Amson, J.C. 1975. “Catastrophe Theory: A Contribution to the Study of Urban Systems?” Environment and Planning B 2, pp. 177-221.
Andersson, Åke E. 1986. “The Four Logistical Revolutions,” Papers of the Regional Science Association 59:1, pp. 1-12.
Andronov, A.A., E.A. Leontovich, I.I. Gordon, and A.G. Maier. 1966. Qualitative Theory of Second-Order Dynamical Systems. Moscow: Nauk.
Arnol’d, Vladimir I. 1992. Catastrophe Theory, 3rd edition. Berlin: Springer-Verlag.
Arnol’d, Vladimir I., S.M. Gusein-Zade, and A.N. Varchenko. 1985. Singularities of Differentiable Maps, Volume I. Boston: Birkhäuser.
Arthur, W. Brian, John H. Holland, Blake LeBaron, Richard Palmer, and Paul Tayler. 1997. “Asset Pricing Under Endogenous Expectations in an Artificial Stock Market,” in The Economy as an Evolving Complex System II. W. Brian Arthur, Steven N. Durlauf, and David A. Lane, eds. Reading: Addison-Wesley, pp. 15-44.
Aubin, David. 2001. “From Catastrophe to Chaos: The Modeling Practices of Applied Topologists,” in Changing Images in Mathematics: From the French Revolution to the New Millennium. Umberto Bottazini and Amy Dahan Dalmedico, eds. London: Routledge, pp. 255-279.
Bak, Per, Kan Chen, José Scheinkman, and Michael Woodford. 1993. “Aggregate Fluctuations from Independent Sectoral Shocks,” Ricerche Economiche 47:1, pp. 3-30.
Balasko, Yves. 1978. “Economic Equilbrium and Catastrophe Theory: An Introduction,” Econmetrica 46:3, pp. 557-785.
Baldwin, Richard E., Philippe Martin, and Gianmarco P. Ottaviano. 2001. “Global Income Divergence, Trade, and Industrialization: The Geography of Growth Take-Offs,” Journal of Economic Growth 6:1, pp. 5-37.
Beckmann, Martin J. and Tönu Puu. 1985. Spatial Economics: Density, Potential, and Flow. Amsterdam: North-Holland.
Birkhoff, George D. 1927. Dynamical Systems. Providence: American Mathematical Society.
Black, Fischer. 1986. “Noise,” Journal of Finance 41:3, pp. 529-543.
Bonanno, Giacomo. 1987. “Monopoly Equilibria and Catastrophe Theory,” Australian Economic Papers 26:49, pp. 197-215.
Brock, William A. 1993. “Pathways to Randomness in the Economy: Emergent Nonlinearity in Economics and Finance,” Estudios Económicos 8:1, pp. 3-55.
Brock, William A., Steven Carpenter, and Donald Ludwig. 1999. “Management of Eutrophication for Lakes Subject to Potentially Irreversible Change,” Ecological Applications 9:3, pp. 751-771.
Bruno, Michael. 1967. “Optimal Accumulation in Discrete Models,” in Essays in the theory of optimal economic growth. Karl Shell, ed. Cambridge: MIT Press, pp. 181-218.
Chang. W.W. and D.J. Smyth. 1971. “The Existence and Persistence of Cycles in a Nonlinear Model: Kaldor’s 1940 Model Re-examined,” Review of Economic Studies 38:1, pp. 37-44.
Clark, Colin W. 1976. Mathematical Bioeconomics. New York: Wiley-Interscience.
Cobb, Loren. 1978. “Stochastic Catastrophe Models and Multimodal Distributions,” Behavioral Science 23, pp. 360-374.
Cobb, Loren. 1981. “Parameter Estimation for the Cusp Catastrophe Model,” Behavioral Science 26:1, pp. 75-78.
Cobb, Loren, P. Koppstein, and N.H. Chen. 1983. “Estimation and Moment Recursion Relationships for Multimodal Distributions of the Exponential Family,” Journal of the American Statistical Association 78:381, pp. 124-130.
Cobb, Loren and Shelemyahu Zacks. 1985. “Applications of Catastrophe Theory for Statistical Modeling in the Biosciences,” Journal of the American Statistical Association 80:392, pp. 793-802.
