Curve fitting--5 S. Arlinghaus Feigenbaum's graphical analysis Feigenbaum's graphical analysis (Feigenbaum, 1980) is a tool from mathematical chaos theory that offers a strategy to understand how small geometric changes lead to large geometric differences. This form of analysis rests on an ordering of events that is not necessarily temporal, but in which the output of one stage serves as the input for the next stage. In the figure below, the line y=x is used as an axis in which the output of one stage becomes the input of the next. An input of x, leads to an output of y, which is then used (after shifting horizontally to the line y=x) as an input to produce an output of y'; then y' is used as an input to generate y'', and so forth. Instead of reading input values from the x-axis, they are read from y=x so that the resulting geometric pattern, in this case, is a rising staircase. The initial value, (x,0), is called the "seed" value of the analysis. The geometric pattern, forced on the trajectory of the seed by the relative positions of the curve and the line y=x, is called the orbit of the seed value. In this case, seed values to the right of (P,0) (but not past the next intersection of the curve with y=x) all generate ascending staircases, which may subsequently exhibit even greater geometric complexity; those to the left of (P,0) (but not to the left of the previous intersection of the curve with y=x) all generate descending staircases. Figure Feigenbaum's graphical analysis applied to a curve; to the right of P orbits are ascending staircases; to the left they are descending. If the descending situation indicates a favorable geometric dynamic--one that is under control--then the point P suggests a threshold of irreversibility; beyond it, the geometric process takes off in an undesired direction (Arlinghaus, Nystuen, and Woldenberg, 1992). However, because P is found as an intersection of a curve and a line, a slowing of the increase, anywhere to the left of P, means that the threshold is shifted further to the right--or that its attainment is delayed and may be delayed indefinitely as long as intervention to the left of P, to control the increase in geometric process, continues to prevent the intersection of the curve and the line y=x. Abstract tools such as this one offer a great deal of promise in suggesting directions for theoretical research--the critical component underlying any form of analysis of data. References 1. Arlinghaus, S. L., Nystuen, J. D., Woldenberg, M. J. An application of graphical analysis to semidesert soils. Geographical Review, 1992, Vol. 82, No. 3, pp. 244-252. 2. Feigenbaum, M. J. 1980. Universal behavior in non-linear systems. Los Alamos Science, summer, 4-27. 3. Kates, R. W. and Burton, I. Geography, Resources, and Environment, 1986, Chicago, University of Chicago Press. 4. Nystuen, J. D. Effects of boundary shape and the concept of local convexity. 1966. Papers, Michigan Inter-University Community of Mathematical Geographers, 10:3-24, Ann Arbor. 5. Tobler, W. R. Personal communication. 1993.