Solving iterated functions using genetic programming

An iterated function f(x) is a function that when composed with itself, produces a given expression f(f(x))=g(x). Iterated functions are essential constructs in fractal theory and dynamical systems, but few analysis techniques exist for solving them analytically. Here we propose using genetic programming to find analytical solutions to iterated functions of arbitrary form. We demonstrate this technique on the notoriously hard iterated function problem of finding f(x) such that f(f(x))=x2–2. While some analytical techniques have been developed to find a specific solution to problems of this form, we show that it can be readily solved using genetic programming without recourse to deep mathematical insight. We find a previously unknown solution to this problem, suggesting that genetic programming may be an essential tool for finding solutions to arbitrary iterated functions