Cobb, Loren and Shelemyahu Zacks. 1988. “Nonlinear Time Series Analysis for Dynamic Systems of Catastrophe Type,” in Nonlinear Time Series and Signal Processing. R.R. Mohler, ed. Berlin: Springer-Verlag, pp. 97-118.
Copes, Parzival. 1970. “The Backward-Bending Supply Curve of the Fishing Industry,” Scottish Journal of Political Economy 17:1, pp. 69-77.
Debreu, Gérard. 1970. “Economies with a Finite Set of Equilibria,” Econometrica 38:3, 387-392.
Dechert, W. Davis and William A. Brock. 2000. “The Lake Game,” mimeo, http://dechert.econ.uh.edu/research/lakegame.pdf.
Dechert, W. Davis and Kazuo Nishimura. 1983. “A Complete Characterization of Optimal Growth Paths in an Aggregated Model with a Non-Concave Production Function,” Journal of Economic Theory 31:2, pp. 332-354.
Deissenberg, Christophe, Gustav Feichtinger, Willi Semmler, and Franz Wirl. 2001. “History Dependence, Multiple Equilibria and Global Dynamics in Efficient Intertemporal Optimization Models,” in Economic Complexity: Non-Linear Dynamics, Multi-agent Economies, and Learning. William A. Barnett, Christophe Deissenberg, and Gustav Feichtinger, eds., Amsterdam: Elsevier, chapter 4.
Dendrinos, Dimitrios S. 1979. “Slums in Capitalist Urban Settings: Some Insights from Catastrophe Theory,” Geographia Polonica 42, pp. 63-75.
Fischer, Edwin O. and Werner Jammernegg. 1986. “Empirical Investigation of a Catastrophe Theory Extension of the Phillips Curve,” Review of Economics and Statistics 68:1, 9-17.
Föllmer, Hans. 1974. “Random Economies with Many Interacting Agents,” Journal of Mathematical Economics 1:1, pp. 51-62.
Fussbudget, H.J. and Snarler, R.S. 1979. “Sagacity Theory: A Critique,” Mathematical Intelligencer 2, 56-59.
Gennotte, Gerard and Hayne Leland. 1990. “Market Liquidity, Hedging, and Crashes,” American Economic Review 80:5, pp. 999-1021.
George, Donald. 1981. “Equilibrium and Catastrophes,” Scottish Journal of Political Economy 28:1, pp. 43-61.
Gilmore, Robert. 1981. Catastrophe Theory for Scientists and Engineers. New York: John Wiley & Sons.
Guastello, Stephen J. 1981. “Catastrophe Modeling of Equity in Organizations,” Behavioral Science 26:1, 63-74.
Guastello, Stephen J. 1995. Chaos, Catastrophe, and Human Affairs: Applications of Nonlinear Dynamics to Work, Organizations, and Social Evolution. Mahwah: Lawrence Erlbaum Associates.
Guckenheimer, John. 1973. “Bifurcation and Catastrophe,” in Dynamical Systems. Manuel M. Peixoto, ed. New York: Academic Press, pp. 111-128.
Guckenheimer, John. 1978. “The Catastrophe Controversy,” Mathematical Intelligencer 1, pp. 15-20.
Guckenheimer, John and Philip Holmes. 1983. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Berlin: Springer-Verlag.
Ho, Thomas and Anthony Saunders. 1980. “A Catastrophe Model of Bank Failure,” Journal of Finance 35:5, pp. 1189-1207.
Horgan, John. 1995. “From Complexity to Perplexity,” Scientific American 272:6, pp. 104-109.
Horgan, John. 1997. The End of Science: Facing the Limits of Knowledge in the Twilight of the Scientific Age, pbk. edn. New York: Broadway Books.
Jones, Dixon D. and Carl J. Walters. 1976. “Catastrophe Theory and Fisheries Regulation.” Journal of the Fisheries Research Board of Canada 33:12, pp. 2829-2833.
Kaldor, Nicholas. 1940. “A Model of the Trade Cycle,” Economic Journal 50:197, pp. 78-92.
