Catastrophe Models and the Expansion Method : A Review of Issues and an Application to the Econometric Modeling of Economic Growth

Many negative reactions to Catastrophe Theory have been triggered by overly simplistic applications unintended and unsuited for statistical-econometric estimation, inference, and testing. In this paper it is argued that stochastic catastrophe models constructed using the Expansion Method hold the most promise to widen the acceptance of Catastrophe Theory by analytically oriented scholars in the social sciences and elsewhere. The paper presents a typology of catastrophe models, and demonstrates the construction and estimation of an econometric expanded cusp catastrophe model of economic growth.


INTRODUCTION
Catastrophe Theory (CT), that originated with Thom's (1975)  'Structural Stability and Morpho-  genesis', aroused an initial intense interest that was later followed by a spate of criticisms.Today CT is very much alive, but perhaps is not having the impact it could and should.The major factors hampering its progress are (a) that many applica- tions of CT are regarded as much too abstract and simplistic by substantive scholars, and   (b) that CT has not entered yet to a sufficient extent into the modeling phase centered upon statistical econometric estimation and inference.
The focus of this paper is upon the application of CT.The paper discusses and implements a 'Modeling Perspective' intimating that the mathe- matical models of any aspect of reality are the central point of any application of mathematics.This perspective calls for defining what mathe- matical models are and how they are constructed.One of the two fundamental modes of model construction is by 'expansions', namely, by mod- eling the parameters of a preexisting model.The Expansion Method articulates the rationales and the operational specifics for constructing by ex- pansions more complex and realistic models from simpler ones.
In the sections that follow, first the salient traits of CT are briefly reviewed.Then the Mod- eling Perspective and the Expansion Methodology are outlined and applied to catastrophe model- ing, and in the process the scope and variety of catastrophe models is discussed and a typology E. CASETTI of these models is proposed.Finally, to demon- strate the leitmotifs of the paper an expanded econometric cusp catastrophe model of modern economic growth is constructed and estimated.

CATASTROPHE THEORY
Let us begin from some generalities on the nature and significance of catastrophe theory.Over the years, the various components and themes of CT have been emphasized to different degrees.For instance, Thom's classification theorem had a prominent role early on, a lesser one in later years, and virtually disappears from considera- tion in the work of scholars such as Arnol'd.
According to Arnol'd (1992, p. 2) "the origins of catastrophe theory lie in Whitney's theory of sin- gularities of smooth mappings and Poicare and Andronov's theory of bifurcations of dynamical systems".It seems fair to say that together with Bifurcation Theory, Catastrophe Theory is today regarded as a branch of the modern Non-Linear Dynamics (Drazin, 1992; Tu, 1992; Glendinning,  1994).
It is to some extent a matter of interpretation what exactly CT is because of its evolution after its original statement by Thorn.The literature on CT has been substantially influenced by early supporters such as Zeeman (1977) and Poston  and Stewart (1978), by its critics (Zahler and  Sussmann, 1977; Sussmann and Zahler, 1978;  Arnol'd, 1992, p. 102 if), and by the many schol- ars who used it and applied it in substantive fields.The reviews of CT applications such as the ones by Gilmore (1981), Wilson (1981), Lung  (1988), Rosser (1991) and Guastello (1995), attest   to the impact that CT has had.
A key point of CT since its very beginning is that 'systems' are found in 'stable equilibrium states'.Under conditions of 'structural stability' small changes in systemic 'control parameters' bring about small changes in these stable states.However, small changes in control parameters across 'critical' thresholds will cause stable equi- libria either to disappear, or to 'bifurcate' into multiple equilibria, some of which are stable.The appearance of multiple stable equilibria at critical points in control space is a special case of the bifurcations dealt with greater generality by Bi-  furcation Theory (Hale and Kocak, 1991).
In most early articulations of CT the stable equilibrium states are viewed as the optima of a function of the state variable(s), specifically, as the minima of a 'potential function'.The latter terminology follows from applications of CT in physics.However, CT has also a dynamic dimen- sion the early development of which gained sub- stantially from the work of Zeeman.'Gradient'   dynamic formulations corresponding to the ones based on the minimization of a potential function can be easily obtained by setting the time deriva- tive(s) of the state variable(s) equal to minus the gradient of the potential function.This is equiva- lent to assuming that the state variable(s) will move downward on the potential manifold, fol- lowing the direction of steepest descent and seek- ing the local minimum in the domain of attraction of which they are located.These 'gradient' equations link CT to the theory of non- linear dynamics at large.The minima of the potential function corre- spond to stable equilibria of 'gradient differential equation(s)'.At the critical point(s) in control space in correspondence to which local minima of the potential