An efficient method for locating and computing periodic orbits of nonlinear mappings
type: Item
instance of: scholarly article
title: An efficient method for locating and computing periodic orbits of nonlinear mappings
zbMATH Open document ID: 0826.65049
author: Michael N. Vrahatis
published in: Journal of Computational Physics
publication date: 1995-08-02T00:00:00Z
review text: Periodic points of a mapping \(\Phi : \mathbb{R}^N \to \mathbb{R}^N\) of period \(p\) are roots of the nonlinear system \(F(x) = \Phi^p (x) - x\), where \(\Phi^p := \Phi \circ \Phi \circ \cdots \circ \Phi\) (\(p\) times). Instead of applying Newton-like methods, the derivative-free bisection method of \textit{M. N. Vrahatis} and \textit{K. I. Iordanidis} [Numer. Math. 49, 123-138 (1986; Zbl 0604.65031)] is suggested to solve \(F(x) = 0\) to an arbitrarily high accuracy. The bisection method starts with a characteristic \(n\)-polyhedron, where the signs of \(F(Y^k)\) have to fulfill certain combinatorial rules at the \(2^n\) vertices \(Y^k\) of the polyhedron. This property allows a degree theoretical conclusion about the existence of solutions in the polyhedron under suitable conditions on the behavior of the mapping on its boundary. A refinement procedure of the polyhedron leads to an approximation of a root to arbitrary precision. The method is demonstrated for finding periodic points (and the whole periodic orbit) of Henon's two-dimensional, quadratic area preserving map (and its four-dimensional complexification) of smaller and larger periods (up to period 62208). The reviewer would like to refer to similar methods based on completely integer-labeled \(n\)-simplices to solve nonlinear equations [see \textit{M. J. Todd}, Lecture Notes in Economics and Mathematical Systems. 124 (1976; Zbl 0332.54003)].
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 37C25
zbMATH DE Number: 772379
zbMATH Keywords: nonlinear mappings
zbMATH Keywords: topological degree
zbMATH Keywords: simplicial method
zbMATH Keywords: derivative- free bisection method
zbMATH Keywords: characteristic \(n\)-polyhedron
zbMATH Keywords: periodic orbit
reviewed by: Bodo Werner
describes a project that uses: CHABIS
MaRDI profile type: Publication
full work available at URL: jcph.1995.1119
OpenAlex ID: W2077490941
DOI: 10.1006/JCPH.1995.1119
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description: scientific article; zbMATH DE number 772379