Kindelberger, Charles P. 1999. Manias, Panics, and Crashes, 3rd edn. New York: Basic Books.
Kolata, Gina Bari. 1977. “Catastrophe Theory: The Emperor has no Clothes.” Science, 196:4286, pp. 287, 350-351.
Krugman, Paul. 1984. “The International Role of the Dollar: Theory and Prospect,” in Exchange rate theory and practice. John F.O. Bilson and Richard C. Marston, eds. Chicago: University of Chicago Press, pp. 261-278.
Krugman, Paul. 1991. “Increasing Returns and Economic Geography,” Journal of Political Economy 99:3, 483-499.
Krugman, Paul. 1996. The Self-Organizing Economy. Oxford: Blackwell.
Kuznetsova, A. Yu., A.P. Kuznetsov, C. Knudsen, and E. Mosekilde. 2004. “Catastrophe Theoretic Classification of Nonlinear Oscillators,” International Journal of Bifurcations and Chaos 14:4, pp. 1241-1266.
Li, Dianmo, Zhen Zhang, Zufei Ma, Baoyu Xie, and Rui Wang. 2004. “Allee Effect and a Catastrophe Model of Population Dynamics,” Discrete and Continuous Dynamical Systems-Series B 4:3, pp. 629-634.
Lordon, Frédéric. 1997. “Endogenous Structural Change and Crisis in a Multiple Time-Scales Growth Model,” Journal of Evolutionary Economics 7:1, pp. 1-21.
Lorenz, Hans-Walter. 1989. Nonlinear Dynamical Economics and Chaotic Motion. Heidelberg: Springer-Verlag.
Lorenz, Hans-Walter. 1992. “Multiple Attractors, Complex Basin Boundaries, and Transient Motion in Deterministic Economic Systems,” in Dynamic Economic Models and Optimal Control. Gustav Feichtinger, ed. Amsterdam: North-Holland, pp. 411-430.
Ludwig, D., D.D. Jones, and C.S. Holling. 1978. “Qualitative Analysis of Insect Outbreak Systems: The Spruce Budworm and the Forest,” Journal of Animal Ecology 47:1, pp. 315-332.
Lux, Thomas and M. Marchesi. 1999. “Scaling and Criticality in a Stochastic Multi-Agent Model of a Financial Market,” Nature 397:2, pp. 498-500.
Magill, Michael J.P. 1977. “Some New Results on the Local Stability of the Process of Capitalist Accumulation,” Journal of Economic Theory 15:1, pp. 174-210.
Malkiel, Burton. 1975. A Random Walk Down Wall Street. New York: Norton.
Malgrange, Bernard. 1966. Ideals of Differentiable Functions. Oxford: Oxford University Press.
Marshall, Alfred. 1890. Principles of Economics. London: Macmillan.
Mather, John N. 1968. “Stability of CPO Mapping III: Finitely Determined Map-Germs,” Publications Mathématiques IHÉS 35, pp. 127-156.
Mees, Alistair I. 1977. “The Revival of Cities in Medieval Europe,” Regional Science and Urban Economics 5:4, pp. 403-425.
Morse, Marston. 1931. “The Critical Points of a Function of N Variables,” Transactions of the American Mathematical Society 33:1, pp. 72-91.
Oliva, T.A. and C.M. Capdeville. 1980. “Sussman and Zahler: Throwing the Baby Out with the Bathwater,” Behavioral Science 25:3, pp. 229-230.
Oliva, T.A. W.S. Desarbo, D.L. Day, and K. Jedidi. 1987. “GEMCAT: A General Multivariate Methodology for Estimating Catastrophe Models,” Behavioral Science 32:2, 121-137.
Poincaré, Henri. 1880-1890. Mémoire sur les Courbes Définies par les Équations Différentielles I-VI, Oeuvre I. Paris: Gauthier-Villars.
Poston, Tim and Ian Stewart. 1978. Catastrophe Theory and its Applications. London: Pitman.
Puu, Tönu. 1979. “Regional Modeling and Structural Stability,” Environment and Planning A 11, pp. 1431-1438.
