This article is from the sci.fractals FAQ, by Michael C. Taylor and Jean-Pierre Louvet with numerous contributions by others.
It models animal populations. The equation is x -> c x (1 - x),
where x is the population (between 0 and 1) and c is a growth
constant. Iteration of this equation yields the period doubling route
to chaos. For c between 1 and 3, the population will settle to a fixed
value. At 3, the period doubles to 2; one year the population is very
high, causing a low population the next year, causing a high
population the following year. At 3.45, the period doubles again to 4,
meaning the population has a four year cycle. The period keeps
doubling, faster and faster, at 3.54, 3.564, 3.569, and so forth. At
3.57, chaos occurs; the population never settles to a fixed period.
For most c values between 3.57 and 4, the population is chaotic, but
there are also periodic regions. For any fixed period, there is some c
value that will yield that period. See "An Introduction to Chaotic
Dynamical Systems", by R. L. Devaney, for more information.