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catastrophe theory |
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A branch of mathematics, developed by Rene Thom (1975), concerned with discontinuous relationships (see also bifurcation). In a two-dimensional situation, as in figure 1, there is a portion of the relationship between the two variables (the end-points of which — A and B — are called the fold-points) where the value of a is associated with two separate values of x. Isnard and Zeeman (1976) illustrated this with the relationship between threat and military action, their graph showing that although in general as the intensity of a threat increases so does the probability of military action between the fold-points there are two values for the probability of a military action (i.e. at a0 the probability may be either x4 or x6). These represent two reactions to the threat — of the dove and of the hawk. There is both an upper level of threat (B on the graph) beyond which public opinion will not accept the doveish option and a lower level (A) below which it will not accept the hawkish reaction. Where the two overlap is what is termed a hysteresis, indicating a range of threat levels within which the likely response is not readily predicted, and the shift from one relationship to the other can be understood in qualitative terms only.
In three-dimensional situations, where the response variable is related to two stimuli (as in figure 2), there is a fold curve (the area shown by M-G) where the probability of the response is indeterminate; the stimuli are the degree of threat and the cost of countering it by military action, the response variable is the probability of military action, and within the cusp the probability of a military action can take several values. The fold curve can take various forms: figure 3 shows a butterfly catastrophe which Isnard and Zeeman identify as a \'pocket of compromise\' for those promoting the hawkish and doveish responses.
The existence of catastrophes creates difficulties in modelling systems, especially in predictive work, since the same conditions can produce different outcomes. Wilson (1981) has illustrated this with examples of the internal structure of cities. (RJJ)
{img src=show_image.php?name=bkhumgeofig6a.gif }
catastrophe theory 1: A threat-action graph for high cost (Isnard and Zeeman, 1976)
{img src=show_image.php?name=bkhumgeofig6b.gif }
catastrophe theory 2: The cusp catastrophe (Isnard and Zeeman, 1976)
{img src=show_image.php?name=bkhumgeofig6c.gif }
catastrophe theory 3: A section of the butterfly catastrophe (Isnard and Zeeman, 1976)
References and Suggested Reading Isnard, C.A. and Zeeman, E.C. 1976: Some models from catastrophe theory in the social sciences. In L. Collins, ed., The use of models in the social sciences. London: Tavistock Publications, 44-99. Thom, R. 1975: Structural stability and morphogenesis. Reading, MA: W.A. Benjamin. Wilson, A.G. 1981: Catastrophe theory and bifurcation: applications to urban and regional systems. London. Croom Helm; Berkeley: University of California Press. |
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