9: What is the logistic equation?
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This article is from the Fractal FAQ, by Ermel Stepp stepp@muvms6.mu.wvnet.edu with numerous contributions by others.
9: What is the logistic equation?
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" for more information.
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