatively, we can divide the points in three classes,
namely,
those in the U.S.A., those in Canada, and the border. The first two
types
are stable, and the catastrophe set is the border. If you choose a
point
'at random' it will most likely be in one of the countries, and you
can
safely forget about the border. But if you choose two points at random
and then (also at random) a route joining them, you can no longer
ignore
the border. A curve from Boston to Montreal must cross the border, and
every curve 'close' to it will also cross it. Thus, in the new set X'
which
consists of all routes (i.e., curves) we have stable elements in which
catas
trophes appear in a stable way.
We now return to our original attempt to explain what catastrophe
theory was about. We have to study objects consisting of a dynamical
system (N9 V)9 a manifold P and a map F.N-+P. The preceding remarks
suggest that we define what it means for such a composite object to be
stable, and that we then study these objects. Unfortunately, such a
general
theory has not yet been developed in a fully satisfactory way. (The
main
difficulty arises in the search for an adequate definition of
stability for
dynamical systems.) For this reason we shall limit ourselves here to
the
study of two particular cases of our general situation.
CATASTROPHE THEORY 235
Case 1 : V= 0, i.e., we just have objects consisting of a pair of
manifolds
N, P and a map F:N-*P. This leads to the study of the stability of
smooth
maps, which was undertaken by Mather with great success. We give a
brief sketch of some parts of this theory in Section 6.
Case 2: (a) N is simply a product Q xP, where Q is some region in Un
(or, more generally, a Riemannian manifold),
(b) the vector field X points, for each yeP, in the direction of the
'fiber' Q x {y}, and
(c) for eachjeP, the vector field on Q defined by Xis the gradient of
a
'potential' fy. Therefore, we have to study 'families of functions
para
metrized by p\ i.e., functions on a set Q which also depend on P. Some
basic facts about stability of families of functions are presented in
Section 7.
Case 2 can be considered as an idealization of some situations which
appear in practice. Consider, for instance, an individual r who
maximizes
a utility function u(x9 y)9 where x is an ?-dimensional parameter
repre
senting the alternatives available to r, while y is an m-dimensional
param
eter over which x has no control. If we assume that the parameter y
changes much more slowly than t makes his decisions, we can then
assume
that, for each y9 % chooses an x(y). The 'mapping' x(y) is defined by
the
requirement that
(3) u (x (y)9 y)
= max u X (x, y)
and it is not a true mapping for several reasons, such as, for
instance, the
fact that there may be more than one x where the maximum is reached,
or
none at all. We will see in Section 9 that, for 'most' w, x(y) is
'nice' as long
as u (considered as a function of x only) is stable. Moreover, the set
of
stable >>'sw ill be 'most' of P, but there will be, 'in general', y's
for which
an 'abrupt change in behavior' will take place. Such changes will
occur
'stably', in the same way as the U.S.-Canada border of our example
appears stably when points which change in time are considered.
3. Genericity and stability: some simple examples
Both in pure mathematics and in applications, one has to deal very
often
with situations in which
236 hector j. sussmann
(a) a certain class of objects Zis being considered, and
(b) all the objects of X have some 'nice' properties, with the
exception
of a certain subclass Xdtz ofX whose elements are 'degenerate9.
A typical example of such a situation is provided by a system
(4) a11x1 + a12x2
=
a13
a2^Xi ~r ??22^2
==
^23
of two linear equations in the two unknowns xu x2. As the reader
knows,
a lot can be said about the solutions, but one has to distinguish
various
cases, depending on the values of the coefficients a^. In particular,
the
'nicest' case occurs when the determinant
A =a??a22
?
ci21al2
is not equal to zero. Indeed, if A ^09 then there is one and only one
solu
tion xl9 x2 of the system (4). On the other hand, when A =0 many
things
can happen. (The list of these possibilities, which is probably well
known
to the reader, need not concern us.) Let us show how this example fits
in
the general framework described above. The set X consists of all
systems
(4). Now, giving such a system is completely equivalent to giving the
six
real numbers aijt Therefore, A" can be identified with the space R6 of
all
sixtuples of real numbers. The 'nice' property that one would expect
from
a system (4) is the existence and uniqueness of a solution. Those
systems
for which this property does not hold are 'degenerate'. They are
charac
terized by the equation
A=0.
In a way which should be intuitively clear to the reader, A is much
more
likely to be different from zero than equal to zero. If the six
numbers a^
are 'chosen at random', A^O will hold 'in general', i.e. except for
some
'exceptional' cases. We describe this by saying that
A # 0 holds 'generically'
or that
a generic system (4) satisfies A # 0.
CATASTROPHE THEORY 237
In general, the word 'generic' is used in mathematics as in this
example.
One says that certain property P holds generically for elements of a
space
X if it holds for 'most' elements of X. An equivalent form of the same
assertion, which is widely used, is
a generic xe^satisfies P.
The reader is warned that this statement should not be interpreted as
referring to the property G of 'genericity' and asserting that every
xeX
which satisfies G necessarily satisfies P. The correct interpretation
is the
one given in the preceding paragraph, i.e., that P holds for 'most'
xeX.
(Remark: strictly speaking, what was just said is only true
'generically'.
There are some cases in which mathematicians have agreed to single out
certain properties Pt of elements of a space X, and call an xeX
generic
if it satisfies the P?.)
Of course, a genericity statement will have a precise mathematical
meaning only where the meaning of the word 'most' has been made
precise. On the other hand, there is no general definition of what
'most'
means which applies to all possible cases. Therefore each situation
will
have to be discussed individually. We shall postpone this discussion
until
Section 4. In the meanwhile, we remark that all the examples to be
discussed before Section 4 will resemble the one that was presented
earlier
in this section, in that the meaning of 'most' (and therefore of
'generic')
will be clear from the context.
We now return to our example. We have already shown that, generically
a system (4) has exactly one solution. Let us call these systems
(i.e., the
ones for which A #0) 'nondegenerate'. We want to discuss another prop
erty of these systems, namely, their stability. Intuitively, this has
the
following meaning: suppose we start with a system (4) and 'perturb' it
slightly by replacing the coefficients atj by fly+ey, where the
numbers ey
are small. If the original system is nondegenerate, then the
perturbation
will only cause a small quantitative change in those aspects of the
system
that are of interest to us, such as the solution of the system.
Moreover,
there will be no 'qualitative jump'. For instance, the new system will
have
one and only one solution, exactly like the old one. When the original
system is degenerate, we encounter a completely different situation.
Here a
small change in the atj can cause a drastic qualitative change in the
proper
238 HECTOR J. SUSSMANN
ties of the solutions. As an illustration of what may happen, we
observe
that the ei7 can always be chosen so that the perturbed system will be
nondegenerate. This is indeed a change! A degenerate system either has
no solution at all or it has infinitely many. And an arbitrarily small
perturbation can change it into a system with exactly one solution. We
summarize all this by saying that nondegenerate systems are stable,
whereas degenerate ones are unstable. Unstable elements are those
where
drastic changes occur, and for that reason the set of unstable
elements is
referred to as the catastrophe set.
In order to formulate a general definition of stability, we shall
begin by
listing those ingredients that are needed to make such a formulation
possible. First, we need a class of objects X as above. Second, we
must
have a concept of 'closeness', so that we can talk about
'perturbations'.
Mathematically, X must have the structure of a topological space.
Third,
we must have a formal definition of what it means for two objects of X
to be 'qualitatively alike'. Mathematically, this will be expressed by
an
equivalence relation.
DEFINITION 1. Let Jf be a topological space, and let R be an equiva
lence relation on X. An element x of X is called R-stable if there is
a
neighborhood Uofx such that every ye Uis P-equivalent to x.
DEFINITION 2. Let X and R be as in Definition 1. The set of P-un
stable elements of X is called the R-catastrophe set of X.
DEFINITION 3. Suppose <f> is a continuous map from a topological
space 7 into X(i.e. {<?){t):teY} is a 'family of elements of X
depending
continuously on the parameter te Y'). The R-catastrophe set of <p is
the
set of all / such that <f){t) is P-unstable.
Remark. We insist on the dependence on R which is made explicit
in the above definitions. The same class X of mathematical objects may
arise from different situations. This may lead to the consideration of
different equivalence relations R, for the question of what we mean we
call
two elements of X 'qualitatively alike' cannot be answered without a
knowledge of what the objects in X are supposed to represent. As an
example, let us return to the set X of 6-tuples of reals that was
considered
above.
CATASTROPHE THEORY 239
We have seen how X can be thought of as the set of linear inhomoge
neous systems of two equations in two unknowns, and how this leads to
a
concept of stability. Suppose now that we consider the set of all
systems of
two linear homogeneous equations in three unknowns. This leads natu
rally to the same space X of 6-tuples {atj:/= 1, 2; j= 1, 2, 3} as
before.
However, the natural equivalence relation for this situation is not
the
same as before, and the concept of stability is different (for
instance,
when a 2 x 3 matrix is thought of as the matrix of a homogeneous
system,
there is no reason to single out the determinant of the first two
columns
for a special role). Clearly, two 6-tuples should be declared
equivalent if
the spaces of solutions of the corresponding systems have the same
dimension. Therefore, there are three equivalence classes, and the
stable
elements are those for which the matrix (fly) has rank two. We now
illustrate Definitions 1,2, and 3 with some examples.
EXAMPLE 1. Consider a linear programming problem in which it is
desired to maximize the linear function
p1x1+p2x2
subject to the constraints
a1x1 + a2x2 < a3
bix1 + b2x2 ^ b3
xx^0, x2^0.
Let us assume that the coefficients ai9 bt are fixed positive numbers,
and
that the 'prices' px are positive but are allowed to vary. Thus, we
actually
have a two-parameter family X of linear programming problems, para
metrized by the points in thepl9p2 plane for which/?! >0 and/?2 >0.
Let us also assume that
aib2 > a2b1, #A > #3^1 ? and b2a3 > a2b3,
so that the feasible set and the constraints look as in Figure 1.
240 HECTOR J. SUSSMANN
0^*02X2=03
TABLE I
Case Condition Solution
ai
?1<
?
P2 b2
p2 bz
?2 P2 a%
P2 02
?2 P2
?l
any point in the segment Q1Q2
?2
any point in the segment ?2?3
03
CATASTROPHE THEORY 241
To describe the solution, we must consider five possible cases, depend
ing on the values of the prices pt and/?2 (cf. Table I).
There is a clear choice of an equivalence relation R on X9 namely, we
declare two elements of X equivalent if they fall in the same case of
Table I. Thus, there are five equivalence classes, of which three are
open
and two are not. The unstable elements are those which belong to one
of
the lines
Pi_bi Pi a,
Pi b2
Now assume that the prices pl9p2 are changing in time, so that we have
a curve t-*(px(t), p2(t)) in X space. The catastrophe points are those
values of t for which (jPi(t), Pi(t)) belongs to one of the lines
described
or ? = ?
(cf. Figure 2)
Pi <*i
Fig. 2.
242 HECTOR J. SUSSMANN
above. As an example, let
Pi(0==Pi + ai> a>0
p2(t) ?p\
? constant
so that the price of commodity 1 is steadily increasing. To describe
the
solution (xi9 x2) of our linear programming problem as a function of/,
it
suffices to describe xl9 for it is clear that this uniquely determines
x2.
A
*1
b2Q3-Q2b3
020-1-02 b1
-.->t
t1 =5L( blP?2-b2p? ) t2 =
?-fa, p?2-a2p? )
Fig. 3.
The graph of xt as a 'function' of t is given in Figure 3. We see that
xx
is a 'good' function of / except at the catastrophe points where
(a) xt has a jump discontinuity, and
(b) xx is not well defined.
CATASTROPHE THEORY 243
EXAMPLE 2. Let ? and p be positive integers, and let Xn'p denote the
set of all real matrices with ? columns and p rows. If we represent
the
vectors in Un (and Up) as columns, then X is in an obvious way
identified
with the set of all linear maps from IRWin to Up. A reasonable
equivalence
relation R is the one that identifies two matrices if they correspond
to the
same transformation except for a linear change of coordinates.
Formally,
two matrices FxeX9F2eX are equivalent if there exist nonsingular ma
trices GeXn>n9 HeXp>p such that
F2
=
HFiG.
It is an easy exercise in linear algebra to prove that two matrices
are
equivalent if and only if they have the same rank. Thus, if we put
m=min
(?,/?), the space Xn,p is partitioned into m+1 equivalence classes X\p
(0^fc<m),where
Xnk'p
=
{M: MeX9 rank(M)
=
k}.
The condition that rank(Af)=m is equivalent to the assertions that the
determinant of some m x m submatrix of M be nonzero. Hence Xn?p
is an open subset of Xn,p. It is not very hard to show that the XI'p
are
submanifolds of Xn,p of codimension (n-k)(p-k). In particular, the
stable
elements ofXn,p are exactly the matrices of rank m. Thus, the set of
stable
elements is dense in Xn'p. Moreover, this set is clearly open.
Therefore,
stability is a generic property. The case ?=2, p = 3 of Example 2 is
basi
cally the one discussed in the remark following Definition 3. We now
describe the analogue of the example presented at the beginning of
this
section.
EXAMPLE 3. We let Xn>p be the sef of all ? xp real matrices, as
before,
but now we think of an AeXn*p as representing an inhomogeneous
system of? linear equations in/?-l unknowns. If AeX"tP9 write A =
(A9 d)9
where a is the last column of A. We declare (A, a) and (B9 b) to be
equiv
alent if there are invertible matrices GeXn'n9 HeXp~1'p~1 such that
S=HAG and b=Ha. It is easy to show that A =
(A9 a) and P=(P, b) are
equivalent if and only if rank A = rank B and rank A = rank B. As in
Example 2, the equivalence classes are submanifolds. The stable
elements
are those A for which rank A = a and rank A = ?, where (7=max (?,/?),
244 HECTOR J. SUSSMANN
<7=max (?,/?? 1). Therefore, stable elements are dense, and stability
is a
generic property.
EXAMPLE 4. Consider the set Xn,n of ? x ? real matrices. If we regard
each matrix in X"'n as the matrix of a linear map between two linear
spaces of dimension ?, then the equivalence relation introduced in the
previous example is obviously the right one. If, however, we regard
the
elements of Xn,n as linear maps of Rn into itself, it becomes
unnatural to
allow for different changes of coordinates in the domain and in the
range.
It is more appropriate to declare Mt and M2 to be equivalent if there
exists a nonsingular matrix P such that
M2=PM1P-1.
Here a new phenomenon appears. The equivalence classes modulo this
relation can no longer be described by means of a discrete family of
inva
riants, such as the five cases of Example 1, or the rank as in Example
2.
In fact, it is easy to show that continuous parameters appear and this
has
the consequence that no stable elements exist ar all. An example of
such a
continuous parameter is the trace of a matrix, i.e., the sum of its
diagonal
elements. It is well known that two matrices which are equivalent in
the
sense of the present discussion necessarily have the same trace. On
the
other hand, ifM eXn,n9 it is clear that we can find matrices M'
arbitrarily
close to M whose trace is not equal to the trace of M. Hence there are
matrices which are arbitrarily close to M but not equivalent to M.
There
fore M is not stable.
4. Genericity of maps and transversality
In Section 3 we discussed the general concept of stability, and we
illu
strated it with a number of examples. All our examples had one feature
in common: the space X was finite-dimensional. In those cases in which
there were sufficiently many stable elements, it turned out that the
set of
such elements was open and dense in X. We then agreed to describe such
a situation by saying that 'stability is a generic property'.
We now want to extend these considerations to some 'infinite dimen
sional' cases. As a simple example, suppose X is the set of all curves
in the
plane R2 (a curve in IR2 is a smooth mapping from the interval [0,1]
to R2).
CATASTROPHE THEORY 245
This leads naturally to two questions, namely, (a) how do we make X
into
a topological space? and (b) what shall we mean by a 'generic'
property?
The answer to (a) is provided by the C?? topology which, in our case,
is
easy to define. We say that a sequence {fn} of curves in R2 converges
tof
in the C00 sense if/? converges to/uniformly together with all the
deriva
tives (i.e., if we put fn(t)=(un(t)9 vn(t))9 and similarly for/, we
are re
quiring that for every e > 0 and every positive integer m there must
exist an
?o depending on e and m, such that
dmun dmu
dtm
v '
df
<?
for all /, and all ?>?0, and a similar inequality for the v's).
The definition of the C
??
topology given above can be extended easily to
the more general case when X is the set of all smooth maps from K to
P,
where K is a compact subset of a manifold N. This is just a matter of
repeating what was done above in the particular case K= [0,1], P=R2.
The only additional difficulty arises from the fact that AT (and P)
need not
be covered by one single coordinate chart. This can be remedied very
easily, and we shall omit the details. The final result is the
topology ofC
convergence on K.
Now let Cco(N9 P) denote the set of all smooth maps from N to P. How
do we topologize C??(iV,P)? When N itself is compact, the preceding
paragraph provides the answer. lfNis not compact, there are at least
two
'natural' topologies on C??(N9P)9 namely, the C00 topology and the
fine
C00 topology, also known as the Whitney topology. We shall define them
in the simple case when N=Rn9 P=R. The general construction is con
ceptually the same, except for the fact that one must choose
coordinates
appropriately.
The C00 topology is easier to define. We say that/,, ->/if/m converges
to
/on K in the C00 sense for every compact subset K of Rn. To define the
Whitney topology it is better to describe what we mean by a neigh
borhood of an/eC??((Rn, R). Suppose x->e(x) is a continuous function
on IRn, such that e(x)>0 for all x. Let k^O be an integer. We consider
the
set V(f;k9 e) of all ^eC??(Rn, R) such that, for all xeRn9 the
inequality
dai
+ ...+??
dxl'-dxl M-9){*) <e(x)
246 HECTOR J. SUSSMANN
holds for all x9 whenever otl-]-han^/:. We illustrate the difference
between the two topologies by an example: suppose/:(Rn-> R is a func
tion such that/(x)#0 for all x. Then there is a neighborhood V off
such
that every ge V vanishes nowhere (e.g. take V= V(f; 0,/)). So, in the
Whitney topology, if/never vanishes then a function which is
sufficiently
close to/cannot vanish anywhere. This is not the case for the C00
topol
ogy. Indeed, if we let <j>nb e smooth functions such that <?>n(x)=\
for
\x\^ ? and (?)n(x )=0 for \x\^ ? +1, then <j)nf -> / in the C??
topology. The
preceding considerations show that the Whitney topology is the most
natural one. From now on, we shall always consider Cco(N9P) as a
topological space with the Whitney topology.