function disappear or multiply, in the phase diagrams of gradient systems stable equilibria disappear or multiply.This links CT to the qualitative analysis of non-linear differential equations that predates it by several decades.Thorn's mathematical contribution that moti- vated and started CT is the 'classification theo- rem'.Essentially, Thorn proved for 'seven elementary catastrophes' that for a wide class of functions, in the neighborhood of their 'degenerate singularities', the types of catastrophes that can occur are the same that characterize the cor- responding canonical catastrophe equations.For example, a broad class of functions involving one state variable and two control parameters will have near a degenerate singularity either a fold multiple stable equilibria, as well as the possibil- or a cusp catastrophe, same as the canonical catas- ity for a system to switch across topologies chartrophe equation with one state variable and two acterized by different constellations of such control parameters, equilibria as a result of changes in systemic pa- This brief articulation of the major points of rameters, fall within the scope of non-linear CT begs the question of whether there are dis- dynamics, as a subset of the state variables' tinctive contributions that should be credited to behaviors considered within it.It could be also CT, and on which, ultimately, the applications of argued that in the literature there are applications CT can rest.
along these and similar lines that predate CT or Its critics deny that CT represents a distinctive were developed independently from it.For in- contribution, or at least a substantial one.They stance the critical minimum effort thesis (Oshima, contend that the classification theorem is only 1959; Leibenstein, 1957), and Nelson's (1960,   valid 'locally', in the neighborhood of degenerate 1965) contribution to the modeling of the escape singularities, and consequently does not justify from the Malthusian trap were related to the attributing a generality of scope to the elemen- qualitative analysis of differential equations but tary catastrophes.Also, they would argue that can be easily viewed in terms of CT. the discontinuities in the elementary catastrophes These and similar examples notwithstanding, it   are not unique to CT, but can be produced within is to the credit of CT's originator and of the Bifurcation and Non-Linear Dynamics theories many scholars who have contributed to CT over as special cases.Furthermore, they have con- the years, to have generated a collective con- tended that the examples of applications of CT sciuousnes of catastrophes and catastrophe mod- are contrived, and better models with same ef- eling.The notion that qualitative jumps across fects can be put together outside CT.
topologies may result from the continuous change However, it can be argued that CT has been of control parameters across critical thresholds having an important role along at least three may very well have been implicit in other schol- dimensions.Namely, (a) it has etched into the arly traditions.Yet it is primarily due to CT if collective consciousness of the scholars interested today, in the investigation of social biological in modeling dynamic phenomena that systemic and physical phenomena, continuous change as equilibria may appear disappear or multiply well as discontinuous qualitative change represent when control parameters move across critical alternatives to be both considered and modeled.
thresholds; (b) it has identified 'types' of catas- trophes, some of which have been found very useful in many applications; and (c) it has linked MATHEMATICAL MODELING the appearance, disappearance, and bifurcation of equilibrium states simultaneously to dynamic All applications of mathematics, including those formalizations and to formalizations in terms of of catastrophe theory, start from, and/or are systemic optima.Such twin formalizations are based upon, mathematical models of realities.synergistic in their potential to inspire substantive The mathematical models of some phase of real- theory and empirical analyses, ity are not just a particular type of models (a Let us comment briefly only on the first of 'species' within a 'genus' encompassing all types these three points.Before CT, a widespread col- of models).Rather, they are conceptual artifacts lective awareness that discontinuities can follow qualitatively distinct from anything else called the smooth change of systemic parameters across 'models'.A recognition of this distinctiveness, critical thresholds did not exist; after CT it did.
and of the fact that any application of mathe- It can be argued that the possibility of having matics is via mathematical models, are necessary prerequisites for understanding the roles of math- involves extracting a new mathematical model ematical models, and the nature of the boundary from preexisting one(s) by a variety of mathemat- that separates and links mathematics to its ap-ical manipulations.As example of type A model plications.Here, this 'Modeling Perspective' is construction suppose that a substantive discipline briefly articulated in generalities, and subsequently has defined variables and important relations is applied to catastrophe modeling.In this sec- among these variables.A scholar from this disci- tion we focus upon the definition of mathemati- pline who has in his/her mind an inventory of cal models and of the major approaches to their analytical structures, selects one of these struc- construction, tures and links it to one such important relation.

