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Dynamics of the Modified Chebyshev's method to multiple roots

Diego Linares, Carlos Cadenas

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.

Dynamics of the Modified Chebyshev's method to multiple roots

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.
Paper Structure (15 sections, 6 theorems, 50 equations, 6 figures)

This paper contains 15 sections, 6 theorems, 50 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Basic preliminaries
  3. Dynamical behavior of the rational function associated with the method in study
  4. Conjugacy classes
  5. Study of the fixed points
  6. Analysis of the case $K=-2$
  7. Analysis of the case $K=1$
  8. Analysis of the case $K=2$
  9. Analysis of the case $K=3$
  10. Stability of the Fixed Points
  11. Study of the Parameter Space
  12. Dynamical Planes
  13. Basins of attraction
  14. Stability of the fixed points
  15. Result and discussion

Key Result

Theorem 3.1

(The Scaling Theorem). Let $f(z)$ be an analytic function on the Riemann sphere, and let $T(z)=\alpha z+\beta$ with $\alpha\neq0$ be an affine map. If $g(z)=(f \circ T)(z)$, then $R_g = T^{-1} \circ R_f \circ T$. That is, the rational map $R_f$ is analytically conjugate to $R_g$ through $T$.

Figures (6)

  • Figure 1: Stability functions: left panel corresponds to $z_1$, middle panel to $z_{2}$, and right panel to $z_{3}$
  • Figure 2: Parameter space associated with the critical points $C_1$, $C_2$, and $C_3$ of $S(z)$.
  • Figure 3: Dynamical planes. $K=0.2:0.2:1.8$
  • Figure 4: Dynamical planes. $K=0.1I:0.1:0.2+0.1I$
  • Figure 5: Dynamical planes. $K= \left\lbrace -\frac{1}{2}i, -\frac{1}{5}i, -\frac{1}{10}i, -\frac{1}{20}i, \frac{2}{5}i, \frac{3}{5}i \right\rbrace$
  • ...and 1 more figures

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 14 more