We now must choose a definition of genericity. The choice which has
become unanimously accepted by the practitioners of this field is as
follows: we call a property &> of maps/eC??(JV, P) generic if the set
of
those/for which P holds contains a countable intersection of open
dense
subsets of C??(7V, P). The main motivation for this definition is that
it is
the one which enables us to prove nice theorems. If the reader does
not
find this satisfactory, the following justification may perhaps seem
prefer
able. Genericity of a set A of points of a space X should mean,
roughly,
that A is very 'thick' and its complement X?A very 'thin', so that
'prac
tically all points of X are in A\ One possible definition
of'thickness' is:
A is thick if it is open. Similarly, we might define B to be thin if
it contains
no open sets. With this choice, we arrive at the definition: A is
generic if it
is open and dense (i.e., if it is open and its complement does not
contain
any open set). However, we can be somewhat less demanding. Suppose
that the class G? of open dense subsets of X is such that, whenever
{Am} is a sequence of elements of 02 9 then the intersection
m
is still dense. If X has this property, we call it a Baire space. In
this case,
suppose {Am} is as above, and let Bm=X? Am. The Bm are clearly 'thin'.
If Zis a Baire space, the Bm are so 'thin' that their union B still
contains no
open sets. Now, the sets B obtained in this way still have the
property that
a countable union of such P's cannot contain any open set. We need
more
than a countable infinity of P's to fill a whole open set. It is
therefore
understandable why such B can still be called 'thin' and their
complements
CATASTROPHE THEORY 247
'thick'. Now, the complement of a B is a set A which is a countable
inter
section of open dense sets. Moreover, if such A9 s are going to be
called
generic then certainly any set which contaits one A should be called
generic as well. This yields the definition that was given above.
The main fact which makes such a definition reasonable for a space Zis
the property of being a Baire space. It is a classical result that the
real line
R is Baire (the category theorem). An example of a property of real
num
bers which is generic in our sense is irrationality (but not
rationality).
It is also true that the space Cco(N9 P) is Baire, and we will there
foreuse,
from now on, the concept of genericity in the very precise sense that
was
introduced above.
The basic facts about genericity are therefore :
(a) If ^ is a generic property then the set of/for which & holds is
dense, i.e., every/can be transformed into an/' for which & holds
by an arbitrarily small perturbation, and
(b) if ^1 and ^2 are generic properties, so is their conjunction 0>l a
&*2
(in fact, this is true for arbitrary finite and even countably
infinite
conjunctions).
We remark that (b) is a consequence of the 'thickness' requirement
that
was made a part of the definition of genericity. If we had allowed 0*
to be
called generic if only (a) holds (i.e., if {/: 0 holds for/} is
dense), then (b)
would not follow (e.g. every real number can be approximated by ratio
nal and by irrationals, but certainly not by numbers which are both
rational and irrational).
We conclude this section with an important example of a generic
property. This is the (simplest form of) the transversality theorem of
Thorn.
We first illustrate it with one example. Suppose two curves yt and y2
are
given in the plane, such as the ones in Figure 4. Then it is clear
that, even
if y? and y2 are tangent, we can perturb one of them (say y2) slightly
so as
to make them nontangent. This suggests that the set of curves y2 which
are not tangent to y ! is dense in the set of all curves. Moreover, it
is clear
that, if y2 is not tangent to yu then any curve which is close enough
to y2
is also not tangent to yt. Therefore, the property of a curve y2 of
not being
tangent to y x is generic.
The transversality theorem is basically an assertion like the
proceeding
one, but for the following more general situation: we have a
submanifold
W of P (such as yt in R2) and we consider smooth maps f:N-*P. We
248 HECTOR J. SUSSMANN
Before Perturbation After Perturbation
Fig. 4.
would like to calif not tangent to W(zt a point xeN such that/(x)e W)
if
no direction v tangent at/(x) to the image of/is also tangent to W.
This,
however, is too strong a requirement, for if the number of linearly
inde
pendent t;'s is k9 and if
k + d\mW>dimP
then the condition will never be satisfied (i.e. there is not 'enough
room'
in P to accommodate so many directions; as an example, consider two
surfaces in R3). We therefore replace the above condition by a weaker
one,
namely, transversality.
If xeNandf(x)e W9 we call/ transversal to Watx if the space A of all
directions tangent to the image at/(x) and the tangent space Wf(x) to
Watf(x) are 'as far apart as possible' (formally: A+ Wf(x)=Pf(x); the
definition of A is A =
{f*(v)9 veNx}).
We call/transversal to Wif it is transversal to W&tx for every x such
that f(x) e W. The transversality theorem of Thorn says that, for
arbitrary
N9P9 Was above, transversality to W is a generic property of maps f:N-
+P.
The reader will observe that, if d\mN+d\rnW<dimP9 and if xeN9
f(x)e W9 then it is impossible for/to be transversal to W at x.
Therefore
/will be transversal to Wif and only if the image of/does not meet W.
So,
in this case, the theorem asserts that every/can be deformed slightly
so
CATASTROPHE THEORY 249
that its image will not meet W. As an example, if the curves yl9 y2 of
Figure 4 are thought of as curves in ?R3, then it follows from Thorn's
theorem that y2 can be perturbed slightly so that it will not
intersect yx.
This is, of course, in perfect agreement with our intuition.
We conclude this section with three important statements.
FACT 1. Iff:N-*P is a smooth map, and if W is a submanifold of P
such that/is transversal to W9 then the set
f^(W)^{x:xeN9f(x)eW}
is a submanifold of N9 and
codim/-1(iF)
= codimJ^ if /-1(FF)#0.
(Note that, if dimJV + dimfl^P so that codimPF>dimiV, this gives a
negative dimension for/
" *
(W )9 so/
" *
(W )=0 as was said above.)
FACT 2. If f:N-*P is smooth and transversal to W9 and if xeN9
f(x)e Wthen for every/
'
N: -+P which is close to/there is an x' eN close
to x such that/'{x')eW.
FACT 3. If W is closed then the set of all/: N-*P which are
transversal
to Wis open in C00 (N9 P). When Wis not closed this need not be true.
5. Stable catastrophes
We can now clarify what is meant by a catastrophe to occur in a stable
way, in spite of the fact that a catastrophe point is, by definition,
an
unstable one. Let us return to Example 1 of Section 3. The catastrophe
set
is depicted in Figure 2, and it consists of two lines. Suppose that we
have
an element of X which depends on one parameter t (i.e., we have a
curve
y-*'-KPi(0> Pii*))in x- The set c of the t for which 0>i(0> pAO) is
unstable is the catastrophe set. If y is an arbitrary curve in X, then
we can
say very little about C (actually, it is easy to show that every
closed subset
of the real line can be obtained as the catastrophe set of some such
curve
y). If, however, we limit ourselves to a generic class of curves, we
can say
much more. First, by the transversality theorem, a generic y is
transverse
250 HECTOR j. sussmann
to the catastrophe set. Then, by Fact 1 of Section 4, we can conclude
that, for generic y, the catastrophe set C is a submanifold of the
real line
of codimension one. So C actually consists of a discrete set of
points.
Finally, if y is a curve which is transverse to the catastrophe set,
and
actually needs it, then Facts 2 and 3 of Section 4 imply that every
curve
y' in a neighborhood of y is also transverse and meets the catastrophe
set. Therefore, the catastrophes of y appear in a stable way.
The preceding remarks illustrate some aspects of the general
situation.
To get a still better picture, let us suppose that we had a situation
in
which X is two dimensional and the catastrophe set consists of the
union
of two intersecting lines. In this case, the preceding discussion
carries over.
Indeed, it suffices to remark that the point x0 where the lines cross
con
stitutes a zero-dimensional submanifold of X. Therefore, the
transversa
lity theorem implies that a generic curve y will not meet x0, and
then, as
long as we are only interested in the generic case, x0 can be ignored.
If we
now consider elements of X which depend on two parameters, a new
feature appears. We now have a smooth map y(tl912) from some region
of R2 into X. Generically, y will be transverse to both lines and to
the
manifold {x0} (observe that we are making essential use of Fact (b)
about
genericity, cf. Section 4). Therefore, we can assert that, for each of
the
two lines Lt (/=1, 2), the set of (tl912) for which y(tl9 t2)eLi is a
one
dimensional manifold. Also, those (tl912) for which y(tl9 t2)
= x0 form
a discrete set of points, but we can no longer assert that it is
empty.
The preceding examples suggest the following picture :
(i) The catastrophe set is a 'stratified' subset of X9 i.e., roughly,
a
finite (or, at least, locally finite) union of a disjoint family of
submanifolds
(the 'strata') such that the closure of each stratum S is the union of
S and
of strata of lower dimension. In our second example, the strata are
four
open half-lines and {x0}.
(ii) To every point x of Iwe can attach an integer, the codimension of
x9 with the property that codimx^O, with equality only when x is
stable.
In some sense, this integer measures how complicated a catastrophe x
is.
More precisely, codimx is the minimum number k of parameters which
are needed for x to be 'seen' (i.e., to appear stably as part of a ^-
param
eter family).
(iii) If x is a smooth family of elements of X depending on k para
meters t=(tl9...9 tk)9 then, for generic y9 the catastrophe set of y
will be
CATASTROPHE THEORY 251
stratified. The strata of codimension k will consist of points t for
which
y (t) is a catastrophe of codimension k.
In addition to these three properties, there is a fourth one which is
easily verified in Examples 1,2,3 of Section 3, namely
(iv) If xeX9 then the stratum of x is precisely the equivalence class
of x.
In the following sections we shall discuss how what has been said so
far
can be extended to the case when A"is the space C00 (TVP, ). This will
lead
us to study the stability of a map/:N-*P9 and then to consider
families
of maps depending smoothly on parameters. It will turn out that (ii)
requires an obvious modification, namely, that we will have to accept
points of 'infinite codimension'. It seems clear (though it has not
been
completely proved, as far as we know) that (i), (ii) and (iii) will
hold,
except for the modification of (ii) mentioned above. However, (iv)
will
definitely not hold. The reason for this is the appearance of families
of
equivalence classes depending on a continuous parameter. (Such
families
are called 'moduli'). The following trivial example gives a finite
dimension
al analogue.
EXAMPLE. X=R29 P=the relation whose equivalence classes are
the upper and lower half-planes, and the one-point subsets of the x
axis.
Here the catastrophe set is the x axis, which is stratified in an
obvious way
with one stratum of dimension one. However, the equivalence classes of
all the points in this stratum are zero dimensional.
6. Stability of smooth maps
We now give a very brief discussion of some aspects of the theory of
singularities of smooth maps of Thorn and Mather. The reader who is
not
familiar with the basic concepts of the theory should keep Example 2
of
Section 3 in mind. The concept of stability to be introduced here is
basi
cally a nonlinear analogue of this example (though, of course, it
leads to
much more interesting mathematics).
The basic object is the class C (N9 P) of all C00 mappings from N to
P. Here N and P are C00 manifolds of dimensions ? and p9 respectively.
We have already explained how to make CCC(N9P) into a topological
space. The second ingredient that is needed in order to talk about
stability
is an equivalence relation. The most natural one (since we are working
252 HECTOR J. SUSSMANN
with C?? maps) seems to be C00 equivalence, which is defined as
follows:
two maps/and g from N to P are C00 equivalent if there exist C00
diffeo
morphisms
such that
(5)
r-.N-^N and I: P-> P
f=igr.
Another possible relation is C? equivalence. The maps/and g are C?
equivalent if there are homeomorphisms r and / such that (5) holds.
Corresponding to these two equivalence relations we have the concepts
of C00 stability and C? stability. It is clear from the definition
that the
sets St^(N9 P), St0(N9 P) of C00 stable and of C0 stable maps are open
in
C ??
(N, P), and that Sr?, (#, P) s St0 (N9 P).
Fig. 5. The shaded region (boundary included) is the complement of the
nice range.
CATASTROPHE THEORY 253
By analogy with the discussion of the preceeding sections, we ask
whether C00 stability is a generic property or, if this is not the
case,
whether C? stability is generic. It turns out that C? stability is
generic for
arbitrary N and P, if N is compact. As for C00 stability, the
situation is
more complicated. The result is that whether or not St^N, P) is dense
in
C00 (N9 P) depends only on the dimensions ? andpofN and P. The set of
values of ? and p for which St^(N9 P) is dense in Cco(N9 P) has been
called the 'nice range' by Mather. It is described in Figure 5. As an
exam
ple, St^ (N9 P) is always dense in C^?N, P) if p^5. Also when/?=6,
except for ? = 8. As for the case n=p, C00 stability is generic if,
and only
if,?<8.
We illustrate the definition of stability by considering the three
func
tions
A(x)
= x29 f2(x)
= x39 f3(x) = x4r-x2
whose graphs are shown in Figure 6. We will show that/ is stable while
f2 and/3 are not.
Let U be the set of all functions g such that
(1) ^'(x)>lforalljc
(2) ?'/(-1)<0and
(3) ff'(l)><>.
It is clear that U is an open set in C??(R, R), and that/e 17. If g
eU9
then the derivative g'{x) is strictly increasing and it has a unique
zero x0.
Moreover,
? 1 < x0 < 1. It follows easily that g is decreasing for x<x0 and
increasing for x>x0. Moreover Jim^.^ g(x)=limx^ + a0 g{x)= + oo.
Thus, the graph of g has the same qualitative properties as the graph
of/.
We shall use this to prove that g is equivalent to/ First, g is C00
equiva
lent to g9 where
g{x)
= g{x + x0)-g{x0)
(indeed, g =
l?g ?r, where r(x)=x+x09 l{y)=y?g{x?)) so we can assume
that x0=0 and that #(0)=0. Second, since g(x)^0 we can define a
function r(x) by r(x)2
=
g(x)9 r(x)^0 for x<0 and r(x)^0 for x^O.
Then g(x)=(fior)(x) so that we will have proved that/and g are C00
equivalent if we show that r is a C00 diffeomorphism. Since #(*)>()
except for x=09 it is clear that r is C00 for x^O. Moreover, since
#(0)
=
=0'(O)=O, 0*(O)>O, we can write g(x)=x2k(x) where A(0)>0. Near
254 HECTOR J. SUSSMANN
-v2
y=x
-> (a) stable
y=xJ
/
^ ^aU
N_^
/
^-s* (b) unstable
/y
A
/
-/?> (c) unstable
Fig. 6.
CATASTROPHE THEORY 255
x=0 the function h(x) has a unique positive square root k(x)9 which is
C00. Then r(x)=xk(x)9 so r is C00. To show that r is a diffeomorphism
we
must prove that r
'
(x) > 0 for all x. If x # 0, then [ r (*)] 2=g(x)so2r(x) r'
(x)
= #
'
(*). Since 0'(x)^O, it follows thatr '(x)#0. Also r'(0)=?(0)^0,
where k is the function defined above. So r is indeed a C00
diffeomorphism,
and the proof that/ is stable is complete.
To show that/2 is not stable, it is better to look at Figure 6(b). The
function f2 is one-to-one (though not a diffeomorphism) and therefore
every function which is equivalent to/2 is also one-to-one. On the
other
hand, there are functions which are arbitrarily close to f2 but are
not
one-to-one, such as the one whose graph is the dotted line of Figure
6(b).
So in every neighborhood of/2 there are functions not equivalent to/2.
Therefore/2 is not stable.
The proof that/3 is not stable follows the same pattern. We must find
a
property of/3 which every function equivalent to/3 must also have, but
which can be destroyed by a small perturbation. In this case, the
desired
property is easy to find:/3 has two local minima and it takes the same
value at both of them.
There is a useful criterion for determining whether a map/eC??(JV,P)
is C?? stable. Assume that N is compact (or that N is arbitrary but /
is
proper). Then fis stable if and only if it is 'infinitesimally
stable*. To define
infinitesimal stability we must introduce some notation. A map which
to
every point x of N assigns a tangent vector to P at/(x) is called a
vector
field along f. Use 0(f) to denote the set of all smooth vector fields
along
/ In particular, the vector fields on N are the vector fields along
the identity
map iN of N. We use an obvious abuse of notation and write 6(N) in
stead of 6(iN). Every YeO(P) gives rise to a feO(f) in an obvious way,
namely, we define Y(x)
=
Y(f(x)). Also, every XeO(N) gives rise to an
XeO(f) by: %(x)
=
df(X(x)). Finally, we define/to be infinitesimally
stable if every vector field along/is of the form %+ Y for some
Xe6(N)9
YeO(P). Equivalently, use tf to denote the map X-+?9 and co/for the
map F-* Y. Then/is infinitesimally stable if and only if
0(f)
=
tf6(N) + fe(P).
The meaning of the infinitesimal stability criterion can be understood
by means of the following heuristic argument. Consider feCco(N9 P).
256 HECTOR J. SUSSMANN
Suppose t-+ft is a smooth curve in C??(iV, P) such that/0=/. Then, if
we
put
X(x)
=
-ft(x)
it follows that X(x) is a tangent vector at/(x), so that XeQ(f).
Moreover,
every XeO(f) clearly arises in this fashion. So 6(f) can be thought of
as
the set of all 'directions' in which/can be deformed. Now suppose that
{/J is a smooth family of diffeomorphisms of P. Then t -
/f/is a curve in
Cco(N9 P) of a particular kind because, at each time t9 IJis C??
equivalent
to/ The 'direction' of such a curve at/is given by
X(x) =
?lt(f(x))
= U(f(x))9
where U(y)=(dldt)?lt(y). So *7e0(P) and Xeco/0(P). A similar reson
ing shows that the elements of tfO(N) can also be interpreted as
'direc
tions' in which / can be changed while keeping it equivalent to
itself.
Therefore, the infinitesimal stability condition simply means that
every infinitesimal change in/can be achieved so as to keep/equivalent
to itself. The reason why this is equivalent to stability is now
clear.
We now illustrate the infinitesimal stability criterion by using it to
characterize the stable proper real-valued functions on the real line.
Let
/: R -> R be C00 and proper (i.e. lim^i-^ \f(x)\
=
oo). A vector field along
/is simply a C00 function a: R -> R (but we think of a as assigning to
each
xeR the vector a(x)(d/dx) at/(x)). Similarly, the vector fields on R
are
the smooth functions R -> R. The maps tf9 ca/are given by
tf(b)(x)=f'(x)b(x)
cof(c)(x)
=
c(f(x)).