Definition
The mathematical modeling of some phase of reality ('mathematical modeling' for short), in- volves the linking of a substantive conceptual frame of reference to analytical mathematical structures.Mathematics defines analytical struc- tures such as equations, inequalities, probability distributions, stochastic processes and so on, in which variables, random variables, and param- eters appear.The substantive scientific disciplines concerned with the study of any aspects of rea- lity, in the social sciences and elsewhere, define entities ('objects') with which they are concerned, variables that take specific values for these ob- jects, and relations among these variables.The mathematical models of any realities consist of analytical structures with some or all of the vari- ables, variates, and parameters in them linked to a substantive conceptual frame of reference, namely, to substantive variables and variates.
The mathematical models can be deterministic, stochastic, or mixed.There are two major ap- proaches to the construction of mathematical models: the conventional modeling and the ex- pansion modeling.

Conventional Modeling
When this link is established, a mathematical model of some aspect of reality is born.
The conventional model building of type B consists in extracting models from other models by mathematical manipulations such as 'solving' or 'optimizing'.Suppose for example that we ob- tain a demand function by maximizing a utility function subject to a budget constraint.The de- mand function is a mathematical model, but so are also the utility function and the budget con- straint.Another example is solving a differential equation that relates the rate of change of a country's population to its population size.The solution of the differential equation and the dif- ferential equation itself are both mathematical models.
It is important to note that the type B conven- tional modeling can be viewed in distinct but equivalent ways, depending upon whether the mathematical manipulation(s) it involves are car- ried out on a mathematical structure or upon a model.Consider the differential equation relating rates of change and levels of population.Here the mathematical manipulation involved consists in solving a differential equation.If the differen- tial equation is a model because a substantive (demographic) frame of reference has been linked to it, then solving the differential equation yields a second model that, we can say, has been cre- ated by a type B conventional modeling.If in- stead the differential equation, taken as a We can recognize at least two types of conven- mathematical structure, is solved to produce a tional model construction, which will be referred second mathematical structure that is then linked to as A and B. The type A consists in the to a demographic frame of reference, the result- straightforward linking of an analytical structure ing model is produced by a conventional model- to a substantive frame of reference.The type B ing of type A.
Both these DEs and their solutions qualify as expansion equations.However, if the solutions of the DEs are selected as expansion equations, clearly they are arrived at by an intermediate manipulation that in this case consists in a 'solv- (a) An 'initial model' is specified.This model ing'.In passing, let us note that in this particular may involve variables and/or variates, and at example the parameters appearing in the initial least some of its parameters are in letter conditions from solving the DE can be also ex- form. panded.
(b) Some or all of the letter parameters of the The redefinition of an initial model's paraminitial model are modeled by redefining them eters into functions of expansion variables can be into functions of expansion variables or vari- implemented in two equivalent ways, depending ates by 'expansion equations'.The expansion upon whether an analytical mathematical struc- variables identify substantively relevant di- ture or a model are expanded.We can start from mensions in terms of which the initial model's a mathematical structure, expand some or all the drifts, and the expansion equations are a spe- parameters in it, thus generating an expanded cification of this drift, structure, and then link a substantive frame of reference to this expanded structure.Alterna- (c) The initial model and the expansion equa- tions constitute a 'terminal' model in struc- tively, we can start from a mathematical model tural form.If the right-hand sides of the and expand some or all of its parameters into expansion equations are substituted for the functions of expansion variables.Both of these corresponding parameters in the initial model qualify as expansion modeling, and yield the a reduced form 'terminal' model is obtained, same terminal model.In both cases the terminal model arrived at consists of an initial model A terminal model in either structural or re- duced form encompasses simultaneously the complemented by models relating some or all of its parameters to expansion variables or variates.initial model and a specification of its parametric drift across the space spanned by the Most models and analytical mathematical expansion variables, structures can be conceptualized as resulting (d) This process can be iterated, with the term- from previous expansion(s), which in itself opens inal model produced by a previous cycle vistas useful to interpret existing models, and to becoming the initial model of the next.view them in terms of a unifying perspective.
However, any models or structures can be also Since the expansion modeling involves the con- regarded as the potential building blocks of more ventional modeling of an initial model's param- complex 'expanded' models and structures.This eters that can be of types A or B, also the latter perspective proves especially useful when .expansions can be of types A or B. In type A the models or structures so viewed possess a dis- cases an expansion equation results from the tinctive identity in the literature and in the con- linking of substantive variables (some of which sciousness of communities of scholars.In fact, are the parameters of the initial models redefined this perspective provides one of the motivations as variables) to a suitable analytical structure, for defining classes of 'initial' catastrophe struc- Instead, in a type B situation the expansion equa- tures or models and of more complex structures tions are arrived at via multiple steps.To exem- or models generated from these by expansions.
plify, suppose that the parameters c and /3 of a Several rationales of the expansion modeling model y c +/3x are expanded by the differential are discussed in Casetti (1997).Here let it suffice E. CASETTI to mention the model context rationale, which suggests that models can be related by expan- sions to external environments or contexts de- fined by contextual variables.If a model's parameters are expanded into these contextual variables, the terminal model obtained encom- passes both the initial model and a specification of its drift across the context.There is a very broad range of situations to which this model context frame of reference can be applied.

CATASTROPHE MODELS
Mathematical models were defined as the resul- tant of linking analytical mathematical structures to substantive frames of reference.Let us now consider which types of models have arisen or can arise from linking catastrophe structures to substantive theory.Typologies of catastrophe models can be based on the analytical structures involved, on the substantive frames of reference attached to these structures, on the manner in which these linkages are established, and finally on the roles that these models have played or can play in the conduct of enquiry and in model- ing literatures.
Let us focus here on typologies based on ana- lytical catastrophe structures, that apply also to the models in which these structures appear.In the process, we will touch upon the possible rela- tion of supportive or critical reactions to CT that can be traced to the comparative abundance or scarcity in the catastrophe literature of certain types of models/structures.
The following typologies will be discussed: (a) the elementary catastrophe models and their duals; (b) gradient and potential models; (c)canonical, generalized canonical, and non-canonical models; (d) expanded and non-expanded models; (e) deterministic and stochastic models.These typologies are concurrent, so that a catas- trophe model or structure can be classified in terms of all the five dimensions outlined above.Thus, for instance, we can have a dual-cusp canonical unexpanded deterministic gradient catastrophe model.It is useful to note that a large number of different catastrophe models exist or can be con- structed.It is not necessarily true that all catas- trophe scholars are aware of all of them.The scope and usefulness of applied catastrophe work can benefit from a greater awareness of the many options available.Let us now proceed with a dis- cussion of these typologies taken in sequence.

Primal and Dual Catastrophe Models
In this paper we will confine ourselves to the 'elementary catastrophes', which are the fold, cusp, swallowtail, butterfly, plus the hyperbolic elliptic and parabolic umbilics.The first four in- volve one state variable, the umbilics, two.The identification of specific types of catastrophes and of their 'duals' (Wilson, 1981, p. 28; Poston  and Stewart, 1978, p. 116 if) is a major contribu- tion.It goes beyond the mere recognition that 'qualitative changes' in dynamics, for instance in differential equations, may be produced by the continuous change of some parameters across cri- tical thresholds.The elementary catastrophes re- present specific types of qualitative change, some of which have been found very useful to under- stand many diverse social biological and physical phenomena.
To clarify, consider the cusp catastrophe, which in this paper is singled out for use in every example and in the demonstration.The cusp catas- trophe and its dual involve one state variable and two control parameters, and corresponds to a well-specified constellation of stable and unstable equilibria.The cusp catastrophe includes a topology with a single stable equilibrium, and a topology wih a low and a high stable equilibria and an intermediate unstable one.For suitable smooth changes in parameters a cusp catastrophe struc- ture/model can switch from a single low level stable equilibrium condition, to a condition char- acterized by two coexisting stable equilibria, one low and one high, to a subsequent condition with a single high level stable equilibrium, or vice versa.
The dual cusp catastrophe (Gilmore, 1981,  p. 267-270) includes a topology with a single unstable equilibrium, and a topology with a low and a high unstable equilibria with an intermedi- ate stable one.Smooth changes in parameters can produce sequential transitions from a single low level unstable equilibrium, to a condition charac- terized by an intermediate stable equilibrium po- sitioned between two unstable ones, to a condition with a single unstable equilibrium, and backward.In a dual cusp catastrophy (as in any dual catastrophe) the equilibria occur at the same values of the state variable that yield equilibria in the primal, however what are stable equilibria in the primal become unstable in the dual, and vice versa.The primal cusp models (but not the dual cusp models) proved very useful in a great many fields and applications.

Potential and Gradient Models
Each primal and dual catastrophy can be formal- ized by potential or gradient structures.To exem- plify, the canonical potential structures of the primal and dual cusps are (1) and FD(X __(1/4X 4 _1_ 1/2b/X 2 qt_ FX), (2) In general, the relations between primal and dual catastrophes are (5) and the relations between potential and gradient formulations are k -grad(F), ( 6) These analytical structures become mathematical models when the variables and parameters in them are linked to a substantive frame of reference.Since these typologies differentiate types of analytical mathematical structures, they are also typologies of catastrophe models.
That a catastrophe can be formalized using either potential or gradient structures is a strong point of catastrophe modeling.Within the frame of reference of CT the systemic equilibria corre- spond to the optima of a potential function, that are also the stable equilibria of the gradient dif- ferential equations implied by the potential func- tion.These twin formalizations open the way to developing substantive theory in terms of both systemic optima and their related dynamics (Casetti, 1991).This possibility, however, does not appear to have received as much attention as it deserves.
where the subscripts P and D stand, respectively, for primal and dual.
The canonical gradient cusp structures are 0 --(X a t-btX q-1:) for the primal cusp and for the dual cusp, where k denotes the derivative of x with respect to time.

Canonical and Non-canonical Models
Examples of the canonical analytical catastrophe structures are the cusp potential equation (1) and the cusp gradient equation (3).All the canonical equations of the elementary catastrophes (cf.for instance Wilson, 1981, p. 29) are characterized on their right-hand sides by polynomials with some parameters set at numerical values and some terms missing.In fact, the missing terms can be regarded as having parameters set to a value of zero.The letter parameters in the canonical equa- tions are the 'control parameters' that determine E. CASETTI the topology of the equation.Let us call 'general- ized canonical equations' the equations obtained by replacing the polynomials in the canonical structures by polynomials of the same degree but with all the parameters in letter form.
As an example, the generalized gradient canon- ical equation for the cusp is , OZ3 Z3 -1 t. 02 Z2 --OZlZ n t-OZ O. ( 8) It should be noted that Eq. ( 8) encompasses the primal and the dual cusp canonical equations as special cases.
In order to link the c's of the generalized ca- nonical equation ( 8) to the control parameters u and v of the canonical equation (3) let us procede in two steps, as follows.First, let us partially generalize the canonical equations so that it will encompass the primal and dual cusps as special cases.To this effect, write 2 h(x + ux + v). ( 9) For h =-1, Eq. ( 9) specializes to the primal or dual canonical gradient cusp equation.The sec- ond step defines the shift transformation z=x-w, which leads to h((z + w) + u(z + w) + v).
(11) Equation ( 11) defines a link between the c's of the generalized equation ( 8) and the control parameters of the primal and dual cusp canonical equations (3) and (4).This link will be revisited and elaborated upon later in this paper.The c's can be regarded as reduced form parameters, while the link parameters h and w plus the con- trol parameters u and v will be referred to as structural parameters of the generalized cusp cat- astrophe equation.Alternative approaches to generalizing canonical models are discussed for instance in Brown (1995, p. 61) and in Cobb and  Zacks (1985, p. 798).
A difference between generalized canonical and canonical models is in that the former influ- ence through h the speed at which the state vari- able(s) approach their stable equilibria.This is in contrast with conventional CT, which presup- poses that systems are in stable equilibrium states.
The perfect delay and the Maxwell conventions are rules for determining in which equilibrium states the system is found when multiple equilibria materialize.According to the first convention the system remains in an equilibrium state until its disappearance.In terms of the Maxwell con- vention the system jumps from an existing equi- librium state to a better one as soon as the better equilibrium appears or becomes better.Both con- ventions, however, presuppose that the system instantly reaches an equilibrium and follows it as it changes.
In the generalized canonical gradient models, when the values of the control parameters and of the shift parameter w are fixed, the speed at which stable equilibria are approached is deter- mined by the parameter h.Thus, h identifies a measurable systemic attribute.
The non-canonical catastrophe models are any 'other' models (namely, neither canonical nor generalized canonical) possessing the topologies that characterize any given catastrophe (cf.for example Wilson and Kirby 1980 p. 344 if).

