Iterative roots of multidimensional operators and applications to dynamical systems

Abstract

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 modeling of certain industrial and financial processes. The problem of computing this “square root” of a function or operator remains a hard task. While the theory of functional equations provides some insight for real and complex valued functions, iterative roots of nonlinear mappings from ${\mathbb{R}^n}$ to ${\mathbb{R}^n}$ are less studied from a theoretical and computational point of view. Here we prove existence of iterative roots of a certain class of monotone mappings in ${\mathbb{R}^n}$ spaces and demonstrate how a method based on neural networks can find solutions to some examples that arise from simple physical dynamical systems.