Thus/is infinitesimally stable if and only if every smooth function
a(x)
can be expressed as f '(x)-b(x) + c(f(x)) for suitable smooth
functions
b and c. From this we can derive a first necessary condition for /to
be
stable. Let x0 be a critical point, i.e. a point such that/'(;to)=0.
Choose
the function a so that a'(xo)^0. If/is stable, it is infinitesimally
stable
and therefore there are functions b and c such that
(6) a(x)=f'(x)b(x) + c(f(x)).
If we differentiate both sides and evaluate at x=x0 we get
a'(x0)=f"(x0)b(x0).
CATASTROPHE THEORY 257
Therefore /"(*o)t*0- A critical point for which f,f(xo)?:0 is called
nondegenerate. We have shown that if/is stable then all the critical
points
of/are nondegenerate.
A second necessary condition can be obtained by comparing the critical
values v1=f(xl)9 v2=f(x2) for two different critical points xx and x2.
Choose the function a so that a(x1)^a(x2). If/is stable we can find b
and
c so that (6) holds. Evaluation at xx and x2 gives
a(xx)
=
c(vi)9 a(x2)
=
c(v2)
so that c(vt) t?c(v2) and therefore vx^v2.
Summarizing, we have shown that, if/is stable, then
(i) all the critical points of/are nondegenerate and
(ii) the critical values for different critical points are different.
Conversely, it is easy to show that conditions (i) and (ii) are
sufficient
for/to be stable. Thus we have proved that
A C00function f: R -> R is stable if and only if:
(i) wheneverf
'
(x0)
= 0 thenf" (x0)^0 and
(ii) wheneverf
'
(x^ =/
'
(x2)=0 andx? # x29 thenf(x1) ?> f(x2).
7. Families of functions and unfoldings
We now restrict ourselves to real valued functions, so that from now
on
P is the real line. It follows that we are in the nice range of
dimensions,
and therefore the C?? stable functions are dense. In fact, it is very
easy to
give a complete characterization of the C00 stable functions. As in
the case
of real functions on the real line (discussed in Section 6) one can
prove
that a proper C
??
function f: N -> R is C
??
stable if and only if
(i) All the critical points of/ are nondegenerate (cf. definition
below)
and
(ii) for any two different critical points, the values of/are
different.
The definition of a nondegenerate critical point is the natural
generali
zation to several variables of the one-dimensional case discussed in
Section 6. We say that/has a critical point at/7 if, in coordinates,
dxtK
'
dxn
Kyj
258 HECTOR J. SUSSMANN
If/? is a critical point of/, we say that/7 is nondegenerate if the
matrix
of the second derivatives (d2f/dXjdXi) (p) has a nonzero determinant.
Functions which satisfy condition (i) above are called Morse
functions.
The fact that Morse functions that satisfy (ii) are dense in C??(N, R)
is a
well known easy consequence of Thorn's transversality theorem.
We now ask whether the general considerations of Section 5, which
were based on our analysis of some finite-dimensional cases, carry
over to
the space C (N9 R). We have already indentified an equivalence
relation
(namely, C00 equivalence) with respect to which the stable elements
form
a dense set. We would like to show that the set of unstable functions
(or,
more precisely, a very large subset of it) can be divided into
'submani
folds' of 'finite codimension'. It turns out that this can be done,
but that
some technical complications occur. It is possible to give a
definition of
what is meant for a function to be of 'finite condimension' and we can
actually define the codimension of such a function. In a sense which
can
be made very precise, the codimension of/is the codimension of the
equivalence class of/with respect to C00 equivalence. One would then
expect that, if {Fx} is a family of functions from N to R depending on
k
parameters (xl9...9xk) then, generically, only functions of
codimension
<& will occur and that, moreover, the catastrophe set (i.e. the set of
values of x for which Fx is not stable) will be stratified as a union
of sub
manifolds Cj of codimension j(Kj<?:), where Cj={x: codimension
(?y=j}.
Actually, these expectations are not totally fulfilled. The reason is
that
there occur 'moduli', i.e. families of equivalence classes of
functions of a
given codimension which depend on a number of continuous parameters,
in a way which is analogous in principle (but, of course, much more
com
plicated) to the example of Section 5. However, moduli do not occur in
low dimensions, and for these cases reality agrees with the intuition
that
can be derived from the discussion of Section 5.
For reasons that will become apparent later, the dimensions k^4 are
the most relevant to.Thorn's theory. And moduli do not occur in this
range.
We now explain how to define the codimension of a function/eC00
(N9 R). We recall from Section 6 the notations 6(f), 6(N), 6(R), tf
and
cofi The set 6(f) of all vector fields along/is identified, as we
observed
before, with the set of all C00 functions N-+ R. Since stability is
equivalent
to the equality 6(f)
=
tf[6(N)'] + cof[d(R)'], it seems reasonable to
CATASTROPHE THEORY 259
measure the deviation from stability by finding out how much larger
6(f), is than its subset i/[0(tf)] + a>/[0(R)]. Now 6(f), 6(N) and
0(R)
are linear spaces over the reals. Moreover, tfa.nd cof are linear maps
from
6(N), 0(R) into 6(f). Even though all these spaces are infinite
dimension
al, it may very well happen that the quotient
v/
tf(d(N)) + cof(6(U))
is finite dimensional. We adopt this condition as our definition. A
func
tion/is said to be offinite codimension if the vector space A(f) is
finite
dimensional. The dimension of A(f) is then, by definition, the codimen
sion off. Clearly, we have codim (/)=0 if and only if/is stable.
The motivation for the above definition is the same as in our dis
cussion of infinitesimal stability in Section 6. If X is finite
dimensional,
W a submanifold of X, and x0e W9 then the tangent space WXo is the set
of all directions of infinitesimal changes that keep x0 in W. By
analogy
we must think of tf6(N)+cofd(R) as the 'tangent space' at/to the equi
valence class of/, which is a 'submanifold' of Cco(N9 R). Also, 6(f)
is
the 'tangent space' to C??(N9 R) at/ In the finite dimensional case,
codim W=dimXxJWXo. Our definition of codim (/) is therefore the
natural infinite dimensional analogue.
We illustrate this definition by computing the codimension of some
unstable functions. First, consider/(x)=x*?2x2 (which was shown to be
unstable in Section 6). It is easy to see that / has three critical
points,
namely ?1, 0 and 1. Moreover, these critical points are nondegenerate.
The source of the instability of/is the fact that/(?1)=/(1). Thus, we
expect the codimension of/to be one, because '/has a simple degene
racy'. To prove this rigorously, we observe that the proof given in
Section
6 that (i) and (ii) imply infinitesimal stability actually shows that,
if all
the critical points of/are non-degenerate, then a function aeC??(R, R)
is in C/T0(R)+] co/[0(R)] if and only if (?(xj^ocfa) for every pair of
critical points such that f(x1)=f(x2). Thus, in our case, tf[?(R)~] +
co/[0(R)] is simply the set of all C00 functions a : R -> R such that
a( -1)
=
=
a(l). This is a subspace of 6(f) (i.e. of C??(R, R)) of codimension
one.
Thus codim (J)= 1 as asserted.
Now consider a second example. Letf(x)=xn9 where ? is an integer.
If ?=1 or ?=2 then/is stable. Suppose ?>2. Suppose oced(f) belongs
260 HECTOR J. SUSSMANN
to i/[0(R)] + cof [0(R)]. This means that a is of the form
*(x) =
d?(x)?(x)
+ y(f(x))
i.e.,
oc(x)
=
nxn~1?(x)+y(xn)
for suitable functions ?9 y. Now y(t)
=
y(0) + 0(t) so y(xn)
=
y(0)+0(xn).
From this we get a=constant+0(jcn_1) so that
(7)
?(?)==0
for i = l,.-??-2
Conversely, every function a which satisfies (7) can be expressed as
a(x)=?i(0)+xn~1?(x)9 from which it follows easily that a is in //
[0(R)] +
a>/[0(R)]. Therefore this set is exactly the set of a's for which (7)
holds.
It follows that codim(/)=??2.
The reader will have no difficulty in showing that a function which
has
a critical point of infinite order (i.e., a point where all the
derivatives of/
vanish) has infinite codimension.
We now give the general rule for determining the codimension of an
arbitrary function/e C00 (N9 R). First, to each critical point a off
assign a
codimension c(f9 a) as follows: choose coordinates (xl9...9 xn) near a
and
let c(f9 a) denote the minimum integer k such that there are smooth
func
tions gl9...9 gk with the property that every smooth function g can be
expressed, in a neighborhood of a9 as
g(x)
= a1g1 (x) + ? + cckgk(x+) ?t (x)-~ (x) + ?+ ?n(x)?- (x),
OX? oxn
for some constants al5..., <xk and smooth functions ?l9...9?n. (As
an
example, let us compute c(f9 a) when N=R9f(x)=xn9 a=0. If g is any
smooth function, then near 0 we can write (Taylor)
9(x)
=
9(0) + g'(0)x+... +
9?^xn-2
+ ?(x)nxn-1
which shows that the ?? 1 functions 1, x9..., xn-.x can serve as our
gt. It
is easy to show that no fewer than ?-1 functions will do. Therefore
c(/, *)=?-!.)
CATASTROPHE THEORY 261
Second, we assign a dimension d(f, a, b) to each pair a, b consisting
of
a critical point a and a number b such that/(a)=6. We let d(f9 a9 b)
be
the smallest integer/ such that the function (f(x)?b)j is of the form
ftM^(.)+...+??|w
near a. (For instance, iff(x)=xn9 a=b = 09 ?>0, then/(x)=(*/?)(3//
d;*;)
so</(/, 0, 0)
= 1, if?> 1. If? = 1, then bf/dx= 1, so d(f 0, 0)=1.)
Third, we assign a dimension to each critical value b by
d(f9 b)
=
sup{</(/, a, b):aeN9f(a)
=
?}.
Finally, we let
(9) </>?c(/, a)-?</(/,?)
the sums being over all critical points and all critical values,
respectively.
EXAMPLES
A. If N= R9f(x)=xn9 so that a=0 is the only critical point, we have
seen
that c(f9 0)=?-1 and that d(f 0, 0)= 1 if ?> 1, d(f, 0, 0)=0 if ? = 1.
Therefore c(/)=0if?= lor 2, and c(/)=?-2 for ?>2.
B. Suppose/: JV-? R is a Morse function (possibly with repeated
critical
values). If a is a critical point, then the functions ij/^df/dXi form
a
coordinate chart near a (because det (dij/i/oxj) # 0). It follows that
every
smooth function g near a can be written, near a9 as
g(x)
= oc+ g1 (x)? (x)+- + gn(x)? (x).
ox1 oxn
Therefore A:< 1. By taking a function g such that g(a)^09 we see that
k cannot be zero. Therefore c(fa)
= l. If we let b=f(a)9 we then have
i=l CXi
near x (Taylor). On the other hand (f(x)?b)?== 1 which is not of the
form Yj ?i(df/dXi) because the df?dxt vanish at a. Therefore d(f9 a9b)
= L
It follows that
c(f)
= number of critical points
? number of critical values.
262 HECTOR J. SUSSMANN
For instance, if/ has 8 critical points, 3 of them with a critical
value
b0, and 5 with a value bl9 b^b0, then c(/)
= 6.
C. Let N=R2,f(x,y)=x3+y3. There is only one critical point, namely
(0, 0) with critical value 0. An arbitrary C00 function g defined near
the
origin can be written as
? ayxV+ ? htj(x9y)xy
i+j^2 ?+7=3
for suitable constants aiy- and smooth functions hti (Taylor's
formula).
Now
? = 3x and ? =
3y .
ox dy
Therefore, all the terms Ay*V are of the form ?(x9 y)(df/dz)(x9 y) for
z=x or z=y. This is also true for the terms cc20x2 and oc02y29 so that
we have shown that every g can be written, near (0, 0), as
g (x9 y)
= a00 + a10x + a0iy + oc^xy +
+ ?i (x, y) jx (*. y) + ?i (x9 y)
?
(x9 y).
Therefore the four functions \9x9y and xy can serve as the gl9..., gk
of
the definition of c(f9 a). We show that k cannot be less than four.
Indeed, suppose that three functions gl9 gl9 g3 were enough to express
every g as
9 =
K9i + X2g2 + X3g3 + ?xx2 + ?2 y2.
One then sees easily that
9 (0)
= X?gt (0) + X2g2 (0) + X3g3 (0)
g(?)
=
^(o)
+
^(?)
+
^(0)
and similarly for dg/dy and d2g/dxdy. Therefore the row vector
4<>>IN
is obtained from the row vector X =
(Xi, X2, X3) by multiplication on the
right by a fixed 3x4 matrix M. By simple linear algebra, not every e
CATASTROPHE THEORY 263
can be of the form XM. But this contradicts the fact that for every
vector e with four entries we can find a g such that e=e0. Therefore
c(/,(0,0))=4.
The calculation of d(f, (0, 0), 0) is easy. Indeed,/itself is
expressible
as ?i(dfldx) + ?2(dfldy)9vfhi\c the function 1 is not. Therefore d(f;
(0,0),0)=1. Finally, the codimension of /is c(/)
=
c(/,(0,0))
-</(/; (0,0),0) so that c(/)
= 3.
Having defined the codimension of a function, we can now study
families of functions and unfoldings. A family of functions from N to
R
with parameters in M is, simply, a family {ft:teM} where, for each
t9ft
is a smooth map N-+ R. Moreover, we assume that/ depends smoothly
on t9 i.e. that the map
(t9x)->ft(x)
from M x N to R is smooth.
As for maps, we can study the stability of families of functions. We
first define equivalence. Two families {/,'}, {/} of maps iV->R, para
metrized by M, will be called equivalent if there exist a
diffeomorphism
g\M-*M and families {rt}, {/,} of diffeomorphisms N-+N, R-?R
respectively, depending smoothly on teM, such that
f?(t)
=
ltf<rt for all teM.
Using this equivalence relation we can define the concept of stability
of
a family of functions as in Section 3. It turns out that stability is
a generic
property, at least if dim AT <4. For higher dimensions the theory
becomes
more complicated, because of the presence of moduli, as explained
above.
Now, if F= {ft:teM} is a family of functions N-> R, we can consider
the catastrophe set of F9 which we denote by C(F). The definition is
ob
tained by particularizing Definition 3 of Section 3. Therefore
C(F)
=
{t :ft is not stable}.
When F is stable, a lot more can be said about the structure of C(F)
Suppose that t0eC(F), and let/=/0. Then /is an unstable element of
Cco(N, R) whose codimension we shall call k. Thinking by analogy with
the finite-dimensional discussion of Section 5, we let Ef denote the
equiv
alence class off modulo C00 equivalence, so that we must think of Ef
as
a 'submanifold' of Cco(N, R) of codimension k. Then Pis a smooth map
264 HECTOR J. SUSSMANN
from the parameter space to Cco(N9 R), which takes the value /when
/=t0. The stability of F implies that Pis a versal deformation off.
Precisely
we define a versal deformation of an/of finite codimension k to be a
family P={/} of functions such that ft0=f and that the map ?->/f is
transversal to Ef at t0. The precise meaning of transversality in this
infinite-dimensional situation is the same as in the finite-
dimensional
case. Recall the notations 0(/), tfd(N)9 cofd(P) introduced in Section
6.
Recall also that, when dealing with functions, 6(f) can be thought of
simply as Cco(N9 R) itself, and that tf6(N) and cofd(P) also have con
crete interpretations. Since 6(f) is 'the set of all directions
tangent to C00
(N9 R) at/', while tfl(N) + cof6(P) is 'the set of those directions
tangent
to P'/5 the transversality of F to Ef should be expressed by the fact
that
(10) 6(f) =
tfd(N) + f6(P) + Bt0(F)9
where Bt0(F) is the linear hull of the derivatives
dt1' 'dtm { \}*? *?
evaluated at t0 (which are therefore functions from N to U, i.e.
elements
of?(/)).
We now forget about the preceding heuristic considerations, and take
(10) to be the definition of transversality.
It is clear that, in a versal deformation off the number of parameters
(called the dimension of the deformation) cannot be smaller than the
codimension off. Equivalently, if we consider stable families F
depending
on k parameters t1,..., tk then, locally, the catastrophe set C(F)
will be
the catastrophe set of a versal deformation of a function of
codimension
^k.
To obtain versal deformations it is convenient to consider universal
unfoldings. A universal unfolding off is a versal deformation off
having
the smallest possible dimension. It then follows easily that this
dimension
is precisely the codimension off One can get all versal deformations
off
by simply starting with a universal unfolding, and then throwing in
new
parameters and making a nonlinear change of coordinates in parameter
space. That is, we let P0(*\..., tk9 x)(xeN) be a universal unfolding
of
/, and then we define Fi(t19...9tl9x)(l>k) arbitrarily, with the only
CATASTROPHE THEORY 265
restriction that
F?(t?9...9tk909...909x)=EFo(t?9 ...9t\x).
We then put
F(t9x)
=
Ft(g(t)9x)
where t=(t19..., tl) and g is a C00 diffeomorphism.
8. Th?m's list of elementary catastrophes
In Section 7 we explained how to obtain the catastrophe set of an arbi
trary stable family F of functions depending on k parameters by taking
universal unfoldings of functions of codimension ^k. If we actually
want to describe all possible catastrophe sets which may appear in
this
fashion, it seems as though we would have to consider infinitely many
cases, because we might have functions of codimension <fc on mani
folds N of arbitrary large dimension. It turns out, however, that (for
fc<4), we actually get af, inite list, because every universal
unfolding of
anfiN^R of codimension ^k can be 'reduced', independently of what
N might be, to one of a small number of 'canonical forms'. The precise
meaning of this reduction need not concern us, except for the fact
that it
is an operation which does not change the catastrophe set.
Following Thorn, we shall give a complete description of the
catastrophe
sets which arise from stable unfoldings of functions of codimension <4
with one critical point. It turns out that there are exactly seven
possible
catastrophe sets, obtained from seven functions of codimension <4. The
complete list (together with their picturesque names) is given in
Table II.
