```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.

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