Catastrophe Theory and Urban Processes

Abstract

Phenomena exhibiting discontinuous change, divergent processes, and hysteresis can be modelled w i t h catastrophe theory, a recent development in differential topology. Exposition of t h e theory is illustrated by qualitative interpretations of the appearance of functions i n central place systems, and of price cycles for urban housing. Catastrophe Theory and Urban Processes John Casti and Harry Swain* Introduction A mathematical theory of "catastrophes" has recently been developed by the French mathematician ~ 6 n 6 Thom I6, 7 1 in an attempt to rationally account for the phenomenon of discontinuous change in behaviors (outputs) resulting from continuous change in parameters (inputs) in a given system. The power and scope of Thomls ideas have been exploited by others, notably Zeeman [lo, 111, to give a mathematical account of various observed discontinuous phenomena in physics, economics, biology, [ 4 1 and psychology. We particularly note the work of Arnson [11 on equilibrium models of cities, which is most closely associated with the work presented here. With the notable exception of Amson's work, little use has been made of the powerful tools of catastrophe theory in the study of urban problems. Perhaps this is not surprisirig since the the0r.y is only now becoming generally known in mathematical circles. However, despite the formidable mathematical appearance of the basic theorems of the theory, the application of catastrophe theory to a given situation is often quite simple, requiring only a modest understanding of simple geometric notions. In this regard, catastrophe theory is much like linear programming in the sense that it is not necessary to understand t,he mechanism in order to make it work--a fairly typical requirement of the working scientist when faced with a new mathematical tool. Thus, our objective in this article is twofold: first, * The authors are Research Scholars at the International Institute for Applied Systems Analysis, Schloss Laxenburg, A-2361 Laxenburg, Austria. t o supp ly a b r s e f $.ntroducti ,on t o t h e bas$c ph i losophy of c a t a s t r o p h e t h e o r y I n a form which w e hope w i l l be congen i a l t o workers i n t h e urban f i e l d , and second, t o i l l u s t r a t e t h e a p p l i c a t i o n s of t h e t heo ry t o some c l a s s i c a l problems i n urban economic geography. S p e c i f i c a l l y , w e c o n s i d e r a n example from c e n t r a l p l a c e t h e o r y i n which t h e s i m p l e s t t y p e of n o n t r i v i a l c a t a s t r o p h e p rov ide s a s a t i s f a c t o r y q l o b a l p i c t u r e of t h e observed developmental p a t t e r n s of f u n c t i o n s provided t o t h e popu l a t i on . A second example i l l u s t r a t e s a p p l i c a t i o n of one of t h e more complex e lementary c a t a s t r o p h e s t o t h e i s s u e of e q u i l i b r i u m r e s i d e n t i a l p r o p e r t y p r i c e s i n urban l and markets . Although t h e s e examples a r e p rov ided p r i m a r i l y a s q u a l i t a t i v e i l l u s t r a t i o n s o f t h e t h e o r y , it i s hoped t h a t t h e y may be of i n t e r e s t i n t h e i r own r i g h t a s p r ov id ing a n a l t e r n a t i v e and p o s s i b l y more comprehensive account of t h e dynamics of t h e s e problems t h a n t h o s e ob t a ined by o t h e r methods. C a t a s t r o ~ h e Theorv I n t h i s s e c t i o n , w e p r e s e n t a b r i e f d i s c u s s i o n of t h e b a s i c assumpt ions and r e s u l t s of c a t a s t r o p h e t h e o r y i n a form u s e f u l f o r a p p l i c a t i o n s . For d e t a i l s and p roo f s , w e r e f e r t o [8, 91. k L e t f : R x R" + R be a smooth ( i n f i n i t e l y d i f f e r e n t i a b l e ) f u n c t i o n r e p r e s e n t i n g a dynamical system Z i n t h e s e n s e t h a t R k i s t h e space of i n p u t v a r i a b l e s ( c o n t r o l s , pa ramete r s ) wh i l e R" represents the space of output variables (res~onses, behaviors). We assume that k ~ 5 , while n is unrestricted. The fundamental assumption is that C attempts to locally mlnlmize f, We hasten to point out that in applications of catastrophe theory, it is not necessary to know the function f. In fact, in most cases f will be a very complicated