# 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, xn, into a succeeding generation, xn+1.

xn+1 = 4 r xn (1-xn)

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.