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Discrete Dynamics in Nature and Society
Volume 2018, Article ID 7486125, 19 pages
https://doi.org/10.1155/2018/7486125
Research Article

Dynamical Analysis via Möbius Conjugacy Map on a Uniparametric Family of Optimal Fourth-Order Multiple-Zero Solvers with Rational Weight Functions

Department of Applied Mathematics, Dankook University, Cheonan 330-714, Republic of Korea

Correspondence should be addressed to Young Hee Geum; moc.lapme@anapnoc

Received 30 July 2017; Accepted 1 January 2018; Published 29 March 2018

Academic Editor: Yong Zhou

Copyright © 2018 Young Ik Kim and Young Hee Geum. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A triparametric family of fourth-order multiple-zero solvers have been proposed. In this paper, we select among them a uniparametric family of optimal fourth-order multiple-zero solvers with rational weight functions and pursue their dynamics by exploring the relevant parameter spaces and dynamical planes, by means of Möbius conjugacy map applied to a prototype polynomial of the form . The resulting dynamics is best illustrated through various stability surfaces and parameter spaces as well as dynamical planes.

1. Introduction

The root-finding problem [1] plays a significant role in many branches of computational sciences. It arises in various fields such as physics, biochemistry, applied engineering, and earth sciences, including industrial mathematics. A considerable number of literatures [28] can be found describing the dynamical behavior of iterative methods to locate the multiple root of a nonlinear equation under consideration.

Many of such existing literatures have studied and emphasized local convergence behavior of the iterative root-finders for nonlinear governing equations under consideration through the viewpoints of the relevant basins of attraction. It is, however, not only worthwhile but also important to investigate the convergence behavior of the root-finders in a global sense. Dealing with such a global convergence behavior certainly motivates our current investigation via the concept of the parameter space where the relevant dynamics of the iterative root-finders under their free critical points continuously evolves as the values of parameter vary along the axes of the extended complex plane.

Homeomorphic conjugacy maps will be introduced in Section 2 in order to better understand the dynamics of the given iterative zero solvers in view of the fact that a topologically conjugated dynamical system preserves its orbit behavior as well as fixed point properties. As a convenient homeomorphic conjugacy map, we will make use of Möbius conjugacy map that enables us to effectively treat the relevant dynamics, by observing that the inverse of Möbius conjugacy map is also of Möbius-type. Indeed, Theorem 3 will show the desired resulting conjugacy map to be studied under current investigation. Additional results on the dynamical behavior including parameter spaces and dynamical planes will be shown in later sections.

To proceed with our investigation, we will employ the triparametric family of fourth-order multiple-zero methods developed by Kim and Geum [9] and introduce a uniparametric family of fourth-order multiple-zero solvers with rational weight functions as follows: where is a multiplicity of the sought zero, ,  ,  ,  , and and is a free parameter.

The aim of this paper is to investigate the complex dynamics on the Riemann sphere by analyzing the parameter spaces associated with the free critical points and the dynamical planes related to the uniparametric family of fourth-order multiple-zero solvers. Such investigation from a viewpoint of complex dynamics may have a drawback restricting us from treating the real dynamics for real nonlinear equations. Nonetheless, our primary motivation for analyzing the relevant complex dynamics lies in seeking the dynamical behavior of a family of iterative methods (1) via Möbius conjugacy map by presenting -parameter spaces and the corresponding dynamical planes.

The rest of this paper is made up of three sections. In Section 2, conjugacy maps along with the property of dynamical analysis for the aforementioned numerical methods are studied and the stability surfaces of the strange fixed points for the conjugacy map are displayed. Section 3 shows the relevant complex dynamics including the parameter spaces and the dynamical planes associated with the basins of attraction. In the last section, we draw a conclusion and suggest the future study by extending the current analysis.

2. Conjugacy Maps and Dynamics

A nonlinear equation (1) can be written in a generic form as a discrete dynamical system:where is the iteration function. As a result, we obtain a complex discrete dynamical system: where ,  , and .

The following definition and remark are useful to construct the conjugacy map and to investigate the relevant dynamics.

Definition 1. Let and be two functions (dynamical systems). We say that and are conjugate if there is a function such that . Then the map is called a conjugacy [10].

