Fitting the Cusp Catastrophe inR: AcuspPackage Primer

Abstract

Of the seven elementary catastrophes in catastrophe theory, the "cusp" model is the most widely applied. Most applications are however qualitative. Quantitative techniques for catastrophe modeling have been developed, but so far the limited availability of flexible software has hindered quantitative assessment. We present a package that implements and extends the method of Cobb (Cobb and Watson 1980;Cobb, Koppstein, and Chen 1983), and makes it easy to quantitatively fit and compare different cusp catastrophe models in a statistically principled way. After a short introduction to the cusp catastrophe, we demonstrate the package with two instructive examples.Keywords: catastrophe theory, cusp, modeling.Catastrophe theory studies qualitative changes in behavior of systems under smooth gradual changes of control factors that determine their behavioral state. In doing so, catastrophe theory has been able to explain for example, how sudden abrupt changes in behavior can result from minute changes in the controlling factors, and why such changes occur at different control factor configurations depending on the past states of the system. Specifically, catastrophe theory predicts all the types of qualitative behavioral changes that can occur in the class of so called gradient systems, under influence of gradual changes in controlling factors (Poston and Stewart 1996).Catastrophe theory was popularized in the 1970's (Thom 1973;Thom and Fowler 1975;Gilmore 1993), and was suggested as a strategy for modeling in various disciplines, like physics, biology, psychology, and economics, by Zeeman (1971)-as cited in Stewart and Peregoy (1983), Zeeman (1973), Zeeman (1974), Isnard and Zeeman (1976), and Poston and Stewart (1996. Although catastrophe theory is well established and applied in the physical sciences, its applications in the biological sciences, and especially in the social and behavioral sciences, has been criticized (Sussmann and Zahler 1978;Rosser 2007). Applications in the latter fields range from modeling of exchange market crashes in economics (Zeeman 1974), to

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