Expanded and Non-Expanded Models
In order to clarify the significance and impor- tance of this typology let us start by applying it to 'canonical' models.For each n-tuplets of con- trol parameter values, a canonical catastrophe structure or model corresponds to a specific to- pology characterized by a constellation of one or more stable equilibria with an appropriate com- plement of unstable equilibria.It does not matter whether these models are of the gradient or po- tential type, and to which elementary catastrophe they correspond: for one set of parameters they will all correspond to one specific constellation of stable and unstable equilibria.In order to be able to express the transition across topologies and/or the change of their equilibrium values across 'con- texts' these parameters have to change.In keep- ing with the 'modeling' section of this paper, we can formalize this change by expanding the con- trol parameters into functions of other variables.
A large number of expansions are possible.
Here, let us focus upon expansions with respect to time, and with respect to one or more vari- ables indexing some appropriate context of the model other than time.To clarify the rationale and usefulness of the expanded catastrophe mod- els, consider a model that for different values of its parameters is characterized by one of the topologies typical of a particular catastrophe.In itself this model is suited for 'comparative statics' analyses, but unsuited to model a switch in topologies over time, or across a 'context'.How- ever, if this model's parameters are expanded with respect to time or with respect to substan- tive contextual variables, the resulting terminal model can portray and resolve a switch in topologies over time or across the substantive context.
In the case of the gradient cusp catastrophe model (3) a duplet of numerical values of u and v corresponds to a specific cusp topology.Conse- quently, if u and v are estimated from empirical data we can determine the topology implied by the data.A comparative statics analysis involves comparing the topologies corresponding to alter- native data sets.However, a transition across topologies is outside the scope of this model.Instead, if u and v are redefined into functions of time t, u cuo + cut + cu2 t2 +-.., (12) P CV0 -+-CV1 n t-Cv2t 2 +''', the terminal model obtained by replacing the u and v in (3) with the right-hand sides of (12) and ( 13), for appropriate values of the ctj and Cv parameters, can produce a switch across the cusp catastrophe topologies over time.
The catastrophe literature on the fast and slow dynamics deals with an interesting class of expan- sions of canonical catastrophe models.This lit- erature differentiates between the dynamics of the fast variables (the state variables), and the dynamics of the slow variables that span the con- trol space.The slow dynamics is formalized by differential equations which specify the rate of change over time of control parameters such as u and v in the example above, as a function of time, of control parameter 'levels', or of slow variables.The fast variables adjust rapidly to their stable equilibrium levels, so that a system characterized by a fast and slow dynamics will reflect the changes in the stable equilibria of the fast variables resulting from the changes in con- trol parameters produced by the slow dynamics.
An early example is given in Zeeman (1972,  1973).In the fast/slow dynamics formalisms, the slow-dynamics equations are expansion equations of the initial model's (control) parameters, while the initial model is represented by the fast vari- ables' equations.
The differential handling of the fast and slow variables constitutes an important methodological contribution implicit in Thorn's initial formu- lation of the catastrophe theory (Thorn, 1975), but made explicit and placed into focus by Zeeman (1977, p. 65

if). Its use in connection
with the application of non-linear dynamics in the spatial sciences has been advocated and theo- rized by Dendrinos and Mullally (1981, 1985)  and Dendrinos and Sonis (1990).A related frame of reference in which fast and slow dynamics concepts appear is Haken's 'synergetics'.Haken   (1983) views dynamic systems in terms of slowly moving 'order parameters' and fast moving 'slave' variables or subsystems.
In the catastrophe and non-linear dynamics lit- eratures, we encounter variables that are fast, variables that are slow, and constants.Also, the fast-slow dychotomy itself may be replaced by multiple 'relative speeds of change' (Wilson,  1981; Dendrinos, 1989).A number of studies modeling multilevel time scales are reviewed in E. CASETTI Rosser (1991, p. 212).These multilevel time scales can be conceptualized as involving iterated expansions.
In the expanded canonical catastrophe models discussed in the previous paragraphs it is one or more control parameters that are expanded.If we expand the generalized canonical models, expan- sions can be carried out on the control parameters and/or on the link parameters such as h and w in the case of the cusp catastrophe (Eq.( 11)).Sup- pose that only h and w in (11) are expanded, say, with respect to time.The resulting terminal mod- el cannot display a catastrophic switch in topology, but is instead capable of accomodating temporal shifts in the values of the state variable corresponding to stable or unstable equilibria via changes in w, and a transition from a catastrophe to its dual via shifts in h carrying this parameter through a change in sign.If also u and v are expanded, the resulting terminal model can also accomodate catastrophic changes in topology.
The expansion of a generalized canonical mod- el can be carried out with respect to structural parameters such as h, w, u, and v for the cusp, but also with respect to the reduced form param- eters such as the c's in (8).If the latter are ex- panded, at each point in expansion space a set of c's becomes defined from which the values of the structural parameters for that point can be com- puted.This is the approach applied in the de- monstration presented later in this paper.The implications of expanding some or all the param- eters of a non-canonical catastrophe model are likely to be model specific and no attempt is made here to address them in generalities.

