Iterative roots of multidimensional operators and applications to dynamical systems

Iterative roots of multidimensional operators and applications to dynamical systems

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Article ID: iaor2014287
Volume: 7
Issue: 8
Start Page Number: 1701
End Page Number: 1710
Publication Date: Dec 2013
Journal: Optimization Letters
Authors: , ,
Keywords: functional differential equation, system dynamics
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 n equ1 to n equ2 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 n equ3 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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