TABLE II
Name Function Codimension Universal unfolding
fold
cusp
swallow tail
butterfly
wave crest
hair
mushroom
x3+y3
x2?xy2
x2y+y*
x3+ux
xA+ux2jt-vx
x5+ux3+vx2 -+w- x
JC+6 ux4 + vx3 + wx2 + tx
x3 +y3+uxy+vx+wy
xs?xy2+u (x2 +y2)+vx+wy
x2y+y4+ux2+vy2+wx+ty
266 HECTOR J. SUSSMANN
The letters u, v, w, t are the unfolding parameters. Using the
considera
tions of Section 7 it is easy to justify the entries of Table II
(except for the
names). The codimension of the first five functions that appear in the
table has already been computed in the examples of Section 7, and the
reader will find no difficulty computing the other two. To derive a
univer
sal unfolding one then proceeds as follows: suppose/has codimension k.
Recall that k is then the dimension of the quotient A (/)
=
6(f)/[t?(N)+
+ co/0(P)], and that 6(f) can be thought of as the space of all C00
functions in N. Choose functions gl9...9gk with the property that
their
equivalence classes modulo tf6(N) + cof6(P) form a basis of A(f). Then
define, for xe N9
k
F(tl9...9tk9x)=f(x)+ ? tigi(x).
i=l
It follows from the definition given in Section 7 that F is a
universal un
folding of/ Using this method, the last column of Table II is easily
obtained.
Using Thorn's list, it is not hard to study the unfoldings and
catastrophe
sets which arise from more general functions. If a function /has
several
critical points, formula (9) of Section 7 enables us to calculate its
codi
mension. The result is a sum of contributions of the individual
critical
points, except for the fact that, whenever we have several critical
points
al9...,ak with the same critical value we only subtract the largest of
the
numbers d(f, ai9 b)9 instead of subtracting their sum. Equivalently,
the
codimension of /is the sum of the codimensions of the individual
critical
points plus, for each critical value b9 an integer k(b)^0, equal to ?
d(f,
ai9 b)
?
sup d(fi ai9 b). It follows that a function of codimension <fc will
have no more than k degenerate critical points, and that their codimen
sions will have to add up to at most k. We can describe all the
possibilities
by using letters to name the various types of degenerate critical
points,
and the letter N for nondegenerate ones. We then group together all
the
points with the same critical value. For instance (NNBW)(NBM) denotes
a function/having four degenerate critical points, namely two
butterflies,
one wave crest and one mushroom and, in addition, at least three non
degenerate critical points. Moreover, two of these three points,
together
with the wave crest and one of the butterflies have the same critical
value
bl9 whereas the other nondegenerate point, together with the mushroom
CATASTROPHE THEORY 267
and the other butterfly, also have a common critical value b2, such
that
b2^b1 (/may have other nondegenerate critical points, but as long as
the
corresponding critical values are different from each other and from
bt
and b2, their contribution to the codimension of /is zero). One sees
easily that, for all the critical points N, B9 W9 M the number d(f9 a9
b) is
one. Therefore k (?>i)=3 and k(b2)=2. The codimension off is then 20.
As an example, we describe all possibilities with codimension <2.
There are exactly six of them, namely (F), (AW), (C), (F) (F), (NNN),
(NF). The first two have codimension one, and the last four have codi
mension two. For a generic family F of functions depending on two
parameters (so that the parameter space is a region of the plane) the
catastrophe set C(F) will consist of a number of curves (corresponding
to
those t for which Ft has codimension one) and points (where C(P,)=2).
Moreover, one sees easily that the closure of each curve must consist
entirely of unstable points and therefore it will be the union of the
curve
itself plus some points of codimension 2.
9. AN EXAMPLE AND CONCLUDING REMARKS
We return to our utility maximizer of Section 3. The function u(x9y)
which he tries to maximize depends on xeQ and yeP9 but our individual
only controls x. This determines a 'function' x(y) by formula (3) of
Section 3. If the parameter y is fc-dimensional, with fc^4, we know
that,
generically, u will be stable. Therefore, for all y in an open dense
set of
P the function uy(x)
=
u(x9 y) will be stable, and we will have a stratifica
tion of the catastrophe set by manifolds of various codimensions lt of
points for which codim (uy)
=
l?. It is not hard to prove that, as long as y
is not in the catastrophe set, the point where u reaches its maximum
(if it
exists) is unique, and that the mapping y ->
x(y) is smooth. Therefore the
catastrophe set of u represents the points where a drastic change in
behav
ior takes place, and Section 8 gives us some information about this
set.
We emphasize one remarkable aspect of this information, namely, its
independence on any detailed information on u9 including the number of
x-variables involved.
The reader will certainly want to argue that, on the other hand, this
information is exceedingly poor. For instance, we are not able to
predict
which values of y will give catastrophic changes. The answer to such
an
268 HECTOR J. SUSSMANN
objection is that, indeed, we could not possibly have expected to get
such
prediction without feeding in as assumptions some information about u.
What we have done here is, as was said in Section 1, no closer to the
real
world than the general theory of ordinary differential equations. To
actually apply the theory it will become necessary to bring in
additional
structure, in the same way as, in Mechanics, one assumes a particular
form of the differential equations. The search for that additional
structure
is a task for the applied mathematician and the natural and social
scien
tist. It is perhaps too soon to form an opinion on the extent to which
such
a task may succeed in the future. As for the extent of its success so
far, the
reader must make this evaluation by himself, by reading the work of
those
who have applied the theory (e.g. Thorn and Zeeman). Whatever the
result of this evaluation might be, this paper will have served a
purpose,
we hope, by helping the reader to make an unbiased judgment based on
an understanding of what the theory really is (or is not).
10. Appendix
For an introduction to the theory of manifolds, the reader may consult
[IDM]. For the theory of singularities of smooth maps, the book [SMS]
provides an accessible presentation. The classic works in this theory
are
(a) Whitney's pioneering paper [22], (b) the papers of Thorn, Mather
and Arnol'd, especially [14], [15], [16], [19], [10] and [1], (c)
Board
man's work [4].
The definition of 'catastrophes' in the general context of a space
with
an equivalence relation appears, for instance, in [6]. Proofs of
Thorn's
transversality theorem can be found in many places, such as [SMS],
[TMF] or [15]. The study of C00 stability and its relation with
infinites
imal stability was undertaken by Mather in [9]. These results require
the
use of tools such as Malgrange's division theorem (cf. Mather's papers
or
[AD]). There are analogues of infinitesimal stability for other
stability
concepts, such as families of functions, or unfoldings (cf. [8] and
[SU]).
The heuristic justification given in the text (i.e. the idea of
viewing 6(f)
as the 'tangent space' to C (N9P) at/) should be translatable into a
formal proof. The tool for such a proof would be a reasonable implicit
function theorem for infinite dimensional non-Banach spaces. Two such
theorems are given in [12] and [13]. The theorem that C? stability is
CATASTROPHE THEORY 269
generic was also proved by Mather (cf. [10]). Condensed presentations
of this proof appear in [11] and [5], For families of functions and un
foldings, see [SU] and [6]. The definition of the codimension of a
func
tion as given here is taken from Sergeraert's paper [13]. An
illustration of
the difficulties that appear when attempting to stratify the space of
func
tions can be found in [7]. Thorn's list of the seven catastrophes is,
of
course, due to Thorn. For a recent, detailed presentation with proofs,
see
[SU].
There is a broad area which has not been touched upon in this paper,
but is closely related to it, namely, the study of generic properties
of
dynamical systems (cf. [TMF] and the papers by Smale, Peixoto, Soto
mayor, etc. in [DS]).
Possible applications of the theory are discussed by Thorn in [SSM],
[18], [19] and [20] and by Zeeman in, e.g., [23], (cf. also Baas [3]).
Rutgers University
NOTE
* Work partially supported by NSF Grant No. GP-37488.
BIBLIOGRAPHY
(i) Books
[SMS] M. Golubitsky and V. Guillemin, Stable Maps and Their
Singularities,
Springer-Verlag, New York, 1974.
[DS] M. M. Peixoto (ed.), Dynamical Systems, Proceedings of a
symposium held
Salvador, Brazil; Academic Press, New York and London, 1973.
[PSSLI] Proceedings of the Liverpool Singularities Symposium I, 1971
(Springer
Lecture Notes No. 192).
[MA1970] Manifolds-Amsterdam 1970, Springer Lecture Notes No. 197.
[SU] G. Wasserman, Stability of Unfoldings, Springer Lecture Notes No.
393.
[AD] V. Poenaru, Analyse Diff?rentielle, Springer Lecture Notes No.
371.
[TMF) R. Abraham and J. Robbin, Transversal Mappings and Flows,
Benjamin,
New York 1967.
[IDM] S. Lang, Introduction to Differentiable Manifolds, Interscience,
New York,
1962.
[SSM] R. Thorn, Stabilit? structurelle et morphog?n?se, Benjamin,
1972.
(ii) Articles (Initials such as DS or PSSLI refer to the books listed
in (i).)
[1] V. I. Arnol'd, 'Singularities of Smooth Maps', Uspehi Mat. Nauk 23
(1968), 3-44,
translated in Russian Math. Surveys (1969), pp. 1-43.
[2] V. I. Arnol'd, (a) 'Integrals of Rapidly Oscillating Functions and
Singularities of
Projections of Lagrangian Manifolds', and (b) 'Normal Forms for
Functions Near
Degenerate Critical Points, the Weyl Groups of Ak, Dkt Ek, and
Lagrangian Sin
gularities', Funktional*nyi Analiz i Ego Prilozheniya (1972) 6; (a) in
No. 3, pp.
270 HECTOR J. SUSSMANN
61-62 and (b) in No. 4, pp. 3-25. Translation in Funct. Anal, and
Applications
(1973), pp. 222-223 and 254-272.
[3] N. A. Baas, 'On the models of Thorn in Biology and Morphogenesis',
Math. Biosci.
17(1973), 173-187.
[4] J. M. Boardman, 'Singularities of Differentiable Mappings', Publ.
I.H.E.S., No. 33
(1967), pp. 21-57.
[5] A. Chenciner, 'Travaux de Mather sur la stabilit? topologique', S?
m. Bourbaki,
expos? 424, Vol. 1972/1973, Springer Lecture Notes No. 382.
[6] J. Guckenheimer, 'Catastrophes and Partial Differential
Equations', Ann. Inst.
Fourier Grenoble 23,2(1973), 31-59.
[7] H. Hendriks, 'La stratification "naturelle" de l'espace des
fonctions diff?rentiables
reelles n'est pas la bonne', C.R. Acad. Sei. Paris 274, s?rie A(1972),
pp. 618-620.
[8] F. Latour, 'Stabilit? des champs d'applications diff?rentiables;
generalization
d'un th?or?me de J. Mather', C. R. Acad. Sei. Paris 268, s?rie
A(1969), pp. 1331
1334.
[9] J. N. Mather, 'Stability of C00 Mappings Y, Ann. of Math. 87
(1968), pp. 89-104;
II ibid. 89 (1969), pp. 254-291 ; III Publ. I.H.E.S. 35 (1969),
127-156; IV ibid. 37
(1070), pp. 223-248; V Advances in Math. 4 (1970), 301-335; and VI in
PSSLI
pp. 207-253.
[10] J. N. Mather, 'Notes on Topological Stability', preprint, Harvard
Univ., Cam
bridge, Massachusetts, 1970.
[11] J. N. Mather, 'Stratifications and Mappings', in DS pp. 195-232.
[12] V. Po?naru, 'Un th?or?me des fonctions implicites pour les
espaces d'applica
tions C?>\Publ. I.H.E.S. 38 (1970), 93-124.
[13] F. Sergeraert, 'Un th?or?me de fonctions implicites sur certains
espaces de
Fr?chet et quelques applications', Ann. Scient. Ec. Norm. Sup., 4th
series, 5 (1972),
599-660.
[14] R. Thom, 'Les singularit?s des applications diff?rentiables',
Ann. Inst. Fourier
Grenoble 6 (1955-1956), 43-87.
[15] R. Thom and H. I. Levine, 'Singularities of Differentiable
Mappings', in PSSLI
pp. 1-89.
[16] R. Thom, 'Ensembles et morphismes stratifi?s', Bull. Amer. Math.
Soc. 75 (1969),
pp. 240-284.
[17] R. Thom, 'The Bifurcation Subset of a Space of Mappings', in MA
1970 pp.
202-208.
[18] R. Thom, 'Topological Models in Biology', Topology 8 (1969),
313-335.
[19] R. Thom, 'Topologie et linguistique', in Essays on Topology, Vol.
dedicated to G.
de Rham, Springer-Verlag, Berlin and New York, 1970.
[20] R. Thom, 'Language et catastrophes', in DS pp. 619-654.
[21] C. T. C. Wail, 'Lectures on C00 Stability and Classification', in
PLSSI pp. 178-206.
[22] H. Whitney, 'On Singularities of Mappings of Euclidean Spaces I,
Mappings of the
Plane Into the Plane, Ann. of Math. 62 (1965), 374-410.
[23] E. C. Zeeman, 'Differential Equations for the Heartbeat and Nerve
Impulse', in
DS pp. 683-741.
E. C. Zeman had a thought about catastrophe theory:
http://zakuski.math.utsa.edu/~gokhman/ecz/c.html
Catastrophe Theory
Author(s): Hector J. Sussman
Source: Synthese, Vol. 31, No. 2, Mathematical Methods of the Social
Sciences (Aug., 1975), pp.
229-270
Published by: Springer
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HECTOR J. SUSSMANN
CATASTROPHE THEORY*
1. Introduction
The purpose of this paper is to give an account of the basic
mathematical
facts pertaining to 'catastrophe theory'. No applications to concrete
problems will be discussed here. However, some consideration will be
given in Section 2 to a number of questions which can be Regarded as
'applications' in a broad sense, for they are of a philosophical
nature. In
particular, we will attempt to show how some of the basic ideas of
cata
strophe theory and systems theory are related.
In deciding what to include in this work, we took the view that 'cata
strophe theory', as mathematics, is a not very well defined part of
the
theory of singularities of smooth maps. It is not clear to us what
specific
problems, or points of view, are the ones which give catastrophe
theory
its own identity. In Section 2 we attempt to isolate one possible
charac
terization, but the reader is advised not to concentrate on this
point.
Indeed, we believe that the main fact about the theory of
singularities of
smooth maps, which the reader may find relevant, is that it leads
naturally
to a number of concepts that are not frequently encountered in Applied
Mathematics: classification of objects according to their qualitative
properties, genericity, stability (in the 'second sense', of Section
2). This
belief has lead us to include a somewhat larger amount of information
than might be considered to be of direct relevance to the theory. On
the
other hand, we give no proofs. (For references to where those proofs
can
be found, see the Appendix.)
Finally, a disclaimer. This author takes no position on the hotly
debated
question as to whether catastrophe theory really has important
applications.
We believe, however, that it is important for a mathematically
oriented
social scientist to be acquainted with the theory, so as to be able to
use it
if he can, and to criticize it if it cannot be used. In the meanwhile,
it must
be remembered that catastrophe theory resembles the classical theory
of
differential equations in that it only claims to provide a general
frame
Synthese 31 (1975) 229-270. All Rights Reserved
Copyright ? 1975 by D. Reidel Publishing Company, Dordrecht-Holland
230 HECTOR J. SUSSMANN
work for modelling certain types of phenomena. Good models for parti
cular phenomena are as unlikely to be derived from the general theory,
as
it is for Newtonian mechanics to be derivable from the existence
theorem
of ordinary differential equations. But, in both cases, the fact that
such
models cannot be derived from the general theory does not qualify as a
valid objection against it.
2. What is catastrophe theory?
The framework which has been most widely used for the mathematical
modeling of processes evolving in time is, undoubtedly, that of the
theory
of ordinary differential equations (O.D.E.'s). In this approach, one
selects a number n of 'state variables' xl9..., xn9 and thinks of each
'state'
of the system as characterized by the values of the state variables
for the
given state. The time evolution of the system is then described by a
system of O.D.E.'s
(1)
?'
=fi(xi9..., x?), i =
1,..., n.
(A number of situations lead to states that are represented by points
in
infinite dimensional spaces. This is the case, for instance, of
Maxwell's
equations and of Quantum Mechanics. In this way, there appear new and
extremely interesting mathematical problems. However, the basic
heuristics for such infinite dimensional systems resembles the finite
dimensional case.)
A system (1) functions very much like a set of traffic signs
consisting of
arrows which, at each point, indicate which way to go (and how fast).
Clearly, such signs will determine the trajectory of any motorist who
follows them, provided only that we know the point from which he
departed. Similarly, the system (1) tells us, for any given state
xl9...9 xn9
what the state will be after an 'infinitesimal' time dt has elapsed.
Therefore,
it seems intuitively clear that a system such as (1) can be solved, in
the
sense that, given any 'initial state' (*?,...,*?), (1) determines
unique
functions of time xt (t)9 ...9xn(t) such that xt (0)=x?l9..., xn(0)=x?
n. This
intuition is amply justified by the existence and uniqueness theorem
of
ordinary differential equations, provided the functions f satisfy some
reasonable technical conditions.
CATASTROPHE THEORY 231
The state-space-O.D.E. approach has certainly had such an impressive
number of successes, that it is both impossible and unnecessary to
list
them. On the other hand, there are limitations as well, and we wish to
discuss two of them which are of significance for catastrophe theory.
First, the method does not lend itself very easily to the study of
situa
tions which essentially involve objects which remain unchanged for
long
intervals of time and then experience abrupt changes. The state of a
system described by O.D.E.'s is either always changing or, if it is
constant
for a very short time, it remains constant forever, unless it is
disturbed by
an agent extraneous to the system. It is true that some aspects of
this
'permanence and sudden change' can be captured by the O.D.E. approach,
but it would be desirable to have a framework in which it appears
natur
rally.
A second and much more serious difficulty arises from the inadequacy
of the state-space O.D.E. approach for the study of complex systems.
This inadequacy is seen whenever one has to deal with a very large
number of variables which cannot easily be 'aggregated' into a small
number of 'macroscopic' ones. Examples of this situation appear when
ever the variables cannot be grouped into a small number of classes of
variables of the same type. Moreover, they also appear in cases in
which
such grouping is possible, but there is no reasonable way of 'adding'
or
'averaging' the variables within each class (e.g., the well known
difficulties
that arise in any attempt to aggregate individual utility functions
into a
'social welfare function').
Mathematically, the reason why the state-space-O.D.E. approach is
inadequate to handle complex systems is the bad behavior of O.D.E.
systems under the operation of 'pushing forward'. Indeed, suppose we
have a system (1), with n very large, and suppose that there is a
small
number m of 'important' variables yl9..., ym which are functions of
the
xi9 namely,
(2) Ji =
0i(*i,...,*?), i=l,...9m.