Deterministic and Stochastic Models
With some notable exceptions such as for in- stance Guastello (1982, 1987, 1988), CT has not entered yet to a sufficient degree into the inferen- tial stage, and tends to be identified with abstract deterministic models by scholars from fields in which preferences for models intended for infer- ence are firmly established.
Yet, though, catastrophe models and structures can be deterministic or stochastic.The determin- istic models are useful to formalize theory and to identify modalities of phenomena, but cannot be used for validation based on estimation and in- ference.For these, the deterministic models have to be converted into stochastic models by refor- mulating them as stochastic processes and/or by adding error terms to them.
The stochastic models can be differentiated into statistical and econometric, although the dif- ference between these is not clear cut.The sta- tistical models tend to be constructed by reformulating a deterministic catastrophe struc- ture as a stochastic non-linear difference or differential equation.These equations can be in- vestigated analytically, numerically, or by simula- tions, to obtain the probability density functions of the state variables that they imply, and in order to determine appropriate estimation ap- proaches (Cobb, 1978; 1992; Cobb et al., 1983;  Cobb and Zacks, 1985).
The econometric catastrophe models can be constructed by adding error terms to determinis- tic models while at the same time redefining some or all of their variables into random variables.Econometric modeling is employed in the exam- ple discussed later in this paper.First, though, let us touch upon some aspects and themes of econometrics that are relevant to catastrophe modeling.
A major portion of econometrics centers on the estimation based on empirical data of deter- ministic relationships that originated in mathe- matical economics.In fact, some authors have identified econometrics with this tradition.For example Johnston (1963, p. 3) writes "Economic theory consists of the study of various groups or sets of relations which are supposed to describe the functioning of a part or the whole of an economic system.The task of econometric work is to estimate these relationships statistically...".At p. 4 he adds "...for measurement and testing purposes, [deterministic] formulations are inadequate.The extension employed is the introduction of a stochastic term into economic discussed here can be converted into stochastic relationships."While econometrics had its origin models.Specifically, models and mathematical with the conversion of the deterministic models structures of the gradient or potential types, from mathematical economics into stochastic canonical or non-canonical, expanded or non- models suited for estimation and inference, over expanded, of catastropheselementary or otherwise, time its scope has become much wider.Any de- can be converted into statistical or econometric terministic relation with theoretical foundations models.not from economics, or without substantive theo- retical foundations, can be converted into econo- metric models by the addition of error terms.
A DEMONSTRATION Consequently, also deterministic catastrophe models can be transformed into econometric for- The catastrophe models prevalent in the earlier mulations, catastrophe literature are deterministic, unex- In the earlier econometrics the error terms panded, and are often based on canonical struc- were assumed to be well behaved RVs, normally tures.These types of models tended to be and independently distributed and with expectaassociated with an abstract and oversimplified tion zero and identical variances.Today, how- substantive modeling, based on variables unre- ever, the assumptions of independence (temporal lated to empirical referents and on relationships and spatial) and of homoschedasticity are rou- inadequately anchored to the causative presuppo- tinely tested, and when the null hypotheses of sitions and mechanisms that are so prominent in independence and homoschedasticity are rejected substantive literatures.Possibly, the future prosthe conversion of the deterministic model into an pects of CT's applications rest on the types of econometric one may involve not only suitably models that are better suited to fit within estab- specified error terms, but also temporally or spa- lished substantive analytical literatures.These tially lagged dependent and or independent vari- models are more likely to be based on generalables.The 'spatial' econometric developments in ized canonical or non-canonical catastrophe this general area represent a research frontier, structures, and to be expanded and stochastic.and have been extensively reviewed and devel- The demonstration that follows centers on the oped in Anselin (1988, 1992a, b).
construction and estimation of an expanded eco- Finally, let us note that in the more recent nometric cusp catastrophe model of modern eco- econometrics, the concept of 'data generating nomic growth.It involves the modeling of economic process' (Spanos, 1986; Darnell and Evans, 1990; growth over the 1700-1910 time span.The ear- Davidson and MacKinnon, 1993) has been used lier portion of this time horizon was still characto justify including 'additional variables' at the terized by a premodern dynamic.In premodern stage when an econometric model is constructed times the product per capita grew very slowly.
from a deterministic one.Such additional vari- With the industrial revolution, in the countries ables are not part of the theoretical deterministic that experienced it, the product per capita went model, but are required by the data generating through a phase of accelerated 'explosive' growth, process which produced the data in which the later followed by retardation.The question is: theoretical relationship under consideration is how, and on the basis of which reasoning, can embedded.All the developments touched upon we formulate a single mathematical model cap- here are .potentiallyrelevant to the construction able of representing these behaviors? of econometric catastrophe models.
In the sections that follow, first the aspects of It is important to point out that every one of the cusp catastrophe that are relevant to the the types of catastrophe models and structures modeling of modern economic growth are brought E. CASETTI into focus.Then an econometric expanded cusp catastrophe model of modern economic growth is constructed, estimated, and evaluated.condition 3x 2 4-u 0. (15) The Cusp Catastrophe The canonical cusp catastrophe equations involve one state variable and two control parameters.
Depending upon the values of the control param- eters, the 'topology' of the system defined by these equations is characterized by one or two stable equilibria.The smooth change of the con- trol parameters across critical thresholds can bring about the transition of the system from a 'low' stable equilibrium, to two stable equilibria, and again to a single 'high' stable equilibrium.As noted earlier, there are two equivalent canonical equations of the cusp catastrophe (and in general of all 'elementary' catastrophes), one in terms of a 'potential' function, and the second in the form of a gradient differential equation.The gradient equation is the one dealt with here.
In the gradient canonical equation of the cusp catastrophe (3) x is the state variable, and u, v are control parameters.In Zeeman's terminology u is a 'splitting factor' and v is a 'normal factor'.The parameter u determines whether the system has one or can have two stable equilibria.When u > 0 only one stable equilibrium can exist what- ever the value of v.When u < 0 it depends upon the values of v whether the system has a single low level stable equilibrium, or a low level and a high level stable equilibria, or a single high level equilibrium.Suppose that v=0 and that u changes from a positive value to a negative one.
At u=0 the stable equilibrium that exists for u >0 bifurcates into a low and an high stable equilibria.
The equilibria of (3) are the values of x for which 2--0, namely for which x + ux + v O.

