A mathematical catastrophe is the abrupt or discontinuous transition between alternative states of a system in response to a smooth change in an underlying control parameter. Catastrophe theory is the mathematical framework, developed largely by the French mathematician Rene Thom, for dealing with such discontinuous processes. The theory has its roots in singularity theory and the smooth manifold mapping work of the American mathematician Hassler Whitney. A manifold is a space in which the local geometry is approximately Euclidean, but whose global geometry can be more complicated. Catastrophe theory has been applied with great quantitative success in the mathematical and physical sciences to problems such as structural stability and optics, and in the humanities using more heuristic approaches. It has been applied in ecology to describe discontinuous demographic processes and phenomena, such as the sudden onset and cessation of epidemic outbreaks, and the transition of communities and ecosystems between alternative stable states. It should be noted clearly that the term catastrophe does not equate to the magnitude of a measurable difference between alternative states, nor does it necessarily imply a perceived decline in the quality of the system (e.g., the eutrophication of a lake). Thom employed the word to emphasize abrupt transition between states, without the negative connotations of its meaning in the English language.
The application of 'catastrophe theory' to the description or prediction of alternative states and their transitions within any system depends on an adequate ability to identify underlying control parameters, and to define the relationships among them. The difficulty of realizing this for many ecological, and indeed nonphysical systems in general, has often resulted in the substitution of heuristic relationships for mathematical ones, and a lack of clarity between systems that fall strictly within a catastrophic framework as compared to other nonlinear or chaotic systems (though there are deep, nonelementary relationships among these). Catastrophic phenomena, however, do exhibit several characteristic properties, including and in addition to catastrophic transitions, such as stable alternative states, hysteresis in state transitions or reversals (i.e., slow response to parameter change, and slow or incomplete return to the initial state upon reversal of the parameter change), and the future divergence between systems given only slight differences in current states, when those systems are in the parameter neighborhood of a catastrophe. While the application of the theory to any natural system can therefore proceed with some qualitative knowledge of the underlying controls and their interrelationships, rigorous application is maximized by an appreciation of the founda-tional mathematical theory, and its practical statements, known generally as 'Thom's classification theorem'. One of Thom's central assertions is, however, that one may assume a priori the existence of a differentiable model of the system, and without explicit knowledge of the model, infer from that assumption the nature of singularities in the system. Conversely, then, the analysis of the observed behavior of a system's states and catastrophes can lead to a qualitative reconstruction of the underlying dynamics. Perhaps 'catastrophe theory's' strongest assertion is that the underlying dynamics of a large array of systems which display alternative states and singularities can be described by a restricted set of underlying manifold models, including Thom's original seven elementary catastrophes. These are, in order of increasing numbers of control parameters and topological complexity, the fold, cusp, swallowtail, butterfly, hyperbolic umbilic, elliptic umbilic, and parabolic umbilic catastrophes.
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