Iterative roots of multidimensional operators and applications to dynamical systems |
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Authors: | Email author" target="_blank">Pando?GeorgievEmail author Lars?Kindermann Panos?M?Pardalos |
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Institution: | 1.University of Florida,Gainesville,USA;2.Alfred Wegener Institute for Polar and Marine Research (AWI),Bremerhaven,Germany |
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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. |
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