# 9: What is the logistic equation?

## Description

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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