(14)
The set of values of x that satisfy simultaneously ( 14) and (15) denote those equilibrium x's at which the extrema of 2(x) touch the zero axis.Eliminating x from ( 14) and (15) yields the cusp curve 4u 27v 2. (16) A switch in topology takes place at the values of u and v satisfying ( 16), that constitute the 'catas- trophe set'.
The canonical cusp equation ( 3) can be, and has been used for modeling substantive phenomena, but in many circumstances it is preferable to employ the more flexible generalized canonical cusp structure (8) that adjusts better to substan- tive variables and data.
By 'comparing' Eq. ( 8) and ( 11) the c's in (8)   can be related to the structural parameters u, v, h, and w, as follows: Oz 0 h(w 4glw 4-1).( 20) If the c's are given, from ( 17) through (20), we can easily obtain the linkage parameters h and w and the control parameters u and v. Namely V-(Oo/OZ3) (2/33)((1/3) 2(2/33)2). ( The values of x in correspondence to which attains a local maximum or minimum satisfy the Thus, if the structural parameters h, w,u, v are given the c's can be obtained; and if the c's are given the structural catastrophe parameters can be obtained. Suppose that ( 8) is converted into an econo- metric model by adding to its right-hand side an error term e: Oz3 z3 -+-Oz2 Z2 n t-CtlZ q-OZ 0 q-6. (25) We can then estimate the c's using empirical data, and then using ( 21)-( 24) obtain estimates of the structural catastrophe parameters.
The unexpanded econometric model ( 25) is useful, but limited in scope.It can only establish whether for given empirical data one or two stable equilibria occur, and thus it opens the way to 'comparative statics' types of analyses.
If some or all of the c's in ( 25) are expanded into variables indexing some suitable substantive context, the terminal model obtained can be used to implement comparative statics analyses.Upon estimation, one such terminal model will yield estimates of the c's at each point in expansion space, and consequently it allows also estimating the structural catastrophe parameters at each point in expansion space.
If we wish instead to establish whether a catas- trophic switch across dynamics has occurred the cgs of (25) should be expanded into deterministic or stochastic functions of time.To exemplify, let us expand the parameters of (25) into linear sto- chastic functions of time t: O AiO --]--Ail + 7i, (26) where T]i is a RV associated with the ith expan- sion equation.
Upon estimation (27) can establish whether over time the systemic equilibria changed in value, and whether a switch across topologies occurred when.
In closing on this point let us note that the parameters of ( 27) could be also expanded into variables indexing a substantive context to pro- duce a terminal model suited for a 'comparative dynamics' analysis.Such model could establish, for instance, whether changes in dynamics occur- red when and where across the substantive con- text considered.
Let us now consider why and how these concepts and related mathematical structures can be applied to the modeling of modern economic growth.Specifically, let us discuss briefly 'modern economic growth', then bring into focus why the cusp catas- trophe notions can give a useful insight into its dynamic, and articulate how a cusp catastrophe model ofmodern economic growth can be arrived at.

