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  A change in the solution to a differential or difference equation at a critical value of a model\'s parameter. Three types of change are common (see figure): (a) a \'jump\' in the relationship between x and t at a critical value of t(tc), shown in the figure as a discontinuous \'step function\'; (b) a shift from a linear relationship at tc to a periodic one; and (c) a shift at tc from a linear to a chaotic relationship. (See also chaos and catastrophe theory: the latter covers a special case of the general features of bifurcation.)

{img src=show_image.php?name=bkhumgeofig4a.gif }

{img src=show_image.php?name=bkhumgeofig4b.gif }

{img src=show_image.php?name=bkhumgeofig4c.gif }

bifurcation Three possibilities

Bifurcations are common in models of systems involving interdependence among the variables, especially if those interrelationships are nonlinear. These are common in environmental science but are also typical of some aspects of human geography: \'jumps\' may characterize the crossing of critical thresholds in the relationship between the percentage of the votes cast and the percentage of the seats won by a political party in first-past-the-post electoral systems, for example, whereas transitions to chaos may occur in the process of population change over time in a society which has reached the carrying capacity of its land. (RJJ)

Suggested Reading 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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