Dynamics of the Modified Chebyshev's method to multiple roots

TL;DR

This paper investigates the complex dynamical behavior of the Modified Chebyshev's root-finding method, analyzing stability, basins of attraction, and chaotic regions through dynamical systems theory.

Contribution

It provides a detailed dynamical systems analysis of the Modified Chebyshev's method, including stability and convergence properties, which were not previously characterized.

Findings

Fixed points stability classified

Basins of attraction mapped

Chaotic regions identified

Abstract

This study explores the complex dynamics of the rational function associated with the Modified Chebyshev's root-finding method. After introducing the basic preliminaries of discrete dynamical systems, we analyze the dynamical behavior of the method, classifying the stability of its fixed points and critical orbits. These theoretical findings are then illustrated through dynamical planes, which map the basins of attraction and reveal the convergence characteristics and potential chaotic regions of the method.