Suppose that the yt are the only variables that really matter to us.
It
would then be desirable to use (1) to get a similar system of O.D.E.
for
the yt. Now, it is easy to see that, in general, such a system will
not exist.
For instance, (a) the future behavior of the y? will not necessarily
be
232 HECTOR J. SUSSMANN
determined by their values y? at a particular time t09 as would be the
case
if thej^ evolved according to O.D.E.'s.
Now, systems exhibiting feature (a) are far from a mathematical
novelty,
For instance, if the yt are the coordinates of the position of a
moving
particle, then (a) holds. The traditional way out of this difficulty
is approxi
mately three centuries old, and it consists of adding new state
variables,
such as the components of the moving particle's velocity vector.
However,
in recent years we have witnessed developments for which this
traditional
way does not work. In the social sciences, in biology, in geology, the
number of variables becomes too large and the models almost impossible
to handle.
We are now in a position to describe the aim of catastrophe theory. //
is
an attempt to study systems of the type (1), together with functions
(2), with
the aim of saying as much as possible about what happens in the space
of the
y?. The space of the x variables, and the properties of the O.D.E.'s
(1) are
regarded as being of lesser interest, except for those aspects which
have
'observable' manifestations in y-space. In particular, it is desirable
to
isolate statements about the dynamics in j>-space which are
independent
of particular assumptions on what happens in x-space.
We now restate the above considerations in a more modern language.
The x-space referred to above was, presumably, the set of all ?-tuples
of
values (#!,..., xn)9 i.e., the space Un. Applications make it
necessary to
consider more general spaces, such as open domains of IRno r even sets
of
points in Un which are defined by equality constraints. This leads
naturally
to the use of arbitrary smooth manifolds as state spaces. If iVis an ?-
dimen
sional smooth manifold, a system of ordinary diflerential equations on
N is,
by definition, the same as a smooth vector field V on N. A dynamical
system is a pair (N, V)9 where N is a smooth manifold and F is a
smooth
vector field on N. Catastrophe theory deals with the study of
situations in
which we are given :
(i) a dynamical system (N, V)
(ii) a smooth manifold P9 and
(iii) a smooth map FrN-* P.
As was explained above, the aim is to describe what happens in the
space
P of the 'observed' variables.
It would now be desirable to give a sketch of the important theorems
of
the theory, and examples of applications. As was explained in the
intro
CATASTROPHE THEORY 233
duction, this paper will not discuss applications. As for the
theorems, it is
a feature of the theory that there are very few whose statement can be
explained without the use of sophisticated mathematical machinery. On
the other hand, in most applications what is used is not catastrophe
theory as a set of results, but catastrophe theory as a conceptual
frame
work (for an example, see Zeeman's beautiful paper [23] on heartbeat
and nerve impulse). Therefore, we shall devote the remainder of this
sec
tion to a discussion of this way of thinking. In subsequent sections
we will
give a more detailed presentation of the mathematics.
The main aspects which we will concentrate on here are the ideas of
stability and catastrophes. The concept of stability has emerged in
modern
science as one of central importance, especially where a 'systems' ap
proach is utilized. Roughly, a stable object, or system, is one that
reacts to
small changes by returning to its original state. This idea of
stability can
be modelled mathematically within the context of dynamical systems
theory by means of the concepts of 'stable equilibrium' and
'attractor'.
If (N9 V) is a dynamical system, a closed subset C of N is said to be
an
attractor if every trajectory of the system which begins sufficiently
close
to C converges to C. A stable equilibrium is an attractor which
consists of a
a single point. This idea of stability exhibits some features in
common with
the 'real life' concept. But it has the drawback it does not allow for
'jumps'
from one attractor to another to be incorporated into the dynamics of
the
system. As we will see below, when this idea is combined with the
frame
work of catastrophe theory, the treatment of such jumps becomes
possible.
In addition to what was explained above, there is a second sense of
the
word 'stability' which is of paramount importance in contemporary
mathematics. Suppose that a class X of objects is given, and that we
have
a notion of 'nearness' and a concept of what it means for two objects
of
Zto be 'qualitatively the same'. We then call an x of X stable if
every y in
X which is sufficiently near x is qualitatively the same as x (cf.
Section 3
for a precise definition). Stable objects in this sense have appeared
in
mathematics in several contexts. Here we shall mention one of them:
consider the class of all smooth maps from N to P9 where N and P are
manifolds. There is a notion of nearness and one of 'being
qualitatively
alike' (cf. Sections 4 and 6). When one attempts to classify the maps,
by
giving a description of the various kinds of qualitative behavior
which
they might exhibit, it is found that such a description is practically
im
234 HECTOR J. SUSSMANN
possible, because some maps may be highly 'pathological'. On the other
hand, the behavior of stable maps is much easier to describe.
Therefore,
stability appears naturally as a technical assumption which can be
made on
a map and which makes it possible to prove interesting theorems.
Moreover,
the point has been made by some, and especially by Thorn, that the no
tion of stability has philosophical significance, and that it is
essential for
the understanding of what we mean by an 'object'. A real object is
charac
terized by the values of a large number of variables, and these values
are
constantly experiencing 'random' changes. Yet we perceive the object
as
remaining the same. This presumably means that the object is stable in
our more technical'sense.
Having explained what shall be meant by stability, we can describe the
technical meaning of 'catastrophes'. An object x will be said to
belong to
the catastrophe set if it is not stable. The following trivial example
will
illustrate the above notions and, in particular, how a catastrophe can
be
stable or, more precisely, how it can occur in a stable way. (In view
of our
definitions, the reader should find this mysterious). The answer is
indeed
simple. Say, for instance, that X is the set of all points in the
U.S.A. and
Canada. Qualitatively, we can divide the points in three classes,
namely,
those in the U.S.A., those in Canada, and the border. The first two
types
are stable, and the catastrophe set is the border. If you choose a
point
'at random' it will most likely be in one of the countries, and you
can
safely forget about the border. But if you choose two points at random
and then (also at random) a route joining them, you can no longer
ignore
the border. A curve from Boston to Montreal must cross the border, and
every curve 'close' to it will also cross it. Thus, in the new set X'
which
consists of all routes (i.e., curves) we have stable elements in which
catas
trophes appear in a stable way.
We now return to our original attempt to explain what catastrophe
theory was about. We have to study objects consisting of a dynamical
system (N9 V)9 a manifold P and a map F.N-+P. The preceding remarks
suggest that we define what it means for such a composite object to be
stable, and that we then study these objects. Unfortunately, such a
general
theory has not yet been developed in a fully satisfactory way. (The
main
difficulty arises in the search for an adequate definition of
stability for
dynamical systems.) For this reason we shall limit ourselves here to
the
study of two particular cases of our general situation.
CATASTROPHE THEORY 235
Case 1 : V= 0, i.e., we just have objects consisting of a pair of
manifolds
N, P and a map F:N-*P. This leads to the study of the stability of
smooth
maps, which was undertaken by Mather with great success. We give a
brief sketch of some parts of this theory in Section 6.
Case 2: (a) N is simply a product Q xP, where Q is some region in Un
(or, more generally, a Riemannian manifold),
(b) the vector field X points, for each yeP, in the direction of the
'fiber' Q x {y}, and
(c) for eachjeP, the vector field on Q defined by Xis the gradient of
a
'potential' fy. Therefore, we have to study 'families of functions
para
metrized by p\ i.e., functions on a set Q which also depend on P. Some
basic facts about stability of families of functions are presented in
Section 7.
Case 2 can be considered as an idealization of some situations which
appear in practice. Consider, for instance, an individual r who
maximizes
a utility function u(x9 y)9 where x is an ?-dimensional parameter
repre
senting the alternatives available to r, while y is an m-dimensional
param
eter over which x has no control. If we assume that the parameter y
changes much more slowly than t makes his decisions, we can then
assume
that, for each y9 % chooses an x(y). The 'mapping' x(y) is defined by
the
requirement that
(3) u (x (y)9 y)
= max u X (x, y)
and it is not a true mapping for several reasons, such as, for
instance, the
fact that there may be more than one x where the maximum is reached,
or
none at all. We will see in Section 9 that, for 'most' w, x(y) is
'nice' as long
as u (considered as a function of x only) is stable. Moreover, the set
of
stable >>'sw ill be 'most' of P, but there will be, 'in general', y's
for which
an 'abrupt change in behavior' will take place. Such changes will
occur
'stably', in the same way as the U.S.-Canada border of our example
appears stably when points which change in time are considered.
3. Genericity and stability: some simple examples
Both in pure mathematics and in applications, one has to deal very
often
with situations in which
236 hector j. sussmann
(a) a certain class of objects Zis being considered, and
(b) all the objects of X have some 'nice' properties, with the
exception
of a certain subclass Xdtz ofX whose elements are 'degenerate9.
A typical example of such a situation is provided by a system
(4) a11x1 + a12x2
=
a13
a2^Xi ~r ??22^2
==
^23
of two linear equations in the two unknowns xu x2. As the reader
knows,
a lot can be said about the solutions, but one has to distinguish
various
cases, depending on the values of the coefficients a^. In particular,
the
'nicest' case occurs when the determinant
A =a??a22
?
ci21al2
is not equal to zero. Indeed, if A ^09 then there is one and only one
solu
tion xl9 x2 of the system (4). On the other hand, when A =0 many
things
can happen. (The list of these possibilities, which is probably well
known
to the reader, need not concern us.) Let us show how this example fits
in
the general framework described above. The set X consists of all
systems
(4). Now, giving such a system is completely equivalent to giving the
six
real numbers aijt Therefore, A" can be identified with the space R6 of
all
sixtuples of real numbers. The 'nice' property that one would expect
from
a system (4) is the existence and uniqueness of a solution. Those
systems
for which this property does not hold are 'degenerate'. They are
charac
terized by the equation
A=0.
In a way which should be intuitively clear to the reader, A is much
more
likely to be different from zero than equal to zero. If the six
numbers a^
are 'chosen at random', A^O will hold 'in general', i.e. except for
some
'exceptional' cases. We describe this by saying that
A # 0 holds 'generically'
or that
a generic system (4) satisfies A # 0.
CATASTROPHE THEORY 237
In general, the word 'generic' is used in mathematics as in this
example.
One says that certain property P holds generically for elements of a
space
X if it holds for 'most' elements of X. An equivalent form of the same
assertion, which is widely used, is
a generic xe^satisfies P.
The reader is warned that this statement should not be interpreted as
referring to the property G of 'genericity' and asserting that every
xeX
which satisfies G necessarily satisfies P. The correct interpretation
is the
one given in the preceding paragraph, i.e., that P holds for 'most'
xeX.
(Remark: strictly speaking, what was just said is only true
'generically'.
There are some cases in which mathematicians have agreed to single out
certain properties Pt of elements of a space X, and call an xeX
generic
if it satisfies the P?.)
Of course, a genericity statement will have a precise mathematical
meaning only where the meaning of the word 'most' has been made
precise. On the other hand, there is no general definition of what
'most'
means which applies to all possible cases. Therefore each situation
will
have to be discussed individually. We shall postpone this discussion
until
Section 4. In the meanwhile, we remark that all the examples to be
discussed before Section 4 will resemble the one that was presented
earlier
in this section, in that the meaning of 'most' (and therefore of
'generic')
will be clear from the context.
We now return to our example. We have already shown that, generically
a system (4) has exactly one solution. Let us call these systems
(i.e., the
ones for which A #0) 'nondegenerate'. We want to discuss another prop
erty of these systems, namely, their stability. Intuitively, this has
the
following meaning: suppose we start with a system (4) and 'perturb' it
slightly by replacing the coefficients atj by fly+ey, where the
numbers ey
are small. If the original system is nondegenerate, then the
perturbation
will only cause a small quantitative change in those aspects of the
system
that are of interest to us, such as the solution of the system.
Moreover,
there will be no 'qualitative jump'. For instance, the new system will
have
one and only one solution, exactly like the old one. When the original
system is degenerate, we encounter a completely different situation.
Here a
small change in the atj can cause a drastic qualitative change in the
proper
238 HECTOR J. SUSSMANN
ties of the solutions. As an illustration of what may happen, we
observe
that the ei7 can always be chosen so that the perturbed system will be
nondegenerate. This is indeed a change! A degenerate system either has
no solution at all or it has infinitely many. And an arbitrarily small
perturbation can change it into a system with exactly one solution. We
summarize all this by saying that nondegenerate systems are stable,
whereas degenerate ones are unstable. Unstable elements are those
where
drastic changes occur, and for that reason the set of unstable
elements is
referred to as the catastrophe set.
In order to formulate a general definition of stability, we shall
begin by
listing those ingredients that are needed to make such a formulation
possible. First, we need a class of objects X as above. Second, we
must
have a concept of 'closeness', so that we can talk about
'perturbations'.
Mathematically, X must have the structure of a topological space.
Third,
we must have a formal definition of what it means for two objects of X
to be 'qualitatively alike'. Mathematically, this will be expressed by
an
equivalence relation.
DEFINITION 1. Let Jf be a topological space, and let R be an equiva
lence relation on X. An element x of X is called R-stable if there is
a
neighborhood Uofx such that every ye Uis P-equivalent to x.
DEFINITION 2. Let X and R be as in Definition 1. The set of P-un
stable elements of X is called the R-catastrophe set of X.
DEFINITION 3. Suppose <f> is a continuous map from a topological
space 7 into X(i.e. {<?){t):teY} is a 'family of elements of X
depending
continuously on the parameter te Y'). The R-catastrophe set of <p is
the
set of all / such that <f){t) is P-unstable.
Remark. We insist on the dependence on R which is made explicit
in the above definitions. The same class X of mathematical objects may
arise from different situations. This may lead to the consideration of
different equivalence relations R, for the question of what we mean we
call
two elements of X 'qualitatively alike' cannot be answered without a
knowledge of what the objects in X are supposed to represent. As an
example, let us return to the set X of 6-tuples of reals that was
considered
above.
CATASTROPHE THEORY 239
We have seen how X can be thought of as the set of linear inhomoge
neous systems of two equations in two unknowns, and how this leads to
a
concept of stability. Suppose now that we consider the set of all
systems of
two linear homogeneous equations in three unknowns. This leads natu
rally to the same space X of 6-tuples {atj:/= 1, 2; j= 1, 2, 3} as
before.
However, the natural equivalence relation for this situation is not
the
same as before, and the concept of stability is different (for
instance,
when a 2 x 3 matrix is thought of as the matrix of a homogeneous
system,
there is no reason to single out the determinant of the first two
columns
for a special role). Clearly, two 6-tuples should be declared
equivalent if
the spaces of solutions of the corresponding systems have the same
dimension. Therefore, there are three equivalence classes, and the
stable
elements are those for which the matrix (fly) has rank two. We now
illustrate Definitions 1,2, and 3 with some examples.
EXAMPLE 1. Consider a linear programming problem in which it is
desired to maximize the linear function
p1x1+p2x2
subject to the constraints
a1x1 + a2x2 < a3
bix1 + b2x2 ^ b3
xx^0, x2^0.
Let us assume that the coefficients ai9 bt are fixed positive numbers,
and
that the 'prices' px are positive but are allowed to vary. Thus, we
actually
have a two-parameter family X of linear programming problems, para
metrized by the points in thepl9p2 plane for which/?! >0 and/?2 >0.
Let us also assume that
aib2 > a2b1, #A > #3^1 ? and b2a3 > a2b3,
so that the feasible set and the constraints look as in Figure 1.
240 HECTOR J. SUSSMANN
0^*02X2=03
TABLE I
Case Condition Solution
ai
?1<
?
P2 b2
p2 bz
?2 P2 a%
P2 02
?2 P2
?l
any point in the segment Q1Q2
?2
any point in the segment ?2?3
03
CATASTROPHE THEORY 241
To describe the solution, we must consider five possible cases, depend
ing on the values of the prices pt and/?2 (cf. Table I).
There is a clear choice of an equivalence relation R on X9 namely, we
declare two elements of X equivalent if they fall in the same case of
Table I. Thus, there are five equivalence classes, of which three are
open
and two are not. The unstable elements are those which belong to one
of
the lines
Pi_bi Pi a,
Pi b2
Now assume that the prices pl9p2 are changing in time, so that we have
a curve t-*(px(t), p2(t)) in X space. The catastrophe points are those
values of t for which (jPi(t), Pi(t)) belongs to one of the lines
described
or ? = ?
(cf. Figure 2)
Pi <*i
Fig. 2.
242 HECTOR J. SUSSMANN
above. As an example, let
Pi(0==Pi + ai> a>0
p2(t) ?p\
? constant
so that the price of commodity 1 is steadily increasing. To describe
the
solution (xi9 x2) of our linear programming problem as a function of/,
it
suffices to describe xl9 for it is clear that this uniquely determines
x2.
A
*1
b2Q3-Q2b3
020-1-02 b1
-.->t
t1 =5L( blP?2-b2p? ) t2 =
?-fa, p?2-a2p? )
Fig. 3.
The graph of xt as a 'function' of t is given in Figure 3. We see that
xx
is a 'good' function of / except at the catastrophe points where
(a) xt has a jump discontinuity, and
(b) xx is not well defined.
CATASTROPHE THEORY 243
EXAMPLE 2. Let ? and p be positive integers, and let Xn'p denote the
set of all real matrices with ? columns and p rows. If we represent
the
vectors in Un (and Up) as columns, then X is in an obvious way
identified
with the set of all linear maps from IRWin to Up. A reasonable
equivalence
relation R is the one that identifies two matrices if they correspond
to the
same transformation except for a linear change of coordinates.
Formally,
two matrices FxeX9F2eX are equivalent if there exist nonsingular ma
trices GeXn>n9 HeXp>p such that
F2
=
HFiG.
It is an easy exercise in linear algebra to prove that two matrices
are
equivalent if and only if they have the same rank. Thus, if we put
m=min
(?,/?), the space Xn,p is partitioned into m+1 equivalence classes X\p
(0^fc<m),where
Xnk'p
=
{M: MeX9 rank(M)
=
k}.
The condition that rank(Af)=m is equivalent to the assertions that the
determinant of some m x m submatrix of M be nonzero. Hence Xn?p
is an open subset of Xn,p. It is not very hard to show that the XI'p
are
submanifolds of Xn,p of codimension (n-k)(p-k). In particular, the
stable
elements ofXn,p are exactly the matrices of rank m. Thus, the set of
stable
elements is dense in Xn'p. Moreover, this set is clearly open.
