An efficient method for locating and computing periodic orbits of nonlinear mappings
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)].
scientific article; zbMATH DE number 772379