The logistic map
The logistic equation was first proposed by Robert May as a simple model of population dynamics. This equation can be written as a one-dimensional difference equation that transforms the population in one generation, x[sub]n[/sub], into a succeeding generation, x[sub]n+1[/sub].
x[sub]n+1[/sub] = 4 r x[sub]n[/sub] (1-x[sub]n[/sub])
Because the population is scaled so that the maximum value is one, the domain of x falls on the interval [0; 1].
The behavior of the logistic equation depends on the value of the growth parameter, r. If the growth parameter is less than a critical value r<0.75.., then x approaches a stable fixed value. Above this value for r, the behavior of x begins to change. First the population begins to oscillate between two values. If r increases further, then x oscillates between four values. Then eight values. This doubling ends when r > 0.8924864... after which almost any x value is possible.
Author : Francisco Esquembre and Wolfgang Christian.
Text and original idea from the Open Source Physics project manual
Date : July 2003