Therefore,
stability is a generic property. The case ?=2, p = 3 of Example 2 is
basi
cally the one discussed in the remark following Definition 3. We now
describe the analogue of the example presented at the beginning of
this
section.
EXAMPLE 3. We let Xn>p be the sef of all ? xp real matrices, as
before,
but now we think of an AeXn*p as representing an inhomogeneous
system of? linear equations in/?-l unknowns. If AeX"tP9 write A =
(A9 d)9
where a is the last column of A. We declare (A, a) and (B9 b) to be
equiv
alent if there are invertible matrices GeXn'n9 HeXp~1'p~1 such that
S=HAG and b=Ha. It is easy to show that A =
(A9 a) and P=(P, b) are
equivalent if and only if rank A = rank B and rank A = rank B. As in
Example 2, the equivalence classes are submanifolds. The stable
elements
are those A for which rank A = a and rank A = ?, where (7=max (?,/?),
244 HECTOR J. SUSSMANN
<7=max (?,/?? 1). Therefore, stable elements are dense, and stability
is a
generic property.
EXAMPLE 4. Consider the set Xn,n of ? x ? real matrices. If we regard
each matrix in X"'n as the matrix of a linear map between two linear
spaces of dimension ?, then the equivalence relation introduced in the
previous example is obviously the right one. If, however, we regard
the
elements of Xn,n as linear maps of Rn into itself, it becomes
unnatural to
allow for different changes of coordinates in the domain and in the
range.
It is more appropriate to declare Mt and M2 to be equivalent if there
exists a nonsingular matrix P such that
M2=PM1P-1.
Here a new phenomenon appears. The equivalence classes modulo this
relation can no longer be described by means of a discrete family of
inva
riants, such as the five cases of Example 1, or the rank as in Example
2.
In fact, it is easy to show that continuous parameters appear and this
has
the consequence that no stable elements exist ar all. An example of
such a
continuous parameter is the trace of a matrix, i.e., the sum of its
diagonal
elements. It is well known that two matrices which are equivalent in
the
sense of the present discussion necessarily have the same trace. On
the
other hand, ifM eXn,n9 it is clear that we can find matrices M'
arbitrarily
close to M whose trace is not equal to the trace of M. Hence there are
matrices which are arbitrarily close to M but not equivalent to M.
There
fore M is not stable.
4. Genericity of maps and transversality
In Section 3 we discussed the general concept of stability, and we
illu
strated it with a number of examples. All our examples had one feature
in common: the space X was finite-dimensional. In those cases in which
there were sufficiently many stable elements, it turned out that the
set of
such elements was open and dense in X. We then agreed to describe such
a situation by saying that 'stability is a generic property'.
We now want to extend these considerations to some 'infinite dimen
sional' cases. As a simple example, suppose X is the set of all curves
in the
plane R2 (a curve in IR2 is a smooth mapping from the interval [0,1]
to R2).
CATASTROPHE THEORY 245
This leads naturally to two questions, namely, (a) how do we make X
into
a topological space? and (b) what shall we mean by a 'generic'
property?
The answer to (a) is provided by the C?? topology which, in our case,
is
easy to define. We say that a sequence {fn} of curves in R2 converges
tof
in the C00 sense if/? converges to/uniformly together with all the
deriva
tives (i.e., if we put fn(t)=(un(t)9 vn(t))9 and similarly for/, we
are re
quiring that for every e > 0 and every positive integer m there must
exist an
?o depending on e and m, such that
dmun dmu
dtm
v '
df
<?
for all /, and all ?>?0, and a similar inequality for the v's).
The definition of the C
??
topology given above can be extended easily to
the more general case when X is the set of all smooth maps from K to
P,
where K is a compact subset of a manifold N. This is just a matter of
repeating what was done above in the particular case K= [0,1], P=R2.
The only additional difficulty arises from the fact that AT (and P)
need not
be covered by one single coordinate chart. This can be remedied very
easily, and we shall omit the details. The final result is the
topology ofC
convergence on K.
Now let Cco(N9 P) denote the set of all smooth maps from N to P. How
do we topologize C??(iV,P)? When N itself is compact, the preceding
paragraph provides the answer. lfNis not compact, there are at least
two
'natural' topologies on C??(N9P)9 namely, the C00 topology and the
fine
C00 topology, also known as the Whitney topology. We shall define them
in the simple case when N=Rn9 P=R. The general construction is con
ceptually the same, except for the fact that one must choose
coordinates
appropriately.
The C00 topology is easier to define. We say that/,, ->/if/m converges
to
/on K in the C00 sense for every compact subset K of Rn. To define the
Whitney topology it is better to describe what we mean by a neigh
borhood of an/eC??((Rn, R). Suppose x->e(x) is a continuous function
on IRn, such that e(x)>0 for all x. Let k^O be an integer. We consider
the
set V(f;k9 e) of all ^eC??(Rn, R) such that, for all xeRn9 the
inequality
dai
+ ...+??
dxl'-dxl M-9){*) <e(x)
246 HECTOR J. SUSSMANN
holds for all x9 whenever otl-]-han^/:. We illustrate the difference
between the two topologies by an example: suppose/:(Rn-> R is a func
tion such that/(x)#0 for all x. Then there is a neighborhood V off
such
that every ge V vanishes nowhere (e.g. take V= V(f; 0,/)). So, in the
Whitney topology, if/never vanishes then a function which is
sufficiently
close to/cannot vanish anywhere. This is not the case for the C00
topol
ogy. Indeed, if we let <j>nb e smooth functions such that <?>n(x)=\
for
\x\^ ? and (?)n(x )=0 for \x\^ ? +1, then <j)nf -> / in the C??
topology. The
preceding considerations show that the Whitney topology is the most
natural one. From now on, we shall always consider Cco(N9P) as a
topological space with the Whitney topology.
We now must choose a definition of genericity. The choice which has
become unanimously accepted by the practitioners of this field is as
follows: we call a property &> of maps/eC??(JV, P) generic if the set
of
those/for which P holds contains a countable intersection of open
dense
subsets of C??(7V, P). The main motivation for this definition is that
it is
the one which enables us to prove nice theorems. If the reader does
not
find this satisfactory, the following justification may perhaps seem
prefer
able. Genericity of a set A of points of a space X should mean,
roughly,
that A is very 'thick' and its complement X?A very 'thin', so that
'prac
tically all points of X are in A\ One possible definition
of'thickness' is:
A is thick if it is open. Similarly, we might define B to be thin if
it contains
no open sets. With this choice, we arrive at the definition: A is
generic if it
is open and dense (i.e., if it is open and its complement does not
contain
any open set). However, we can be somewhat less demanding. Suppose
that the class G? of open dense subsets of X is such that, whenever
{Am} is a sequence of elements of 02 9 then the intersection
m
is still dense. If X has this property, we call it a Baire space. In
this case,
suppose {Am} is as above, and let Bm=X? Am. The Bm are clearly 'thin'.
If Zis a Baire space, the Bm are so 'thin' that their union B still
contains no
open sets. Now, the sets B obtained in this way still have the
property that
a countable union of such P's cannot contain any open set. We need
more
than a countable infinity of P's to fill a whole open set. It is
therefore
understandable why such B can still be called 'thin' and their
complements
CATASTROPHE THEORY 247
'thick'. Now, the complement of a B is a set A which is a countable
inter
section of open dense sets. Moreover, if such A9 s are going to be
called
generic then certainly any set which contaits one A should be called
generic as well. This yields the definition that was given above.
The main fact which makes such a definition reasonable for a space Zis
the property of being a Baire space. It is a classical result that the
real line
R is Baire (the category theorem). An example of a property of real
num
bers which is generic in our sense is irrationality (but not
rationality).
It is also true that the space Cco(N9 P) is Baire, and we will there
foreuse,
from now on, the concept of genericity in the very precise sense that
was
introduced above.
The basic facts about genericity are therefore :
(a) If ^ is a generic property then the set of/for which & holds is
dense, i.e., every/can be transformed into an/' for which & holds
by an arbitrarily small perturbation, and
(b) if ^1 and ^2 are generic properties, so is their conjunction 0>l a
&*2
(in fact, this is true for arbitrary finite and even countably
infinite
conjunctions).
We remark that (b) is a consequence of the 'thickness' requirement
that
was made a part of the definition of genericity. If we had allowed 0*
to be
called generic if only (a) holds (i.e., if {/: 0 holds for/} is
dense), then (b)
would not follow (e.g. every real number can be approximated by ratio
nal and by irrationals, but certainly not by numbers which are both
rational and irrational).
We conclude this section with an important example of a generic
property. This is the (simplest form of) the transversality theorem of
Thorn.
We first illustrate it with one example. Suppose two curves yt and y2
are
given in the plane, such as the ones in Figure 4. Then it is clear
that, even
if y? and y2 are tangent, we can perturb one of them (say y2) slightly
so as
to make them nontangent. This suggests that the set of curves y2 which
are not tangent to y ! is dense in the set of all curves. Moreover, it
is clear
that, if y2 is not tangent to yu then any curve which is close enough
to y2
is also not tangent to yt. Therefore, the property of a curve y2 of
not being
tangent to y x is generic.
The transversality theorem is basically an assertion like the
proceeding
one, but for the following more general situation: we have a
submanifold
W of P (such as yt in R2) and we consider smooth maps f:N-*P. We
248 HECTOR J. SUSSMANN
Before Perturbation After Perturbation
Fig. 4.
would like to calif not tangent to W(zt a point xeN such that/(x)e W)
if
no direction v tangent at/(x) to the image of/is also tangent to W.
This,
however, is too strong a requirement, for if the number of linearly
inde
pendent t;'s is k9 and if
k + d\mW>dimP
then the condition will never be satisfied (i.e. there is not 'enough
room'
in P to accommodate so many directions; as an example, consider two
surfaces in R3). We therefore replace the above condition by a weaker
one,
namely, transversality.
If xeNandf(x)e W9 we call/ transversal to Watx if the space A of all
directions tangent to the image at/(x) and the tangent space Wf(x) to
Watf(x) are 'as far apart as possible' (formally: A+ Wf(x)=Pf(x); the
definition of A is A =
{f*(v)9 veNx}).
We call/transversal to Wif it is transversal to W&tx for every x such
that f(x) e W. The transversality theorem of Thorn says that, for
arbitrary
N9P9 Was above, transversality to W is a generic property of maps f:N-
+P.
The reader will observe that, if d\mN+d\rnW<dimP9 and if xeN9
f(x)e W9 then it is impossible for/to be transversal to W at x.
Therefore
/will be transversal to Wif and only if the image of/does not meet W.
So,
in this case, the theorem asserts that every/can be deformed slightly
so
CATASTROPHE THEORY 249
that its image will not meet W. As an example, if the curves yl9 y2 of
Figure 4 are thought of as curves in ?R3, then it follows from Thorn's
theorem that y2 can be perturbed slightly so that it will not
intersect yx.
This is, of course, in perfect agreement with our intuition.
We conclude this section with three important statements.
FACT 1. Iff:N-*P is a smooth map, and if W is a submanifold of P
such that/is transversal to W9 then the set
f^(W)^{x:xeN9f(x)eW}
is a submanifold of N9 and
codim/-1(iF)
= codimJ^ if /-1(FF)#0.
(Note that, if dimJV + dimfl^P so that codimPF>dimiV, this gives a
negative dimension for/
" *
(W )9 so/
" *
(W )=0 as was said above.)
FACT 2. If f:N-*P is smooth and transversal to W9 and if xeN9
f(x)e Wthen for every/
'
N: -+P which is close to/there is an x' eN close
to x such that/'{x')eW.
FACT 3. If W is closed then the set of all/: N-*P which are
transversal
to Wis open in C00 (N9 P). When Wis not closed this need not be true.
5. Stable catastrophes
We can now clarify what is meant by a catastrophe to occur in a stable
way, in spite of the fact that a catastrophe point is, by definition,
an
unstable one. Let us return to Example 1 of Section 3. The catastrophe
set
is depicted in Figure 2, and it consists of two lines. Suppose that we
have
an element of X which depends on one parameter t (i.e., we have a
curve
y-*'-KPi(0> Pii*))in x- The set c of the t for which 0>i(0> pAO) is
unstable is the catastrophe set. If y is an arbitrary curve in X, then
we can
say very little about C (actually, it is easy to show that every
closed subset
of the real line can be obtained as the catastrophe set of some such
curve
y). If, however, we limit ourselves to a generic class of curves, we
can say
much more. First, by the transversality theorem, a generic y is
transverse
250 HECTOR j. sussmann
to the catastrophe set. Then, by Fact 1 of Section 4, we can conclude
that, for generic y, the catastrophe set C is a submanifold of the
real line
of codimension one. So C actually consists of a discrete set of
points.
Finally, if y is a curve which is transverse to the catastrophe set,
and
actually needs it, then Facts 2 and 3 of Section 4 imply that every
curve
y' in a neighborhood of y is also transverse and meets the catastrophe
set. Therefore, the catastrophes of y appear in a stable way.
The preceding remarks illustrate some aspects of the general
situation.
To get a still better picture, let us suppose that we had a situation
in
which X is two dimensional and the catastrophe set consists of the
union
of two intersecting lines. In this case, the preceding discussion
carries over.
Indeed, it suffices to remark that the point x0 where the lines cross
con
stitutes a zero-dimensional submanifold of X. Therefore, the
transversa
lity theorem implies that a generic curve y will not meet x0, and
then, as
long as we are only interested in the generic case, x0 can be ignored.
If we
now consider elements of X which depend on two parameters, a new
feature appears. We now have a smooth map y(tl912) from some region
of R2 into X. Generically, y will be transverse to both lines and to
the
manifold {x0} (observe that we are making essential use of Fact (b)
about
genericity, cf. Section 4). Therefore, we can assert that, for each of
the
two lines Lt (/=1, 2), the set of (tl912) for which y(tl9 t2)eLi is a
one
dimensional manifold. Also, those (tl912) for which y(tl9 t2)
= x0 form
a discrete set of points, but we can no longer assert that it is
empty.
The preceding examples suggest the following picture :
(i) The catastrophe set is a 'stratified' subset of X9 i.e., roughly,
a
finite (or, at least, locally finite) union of a disjoint family of
submanifolds
(the 'strata') such that the closure of each stratum S is the union of
S and
of strata of lower dimension. In our second example, the strata are
four
open half-lines and {x0}.
(ii) To every point x of Iwe can attach an integer, the codimension of
x9 with the property that codimx^O, with equality only when x is
stable.
In some sense, this integer measures how complicated a catastrophe x
is.
More precisely, codimx is the minimum number k of parameters which
are needed for x to be 'seen' (i.e., to appear stably as part of a ^-
param
eter family).
(iii) If x is a smooth family of elements of X depending on k para
meters t=(tl9...9 tk)9 then, for generic y9 the catastrophe set of y
will be
CATASTROPHE THEORY 251
stratified. The strata of codimension k will consist of points t for
which
y (t) is a catastrophe of codimension k.
In addition to these three properties, there is a fourth one which is
easily verified in Examples 1,2,3 of Section 3, namely
(iv) If xeX9 then the stratum of x is precisely the equivalence class
of x.
In the following sections we shall discuss how what has been said so
far
can be extended to the case when A"is the space C00 (TVP, ). This will
lead
us to study the stability of a map/:N-*P9 and then to consider
families
of maps depending smoothly on parameters. It will turn out that (ii)
requires an obvious modification, namely, that we will have to accept
points of 'infinite codimension'. It seems clear (though it has not
been
completely proved, as far as we know) that (i), (ii) and (iii) will
hold,
except for the modification of (ii) mentioned above. However, (iv)
will
definitely not hold. The reason for this is the appearance of families
of
equivalence classes depending on a continuous parameter. (Such
families
are called 'moduli'). The following trivial example gives a finite
dimension
al analogue.
EXAMPLE. X=R29 P=the relation whose equivalence classes are
the upper and lower half-planes, and the one-point subsets of the x
axis.
Here the catastrophe set is the x axis, which is stratified in an
obvious way
with one stratum of dimension one. However, the equivalence classes of
all the points in this stratum are zero dimensional.
6. Stability of smooth maps
We now give a very brief discussion of some aspects of the theory of
singularities of smooth maps of Thorn and Mather. The reader who is
not
familiar with the basic concepts of the theory should keep Example 2
of
Section 3 in mind. The concept of stability to be introduced here is
basi
cally a nonlinear analogue of this example (though, of course, it
leads to
much more interesting mathematics).
The basic object is the class C (N9 P) of all C00 mappings from N to
P. Here N and P are C00 manifolds of dimensions ? and p9 respectively.
We have already explained how to make CCC(N9P) into a topological
space. The second ingredient that is needed in order to talk about
stability
is an equivalence relation. The most natural one (since we are working
252 HECTOR J. SUSSMANN
with C?? maps) seems to be C00 equivalence, which is defined as
follows:
two maps/and g from N to P are C00 equivalent if there exist C00
diffeo
morphisms
such that
(5)
r-.N-^N and I: P-> P
f=igr.
Another possible relation is C? equivalence. The maps/and g are C?
equivalent if there are homeomorphisms r and / such that (5) holds.
Corresponding to these two equivalence relations we have the concepts
of C00 stability and C? stability. It is clear from the definition
that the
sets St^(N9 P), St0(N9 P) of C00 stable and of C0 stable maps are open
in
C ??
(N, P), and that Sr?, (#, P) s St0 (N9 P).
Fig. 5. The shaded region (boundary included) is the complement of the
nice range.
CATASTROPHE THEORY 253
By analogy with the discussion of the preceeding sections, we ask
whether C00 stability is a generic property or, if this is not the
case,
whether C? stability is generic. It turns out that C? stability is
generic for
arbitrary N and P, if N is compact. As for C00 stability, the
situation is
more complicated. The result is that whether or not St^N, P) is dense
in
C00 (N9 P) depends only on the dimensions ? andpofN and P. The set of
values of ? and p for which St^(N9 P) is dense in Cco(N9 P) has been
called the 'nice range' by Mather. It is described in Figure 5. As an
exam
ple, St^ (N9 P) is always dense in C^?N, P) if p^5. Also when/?=6,
except for ? = 8. As for the case n=p, C00 stability is generic if,
and only
if,?<8.
We illustrate the definition of stability by considering the three
func
tions
A(x)
= x29 f2(x)
= x39 f3(x) = x4r-x2
whose graphs are shown in Figure 6. We will show that/ is stable while
f2 and/3 are not.
