A numerical method for locating stable periodic orbits in chaotic systems
type: Item
instance of: scholarly article
title: A numerical method for locating stable periodic orbits in chaotic systems
zbMATH Open document ID: 0761.58033
DOI: 10.1016/0167-2789(92)90176-N
author: Dov Grobgeld
author: Eli Pollak
author: Jakub Zakrzewski
published in: Physica D
publication date: 1993-01-16T00:00:00Z
review text: The knowledge of the stable and weakly unstable periodic orbits of chaotic systems is a subject of increasing importance, both from the theoretical point of view, as well as for experimental studies. Recent work on the periodic orbit expansion in terms of zeta functions has shown that only a few short orbits are needed in order to construct the full classical spectrum [\textit{P. Cvitanović} and \textit{B. Eckhardt}, J. Phys. A, Math. Gen. 24, L 237--L 241 (1991; Zbl 0729.58042)]. The quadratic Zeeman effect (QZE) for atomic hydrogen [\textit{M. Yu. Sumetskij}, Sov. Phys. JETP 56, 959 (1983)] is a well known case study which has provided interesting examples of analysis of high energy atomic spectra in terms of periodic orbits. A numerical method for locating stable orbits in chaotic systems is proposed in the paper under review. The method is based on the selection of trajectory segments which are characterized by a relatively low positive local Lyapunov exponent (LLE). Once a selection is made, convergence to stable (or weakly unstable) periodic orbits is obtained by a Newton method. Previous experience with the LLE distribution for the standard map [\textit{M. A. Sepulveda}, \textit{R. Badii} and the second author, Phys. Rev. Lett. 63, 1226-1229 (1989)] has shown that all trajectory segments with low exponents gather on the boundaries of the stable islands. The new approach is well-suited for the study of systems with more than two degrees of freedom, as compared with the algorithms given in [\textit{B. Mestel} and \textit{I. Percival}, Physica D 24, 172-178 (1987; Zbl 0616.65134), \textit{M. Baranger}, \textit{K. T. R. Davies} and \textit{J. H. Mahoney}, Ann. Phys. 186, 95-110 (1988; Zbl 0651.58031) and \textit{D. Auerbach}, \textit{P. Cvitanović}, \textit{P. Eckmann}, \textit{J.-P. Gunaratne} and \textit{I. Procaccia}, Phys. Rev. Lett. 58, 2387-2389 (1987)], which are also based on the use of Newton's method, or variants of it. The proposed technique is tested …
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 37C25
Mathematics Subject Classification ID: 37C70
zbMATH DE Number: 91659
zbMATH Keywords: chaotic systems
zbMATH Keywords: stable periodic orbits
zbMATH Keywords: Lyapunov exponents
Wikidata QID: Q59657416
MaRDI profile type: Publication
cites work: Q3353851
cites work: Quantization of chaos
cites work: Quantum eigenvalues from classical periodic orbits
cites work: Existence of stable orbits in the x 2 </mml:m
cites work: Newton method for highly unstable orbits
cites work: The calculation of periodic trajectories
cites work: LYAPUNOV EXPONENTS IN CHAOTIC SYSTEMS: THEIR IMPORTANCE AND THEIR EVALUATION USING OBSERVED DATA
full work available at URL: 0167-2789(92)90176-n
OpenAlex ID: W2169555639
Recommended article: A direct numerical method for quantifying regular and chaotic orbits
Recommended article: Q3548325
Recommended article: Stabilization of unstable periodic orbits of chaotic maps
Recommended article: Periodic orbit quantization of the anisotropic Kepler problem
Recommended article: New variational principles for locating periodic orbits of differential equations
Recommended article: Q4664312
Recommended article: The measure of chaos by the numerical analysis of the fundamental frequencies. Applications to the standard mapping
Recommended article: Detecting unstable periodic orbits in chaotic systems by using an efficient algorithm
Recommended article: New indicators of chaos
description: scientific article; zbMATH DE number 91659