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

Authors: Young Ik Kim; Young Hee Geum
Journal: Discrete Dynamics in Nature and Society
 (2018)

---

## 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(z-A)m(z-B)m. The resulting dynamics is best illustrated through various stability surfaces and parameter spaces as well as dynamical planes.


---

## Body

## 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 [2–8] 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 Section2 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:(1)yn=xn-2mm+2fxnf′xn,xn+1=xn-a+bs+cs1+λsfxnf′xn, where m∈N is a multiplicity of the sought zero, a=1/8m8+2+m-2m1+m-3m2+mκλ-2+m2κ2λ2,  b=1/8m(2+m)3κ(1+κλ),  c=m(m+(2+m)κλ)3/8κ,  s=f′(yn)/f′(xn), and κ=m/2+mm and λ∈C 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 λ-parameter spaces and the corresponding dynamical planes.The rest of this paper is made up of three sections. In Section2, 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:(2)xn+1=Rfxn,where Rf is the iteration function. As a result, we obtain a complex discrete dynamical system: (3)zn+1=Rfzn=zn-fznf′znHfzn, where Hf(zn)=a+b/s+c/1/s+λ,  yn=zn-2m/m+2f(zn)/f′(zn), and s=f′(yn)/f′(zn).The following definition and remark are useful to construct the conjugacy map and to investigate the relevant dynamics.Definition 1.
LetF:X→X and G:Y→Y be two functions (dynamical systems). We say that F and G are conjugate if there is a function h:X→Y such that h∘F=G∘h. Then the map h is called a conjugacy [10].Remark 2.
Note that a conjugacy indeed preserves the dynamical behavior between the two dynamical systems; for example, ifF is conjugate to G via h and ξ is a fixed point of F, then h(ξ) is a fixed point of G.
Furthermore, ifh is a homeomorphism, that is, if F is topologically conjugate to G via h, and ζ is a fixed point of G, then h-1(ζ) is a fixed point of F. Also, we find G=h∘F∘h-1 and Gn=h∘F∘h-1∘h∘F∘h-1⋯∘(h∘F∘h-1) = h∘Fn∘h-1. If F and G are invertible, then the topological conjugacy h maps an orbit (4)…,F-2x,F-1x,x,Fx,F2x,…,of F onto an orbit (5)…,G-2y,G-1y,y,Gy,G2y,…, of G, where y=h(x) and the order of points is preserved. Hence, the orbits of the two maps behave similarly under homeomorphism h or h-1.Via Möbius conjugacy mapM(z)=z-A/z-B with z,  A≠B, A,B∈C∪{∞} considered by Blanchard [11], Rf in (3) is conjugated to J satisfying(6)Jz;A,B,λ=Hz;A,B,λDz;A,B,λ.when applied to a quadratic polynomial f(z)=[(z-A)(z-B)]m raised to the power of m, where H and D are polynomials with no common factors whose coefficients are generally dependent upon parameters A, B, and λ. The following theorem shows that J is dependent only on λ but independent of parameters A and B.Theorem 3.
Letf(z)=[(z-A)(z-B)]m with m∈N and M(z)=z-A/z-B,A≠B,A,B∈C∪{∞}. Then Rf(z;λ) is conjugate to J(z;λ) satisfying (7)Jz;λ=z·r1+r2σβ1+zβ2τ1+σ2β1+zβ22τ2r1z+r2σβ1+zβ2ω1+σ2β1+zβ22ω2,where (8)r1=2+m1+4m1+z4mκ21+κλ,r2=-22+m2m-11+z2mκ,σ=β1β2m-1,β1=2+m+mz,β2=m+2+mz,τ1=4z-κλ+m2+m3+m1+κλ,τ2=m3+m3-4m-8zκλ,ω1=zτ1+41-z2,ω2=zτ2+8κλ-1+z2,κ=mm+2m.Proof.
