A classification of slow convergence near parametric periodic points of discrete dynamical systems. Issue 6 (2nd June 2016)

• Record Type: Journal Article
• Authors: Solis, Francisco J.; Chen-Charpentier, Benito M.; Kojouharov, Hristo V.
• Is Part Of: International journal of computer mathematics. Volume 93:Issue 6(2016)
• Page Start: 1011
• Page End: 1021
• Publication Date: 2016-06-02
• Subjects: slow convergence -- discrete dynamical systems -- difference equations -- iteration function -- critical exponent; 37M05 -- 39A30 -- 65P40 -- 65Q10; Computers -- Periodicals; Numerical analysis -- Periodicals; Automation -- Periodicals; 004.0151
• DOI: 10.1080/00207160.2015.1015528
• Languages: English
• ISSNs: 0020-7160

Abstract

Abstract : We study the phenomenon of slow convergence in families of discrete dynamical systems where the iteration function has a Puiseux series representation. Such occurrence consists in the slow convergence of orbits near non-hyperbolic parametric periodic points. We provide a precise new definition of the slowness of convergence which is based on literature results for the critical exponents associated with parametric periodic points. Such exponents establish a general classification for slow systems and provide a measure of rates of convergence. For dynamical systems whose iteration functions have Taylor series expansions, the new definition is natural with wider applicability. However, it can be also used for iteration functions where a more sophisticated approach, such as a Lagrange expansion, is needed. In addition, we show that even for such iteration functions, the critical exponent can be easily computed. The presented theoretical results are illustrated by numerical examples having different rates of convergence.