function whose structure could never be determined. All we assume is that there exists such a function which C seeks to locally minimize. k Given any such function f , if we fix the point c E , 4 , we obtain a local potential function fc : Rn+ R and we may postulate a differential equation 2 = grad f, X af where x E Rn, gradx f = grad f = (.g , . . . ,) . aX1 a xn Thus, the phase trajectory of C will flow toward a minimum of f call it xc. The stable equilibria are given ci by the minima of fc, and, since there are usually several minima, x will be a multivalued function of c; that Is, C x : Rk + Rn is not one-to-one. The objective of catastrophe C theory is to analyze this multivaluedness by means of the theory of singularities of smooth mappings. We first state the fundamental result of catastrophe theory in relatively precise mathematical language. We then interpret each of the conclusions of the main theorem in everyday language to show their reasonableness and applicability for realworld problems. For completeness, and to round out the mathematical theory, we consider not only the minima but also the maxima and other stationary values of f . Define the manifold MfC R C k+n as and let X f : Mf -+Rk be the map induced by the projection of R ~ + ~ + R ~ .X is called the catastrophe map of f . Further, let J be the space of cm-functions of R k+n with the usual Whitney ~ ~ t o ~ o l o ~ ~ . Then the basic theorem of catastrophe theory (due to Thorn) is the following. Theorem: There exists a n open dense set J o C J, called generic functions, such that if f E J o (i) M is a k-manifold; f (ii) any singularity of x is equivalent to one of a finite f n.umber of elementary catastrophes; (iii) Xf is stable under small perturbations of f. Remarks : 1. Here equivalence is understood in the following sense: maps X: M + N and : M + G are equivalent if there exist diffeomorphisms h, g such that the diagram R T W is commutative. If the maps X, X have singularities at x EM, x E 3, respectively, then the singularities are equivalent if the above definition holds locally with 2. S t a b l e means that Xf is equivalent to x for all g in 9 a neighborhood of f in J (in the Whitney topology). 3. The number of elementary catastrophes depends only upon k and is given in the following table: . < number of elementary 2 8 ' co catastrophes A finite classification for k > , 6 maybe obtained under topological, rather than diffeomorphic,, equivalence but the' smooth classification is more important for applications. 4 . Roughly speaking, Jo being open and dense in J simply means that if the potential function ~ E J were to be selected at random, then f E J with probability one. 0 Thus, a given system function f is almost always in J 0 and furthermore, if it is not, an arbitrarily small perturbation will make it so. 5. The importance of Mf being a k-manifold is that Mf is the place where controlling influence is exerted: from the standpoint of the decision maker, Mf is the manifold which he may manipulate. Thus, the dimension of the behavior or output space does not enter into the classification at all. Since n, the dimension of the behavior space, may be very large, this conclusion enables us to focus attention upon a much smaller set in investigating where and when catastrophic changes in behavior wrll occur, To summarize, M 2s where the action 2s. f 6. Conclusion (iil shows that, mathematically speaking, only a very small number of distinctly different catastrophes can occur. Intuitively, catastrophes are equivalent if they differ only by a change of coordinate system. Since the coordinate system chosen to describe a phenomenon is not an intrinsic feature of the system, we may restrict our attention to the analysis of only a small handful of mathematical catastrophes, safe in the knowledge that more complex forms cannot possibly occur. In addition, as indicated below, the elementary catastrophes are all described by simple polynomials which make their analysis and properties particularly simple. 