Remark 2. Note that a conjugacy indeed preserves the dynamical behavior between the two dynamical systems; for example, if is conjugate to via and is a fixed point of , then is a fixed point of .
Furthermore, if is a homeomorphism, that is, if is topologically conjugate to via , and is a fixed point of , then is a fixed point of . Also, we find and = . If and are invertible, then the topological conjugacy maps an orbit of onto an orbit of , where and the order of points is preserved. Hence, the orbits of the two maps behave similarly under homeomorphism or .

Via Möbius conjugacy map with ,  ,  considered by Blanchard [11], in (3) is conjugated to satisfyingwhen applied to a quadratic polynomial raised to the power of , where and are polynomials with no common factors whose coefficients are generally dependent upon parameters , , and . The following theorem shows that is dependent only on but independent of parameters and .

Theorem 3. Let with and . Then is conjugate to satisfying where

Proof. Since the inverse of is easily found to be , we find after a lengthy computation with the aid of Mathematica [12] symbolic capability: where and are polynomials of degree at most in with a single free parameter . This gives the desired result, completing the proof.

The result of Theorem 3 enables us to discover that (corresponding to fixed point of or root of ) and (corresponding to fixed point of or root of are clearly two of their fixed points of the conjugate map , regardless of -values. Besides, by direct computation, we find that is a strange fixed point [1315] of (that is not a root of ) due to the fact that , regardless of -values.

We now seek further strange fixed points including (corresponding to the original convergence to infinity in view of the fact that or ). To do so, we first investigate some properties of stated in the following theorem.

Theorem 4. Let and be given by (9). Then the following hold.
(a) The leading highest-order term of is given by , provided that .
(b) has a factor , provided that .
(c) , and , where with and .
(d) approaches as tends to , provided that .

Proof. After a lengthy computation and careful algebraic treatments with the aid of Mathematica, (a), (c) follow without difficulty. For the proof of (b), we directly compute the values of and . The proof of (d) follows from the fact that , by using (a) along with a highest-order term of having degree at most .

We will find out the fixed points of the iteration function . Let , whose zeros are the desired fixed points of . From (b) and (c) of Theorem 4, we find that and are the roots of . Hence the expression of will take the following form: whereand are given in Theorem 3.

As a result, ,  , and are the fixed points of . Among these fixed points, is a strange fixed point that is not root or . Further strange fixed points are calculated from the roots of . The following theorem describes some properties of .

Theorem 5. Let be given by (10). Then the following hold.
(a) for , regardless of -values.
(b) has double roots at and , that is, has a factor , provided that for and for .
(c) for , regardless of -values, where ,  .
(d) has also double roots at and , that is, has a common factor as shown in , provided that for and for .
(e) , for , regardless of -values.

Proof. Via careful algebraic treatments and symbolic computation with the aid of Mathematica, (a), (c), (e) follow without difficulty. For the proof of (b), we directly compute the values of and In view of the relations, ,  ,  . We find and for with and with . The proof of (d) follows from the fact that and We also find and

With the use of properties of , we now consider some strange fixed points along with their stability for selected values of and .

To continue our investigation of dynamics behind iterative map (3) applied to a quadratic polynomial raised to the power of ,  , we will describe the fixed points of in (9) and their stability. In view of the fact that is a fixed point of for a fixed point of with , we are interested in the explicit form of for as follows:where we conveniently denote

This enables us to discover that (corresponding to fixed point of or root of ) and (corresponding to fixed point of or root of are clearly two of their fixed points regardless of . To find further strange fixed points, we solve in (15) for with typical values of .

We now investigate further strange fixed points including (corresponding to the original convergence to infinity in view of the fact that or ). By direct computation, we will describe the roots of for . To this end, we first check the existence of -values for common factors (divisors) of and . Besides, will be checked if it has a divisor or . The following theorem best describes relevant properties of such existence as well as explicit strange fixed points.

Theorem 6. Let in (15). Then the following hold.
(a) If , then and the strange fixed points are given by and .
(b) If , then and the strange fixed points are given by and .
(c) If , then and the strange fixed points are given by and ,  .
(d) Let . Then holds for . Hence, if is a root of , then so is .

Proof. (a)–(c) Suppose that and for some values of . Observe that parameter exists in a linear fashion in all coefficients of both polynomials. By eliminating from the two polynomials, we obtain the relation: . Hence, any root of is a candidate for a common divisor of and . Substituting all the roots of into and , we find required relations for and, solving them for , we find . The remaining part of the proof is straightforward. (d) If is a divisor of , then yielding , which is already handled in (b). If is a divisor of , then , yielding . Then remaining proof is trivial. (e) By direct substitution, we find without difficulty. Hence if and only if for . This completes the proof.

