Modelling Iterative Roots of Mappings in Multidimensional Spaces

Solutions φ(x) of the functional equation φ(φ(x)) = f(x) are called iterative roots of the given function f(x). They are of interest in dynamical systems, chaos and complexity theory and also in the modelling of certain industrial and financial processes. The problem of computing this ”square root” in function (or operator) spaces remains a hard task and is, for the general case, still unsolved. While the theory of functional equations provides some insight for real and complex valued functions, iterative roots of mappings from R to R are not well understood by theory and there exists no published numerical algorithm for their computation. Here we prove existence of iterative roots of a certain class of monotonic mappings in R spaces and demonstrate how a method based on neural networks can find solutions to some examples that arise from simple physical systems.