7. The last conclusion, stability, means that should the potential f describing C be perturbed slightly, the new potential will also exhibit the same qualitative catastrophic behavior as f. Since no physical system is known precisely, this fact enables us to feel confident about various predictions based upon useof any f €Joe Discontinuity, Divergence, and the Cusp Catastrophe Our critical assumption is that C , the system under study, seeks to minimize the function f: that is, Z is dissipative. Thus, the system behaves in a manner quite different from the Hamiltonian systems of classical physics, In this section we shall mention two striking features displayed by catastrophe theory which are not present in Hamiltonian systems but which are observed in many physical phenomena. The first basic feature is discontinuity. If B is the . . image in R~ of the set of 'singularities of Xf, then B is called the bifurcation set and consists of surfaces bounding regions of qualitatively different behavior similar to surfaces of phase transition. Slowly crossing such a boundary may result in a sudden change in the behavior of 2, giving rise to the term "catastrophe". Since the dimension of the output space does not enter in the classification theorem, all information about where such catastrophic changes in output will occur is carried in the bifurcation set p which, by a corollary of conclusion (i) of the k Theorem, is a subset of the input space R . Hence, even though 2 may have an output space of inconceivably high dimension, the "action" is on a manifold of low dimension which may be analyzed by ordinary geometric and analytical tools. The second basic feature exhibited by catastrophe theory is the phenomenon of divergence. In systems of classical physics a small change in the initial conditions results in only a small change in the future trajectory of the process, one of the classical concepts of stability. However, in catastrophe theory the notion of stability is relative to perturbations of the system itself (the function f), rather than just to perturbations of the initial conditions, and so the Hamiltonian result may not apply. For example, adjacent tissues in a homogeneous embryo will differentiate. Let us now illustrate the aboye ldeas by considering the cusp catastrophe. It will turn out that a minor modification of this catastrophe is also the appropriate catastrophe for one of the main examples of this paper, the problem of central place discontinuities. Let k = 2, n = 1, and let the control and behavior space have coordinates a, b, and x, respectively. 2 Let f : R x RI R be given by The manifold Mf is given by the set of points (.a,b,xl C R 3 where grad f (a,b,x) = 0, X that is, The map xf: Mf + R~ has singularities when two stationary values of f coalesce, that is, Thus, Equations (1) and (2) describe the singularity set S of x. It is not hard to see that S consists of two fold-curves given parametrically by (a,b,xl = (-3h2, 2h3, h ) , h t 0 , and one cusp singularity at the origin. The bifurcation set B is given by 2 which is the cusp 4a3 + 27b = 0. Since Mf and S are smooth Figure 1. The Cusp Catastrophe at the origin, the cusp occurs in L? and not in $, Figure 1 graphically depicts the situation, It is clear from the figure that if the control point (a,b) is fixed outside the cusp, the function f has a unique minimum, while if (a,b) is inside the cusp, f has two minima separated by one maximum. Thus, over the inside of the cusp, M is triple-sheeted. f The phenomenon of smooth changes in (arb) resulting in discontinuous behavior in x is easily seen from Figure 1 by fixing the control parameter a at some negative value', then varying b. On entering the inside of the cusp nothing unusual is observed in x; but upon further change in b, resulting in an exit from the cusp, the system will make a catastrophic jump from the lower sheet of Mf to the upper, or vice versa, depending upon whether b is increasing or decreasing. The cause of the jump is the bifurcation of the differential equation 8 = -gradx f, since the basic assumption is that 1 always moves so as to minimize f. As a result, no position on the middle sheet of maxima can be maintained and C must move from one sheet of minima to the other. A hysteresis effect is observed when moving b in the opposite direction from that which caused the original jump: the jump phenomenon will occur only when leaving the interior of the cusp from the opposite side to the point of entry. To see the previously mentioned divergence effect, consider two control points (a,b) with a > 0, b 30. Maintaining the b values fixed with decreasing