Dynamics of a family of Chebyshev-Halley-type methods

TL;DR

This paper investigates the complex dynamics of Chebyshev-Halley iterative methods on quadratic polynomials, revealing a parameter space structure similar to the Mandelbrot set and identifying convergence failures due to periodic orbits and strange attractors.

Contribution

It introduces the concept of a 'cat set' in the parameter space and analyzes its properties, highlighting new dynamical behaviors of the family of methods.

Findings

Discovery of a 'cat set' in parameter space with Mandelbrot-like features

Identification of parameter regions with non-converging behavior

Presence of periodic orbits and strange attractors in the dynamical plane

Abstract

In this paper, the dynamics of the Chebyshev-Halley family is studied on quadratic polynomials. A singular set, that we call cat set, appears in the parameter space associated to the family. This cat set has interesting similarities with the Mandelbrot set. The parameters space has allowed us to find different elements of the family such that can not converge to any root of the polynomial, since periodic orbits and attractive strange fixed points appear in the dynamical plane of the corresponding method.