Let U be the set of all functions g such that
(1) ^'(x)>lforalljc
(2) ?'/(-1)<0and
(3) ff'(l)><>.
It is clear that U is an open set in C??(R, R), and that/e 17. If g
eU9
then the derivative g'{x) is strictly increasing and it has a unique
zero x0.
Moreover,
? 1 < x0 < 1. It follows easily that g is decreasing for x<x0 and
increasing for x>x0. Moreover Jim^.^ g(x)=limx^ + a0 g{x)= + oo.
Thus, the graph of g has the same qualitative properties as the graph
of/.
We shall use this to prove that g is equivalent to/ First, g is C00
equiva
lent to g9 where
g{x)
= g{x + x0)-g{x0)
(indeed, g =
l?g ?r, where r(x)=x+x09 l{y)=y?g{x?)) so we can assume
that x0=0 and that #(0)=0. Second, since g(x)^0 we can define a
function r(x) by r(x)2
=
g(x)9 r(x)^0 for x<0 and r(x)^0 for x^O.
Then g(x)=(fior)(x) so that we will have proved that/and g are C00
equivalent if we show that r is a C00 diffeomorphism. Since #(*)>()
except for x=09 it is clear that r is C00 for x^O. Moreover, since
#(0)
=
=0'(O)=O, 0*(O)>O, we can write g(x)=x2k(x) where A(0)>0. Near
254 HECTOR J. SUSSMANN
-v2
y=x
-> (a) stable
y=xJ
/
^ ^aU
N_^
/
^-s* (b) unstable
/y
A
/
-/?> (c) unstable
Fig. 6.
CATASTROPHE THEORY 255
x=0 the function h(x) has a unique positive square root k(x)9 which is
C00. Then r(x)=xk(x)9 so r is C00. To show that r is a diffeomorphism
we
must prove that r
'
(x) > 0 for all x. If x # 0, then [ r (*)] 2=g(x)so2r(x) r'
(x)
= #
'
(*). Since 0'(x)^O, it follows thatr '(x)#0. Also r'(0)=?(0)^0,
where k is the function defined above. So r is indeed a C00
diffeomorphism,
and the proof that/ is stable is complete.
To show that/2 is not stable, it is better to look at Figure 6(b). The
function f2 is one-to-one (though not a diffeomorphism) and therefore
every function which is equivalent to/2 is also one-to-one. On the
other
hand, there are functions which are arbitrarily close to f2 but are
not
one-to-one, such as the one whose graph is the dotted line of Figure
6(b).
So in every neighborhood of/2 there are functions not equivalent to/2.
Therefore/2 is not stable.
The proof that/3 is not stable follows the same pattern. We must find
a
property of/3 which every function equivalent to/3 must also have, but
which can be destroyed by a small perturbation. In this case, the
desired
property is easy to find:/3 has two local minima and it takes the same
value at both of them.
There is a useful criterion for determining whether a map/eC??(JV,P)
is C?? stable. Assume that N is compact (or that N is arbitrary but /
is
proper). Then fis stable if and only if it is 'infinitesimally
stable*. To define
infinitesimal stability we must introduce some notation. A map which
to
every point x of N assigns a tangent vector to P at/(x) is called a
vector
field along f. Use 0(f) to denote the set of all smooth vector fields
along
/ In particular, the vector fields on N are the vector fields along
the identity
map iN of N. We use an obvious abuse of notation and write 6(N) in
stead of 6(iN). Every YeO(P) gives rise to a feO(f) in an obvious way,
namely, we define Y(x)
=
Y(f(x)). Also, every XeO(N) gives rise to an
XeO(f) by: %(x)
=
df(X(x)). Finally, we define/to be infinitesimally
stable if every vector field along/is of the form %+ Y for some
Xe6(N)9
YeO(P). Equivalently, use tf to denote the map X-+?9 and co/for the
map F-* Y. Then/is infinitesimally stable if and only if
0(f)
=
tf6(N) + fe(P).
The meaning of the infinitesimal stability criterion can be understood
by means of the following heuristic argument. Consider feCco(N9 P).
256 HECTOR J. SUSSMANN
Suppose t-+ft is a smooth curve in C??(iV, P) such that/0=/. Then, if
we
put
X(x)
=
-ft(x)
it follows that X(x) is a tangent vector at/(x), so that XeQ(f).
Moreover,
every XeO(f) clearly arises in this fashion. So 6(f) can be thought of
as
the set of all 'directions' in which/can be deformed. Now suppose that
{/J is a smooth family of diffeomorphisms of P. Then t -
/f/is a curve in
Cco(N9 P) of a particular kind because, at each time t9 IJis C??
equivalent
to/ The 'direction' of such a curve at/is given by
X(x) =
?lt(f(x))
= U(f(x))9
where U(y)=(dldt)?lt(y). So *7e0(P) and Xeco/0(P). A similar reson
ing shows that the elements of tfO(N) can also be interpreted as
'direc
tions' in which / can be changed while keeping it equivalent to
itself.
Therefore, the infinitesimal stability condition simply means that
every infinitesimal change in/can be achieved so as to keep/equivalent
to itself. The reason why this is equivalent to stability is now
clear.
We now illustrate the infinitesimal stability criterion by using it to
characterize the stable proper real-valued functions on the real line.
Let
/: R -> R be C00 and proper (i.e. lim^i-^ \f(x)\
=
oo). A vector field along
/is simply a C00 function a: R -> R (but we think of a as assigning to
each
xeR the vector a(x)(d/dx) at/(x)). Similarly, the vector fields on R
are
the smooth functions R -> R. The maps tf9 ca/are given by
tf(b)(x)=f'(x)b(x)
cof(c)(x)
=
c(f(x)).
Thus/is infinitesimally stable if and only if every smooth function
a(x)
can be expressed as f '(x)-b(x) + c(f(x)) for suitable smooth
functions
b and c. From this we can derive a first necessary condition for /to
be
stable. Let x0 be a critical point, i.e. a point such that/'(;to)=0.
Choose
the function a so that a'(xo)^0. If/is stable, it is infinitesimally
stable
and therefore there are functions b and c such that
(6) a(x)=f'(x)b(x) + c(f(x)).
If we differentiate both sides and evaluate at x=x0 we get
a'(x0)=f"(x0)b(x0).
CATASTROPHE THEORY 257
Therefore /"(*o)t*0- A critical point for which f,f(xo)?:0 is called
nondegenerate. We have shown that if/is stable then all the critical
points
of/are nondegenerate.
A second necessary condition can be obtained by comparing the critical
values v1=f(xl)9 v2=f(x2) for two different critical points xx and x2.
Choose the function a so that a(x1)^a(x2). If/is stable we can find b
and
c so that (6) holds. Evaluation at xx and x2 gives
a(xx)
=
c(vi)9 a(x2)
=
c(v2)
so that c(vt) t?c(v2) and therefore vx^v2.
Summarizing, we have shown that, if/is stable, then
(i) all the critical points of/are nondegenerate and
(ii) the critical values for different critical points are different.
Conversely, it is easy to show that conditions (i) and (ii) are
sufficient
for/to be stable. Thus we have proved that
A C00function f: R -> R is stable if and only if:
(i) wheneverf
'
(x0)
= 0 thenf" (x0)^0 and
(ii) wheneverf
'
(x^ =/
'
(x2)=0 andx? # x29 thenf(x1) ?> f(x2).
7. Families of functions and unfoldings
We now restrict ourselves to real valued functions, so that from now
on
P is the real line. It follows that we are in the nice range of
dimensions,
and therefore the C?? stable functions are dense. In fact, it is very
easy to
give a complete characterization of the C00 stable functions. As in
the case
of real functions on the real line (discussed in Section 6) one can
prove
that a proper C
??
function f: N -> R is C
??
stable if and only if
(i) All the critical points of/ are nondegenerate (cf. definition
below)
and
(ii) for any two different critical points, the values of/are
different.
The definition of a nondegenerate critical point is the natural
generali
zation to several variables of the one-dimensional case discussed in
Section 6. We say that/has a critical point at/7 if, in coordinates,
dxtK
'
dxn
Kyj
258 HECTOR J. SUSSMANN
If/? is a critical point of/, we say that/7 is nondegenerate if the
matrix
of the second derivatives (d2f/dXjdXi) (p) has a nonzero determinant.
Functions which satisfy condition (i) above are called Morse
functions.
The fact that Morse functions that satisfy (ii) are dense in C??(N, R)
is a
well known easy consequence of Thorn's transversality theorem.
We now ask whether the general considerations of Section 5, which
were based on our analysis of some finite-dimensional cases, carry
over to
the space C (N9 R). We have already indentified an equivalence
relation
(namely, C00 equivalence) with respect to which the stable elements
form
a dense set. We would like to show that the set of unstable functions
(or,
more precisely, a very large subset of it) can be divided into
'submani
folds' of 'finite codimension'. It turns out that this can be done,
but that
some technical complications occur. It is possible to give a
definition of
what is meant for a function to be of 'finite condimension' and we can
actually define the codimension of such a function. In a sense which
can
be made very precise, the codimension of/is the codimension of the
equivalence class of/with respect to C00 equivalence. One would then
expect that, if {Fx} is a family of functions from N to R depending on
k
parameters (xl9...9xk) then, generically, only functions of
codimension
<& will occur and that, moreover, the catastrophe set (i.e. the set of
values of x for which Fx is not stable) will be stratified as a union
of sub
manifolds Cj of codimension j(Kj<?:), where Cj={x: codimension
(?y=j}.
Actually, these expectations are not totally fulfilled. The reason is
that
there occur 'moduli', i.e. families of equivalence classes of
functions of a
given codimension which depend on a number of continuous parameters,
in a way which is analogous in principle (but, of course, much more
com
plicated) to the example of Section 5. However, moduli do not occur in
low dimensions, and for these cases reality agrees with the intuition
that
can be derived from the discussion of Section 5.
For reasons that will become apparent later, the dimensions k^4 are
the most relevant to.Thorn's theory. And moduli do not occur in this
range.
We now explain how to define the codimension of a function/eC00
(N9 R). We recall from Section 6 the notations 6(f), 6(N), 6(R), tf
and
cofi The set 6(f) of all vector fields along/is identified, as we
observed
before, with the set of all C00 functions N-+ R. Since stability is
equivalent
to the equality 6(f)
=
tf[6(N)'] + cof[d(R)'], it seems reasonable to
CATASTROPHE THEORY 259
measure the deviation from stability by finding out how much larger
6(f), is than its subset i/[0(tf)] + a>/[0(R)]. Now 6(f), 6(N) and
0(R)
are linear spaces over the reals. Moreover, tfa.nd cof are linear maps
from
6(N), 0(R) into 6(f). Even though all these spaces are infinite
dimension
al, it may very well happen that the quotient
v/
tf(d(N)) + cof(6(U))
is finite dimensional. We adopt this condition as our definition. A
func
tion/is said to be offinite codimension if the vector space A(f) is
finite
dimensional. The dimension of A(f) is then, by definition, the codimen
sion off. Clearly, we have codim (/)=0 if and only if/is stable.
The motivation for the above definition is the same as in our dis
cussion of infinitesimal stability in Section 6. If X is finite
dimensional,
W a submanifold of X, and x0e W9 then the tangent space WXo is the set
of all directions of infinitesimal changes that keep x0 in W. By
analogy
we must think of tf6(N)+cofd(R) as the 'tangent space' at/to the equi
valence class of/, which is a 'submanifold' of Cco(N9 R). Also, 6(f)
is
the 'tangent space' to C??(N9 R) at/ In the finite dimensional case,
codim W=dimXxJWXo. Our definition of codim (/) is therefore the
natural infinite dimensional analogue.
We illustrate this definition by computing the codimension of some
unstable functions. First, consider/(x)=x*?2x2 (which was shown to be
unstable in Section 6). It is easy to see that / has three critical
points,
namely ?1, 0 and 1. Moreover, these critical points are nondegenerate.
The source of the instability of/is the fact that/(?1)=/(1). Thus, we
expect the codimension of/to be one, because '/has a simple degene
racy'. To prove this rigorously, we observe that the proof given in
Section
6 that (i) and (ii) imply infinitesimal stability actually shows that,
if all
the critical points of/are non-degenerate, then a function aeC??(R, R)
is in C/T0(R)+] co/[0(R)] if and only if (?(xj^ocfa) for every pair of
critical points such that f(x1)=f(x2). Thus, in our case, tf[?(R)~] +
co/[0(R)] is simply the set of all C00 functions a : R -> R such that
a( -1)
=
=
a(l). This is a subspace of 6(f) (i.e. of C??(R, R)) of codimension
one.
Thus codim (J)= 1 as asserted.
Now consider a second example. Letf(x)=xn9 where ? is an integer.
If ?=1 or ?=2 then/is stable. Suppose ?>2. Suppose oced(f) belongs
260 HECTOR J. SUSSMANN
to i/[0(R)] + cof [0(R)]. This means that a is of the form
*(x) =
d?(x)?(x)
+ y(f(x))
i.e.,
oc(x)
=
nxn~1?(x)+y(xn)
for suitable functions ?9 y. Now y(t)
=
y(0) + 0(t) so y(xn)
=
y(0)+0(xn).
From this we get a=constant+0(jcn_1) so that
(7)
?(?)==0
for i = l,.-??-2
Conversely, every function a which satisfies (7) can be expressed as
a(x)=?i(0)+xn~1?(x)9 from which it follows easily that a is in //
[0(R)] +
a>/[0(R)]. Therefore this set is exactly the set of a's for which (7)
holds.
It follows that codim(/)=??2.
The reader will have no difficulty in showing that a function which
has
a critical point of infinite order (i.e., a point where all the
derivatives of/
vanish) has infinite codimension.
We now give the general rule for determining the codimension of an
arbitrary function/e C00 (N9 R). First, to each critical point a off
assign a
codimension c(f9 a) as follows: choose coordinates (xl9...9 xn) near a
and
let c(f9 a) denote the minimum integer k such that there are smooth
func
tions gl9...9 gk with the property that every smooth function g can be
expressed, in a neighborhood of a9 as
g(x)
= a1g1 (x) + ? + cckgk(x+) ?t (x)-~ (x) + ?+ ?n(x)?- (x),
OX? oxn
for some constants al5..., <xk and smooth functions ?l9...9?n. (As
an
example, let us compute c(f9 a) when N=R9f(x)=xn9 a=0. If g is any
smooth function, then near 0 we can write (Taylor)
9(x)
=
9(0) + g'(0)x+... +
9?^xn-2
+ ?(x)nxn-1
which shows that the ?? 1 functions 1, x9..., xn-.x can serve as our
gt. It
is easy to show that no fewer than ?-1 functions will do. Therefore
c(/, *)=?-!.)
CATASTROPHE THEORY 261
Second, we assign a dimension d(f, a, b) to each pair a, b consisting
of
a critical point a and a number b such that/(a)=6. We let d(f9 a9 b)
be
the smallest integer/ such that the function (f(x)?b)j is of the form
ftM^(.)+...+??|w
near a. (For instance, iff(x)=xn9 a=b = 09 ?>0, then/(x)=(*/?)(3//
d;*;)
so</(/, 0, 0)
= 1, if?> 1. If? = 1, then bf/dx= 1, so d(f 0, 0)=1.)
Third, we assign a dimension to each critical value b by
d(f9 b)
=
sup{</(/, a, b):aeN9f(a)
=
?}.
Finally, we let
(9) </>?c(/, a)-?</(/,?)
the sums being over all critical points and all critical values,
respectively.
EXAMPLES
A. If N= R9f(x)=xn9 so that a=0 is the only critical point, we have
seen
that c(f9 0)=?-1 and that d(f 0, 0)= 1 if ?> 1, d(f, 0, 0)=0 if ? = 1.
Therefore c(/)=0if?= lor 2, and c(/)=?-2 for ?>2.
B. Suppose/: JV-? R is a Morse function (possibly with repeated
critical
values). If a is a critical point, then the functions ij/^df/dXi form
a
coordinate chart near a (because det (dij/i/oxj) # 0). It follows that
every
smooth function g near a can be written, near a9 as
g(x)
= oc+ g1 (x)? (x)+- + gn(x)? (x).
ox1 oxn
Therefore A:< 1. By taking a function g such that g(a)^09 we see that
k cannot be zero. Therefore c(fa)
= l. If we let b=f(a)9 we then have
i=l CXi
near x (Taylor). On the other hand (f(x)?b)?== 1 which is not of the
form Yj ?i(df/dXi) because the df?dxt vanish at a. Therefore d(f9 a9b)
= L
It follows that
c(f)
= number of critical points
? number of critical values.
262 HECTOR J. SUSSMANN
For instance, if/ has 8 critical points, 3 of them with a critical
value
b0, and 5 with a value bl9 b^b0, then c(/)
= 6.
C. Let N=R2,f(x,y)=x3+y3. There is only one critical point, namely
(0, 0) with critical value 0. An arbitrary C00 function g defined near
the
origin can be written as
? ayxV+ ? htj(x9y)xy
i+j^2 ?+7=3
for suitable constants aiy- and smooth functions hti (Taylor's
formula).
Now
? = 3x and ? =
3y .
ox dy
Therefore, all the terms Ay*V are of the form ?(x9 y)(df/dz)(x9 y) for
z=x or z=y. This is also true for the terms cc20x2 and oc02y29 so that
we have shown that every g can be written, near (0, 0), as
g (x9 y)
= a00 + a10x + a0iy + oc^xy +
+ ?i (x, y) jx (*. y) + ?i (x9 y)
?
(x9 y).
Therefore the four functions \9x9y and xy can serve as the gl9..., gk
of
the definition of c(f9 a). We show that k cannot be less than four.
Indeed, suppose that three functions gl9 gl9 g3 were enough to express
every g as
9 =
K9i + X2g2 + X3g3 + ?xx2 + ?2 y2.
One then sees easily that
9 (0)
= X?gt (0) + X2g2 (0) + X3g3 (0)
g(?)
=
^(o)
+
^(?)
+
^(0)
and similarly for dg/dy and d2g/dxdy. Therefore the row vector
4<>>IN
is obtained from the row vector X =
(Xi, X2, X3) by multiplication on the
right by a fixed 3x4 matrix M. By simple linear algebra, not every e
CATASTROPHE THEORY 263
can be of the form XM. But this contradicts the fact that for every
vector e with four entries we can find a g such that e=e0. Therefore
c(/,(0,0))=4.