Theorem 7. Let in (15). Then the following hold.
(a) If , then and the strange fixed points are given by and .
(b) If , then and the strange fixed points are given by ,  , and .
(c) If , then and the strange fixed points are given by ,  ,  ,  , and .
(d) Let . Then holds for . Hence, if is a root of , then so is .

Proof. The proofs immediately follow from the same argument as used in the proofs of Theorem 6.

As a result of Theorem 5(a), we find the fixed points of , that is, the roots of explicitly as stated in the following corollary.

Corollary 8. Let be a root of , that is, a root of for in (15). Suppose and have no common factors for some suitable -values. Then the roots of for are explicitly given by the following.
(a) The four roots of are explicitly found to be (b) The eight roots of are explicitly found to be where

Proof. Since is a root of for , so is from the result of Theorem 5(a). For the proof of (a), thus can be written as a product of two factors: By expanding the right side of the above equation and comparing the coefficients of the first and second-order terms, we find two relations: which gives the desired values of , . Then the four roots can be found explicitly from or . Similarly for the proof of (b), can be written as a product of four factors: By the same argument as used in the proof of (a), the desired result follows. This completes the proof.

We find that can be reduced to a fraction of a common denominator as follows: where is a polynomial of degree at most defined bywith ,   and ;   and are described earlier in Theorem 6.

We are now ready to determine the stability of the fixed points. In particular, it is necessary to compute the derivative of from Theorem 19: where

We first check the existence of -values for common factors (divisors) of and . Besides, will be checked if it has divisors and . The following theorem best describes relevant properties of such existence as well as explicit strange fixed points.

Theorem 9. Let in (25). Then the following hold.
(a) If , then .
(b) If , then .
(c) If , then .
(d) If , then .
(e) If , then .
(f) If , then .
(g) Let . Let be a fixed point of satisfying . Then holds for .

Proof. The proofs of (a)–(f) immediately follow from the same argument as used in the proofs of Theorem 19. Eliminating from the two polynomials and plays a key role in obtaining the relation: , whose roots enable us to deduce some desired -values. Additional requirement that are candidates for common divisors of and gives only . For the proof of (g), we use Theorem 19 to find , where . This completes the proof.

Theorem 10. Let in (25). Then the following hold.
(a) If , then .
(b) If , then .
(c) If , then , where
.
(d) Let . Let be a fixed point of satisfying . Then holds for .

Proof. The proofs of (a)–(c) immediately follow from the same argument as used in the proofs of Theorem 19. (d) We use Theorem 19 to find , where + . This completes the proof.

Table 1 summarizes the stability results for the strange fixed points of for special -values with .

Table 1: Stability check from of strange fixed points for special -values with .

We are ready to discuss the stability of the fixed points described in Theorems 6 and 7 in terms of parameter .

Theorem 11. Let and . Then the following hold.
(a) The strange fixed point becomes an attractor, parabolic (indifferent, neutral) point, and a repulser, respectively, when ,  , and .
(b) The strange fixed point is a superattractor if .

Proof. (a) From the case of in (25), we find . Solving for , we obtain circle in the cross-sectional -parameter plane for to be a parabolic point, where and . (b) Solving easily yields .

Theorem 12. Let and . Then the following hold.
(a) The strange fixed point is a parabolic (neutral, indifferent) point, respectively, when ,  , and .
(b) The strange fixed point is a superattractor if .

Proof. From the case of in (25), we find . Solving for , we obtain an ellipse in the cross-sectional -parameter plane for to be a parabolic point, where and . (b) Solving easily yields .

We now proceed to discuss the stability of the strange fixed points for conjugate map with using . As a consequence of Theorems 9(g) and 10(d) together with Corollary 8, the stability can be stated at most five strange fixed points including . Then the stability of these fixed points can be best described by illustrative conical surfaces shown in Figures 1-2. The top row of each figure refers to a stability surface for strange fixed point . The stability surfaces for the remaining fixed points are displayed in order from top to bottom and from left to right in each case of and . The underlying theory is clearly verified via cross-sectional views of the stability surfaces with -parameter domains.