Dynamics of a family of Chebyshev-Halley-type methods

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.

Dynamics of Chebyshev-Halley Type Methods on Quadratic Polynomials

The paper examines the intricate dynamics of the Chebyshev-Halley family of iterative methods, focusing specifically on quadratic polynomials. This family is characterized by its convergence order of three, and includes notable methods like Chebyshev's, Halley's, super-Halley's, and Newton's, based on the choice of a complex parameter $\alpha$. While previous inquiries into iterative methods primarily scrutinized their convergence properties, this paper extends that scholarship by exploring the complex dynamical behavior these methods exhibit when applied to quadratic polynomials.

Fixed Points and Critical Points Analysis

The study begins by identifying the fixed points and critical points associated with the Chebyshev-Halley family when applied to the simplified quadratic polynomial $f(z) = z^2 + c$. These points are essential in understanding the dynamics because they form the core of the associated rational functions. Strikingly, the study establishes that the number and stability of these fixed points are inextricably linked to the parameter $\alpha$. For example, when $\alpha \neq \frac{1}{2}$ and $\alpha \neq \frac{5}{2}$, there are five distinct fixed points. However, specific values, namely $\alpha = \frac{1}{2}$ or $\alpha = \frac{5}{2}$, lead to increased multiplicity and different dynamical behavior. The critical points, determined as roots of the derivative of the operator, $\alpha = 0, \frac{1}{2}, \frac{3}{2}, 2$, mirror the behavior of the fixed points in complexity and diversity.

Parameter Space and the Cat Set

A remarkable contribution of the paper is the introduction of the "cat set" within the parameter space. This set bears visual and structural similarities to the Mandelbrot set, a well-known concept in fractal geometry and complex dynamics. Parameters inside the cat set exhibit connected Julia sets and consistent dynamical properties across elements of the Chebyshev-Halley family. Conversely, parameters outside this set yield disconnected Julia sets, implying disparate dynamical behavior. The cat set's structure, including its head and body, correlates with different dynamical qualities observed in fixed and critical points as $\alpha$ varies.

Implications and Prospective Work

This research provides insights into the stability and reliability of iterative methods by leveraging the dynamic characteristics of rational functions associated with these methods. The intricate relationship between the convergence properties of Chebyshev-Halley methods and their dynamical behavior enables a more comprehensive understanding of their applicability, especially in polynomial root-finding problems.

The theoretical implications suggest potential exploration into more complex polynomials and iterative families, possibly extending to higher-dimensional cases or other polynomial degrees. Practically, these findings can inform algorithm design, enhancing the efficacy of these methods in computational settings.

Future studies could investigate whether the cat set is fully connected and explore its potential fractal boundary, akin to the Mandelbrot set. Additionally, there remains uncharted territory regarding the full range of dynamics shown by these methods, particularly in parameter ranges within the cats' antennas or in similar structures within the cat set bubble clusters.

In essence, the interplay between parameter $\alpha$ and the dynamics of fixed and critical points enriches our understanding of the versatile and complex nature of the Chebyshev-Halley family, providing valuable paths for further research and application in computational and mathematical fields.