Puu, Tönu. 1981a. “Structural Stability and Change in Geographical Space,” Environment and Planning A 13, pp. 979-989.
Puu, Tönu. 1981b. “Catastrophic Structural Change in a Continuous Regional Model,” Regional Science and Urban Economics 11:3, pp. 317-333.
Puu, Tönu. 1989. Nonlinear Economic Dynamics. Heidelberg: Springer-Verlag.
Rand, David. 1976. “Threshold in Pareto Sets,” Journal of Mathematical Economics 3:2, pp. 139-154.
Rand, David. 1978. “Exotic Phenomena in Games and Duopoly Models,” Journal of Mathematical Economics 5:2, pp. 173-184.
Rosser, J. Barkley, Jr. 1983. “Reswitching as a Cusp Catastrophe,” Journal of Economic Theory 31:1, 182-193.
Rosser, J. Barkley, Jr. 1991. From Catastrophe to Chaos: A General Theory of Economic Discontinuities. Boston: Kluwer. (Second edn. 2000a. Volume I: Mathematics, Microeconomics, Macroeconomics, and Finance. Boston: Kluwer)
Rosser, J. Barkley, Jr. 1997. “Speculations on Nonlinear Speculative Bubbles,” Nonlinear Dynamics, Psychology, and Life Sciences 1:4, pp. 275-300.
Rosser, J. Barkley, Jr. 1999. “On the Complexities of Complex Economic Dynamics,” Journal of Economic Perspectives 13:4, pp. 169-192.
Rosser, J. Barkley, Jr. 2000b. “Aspects of Dialectics and Nonlinear Dynamics,” Cambridge Journal of Economics 24:3, pp. 311-324.
Skiba, A.K. 1978. “Optimal Growth with a Convex-Concave Production Function,” Econometrica 46:3, pp. 527-539.
Smale, Steve. 1967. “Differentiable Dynamical Systems,” Bulletin of the American Mathematical Society 73, pp. 747-817.
Smale, Steve. 1974. “Global Dynamics and Economics IIA,” Journal of Mathematical Economics 1:1, pp. 1-14.
Stigler, George. 1948. “Notes on the History of the Giffen paradox,” Journal of Political Economy 56:1, pp. 61-62.
Strotz, Robert H., J.C. MacAnulty, and Joseph B. Naines, Jr. 1953. “Goodwin’s Nonlinear Theory of the Business Cycle: An Electro-Analog Solution.” Econometrica 21:3, pp. 390-411.
Sussman, Hector J. and Raphael Zahler. 1978a. “Catastrophe Theory as Applied to the Social and Biological Sciences,” Synthèse 37:2, pp. 117-216.
Sussman, Hector and Raphael Zahler. 1978b. “A Critique of Applied Catastrophe Theory in Applied Behavioral Sciences.” Behavioral Science 23, pp. 383-389.
Thom, René. 1956. “Les singularités des applications différentiables.” Annales Institute Fourier (Grenoble) 6, 43-87.
Thom, René. 1969. “Topological Models in Biology,” Topology 8, pp. 313-335.
Thom, René. 1972. Stabilité Structurelle et Morphogenèse: Essai d’une Théorie Générale des Modèles. New York: Benjamin (English trans. by D.H. Fowler. 1975. Structural Stability and Morphogenesis: An Outline of a Theory of Models. Reading: Benjamin).
Thom, René, with response by E. Christopher Zeeman. 1975. “CatastropheTtheory: Its Present State and Future Perspectives,” in Dynamical systems-Warwick 1974. Lecture Notes in Mathematics No. 468. Anthony Manning, ed. Berlin: Springer-Verlag, pp. 366-389.
Thom, René. 1983. Mathematical Models of Morphogenesis. Chichester: Ellis Harwood.
Thompson, J.M.T. 1982. Instabilities and Catastrophes in Science and Engineering. New York: John Wiley & Sons.
Trotman, David J.A. and E. Christopher Zeeman. 1976. “Classification of Elementary Catastrophes of Codimension 5,” in Structural stability, the theory of catastrophes and applications in the sciences. Peter J. Hilton, ed. Berlin: Springer-Verlag, pp. 263-327.