a, the point with positive b follows a trajectory on the lower sheet of Mf, while the other point moves on the upper sheet. Thus, two points which may have been arbitrarily close to begin with end up at radically different positions depending upon which side of the cusp point they pass. While the cusp is only one of several elementary catastrophes, it is perhaps the most important for applications. In Table I, we list several other types for k 5 4, but refer the reader to [6] for geometrical details and applications. Table I. The Elementary Catastrophes for k I 4. control behavior space space Name potential function f dimension dimension fold xJ + ux 1 cusp x4 + ux2 + vx 2 swallowtail x5 + ux3 + vx2 + wx 3 butterfly x6 + ux4 + vx3 + wx2 + tx 4 hyperbolic 3 urnbilic x + y3 + uxy + vx + wy elliptic umbilic parabolic umbilic Central Place Catastrophes To illustrate the cusp catastrophe in an urban context, consider the supply of goods and services to an urban-centered market a r ea under a l l t h e n a m a l p o s t u l a t e s of c l a s s i c a l (geometric, s t a t r c , d e t e r m i n i s t i c ) c e n t r a l p lace t h e o r y , Then t h e r e e x i s t s p a t i a l monopoly p r o f i t s , T , i n t h e d i s t r l ~ bu t ion of t h a t v a s t m a j o r i t y of goods whose t h r e s h o l d l i e s between t h e s i z e of t h e e x i s t i n g market and t h a t of t h e market t h a t would be requi red t o induce a competing s u p p l i e r t o l o c a t e t h e r e . The argument i s s i m i l a r f o r t h e number of es tab l i shments handl ing t h a t good, t h e number of f u n c t i o n s i n a given c e n t r a l p l a c e , and t h e order of t h a t c e n t r a l p l ace ( c f . Dacey [3] f o r d e f i n i t i o n of t e r m s ) . But now l e t t h e r e be emigra t ion from t h a t market a r e a , o r some o t h e r p rocess producing a slow leakage of aggrega te l o c a l purchasing power. Then a + O , t h e minimum th reshold , a t which p o i n t t h e good ceases t o be d i s t r i b u t e d . The th re sho ld f o r ( re )appearance of t h e good ( e s t a bl ishment, f u n c t i o n ) i s , however, h igher than T = 0 s i n c e an en t repreneur would choose t h a t combination of good and market a r e a o f f e r i n g maximal s p a t i a l monopoly p r o f i t s ( t h e upper t h r e s h o l d ) . Thus we have t h e c h a r a c t e r i s t i c discont i n u i t y and h y s t e r e s i s e f f e c t s of ca t a s t rophe theo ry . The cusp c a t a s t r a p h e provides a reasonable g l o b a l p i c t u r e f o r t h e s e c e n t r a l p l ace phenomena. Let t h e independent o r c o n t r o l v a r i a b l e s be x , t h e popula t ion of a market a r e a , and y , t h e d i sposab le income p e r c a p i t a . The behavior o r ou tpu t v a r i a b l e can then be i n t e r p r e t e d a s t h e o r d e r of t h e c e n t r a l p l a c e , o r number of func t ions o r goods provided t h e r e ; a l l t h r e e may be generally referred to as the functional level, m, of the central place or market area. (The implicit potential function, for this system is, in contradistinction to the prior discussion, maximized by the action of the central place process. Thus we operate with -f and apply the preceding theory.) The relevant picture is given in Fig. 2. Each point on the manifold M represents a functional level corresponding to given levels of aggregate local purchasing power. But though x and y determine the functional level, the fact that M is triplesheeted within a region near the relevant thresholds means that m can take on two distinct stable equilibrium values; values, moreover, which depend on the trajectory (history or direction of change) in x and y. Thus in Fig. 2 it may be readily seen that, for a fixed level of disposable income per capita, smooth increases in population will have but small effects on the functional level of the central place until the locus of that trajectory crosses the right-hand cusp border into region I1 (see a). At this point the functional level jumps dramatically from the lower sheet of M to the upper (the middle sheet shown in Figure 2 corresponds to relative minima and is of no interest here). The vector b shows the same qualitative result, and clearly various combinations of a and b will do the same