Since the inverse ofM(z) is easily found to be M-1(z)=Bz-A/z-1, we find after a lengthy computation with the aid of Mathematica [12] symbolic capability: (9)Jz;λ=M∘Rf∘M-1z=z·Hz;λDz;λ,where H(z;λ)=r1+r2σ(β1+zβ2)τ1+σ2(β1+zβ2)2τ2 and D(z;λ)=r1z+r2σ(β1+zβ2)ω1+σ2(β1+zβ2)2ω2 are polynomials of degree at most 4m+1 in z with a single free parameter λ∈C. This gives the desired result, completing the proof.The result of Theorem3 enables us to discover that z=0 (corresponding to fixed point A of Rf or root A of f(z)=[(z-A)(z-B)]m) and z=∞ (corresponding to fixed point B of Rf or root B of f(z)) are clearly two of their fixed points of the conjugate map J(z;λ), regardless of λ-values. Besides, by direct computation, we find that z=1 is a strange fixed point [13–15] of J (that is not a root of f(z)=[(z-A)(z-B)]m) due to the fact that J(1;λ)=1, regardless of λ-values.We now seek further strange fixed points includingz=1 (corresponding to the original convergence to infinity in view of the fact that M-1(1)=∞ or M(∞)=1). To do so, we first investigate some properties of J(z;λ)=z·H(z;λ)/D(z;λ) stated in the following theorem.Theorem 4.
LetH(z;λ) and D(z;λ) be given by (9). Then the following hold.
(a) The leading highest-order term ofH(z;λ) is given by -8m3m-2(m+2)mz4m+1λ, provided that λ≠0.
(b)H(z;λ) has a factor z3, provided that λ≠-m16+12m+20m2+15m3/-16+16m+52m2+50m3+15m4κ.
(c)H(1;λ)=D(1;λ), and H(1;λ)/D(1;λ)=1, where λ≠-m3(m+1)4m-2+w2(m+2)2m-1m3+3m2+2m+4+w1/κ(m+1)4m-2m3-4m-8+w2(m-1)(m+2)2m+1+w1 with w1=(m+2)4m+1κ2 and w2=2(m+1)2m-1κ.
(d)J(z;λ) approaches ∞ as z tends to ∞, provided that λ≠-m+2/mm-1.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 ofH(0;λ)=H′(0;λ)=H′′(0;λ)=0 and H′′′(0;λ)≠0. The proof of (d) follows from the fact that J(∞;λ)=∞, by using (a) along with a highest-order term of D(z;λ) having degree at most 4m+1.We will find out the fixed points of the iteration functionJ(z;λ). Let ϕ(z;λ)=J(z;λ)-z, whose zeros are the desired fixed points of J. From (b) and (c) of Theorem 4, we find that z=0 and z=1 are the roots of ϕ. Hence the expression of ϕ(z;λ) will take the following form: (10)ϕz;λ=zz-1·Ψz;λTz;λ,where(11)Ψz;λ=-2+mβ12β22r1+β1+zβ2σr2ν1+β1+zβ2ν2σ,ν1=-41+κλ+m2+m3+m1+κλ,ν2=m3+8-4m+m3κλand β1,β2,r1,r2,σ,κ are given in Theorem 3.As a result,z=0,  z=1, and z=∞ are the fixed points of J. Among these fixed points, z=1 is a strange fixed point that is not root A or B. Further strange fixed points are calculated from the roots of Ψ(z;λ). The following theorem describes some properties of ϕ(z;λ).Theorem 5.
Letϕ(z;λ) be given by (10). Then the following hold.
(a)Ψ(1/z;λ)=1/z4(m+1)Ψ(z;λ) for z≠0, regardless of λ-values.
(b)Ψ(z;λ) has double roots at z=-m+2/m and z=-m/m+2, that is, has a factor (z+m+2/m)2(z+m/m+2)2, provided that λ≠-1/κ for m≠1 and λ≠-13/23 for m=1.
(c)T(1/z;λ)=1/z4m+5T~(z;ρ) for z≠0, regardless of λ-values, where T~(z;λ)=T(z;ρ)-δ0/z,  δ0=4m+2z2-1β12β22β1+zβ2σ{-r2+2(β1+zβ2)κλσ}.