Varian, Hal R. 1979. “Catastrophe Theory and the Business Cycle,” Economic Inquiry 17:1, pp. 14-28.
von Mangoldt, Hans Karl Emil. 1863. Grundriss der Volkswirtschaftslehre. Stuttgart: Engelhorn.
Wagener, Floriaan O.O. 2003. “Skiba Points and Heteroclinic Bifurcations, with Applications to the Shallow Lake System,” Journal of Economic Dynamics and Control
27:9, pp. 1533-1561.
Walras, Léon. 1874. Éléments d’Économie Politique Pure. Lausanne: F. Rouge.
Walters, Carl. 1986. Adaptive Management of Renewable Resources. New York: Macmillan.
Weidlich, Wolfgang and Martin Braun. 1992. “The Master Equation Approach to Nonlinear Economics,” Journal of Evolutionary Economics 2:3, pp. 233-265.
Weintraub, E. Roy. 1983. “Critique and Comment: Zeeman’s Unstable Stock Exchange,” Behavioral Science 28:1, pp. 79-83.
Weintraub, E. Roy. 2002. How Economics Became a Mathematical Science. Durham: Duke University Press.
Weintraub, E. Roy. 2003. Personal Communication.
Whitney, Hassler. 1955. “Mappings of the Plane into the Plane,” Annals of Mathematics 62:3, Second Series, pp. 374-410.
Wilson, Alan G. 1976. “Catastrophe Theory and Urban Modelling: An Application to Modal Choice.” Environment and Planning A 8, pp. 351-356.
Woodcock, Alexander and Monte Davis. 1978. Catastrophe Theory. New York: E.P. Dutton.
Zahler, Raphael and Hector J. Sussman. 1977. “Claims and Accomplishments of Applied Catastrophe Theory,” Nature 269:10, pp. 759-763.
Zeeman, E. Christopher. 1974. “On the Unstable Behavior of the Stock Exchanges,” Journal of Mathematical Economics 1:1, pp. 39-44.
Zeeman, E. Christopher. 1977. CatastropheTtheory: Selected Papers, 1972-1977. Reading: Addison-Wesley.
1Program in Economics, James Madison University. The author is grateful to William Brock, Dee Dechert, Christophe Deissenberg, Gustav Feichtinger, Steven Guastello, Cars Hommes, Judy Klein, Hans-Walter Lorenz, Erick Mosekilde, Tönu Puu, Willi Semmler, Floriaan Wagener, and especially Roy Weintraub for either materials or advice. None of the above are responsible for any errors or questionable interpretations in this paper.
2 Rosser (1999) agrees with Horgan that the four C’s are linked through their common use of nonlinear dynamical systems. But he argues that this is something to be celebrated and appreciated rather than denigrated or dismissed. Just as the term “impressionism” in art was originally bestowed by a critic, so the “chaoplexologists” should accept the label originally provided in derision and wear it in pride. Ironically at the time that catastrophe theory was first criticized in 1977, its main founder, René Thom, worried that it could “have the same fate as cybernetics.” (Aubin, 2001, p. 274).
3 An example of this is Wagener (2003), to be discussed below, which uses catastrophe theory without ever mentioning its name. Of course there has been a non-trivial literature in recent years that has discussed catastrophe insurance. But such discussions rarely, if ever, rely upon catastrophe theory per se. That there seem to be fewer inhibitions about this in some other disciplines can be seen by recent applications in physics (Kuznetsova, Kuznetsov, Knudsen, and Mosekilde, 2004) and biology (Li, Zhang, Ma, Xie, and Wang, 2004).
4 For good overviews of catastrophe theory and its application in many disciplines, see Poston and Stewart (1978), Woodcock and Davis (1978), Gilmore (1981), Thompson (1982), and Arnol’d (1992). Except for the last of these, which was a third edition of a book initially written earlier by this famous Russian mathematician, all of these were written during the heyday of catastrophe theory’s faddishness.
5 For more extensive discussions of such applications in economics, see Rosser (1991, 2000a).
6 Two years later, Rand (1978) would publish the first self-conscious model of chaotic dynamics in economics, although others had provided such examples earlier without realizing what they were, e.g. Strotz, MacAnulty, and Naines (1953).