A Cusp Model of Modern Economic Growth
The dynamics of the product per capita, y, for the countries of North-West Europe over the 1700-1910 time horizon was characterized by the following.Before the industrial revolution, that started circa in 1750 in the UK, the product per capita was stagnant at premodern low values.It has been noted that in premodern times the rate of growth of y was so small to be negligible over any short to medium time interval.Instead, the industrial revolution brought about a phase of accelerated growth of product per capita, that was eventually followed by a phase of retardation (Kuznets, 1966; 1967; 1971).Economic growth theory has been a leading theme in modern eco- nomics (Hamberg, 1971; Burmeister and Dobell,   1970; Wan, 1971; Barro and Sala-I-Martin, 1995).
The premodern stagnation in product per capita has been theorized as the results of a Malthusian trap (Boulding, 1955).The explosive growth of y at the onset of the industrial revolution has been the focus of extensive theoretical and histor- ical literatures (Nelson, 1965; Leibenstein, 1957;  Rostow, 1960; Kuznets, 1971).The subsequent retardation in the growth rates of product per E. CASETTI capita has also been the object of theories and data analyses reviewed in Casetti (1986) and of the more recent literatures on the so-called 'convergence' (Baumol et al., 1994).
Here we are concerned with the more formal aspects of the dynamics of the product per capita, rather than on the economic and social mechanisms suggested to explain it.Within this perspective, the premodern stagnation of y can be conceptualized as the result of a slowly mov- ing low level 'point attractor'.A country's subse- quent explosive economic growth can be then construed as the initial effect of its capture by a high level point attractor possibly resulting from the disappearance of its low level counterpart.And finally the retardation in economic growth can be also explained by an increasing closeness to the high level attractor, that is also in the process of increasing slowly.
The empirical analyses to follow are based on (27), which is an econometric gradient general- ized-canonical cusp equation with all its param- eters expanded with respect to time.This equation, for and z denoting, respectively, per- centage rate of change and logarithm of GNP per capita, is well suited to test whether the hypothesized switch in topology and temporal changes in stable equilibria did occur for the countries and over the time horizon considered.
The analyses are based on the GNPs per capita for the UK, Denmark, Sweden, and Norway, for the years 1830, 1840,..., 1910, published in   Bairoch (1976, Table 6, p. 286).These data are in 1960 US dollars and are based on three year averages.Annual percentage growth rates of GNP per capita for the decades 1830-1840 to 1900-1910, and GNP per capita at the midpoints of these decades were calculated using these data.The countries included in the sample were selected because they are close enough to each other geo- graphically and otherwise.Time is in deviation from the year 1800.The time interval 1700-1910, addressed in the analysis, however, is wider than the data cover- age.The available data extend over the explosive growth phase and over a portion of the growth retardation phase, and ends before the period of convulsions and dislocations from World War to the early 1950s.However, the data available begin with 1830.Thus the data leave the crucial premodern stagnation uncovered.
In order to remedy this substantial shortcom- ing of the data 'prior information' has to be en- tered into the analysis.This could be accomplished by Mixed Estimation, or by Bayesian regression.The approach followed here was based on a 'quick and dirty' constrained regression, that re- presents a limiting case of Mixed Estimation and can be justified by a sufficiently strong confi- dence in the prior information.Specifically, the estimation of ( 27) was carried out subject to the condition that in the year 1700 the product per capita of the countries in the sample was 100 US 19605 per capita, and its rate of growth was zero.
A description of the specification search car- ried out to parametrize Eq. ( 27) is of no interest here.It will suffice to say that it produced the following estimated equation: 246.328 -133.54z+0.002199zt (3.38) (-3.47) (2.54) + 24.0247z 2 -1.43191z (3.55) (-3.62) The values are shown in parentheses under their respective regression coefficients.The equation is associated with an R-square of 0.472 and an ad- justed R-square of 0.415.
The evaluation of this estimated equation cen- ters on determining whether it is consistent with the notion that the dynamics of product per capita for the countries and time horizon selected in- volved a cusp catastrophe.The first step con- sisted in plotting the (y) relationship it implies at a sequence of points in time, and specifically, for the years 1725, 1750, 1775, and 1800.For the sake of clarity, let us be reminded that the (y) relationship, is between the percentage rate of change of product per capita, , and the product per capita, y.The plots are given in Fig. 1  :','.'"""t " ?,,\ _ .,,,,......,..,i .t ,t"""'." "' ""'LTY"'"".Figure shows that between 1725 and 1800 a change in topology did occur.The 1725 curve intersects the zero axis in one point only, thus indicating that before the industrial revolution only a low level 'Malthusian' stable equilibrium existed.The 1750 and 1775 curves show three equilibria: a low level stable equilibrium, a high level stable equilibrium, and an unstable equi- librium between them.Finally, the 1800 curve is characterized by a single stable high level equilibrium.
The second evaluation of the estimated equations is in terms of estimated structural parameters.The regressions are based on substantive variables and parameters.The structural coeffi- cients and variables are the control parameters and state variable that appear in CT, plus the h and w parameters that link the CT to the substan- tive variables.There is a substantial advantage to be gained by obtaining estimated structural param- eters: in this manner we can relate different esti- mates within the same substantive problem, as well as estimates from altogether different analyses in the same and in other substantive areas to the common yardsticks represented by the control parameters u and v.
The approach to obtaining estimated u,v,h, and w, that is employed here is fully general.As soon as we have estimates of the c's appearing in Eq. ( 26), we can calculate from them the struc- tural parameters using Eq. ( 21)-( 24).These esti- mated c's are obtained directly when we are dealing with unexpanded catastrophe models.Whenever instead we .aredealing with expanded models, the estimated expansion equations can be used to evaluate estimated c's at any point in expansion space.Then, from these we can again obtain the estimated structural parameters for that point in expansion space using Eq. ( 21)-( 24).
In this demonstration, the expansion space is time.The estimated c's were computed for the years 1700, 1710,... through 1850, and then esti- mated structural parameters were obtained from them.The trajectory in control space correspond- ing to the estimated u and v is shown in Fig. 2.This trajectory shows that the cusp lines are crossed twice.Trajectories crossing the catas- trophe set twice are the ones that produce the succession of topologies with one, then two, and then again one stable equilibria typical of the cusp catastrophe.

CONCLUSION
The themes discussed in this paper are all con- cerned with the application of catastrophe theory.Specifically, the paper touched upon these questions: what did CT add to our ability to con- struct mathematical models of realities?Why CT was received early on with an enthusiasm later followed by a wave of sharp criticisms?What are the prospects for CT's future?
The applications of mathematics in general, and in this case of CT, center on the mathemati- cal modeling of realities.The mathematical models come into existence when analytical mathematical structures such as equations are linked to a sub- stantive frame of reference by interpreting sub- stantively the variables and parameters in the structures.The application of CT involves linking analytical catastrophe structures to the substan- tive frames of reference of substantive disciplines.The positive response that followed the intro- duction of CT was due to its having generated a widespread awareness of the discontinuities brought about by the smooth change of control parameters across critical threshold, and by hav- ing pinpointed well-defined catastrophe types, some of which proved very useful.However, the catastrophy models based on canonical formulations prevalent in earlier applications were often imperfectly suited to the practices of the substan- tive scholars, especially in the social sciences.This mismatch contributed to the critical appraisals of CT.In this paper it is argued that non-canonical, expanded, stochastic catastrophe models and structures hold considerable promise with respect to the application of CT.The themes of this paper were demonstrated by constructing an econometric expanded gradient generalized- canonical cusp model of modern economic growth, and then by estimating it and evaluating its performance. .

FIGURE 2
FIGURE 2 Plot of estimated u, v trajectory.