We are dealing here with random variables recursively defined by Xn+1 = g(Xn), with X1 being the initial condition. The examples discussed here are simple, discrete and one-dimensional: the purpose is to illustrate the concepts so that it can be understood and useful to a large audience, not just to mathematicians. I wrote many articles about dynamical systems, see for example here. The originality in this article is that the systems discussed are now random, as X1 is a random variable. Applications include the design of non-periodic pseudorandom number generators, and cryptography. Also, such systems, especially more complex ones such as fully stochastic dynamical systems, are routinely used in financial modeling of commodity prices.

We focus on mappings g on the fixed interval [0, 1]. That is, the support domain of Xn is [0, 1], and g is a many-to-one mapping onto [0,1]. The most trivial example, known as the dyadic or Bernoulli map, is when g(x) = 2x – INT(2x) = { 2x } where the curly brackets represent the fractional part function (see here). This is sometimes denoted as g(x) = 2x mod 1. The most well-known and possibly oldest example is the logistic map (see here) with g(x) = 4x(1 – x).

We start with a simple exercise that requires very little mathematical knowledge, but a good amount of out-of-the-box thinking. The solution is provided. The discussion is about a specific, original problem, referred to as the inverse problem, and introduced in section 2. The reasons for being interested in the inverse problem are also discussed. Finally, I provide an Excel spreadsheet with all my simulations, for replication purposes.

1. The standard problem

One of the main problems in dynamical systems is to find if the distribution of Xn converges, and find the limit, called invariant measure, invariant distribution, fixed-point distribution, or attractor. The attractor, depending on g, is typically the same regardless of the initial condition X1, except for some special initial conditions causing problems (this set of bad initial conditions has Lebesgue measure zero, and we ignore it here). As an example, with the Bernoulli map g(x) = { 2x }, all rational numbers (and many other numbers) are bad initial conditions. They are however far outnumbered by good initial conditions. It is typically very difficult to determine if a specific initial condition is a good one. Proving that π/4 is a good initial condition for the Bernoulli map would be a major…