Although not specifically using catastrophe theory, Debreu’s colleague in mathematics at Berkeley, Steve Smale (1974) also studied structural stability of general equilibria, drawing on his earlier work on genericity that also provided a foundation for chaos theory (Smale, 1967). Smale was in close contact with Thom and Zeeman during the early 1970s (Aubin, 2001), and would later encourage Hal Varian to study catastrophe theory, leading him to develop his business cycle model discussed below (I thank Roy Weintraub, 1983, for this information, who interviewed Varian).
7 Balasko ultimately sided with the critics of applications of catastrophe theory in economics by noting that some of the necessary mathematical conditions are rarely fulfilled, especially that of a potential function.
8 For further discussion see Cobb, Koppstein, and Chen (1983) and Cobb and Zacks (1985, 1988). It is striking that few of the critics of catastrophe theory who charge that it cannot be used practically or empirically seem to be aware of this work.
9 Puu (1989) was probably the first to analyze an economic model using both catastrophic and chaotic dynamics in a model of business cycles in which an economy experiences temporary periods of chaotic dynamics immediately after catastrophic jumps occur. Rosser (1991) has labeled such a phenomenon as being “chaotic hysteresis.”
10 Krugman’s (1991) core-periphery model of regional economic structure could also be easily put into a catastrophe theory framework, although he has not done so. However, Baldwin, Martin, and Ottaviano (2001) have used the language of “catastrophic agglomeration” in connection with a closely related model, a continuing sign that in urban and regional economics there has remained more openness to such approaches as there has also within ecological economics, in contrast with most sub-fields of economics. Another paper to specifically mention the possibility of using catastrophe theory for the model studied was Gennotte and Leland’s (1990) model of financial market dynamics. Krugman (1996) in particular once wisecracked that he had “forgotten more catastrophe theory than most people ever knew in the first place.”
11 A satire of the work of the work of Sussman and Zahler appeared under the alleged authorship of “Fussbudget and Snarler” (1979).
12 Psychology is a field that has remained somewhat more open to using catastrophe theory than some others, with good coverage in Guastello (1995).
13 They also made a number of arguments that looked serious at the time but look petty in retrospect, such as that some of the crucial papers in catastrophe theory initially appeared in unrefereed Conference Proceedings.
14 It is perhaps not an accident that there remains a more favorable attitude towards catastrophe theory in Thom’s homeland of France, home of highly abstract thought, with Lordon (1997) providing a recent application in economics.. This observer has speculated that it may also have to do with the less dramatic meaning that the word “catastrophe” has in the French language than it does in English, with minor social faux pas regularly described as “catastrophes.” Weintraub (2002, p. 182) argues that Thom was a “Bourbakist.” Although he was initially trained by French Bourbakist mathematicians, the form of intellectual abstraction he pursued in this later period was very anti-Bourbakist in spirit and abjured formal, axiomatic approaches.
15 More generally Thom would argue that catastrophe theory showed how qualitative changes could arise from quantitative changes as in Hegel’s dialectical formulation. See Rosser (2000b) for further discussion.
16 A much studied model of the stock market with heterogeneous agents who have evolving strategies was developed at the Santa Fe Institute by Arthur, Holland, LeBaron, Palmer, and Tayler (1997). An example of econophysics modeling of heterogeneous agents in financial markets is Lux and Marchesi (1999).
17 Much of the immediate intellectual response to that event was for many economists to try to explain it using the sensitive dependence on initial conditions idea of chaos theory, which was very much near the peak of its intellectual bubble at that time. Almost nobody bothered to use the Zeeman model, which in retrospect looks much more suitable for explaining the kinds of really big discontinuities one observes in a major stock market crash. One who has used studied the 1987 crash using the Zeeman model is Guastello (1995, pp. 292-297). Rosser (1997) extended the Zeeman model using the five dimensional butterfly catastrophe model to explain the phenomenon observed by Kindleberger (1999), that many historical bubbles experienced a period of gradually declining prices during a “period of distress” after the peak and prior to the main crash. Thus in 1987 the market peaked in August and then gradually declined before plunging on October 19.