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functions, the model is either in chaos or (more likely), the choice of DT is too large.
The DT choice in the latter case is a longer time than the critical time of the model.
Shortening the DT will cause such a model to behave with at least some smooth
curves. For the most part, these smooth curves will have continuous derivatives. Pa-
rameter change can bring a return of chaotic behavior, and in the nonchaotic model,
bring back of return of random behavior. The best test here is an examination of the
model in the scatter plot and a check on the sensitivity to initial condition changes.
At this point we should ask if chaos occurs in nature. We find that, indeed, it does.
Chaos is evident in the variations of heartbeats and brainwaves or the irregularity
of water dripping from a faucet. Both living and nonliving systems seem to show
chaos. Why? To what advantage is such a result to these systems? Stuart Kaufman,
1
among others, has proposed that all systems seem to evolve toward higher and higher
efficiencies of operation. Many systems are so highly disturbed by variations in their
environment that their efficiencies are rarely high. However, if these disturbances
can be held to a minimum, then the evolution of the system becomes more complete
and more efficient but closer to the border of chaotic behavior.
Earthquakes and avalanches are examples of energy-storing systems that contin-
uously redistribute the incoming stresses more and more efficiently until a breaking
point is reached and the border to chaos is opened. Does this mean that the brain
and the heart have somehow evolved closer to some maximum efficiency for such
organisms? We do not know the answer. We do know that the scale of measurement
matters. For example, if we were to watch the pattern on a patch of natural forest
over many centuries, we would see the rise and sharp fall of the biomass levels, un-
predictably. Forest fires and insects find ample host in such forest patches once they
have developed a large amount of dry biomass bound up in relatively few species.
The patch evolves or succeeds to greater and greater efficiency of light energy con-
version by getting larger and fewer species. But the patch also becomes more vul-
nerable to fire and pests, and eventually collapses. Yet if we look at the total biomass
on a large collection of such biomasses, whose collapses are not synchronized, this
total biomass remains relatively constant. Thus, chaotic-like behavior in the small
is not seen in the large. Could this mean that natural systems have “found” chaos in
their search for greater efficiencies and have “learned” to stagger the chaotic events,
allowing faster rebound and large-scale stability? We do not know the answers, but
we think the implications are fascinating.
13.1.3 Questions and Tasks
1. Re-create the model described in this chapter, but try to find chaos in a different
way. Instead of increasing the maximum value for the CONTAGION RATE for
subsequent runs in the graph of Figure 13.2, set that maximum to 1.4 and increase
the curvature of the graph. Can you make chaos occur? What does this mean for
the underlying mechanisms behind this epidemic?
1
Kaufman, S. 1993. The Origins of Order: Self-Organization and Selection in Evolution,
New York, Oxford University Press.
