Chapter 6: Starting from Randomness

Section 6: Special Initial Conditions

[Repetition in] systems based on numbers

An iterated map of the kind discussed on page 150 with rule x -> Mod[a x, 1] (with rational a) will yield repetitive behavior when its initial condition is a rational number. The same is true for higher-dimensional generalizations such as so-called Anosov maps {x, y} -> Mod[m . {x, y}, 1]. The continued fraction map x -> Mod[1/x, 1] discussed on page 914 becomes repetitive whenever its initial condition is a solution to a quadratic equation.

For a map x -> f[x] where f[x] is a polynomial such as a x (1 - x) the real initial conditions that yield period p are given by

Select[x /. Solve[Nest[f, x, p] == x, x], Im[#] == 0 &]

For x -> a x (1 - x) the results usually cannot be expressed in terms of explicit radicals beyond period 2. (See page 961.)

From Stephen Wolfram: A New Kind of Science [citation]