provided such combinations pass through the x, y projection of the multi-sheeted part of M. The hysteresis effect can be demonstrated by examining m for, say, fixed income and changing population. Let population increase along a as before; thus the cusp region is entered from I with no discontinuous output; the point then leaves I and enters region I1 with a positive jump in functional level. But then let Figure 2. A Manifold for Central Place Catastrophes population smoothly decrease (-a): the cusp is entered from I1 at the same point as before, and the point exits into I as before. The only difference is that this time the catastrophic jump downwards in functional level takes place when entering I and not 11. Only an exit from the cusp region across a different boundary than the entry branch gives rise to catastrophic change. Thus the cusp catastrophe illustrates the theoretical prediction, and observed fact, that the threshold for (re-)appearance of a function is higher than for its disappearance. Note that this qualitatively nice behavior is obtained even with the highly restrictive and unrealistic postulates of classical central place theory. More realistic models incorporating entrepreneurial inertia (lagged feedback plus conservative behavior in the face of uncertainty), nonzero entry costs, and substantial indivisibilities would only serve to accentuate the hysteresis effect. The third basic feature, divergence, can be appreciated by examining the change in functional level from nearby initial points p and q as disposable income falls for a fixed population. The trajectory in M from p passes to the left of the cusp point C, and consequently m drops smoothly to levels on the lower sheet of M. On the other hand, the point q, which began with a population close to p, has a trajectory which takes it to the right of C; m is thus maintained, for a while at least, at "artificially" or "anomalously" high levels. The critical factor is that slow change of the same sort in real regional systems with similar initial conditions may lead to fundamentally different futures, depending on the location and orientation of cusp points. Moreover, one would expect these m-anomalies to be most glaring at low levels of population and income. Property Prices and the ~utterfl~ Catastrophe The cusp catastrophe is probably useful in many other urban settings. Casual observation suggests that many of the lifestyle definition processes of our proliferating subcultures-processes noted for teenage gangs long before becoming part of the conventional wisdom about the post-industrial middle classes [2] -may exhibit the characteristic non-Hamiltonian divergence of catastrophe theory, and may under special conditions display discontinuities and even hysteresis [ 5 ] . We discuss a more prosaic example, the purchase price of urban dwellings, not so much to exploit the cusp further but to use it as a vehicle to introduce a generalization which is perhaps the second-mostimportant elementary catastrophe for applied work, the so-called butterfly catastrophe. Let r repressnt the real rate of change of housing prices in a particular urban market. In the first approximation, we assume that there are two types of buyers who are interested in this sort of property, and that the d i n e d level of their activities in the property market dictates r. Call.these buyers consumers and speculators. The former are interested in a wide range of attributes of the housing bundle and their demand is strongly price-elastic, especially in volatile or cyclical markets. Speculators, on the other hand, are overwhelmingly concerned with short-term (and often highly leveraged) capital gains. Since the two groups have fundamentally different objectives, time horizons, and price elasticities, they may reasonably be thought of as disjoint sets of investors. If Dc represents the demand F i g u r e 3 . C a t a s t r o p h e M a n i f o l d f o r Urban P r o p q r t y P r i c e s f o r p roper ty by consumers and Ds t h e demand by s p e c u l a t o r s , then t h e g l o b a l behavior of p roper ty p r i c e s may i n t h i s simple c a s e be a s dep ic t ed i n F igure 3 , Inc reas ing e i t h e r D o r Ds t ends t o i n c r e a s e r , but t h e C key t o c a t a s t r o p h i c rises and f a l l s l i e s w i th t h e s p e c u l a