## Logistic Equation

One often looks toward physical systems to find chaos, but it also exhibits itself in biology. Biologists had been studying the variability in populations of various species and they found an equation that predicted animal populations reasonably well. This equation was a simple quadratic equation called the logistic difference equation. On the surface, one would not expect this equation to provide the fantastically complex and chaotic behavior that it exhibits.

The logistic difference equation is given by

where *r* is the so-called driving parameter. The equation
is used in the following manner. Start with a fixed value
of the driving parameter, *r*, and an initial value of
*x*_{0}. One then runs the equation recursively,
obtaining *x*_{1}, *x*_{2},
. . .*x*_{n}. For low values of *r*,
*x*_{n} (as *n* goes to infinity)
eventually converges to a single number. In biology, this number
(*x*_{n} as *n* approaches infinity) represents
the population of the species.

It is when the driving parameter, *r*, is slowly turned
up that interesting things happen. When *r* = 3.0,
*x*_{n} no longer converges — it oscillates
between two values. This characteristic change in behavior
is called a bifurcation. Turn up the driving parameter even
further and *x*_{n} oscillates between not two, but
four values. As one continues to increase the driving parameter,
*x*_{n} goes through bifurcations of period eight,
then sixteen, then chaos! When the value of the driving parameter
*r* equals 3.57, *x*_{n} neither converges or
oscillates — its value becomes completely random. For values
of *r* larger than 3.57, the behavior is largely chaotic.
However, there is a particular value of *r* where the
sequence again oscillates with period of three. The bifurcations
then begin again with period 6, 12, 24, then back to chaos. In fact
it was discovered in James Yorke's famous paper "Period Three Implies
Chaos." that any sequence with a period of three will display regular
cycles of every other period as well as exhibiting chaotic cycles.

The bifurcation diagram of the logistic difference equation is shown below: