and after the next generation (and every subsequent generation) there
will be 50 percent rabbits. Other values of x drift toward .5 as time passes.
For instance, if x .8, then f(.8) 2 .8.2 .32, f(.32) 2 .32 .68
.4352, and f(.4352).49160192; after just three generations, a rabbit popu-
lation of 80 percent has become a rabbit population of almost 50 percent.
At a 3 , there are two equilibrium points. This state of affairs continues
until a3.5, when there are four points; but as a increases to 3.56, the
number of equilibrium points increases to eight, then sixteen, then
thirty-two,.... When a3.569946, something utterly bizarre happens:
there are no equilibrium points at all! As a increases from 3.6 to 4, we
see the development of chaos; the number of equilibrium points vary
unpredictably, with intervals characterized by an absence of equilibrium
points followed by intervals in which the smallest change in the value of
a creates a wildly different number of equilibrium points. It’s a com-
pletely deterministic system, but it’s one in which it’s impossible to pre-
dict the number of equilibrium points. In a chaotic system such as this
one, the equilibrium points are called strange attractors.
The following table gives an indication of how the number of equilib-
rium points of the system changes as a increases. The numbers on the
top line represent the generations; the values in the table indicate the frac-
tion of the population consisting of rabbits. In each case, the first genera-
tion starts off with half of the total population consisting of rabbits; the
rest of the table shows the rabbit fraction for generations 126–134. When
a2.8, the population stabilizes at 64.3 percent rabbits. When a 3.1, the
rabbit population oscillates between 76.5 percent and 55.8 percent. When
a3.5, there are four equilibrium points; and when a3.55, there are
eight equilibrium points (generation 135 repeats the value of generation
127, generation 136 repeats the value of generation 128, and so on).
Generation 1 126 127 128 129 130 131 132 133 134
a 2.8 0.5 0.643 0.643 0.643 0.643 0.643 0.643 0.643 0.643 0.643
a 3.1 0.5 0.765 0.558 0.765 0.558 0.765 0.558 0.765 0.558 0.765
a 3.5 0.5 0.383 0.827 0.501 0.875 0.383 0.827 0.501 0.875 0.383
a 3.55 0.5 0.355 0.813 0.54 0.882 0.37 0.828 0.506 0.887 0.355The Prevalence of Chaos
Chaotic behavior can be seen in a wide variety of phenomena: the relative
populations of predator and prey, the spread pattern of disease epidemics,
the onset of cardiac arrhythmia, prices in energy markets, the f lipping of
182 How Math Explains the World