t o r s ; changes i n Dc f o r c o n s t a n t Ds cause only smooth changes i n r. A l l of t h e f e a t u r e s observed i n t h e prev ious example--divergence, d i s c o n t i n u i t y , and h y s t e r e s i s a r e . a l , s o p r e s e n t here' . Moreover, i n empi r i ca l a p p l i c a t i o n s t h e r e i s f rkquen t ly a r e l a t i o n between t h e l o c a t i o n of t h e cusp p o i n t and t h e time c o n s t a n t s of t h e system, wi th l o c i avo id ing t h e mul t i -sheeted p a r t s of M t end ing t o be slower. I n t h i s example, suppose t h e process s t a r t s a t 0' i n t h e Dc-Ds space. There a r e then two p o s s i b i l i t i e s f o r passage through t h e cusp reg ion and back t o O ' , t h e pa ths O P Q R O and OPQSO. The f i r s t corresponds t o a s p u r t of s p e c u l a t i v e demand caus ing , a f t e r a s h o r t l a g , a jump i n p r i c e s from P t o Q , followed by a p r o f i t t a k i n g s e l l o f f by s p e c u l a t o r s w i th on ly moderate i n c r e a s e i n consumer demand, t r i g g e r i n g a c o l l a p s e of p r i c e s a t R . This s o r t of p rocess i s c h a r a c t e r i s t i c of t h e highfrequency components of r and i s q u i t e t y p i c a l i n s p e c u l a t i v e markets. The demand by consumers f o r market i n t e r v e n t i o n i s r e l a t e d t o both t h e magnitude of r and t h e ampli tude of t h e s e r e l a t i v e l y shortterm "boom-and-bust" cyc l e s . Slowing t h e frequency of t h e OPQRO c y c l e may be an a p p r o p r i a t e response under such c o n d i t i o n s , i f it a l lows Dc t o b u i l d up s u f f i c i e n t l y a t Q t o d r i v e t h e r e t u r n pa th around t h e cusp through S. Rapid and d i s t r e s s i n g , f a l l s i n p r i c e a r e t h u s avoided. This observa t ion i l l u s t r a t e s , i f c rude ly , t h e f a s t t ime, slow t i m e ( . " s i l l y p u t t y t t ) behav io r d i v e r g e n c e which i s c h a r a c t e r i s t i c of dynamic c a t a s t r o p h e models . Governments i n t e r e s t e d i n o r d e r l i n e s s and s t a b i l i t y i n housing markets--low and v i s c o u s r u s u a l l y r e q u l a t e Dc and Ds by t i g h t e n i n g o r loosen ing t h e supp ly of money, t h a t is , by r a i s i n g o r lower ing i n t e r e s t r a t e s . W e now show how t h e b u t t e r f l y c a t a s t r o p h e , a g e n e r a l i z a t i o n of t h e c u s p , e n a b l e s u s t o upgrade t h e urban p r o p e r t y p r i c e example by i n c l u d i n q t i m e dependence a s w e l l a s i n t e r e s t r a t e changes i n t h e c a t a s t r o p h e mani fo ld . I t w i l l b e s e e n t h a t i n c l u s i o n of t h e s e i m p o r t a n t f a c t o r s g e n e r a t e s t h e p o s s i b i l i t y f o r a t h i r d mode of s t a b l e behav io r f o r r , a t y p e of "compromise" r a t e of change of p r i c e s . For t h e b u t t e r f l y k = 4 , n = 1 , t h e c a n o n i c a l form f o r t h e p o t e n t i a l i s g i v e n by X 1 1 1 + C X f ( c , x ) = 6 + 5 c2x3 + 7 c 3 x2 + c 4 x , 4 1where c E R ~ , x E: R. The a s s o c i a t e d c a t a s t r o p h e s u r f a c e M i s t h e four -d imens iona l s u r f a c e g i v e n by 5 The s u r f a c e M C R and t h e b i f u r c a t i o n set B C R ~ . W e draw twod imens iona l s e c t i o n s of @ . t o show how it g e n e r a l i z e s t h e cusp. When t h e butterf Zy factor cl > 0, t h e x 4 t e r m swamps t h e x6 t e r m and w e o b t a i n t h e cusp . The e f f e c t of t h e bias f a c t o r c 2 i s merely t o b i a s t h e p o s i t i o n of t h e cusp . When t h e b u t t e r f l y f a c t o r c l < 9, then t h e x4 term con£ l i c t s wi th t h e x6 t e r m and causes t h e cusp t o b i f u r c a t e i n t o t h r e e cusps enc los ing a pocket . This pocket r e p r e s e n t s t h e e m e r gence of a compromise behavior midway between t h e two extremes r ep re sen ted by t h e upper and lower s u r f a c e s of t h e cusp. c3 =s c3 C, > 0 c1 >O c2= 0 c1 >O

DOI: 10.1007/3-540-07622-0_483

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Cite this paper

@inproceedings{Casti1975CatastropheTA, title={Catastrophe Theory and Urban Processes}, author={John L. Casti and Harry Swain}, booktitle={Optimization Techniques}, year={1975} }