The Inverse Problem in Random Dynamical Systems

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. 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. This is sometimes denoted as?g(x) = 2x?mod 1. The most well-known and possibly oldest example is the logistic map 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. Before discussing the inverse problem, we discuss the standard problem in section 1.

Read full article here.