PREDICTIONS – 10 Years Later - Santa Fe Institute

10. IF I CAN, I WANT
Between two successive curves there is typically a period of instability
characterized by chaotic fluctuations.
Debecker and I tried to reproduce mathematically the chaotic patterns
seen in the coal production data. We knew that in order to
observe chaos we had to pass to the discrete case, but instead of tampering
with Verhulst’s law, the differential equation, we put into a
discrete mathematical form only its solution, the S-curve. By doing that
we obtained instabilities at both ends of the growth process (see Figure
10.4).
In real-life situations we cannot see the early oscillations completely
because negative values have no physical meaning. We may see, however,
a precursor followed by an accelerated growth rate, an overshoot
of the ceiling, and finally erratic fluctuations. These phenomena are interrelated.
Precursors are not simple cases of “infant mortality.” Their
size and frequency can help predict the steepness of the catching-up trajectory,
which in turn will reflect on the importance of the overshoot
when the ceiling is approached. Marketers are familiar with these phenomena
but do not always understand them. They refer to the catchingup
effect as pending demand, and are likely to blame over-shooting the
market on poor planning. But they invariably treat precursors as failures.
An example is the BankAmericard (the predecessor of the Visa
card) that was introduced by the Bank of America in 1958. A year and a
half later the experiment was classified a failure; the bank lost money
and fired many employees. But nine years later the predecessor of
MasterCard was launched and diffused aggressively into society.
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FIGURE 10.4 (next page). These are examples of the patterns we obtained by
putting a natural-growth curve in a discrete mathematical form. The figure at
the top demonstrates the so-called early catching-up effect, and the possibility
of precursor and overshoot. In the middle, we see regular oscillations, and at
the bottom states resembling chaos. In all cases, similar behavior is observed
early as well as late in the growth process. ∗
∗ Adapted from Theodore Modis and Alain Debecker, “Chaoslike States Can Be Expected
Before and After Logistic Growth,” Technological Forecasting and Social
Change, vol. 41, no. 2 (1992).
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