The calculation of d(f, (0, 0), 0) is easy. Indeed,/itself is
expressible
as ?i(dfldx) + ?2(dfldy)9vfhi\c the function 1 is not. Therefore d(f;
(0,0),0)=1. Finally, the codimension of /is c(/)
=
c(/,(0,0))
-</(/; (0,0),0) so that c(/)
= 3.
Having defined the codimension of a function, we can now study
families of functions and unfoldings. A family of functions from N to
R
with parameters in M is, simply, a family {ft:teM} where, for each
t9ft
is a smooth map N-+ R. Moreover, we assume that/ depends smoothly
on t9 i.e. that the map
(t9x)->ft(x)
from M x N to R is smooth.
As for maps, we can study the stability of families of functions. We
first define equivalence. Two families {/,'}, {/} of maps iV->R, para
metrized by M, will be called equivalent if there exist a
diffeomorphism
g\M-*M and families {rt}, {/,} of diffeomorphisms N-+N, R-?R
respectively, depending smoothly on teM, such that
f?(t)
=
ltf<rt for all teM.
Using this equivalence relation we can define the concept of stability
of
a family of functions as in Section 3. It turns out that stability is
a generic
property, at least if dim AT <4. For higher dimensions the theory
becomes
more complicated, because of the presence of moduli, as explained
above.
Now, if F= {ft:teM} is a family of functions N-> R, we can consider
the catastrophe set of F9 which we denote by C(F). The definition is
ob
tained by particularizing Definition 3 of Section 3. Therefore
C(F)
=
{t :ft is not stable}.
When F is stable, a lot more can be said about the structure of C(F)
Suppose that t0eC(F), and let/=/0. Then /is an unstable element of
Cco(N, R) whose codimension we shall call k. Thinking by analogy with
the finite-dimensional discussion of Section 5, we let Ef denote the
equiv
alence class off modulo C00 equivalence, so that we must think of Ef
as
a 'submanifold' of Cco(N, R) of codimension k. Then Pis a smooth map
264 HECTOR J. SUSSMANN
from the parameter space to Cco(N9 R), which takes the value /when
/=t0. The stability of F implies that Pis a versal deformation off.
Precisely
we define a versal deformation of an/of finite codimension k to be a
family P={/} of functions such that ft0=f and that the map ?->/f is
transversal to Ef at t0. The precise meaning of transversality in this
infinite-dimensional situation is the same as in the finite-
dimensional
case. Recall the notations 0(/), tfd(N)9 cofd(P) introduced in Section
6.
Recall also that, when dealing with functions, 6(f) can be thought of
simply as Cco(N9 R) itself, and that tf6(N) and cofd(P) also have con
crete interpretations. Since 6(f) is 'the set of all directions
tangent to C00
(N9 R) at/', while tfl(N) + cof6(P) is 'the set of those directions
tangent
to P'/5 the transversality of F to Ef should be expressed by the fact
that
(10) 6(f) =
tfd(N) + f6(P) + Bt0(F)9
where Bt0(F) is the linear hull of the derivatives
dt1' 'dtm { \}*? *?
evaluated at t0 (which are therefore functions from N to U, i.e.
elements
of?(/)).
We now forget about the preceding heuristic considerations, and take
(10) to be the definition of transversality.
It is clear that, in a versal deformation off the number of parameters
(called the dimension of the deformation) cannot be smaller than the
codimension off. Equivalently, if we consider stable families F
depending
on k parameters t1,..., tk then, locally, the catastrophe set C(F)
will be
the catastrophe set of a versal deformation of a function of
codimension
^k.
To obtain versal deformations it is convenient to consider universal
unfoldings. A universal unfolding off is a versal deformation off
having
the smallest possible dimension. It then follows easily that this
dimension
is precisely the codimension off One can get all versal deformations
off
by simply starting with a universal unfolding, and then throwing in
new
parameters and making a nonlinear change of coordinates in parameter
space. That is, we let P0(*\..., tk9 x)(xeN) be a universal unfolding
of
/, and then we define Fi(t19...9tl9x)(l>k) arbitrarily, with the only
CATASTROPHE THEORY 265
restriction that
F?(t?9...9tk909...909x)=EFo(t?9 ...9t\x).
We then put
F(t9x)
=
Ft(g(t)9x)
where t=(t19..., tl) and g is a C00 diffeomorphism.
8. Th?m's list of elementary catastrophes
In Section 7 we explained how to obtain the catastrophe set of an arbi
trary stable family F of functions depending on k parameters by taking
universal unfoldings of functions of codimension ^k. If we actually
want to describe all possible catastrophe sets which may appear in
this
fashion, it seems as though we would have to consider infinitely many
cases, because we might have functions of codimension <fc on mani
folds N of arbitrary large dimension. It turns out, however, that (for
fc<4), we actually get af, inite list, because every universal
unfolding of
anfiN^R of codimension ^k can be 'reduced', independently of what
N might be, to one of a small number of 'canonical forms'. The precise
meaning of this reduction need not concern us, except for the fact
that it
is an operation which does not change the catastrophe set.
Following Thorn, we shall give a complete description of the
catastrophe
sets which arise from stable unfoldings of functions of codimension <4
with one critical point. It turns out that there are exactly seven
possible
catastrophe sets, obtained from seven functions of codimension <4. The
complete list (together with their picturesque names) is given in
Table II.
TABLE II
Name Function Codimension Universal unfolding
fold
cusp
swallow tail
butterfly
wave crest
hair
mushroom
x3+y3
x2?xy2
x2y+y*
x3+ux
xA+ux2jt-vx
x5+ux3+vx2 -+w- x
JC+6 ux4 + vx3 + wx2 + tx
x3 +y3+uxy+vx+wy
xs?xy2+u (x2 +y2)+vx+wy
x2y+y4+ux2+vy2+wx+ty
266 HECTOR J. SUSSMANN
The letters u, v, w, t are the unfolding parameters. Using the
considera
tions of Section 7 it is easy to justify the entries of Table II
(except for the
names). The codimension of the first five functions that appear in the
table has already been computed in the examples of Section 7, and the
reader will find no difficulty computing the other two. To derive a
univer
sal unfolding one then proceeds as follows: suppose/has codimension k.
Recall that k is then the dimension of the quotient A (/)
=
6(f)/[t?(N)+
+ co/0(P)], and that 6(f) can be thought of as the space of all C00
functions in N. Choose functions gl9...9gk with the property that
their
equivalence classes modulo tf6(N) + cof6(P) form a basis of A(f). Then
define, for xe N9
k
F(tl9...9tk9x)=f(x)+ ? tigi(x).
i=l
It follows from the definition given in Section 7 that F is a
universal un
folding of/ Using this method, the last column of Table II is easily
obtained.
Using Thorn's list, it is not hard to study the unfoldings and
catastrophe
sets which arise from more general functions. If a function /has
several
critical points, formula (9) of Section 7 enables us to calculate its
codi
mension. The result is a sum of contributions of the individual
critical
points, except for the fact that, whenever we have several critical
points
al9...,ak with the same critical value we only subtract the largest of
the
numbers d(f, ai9 b)9 instead of subtracting their sum. Equivalently,
the
codimension of /is the sum of the codimensions of the individual
critical
points plus, for each critical value b9 an integer k(b)^0, equal to ?
d(f,
ai9 b)
?
sup d(fi ai9 b). It follows that a function of codimension <fc will
have no more than k degenerate critical points, and that their codimen
sions will have to add up to at most k. We can describe all the
possibilities
by using letters to name the various types of degenerate critical
points,
and the letter N for nondegenerate ones. We then group together all
the
points with the same critical value. For instance (NNBW)(NBM) denotes
a function/having four degenerate critical points, namely two
butterflies,
one wave crest and one mushroom and, in addition, at least three non
degenerate critical points. Moreover, two of these three points,
together
with the wave crest and one of the butterflies have the same critical
value
bl9 whereas the other nondegenerate point, together with the mushroom
CATASTROPHE THEORY 267
and the other butterfly, also have a common critical value b2, such
that
b2^b1 (/may have other nondegenerate critical points, but as long as
the
corresponding critical values are different from each other and from
bt
and b2, their contribution to the codimension of /is zero). One sees
easily that, for all the critical points N, B9 W9 M the number d(f9 a9
b) is
one. Therefore k (?>i)=3 and k(b2)=2. The codimension off is then 20.
As an example, we describe all possibilities with codimension <2.
There are exactly six of them, namely (F), (AW), (C), (F) (F), (NNN),
(NF). The first two have codimension one, and the last four have codi
mension two. For a generic family F of functions depending on two
parameters (so that the parameter space is a region of the plane) the
catastrophe set C(F) will consist of a number of curves (corresponding
to
those t for which Ft has codimension one) and points (where C(P,)=2).
Moreover, one sees easily that the closure of each curve must consist
entirely of unstable points and therefore it will be the union of the
curve
itself plus some points of codimension 2.
9. AN EXAMPLE AND CONCLUDING REMARKS
We return to our utility maximizer of Section 3. The function u(x9y)
which he tries to maximize depends on xeQ and yeP9 but our individual
only controls x. This determines a 'function' x(y) by formula (3) of
Section 3. If the parameter y is fc-dimensional, with fc^4, we know
that,
generically, u will be stable. Therefore, for all y in an open dense
set of
P the function uy(x)
=
u(x9 y) will be stable, and we will have a stratifica
tion of the catastrophe set by manifolds of various codimensions lt of
points for which codim (uy)
=
l?. It is not hard to prove that, as long as y
is not in the catastrophe set, the point where u reaches its maximum
(if it
exists) is unique, and that the mapping y ->
x(y) is smooth. Therefore the
catastrophe set of u represents the points where a drastic change in
behav
ior takes place, and Section 8 gives us some information about this
set.
We emphasize one remarkable aspect of this information, namely, its
independence on any detailed information on u9 including the number of
x-variables involved.
The reader will certainly want to argue that, on the other hand, this
information is exceedingly poor. For instance, we are not able to
predict
which values of y will give catastrophic changes. The answer to such
an
268 HECTOR J. SUSSMANN
objection is that, indeed, we could not possibly have expected to get
such
prediction without feeding in as assumptions some information about u.
What we have done here is, as was said in Section 1, no closer to the
real
world than the general theory of ordinary differential equations. To
actually apply the theory it will become necessary to bring in
additional
structure, in the same way as, in Mechanics, one assumes a particular
form of the differential equations. The search for that additional
structure
is a task for the applied mathematician and the natural and social
scien
tist. It is perhaps too soon to form an opinion on the extent to which
such
a task may succeed in the future. As for the extent of its success so
far, the
reader must make this evaluation by himself, by reading the work of
those
who have applied the theory (e.g. Thorn and Zeeman). Whatever the
result of this evaluation might be, this paper will have served a
purpose,
we hope, by helping the reader to make an unbiased judgment based on
an understanding of what the theory really is (or is not).
10. Appendix
For an introduction to the theory of manifolds, the reader may consult
[IDM]. For the theory of singularities of smooth maps, the book [SMS]
provides an accessible presentation. The classic works in this theory
are
(a) Whitney's pioneering paper [22], (b) the papers of Thorn, Mather
and Arnol'd, especially [14], [15], [16], [19], [10] and [1], (c)
Board
man's work [4].
The definition of 'catastrophes' in the general context of a space
with
an equivalence relation appears, for instance, in [6]. Proofs of
Thorn's
transversality theorem can be found in many places, such as [SMS],
[TMF] or [15]. The study of C00 stability and its relation with
infinites
imal stability was undertaken by Mather in [9]. These results require
the
use of tools such as Malgrange's division theorem (cf. Mather's papers
or
[AD]). There are analogues of infinitesimal stability for other
stability
concepts, such as families of functions, or unfoldings (cf. [8] and
[SU]).
The heuristic justification given in the text (i.e. the idea of
viewing 6(f)
as the 'tangent space' to C (N9P) at/) should be translatable into a
formal proof. The tool for such a proof would be a reasonable implicit
function theorem for infinite dimensional non-Banach spaces. Two such
theorems are given in [12] and [13]. The theorem that C? stability is
CATASTROPHE THEORY 269
generic was also proved by Mather (cf. [10]). Condensed presentations
of this proof appear in [11] and [5], For families of functions and un
foldings, see [SU] and [6]. The definition of the codimension of a
func
tion as given here is taken from Sergeraert's paper [13]. An
illustration of
the difficulties that appear when attempting to stratify the space of
func
tions can be found in [7]. Thorn's list of the seven catastrophes is,
of
course, due to Thorn. For a recent, detailed presentation with proofs,
see
[SU].
There is a broad area which has not been touched upon in this paper,
but is closely related to it, namely, the study of generic properties
of
dynamical systems (cf. [TMF] and the papers by Smale, Peixoto, Soto
mayor, etc. in [DS]).
Possible applications of the theory are discussed by Thorn in [SSM],
[18], [19] and [20] and by Zeeman in, e.g., [23], (cf. also Baas [3]).
Rutgers University
NOTE
* Work partially supported by NSF Grant No. GP-37488.
BIBLIOGRAPHY
(i) Books
[SMS] M. Golubitsky and V. Guillemin, Stable Maps and Their
Singularities,
Springer-Verlag, New York, 1974.
[DS] M. M. Peixoto (ed.), Dynamical Systems, Proceedings of a
symposium held
Salvador, Brazil; Academic Press, New York and London, 1973.
[PSSLI] Proceedings of the Liverpool Singularities Symposium I, 1971
(Springer
Lecture Notes No. 192).
[MA1970] Manifolds-Amsterdam 1970, Springer Lecture Notes No. 197.
[SU] G. Wasserman, Stability of Unfoldings, Springer Lecture Notes No.
393.
[AD] V. Poenaru, Analyse Diff?rentielle, Springer Lecture Notes No.
371.
[TMF) R. Abraham and J. Robbin, Transversal Mappings and Flows,
Benjamin,
New York 1967.
[IDM] S. Lang, Introduction to Differentiable Manifolds, Interscience,
New York,
1962.
[SSM] R. Thorn, Stabilit? structurelle et morphog?n?se, Benjamin,
1972.
(ii) Articles (Initials such as DS or PSSLI refer to the books listed
in (i).)
[1] V. I. Arnol'd, 'Singularities of Smooth Maps', Uspehi Mat. Nauk 23
(1968), 3-44,
translated in Russian Math. Surveys (1969), pp. 1-43.
[2] V. I. Arnol'd, (a) 'Integrals of Rapidly Oscillating Functions and
Singularities of
Projections of Lagrangian Manifolds', and (b) 'Normal Forms for
Functions Near
Degenerate Critical Points, the Weyl Groups of Ak, Dkt Ek, and
Lagrangian Sin
gularities', Funktional*nyi Analiz i Ego Prilozheniya (1972) 6; (a) in
No. 3, pp.
270 HECTOR J. SUSSMANN
61-62 and (b) in No. 4, pp. 3-25. Translation in Funct. Anal, and
Applications
(1973), pp. 222-223 and 254-272.
[3] N. A. Baas, 'On the models of Thorn in Biology and Morphogenesis',
Math. Biosci.
17(1973), 173-187.
[4] J. M. Boardman, 'Singularities of Differentiable Mappings', Publ.
I.H.E.S., No. 33
(1967), pp. 21-57.
[5] A. Chenciner, 'Travaux de Mather sur la stabilit? topologique', S?
m. Bourbaki,
expos? 424, Vol. 1972/1973, Springer Lecture Notes No. 382.
[6] J. Guckenheimer, 'Catastrophes and Partial Differential
Equations', Ann. Inst.
Fourier Grenoble 23,2(1973), 31-59.
[7] H. Hendriks, 'La stratification "naturelle" de l'espace des
fonctions diff?rentiables
reelles n'est pas la bonne', C.R. Acad. Sei. Paris 274, s?rie A(1972),
pp. 618-620.
[8] F. Latour, 'Stabilit? des champs d'applications diff?rentiables;
generalization
d'un th?or?me de J. Mather', C. R. Acad. Sei. Paris 268, s?rie
A(1969), pp. 1331
1334.
[9] J. N. Mather, 'Stability of C00 Mappings Y, Ann. of Math. 87
(1968), pp. 89-104;
II ibid. 89 (1969), pp. 254-291 ; III Publ. I.H.E.S. 35 (1969),
127-156; IV ibid. 37
(1070), pp. 223-248; V Advances in Math. 4 (1970), 301-335; and VI in
PSSLI
pp. 207-253.
[10] J. N. Mather, 'Notes on Topological Stability', preprint, Harvard
Univ., Cam
bridge, Massachusetts, 1970.
[11] J. N. Mather, 'Stratifications and Mappings', in DS pp. 195-232.
[12] V. Po?naru, 'Un th?or?me des fonctions implicites pour les
espaces d'applica
tions C?>\Publ. I.H.E.S. 38 (1970), 93-124.
[13] F. Sergeraert, 'Un th?or?me de fonctions implicites sur certains
espaces de
Fr?chet et quelques applications', Ann. Scient. Ec. Norm. Sup., 4th
series, 5 (1972),
599-660.
[14] R. Thom, 'Les singularit?s des applications diff?rentiables',
Ann. Inst. Fourier
Grenoble 6 (1955-1956), 43-87.
[15] R. Thom and H. I. Levine, 'Singularities of Differentiable
Mappings', in PSSLI
pp. 1-89.
[16] R. Thom, 'Ensembles et morphismes stratifi?s', Bull. Amer. Math.
Soc. 75 (1969),
pp. 240-284.
[17] R. Thom, 'The Bifurcation Subset of a Space of Mappings', in MA
1970 pp.
202-208.
[18] R. Thom, 'Topological Models in Biology', Topology 8 (1969),
313-335.
[19] R. Thom, 'Topologie et linguistique', in Essays on Topology, Vol.
dedicated to G.
de Rham, Springer-Verlag, Berlin and New York, 1970.
[20] R. Thom, 'Language et catastrophes', in DS pp. 619-654.
[21] C. T. C. Wail, 'Lectures on C00 Stability and Classification', in
PLSSI pp. 178-206.
[22] H. Whitney, 'On Singularities of Mappings of Euclidean Spaces I,
Mappings of the
Plane Into the Plane, Ann. of Math. 62 (1965), 374-410.
[23] E. C. Zeeman, 'Differential Equations for the Heartbeat and Nerve
Impulse', in
DS pp. 683-741.
E. C. Zeman had a thought about catastrophe theory:
http://zakuski.math.utsa.edu/~gokhman/ecz/c.html