(d)T(z;λ) has also double roots at z=-m+2/m and z=-m/m+2, that is, has a common factor (z+m+2/m)2(z+m/m+2)2 as shown in Ψ(z;λ), provided that λ≠-1/κ for m≠1 and λ≠-7/5 for m=1.
(e)ϕ(1/z;λ)=-z-1/z·Ψ(z;λ)/T~(z;λ), for z≠0, regardless of λ-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 ofΨ(-m+2/m;λ)=Ψ′(-m+2/m;λ)=0 and (12)Ψ′′-m+2m;λ=-25+4m1+m22+m2κ-21+κλ≠0,form≠1,λ≠-1κ,-1843213+23λ≠0,form=1,λ≠-1323. In view of the relations, Ψ(z)=z4m+4Ψ(1/z),  Ψ′(z)=z4m+2[4(m+1)zΨ(1/z)-Ψ′(1/z)],  Ψ′′(z)=z4m[4(4m2+7m+3)z2Ψ(1/z)-2(4m+3)zΨ′(1/z)+Ψ′′(1/z)]. We find Ψ(-m/m+2;λ)=Ψ′(-m/m+2;λ)=0 and Ψ′′(-m/m+2;λ)=κ4Ψ′′(-m+2/m;λ)≠0 for m=1 with λ≠-13/23 and m≠1 with λ≠-1/κ. The proof of (d) follows from the fact that T-m+2/m;λ=T′(-m+2/m;λ)=0 and (13)T′′-m+2m;λ=-25+4mm1+m22+mκ21+κλ≠0,form≠1,λ≠-1κ,-184327+5λ≠0,form=1,λ≠-75.
We also findT(-m/m+2;λ)=T′(-m/m+2;λ)=0 and (14)T′′-mm+2;λ=-25+4mm-11+m22+m3κ-21+κλ≠0,form≠1,λ≠-1κ,-2048915+29λ≠0,form=1,λ≠-1529.With the use of properties ofϕ(z;λ), we now consider some strange fixed points along with their stability for selected values of m=1 and m=2.To continue our investigation of dynamics behind iterative map (3) applied to a quadratic polynomial raised to the power of m,  f(z)=p(z)=[(z-A)(z-B)]m, we will describe the fixed points of J in (9) and their stability. In view of the fact that M(z) is a fixed point of J for a fixed point z of Rp with M-1(z)=zB-A/z-1, we are interested in the explicit form of ϕ(z;λ)=J(z;λ)-z for m∈{1,2} as follows:(15)
ϕ
z
;
λ
=
z
z
-
1
Ψ
1
z
q
1
z
,
if
m
=
1
,
z
z
-
1
Ψ
2
z
q
2
z
,
if
m
=
2
,where we conveniently denote (16)Ψ1z=91+λ+9z41+λ+3z11+7λ+3z311+7λ+z254+34λ,q1z=91+λ+12z2+λ+z221+13λ,Ψ2z=42+λ+4z82+λ+4z17+8λ+4z717+8λ+3z4236+97λ+z2256+113λ+z6256+113λ+z3548+230λ+z5548+230λ,q2z=42+λ+10z512+5λ+4z15+7λ+z620+9λ+6z352+21λ+z2188+81λ+z4280+111λ.This enables us to discover thatz=0 (corresponding to fixed point A of Rp or root A of p(z)) and z=∞ (corresponding to fixed point B of Rp or root B of p(z)) are clearly two of their fixed points regardless of m. To find further strange fixed points, we solve ϕ(z;λ)=0 in (15) for z with typical values of m∈{1,2}.We now investigate further strange fixed points includingz=1 (corresponding to the original convergence to infinity in view of the fact that M-1(1)=∞ or M(∞)=1). By direct computation, we will describe the roots of ϕ(z;λ)=0 for m∈{1,2}. To this end, we first check the existence of λ-values for common factors (divisors) of Ψi(z) and qi(z). Besides, qi(z) will be checked if it has a divisor (z-1) or z. The following theorem best describes relevant properties of such existence as well as explicit strange fixed points.Theorem 6.
Letm=1 in (15). Then the following hold.
(a) Ifλ=-3, then ϕ(z;λ)=z(z-1)(z2+z+1) and the strange fixed points z are given by z=1 and z=-1±3i/2.
(b) Ifλ=-1, then ϕ(z;λ)=z(z-1)3+5z+3z2/3+2z and the strange fixed points are given by z=1 and z=1/6(-5-±i11).
(c) Ifλ=-27/17, then ϕ(z;λ)=z(15+z+z2+15z4)/15+z and the strange fixed points are given by z=±1 and z=1/60-1-1801±i1798-21801,  z=1/60(-1+1801±i1798+21801).
(d) Letλ∉{-3,-1,-27/17}. Then Ψ1(1/z)=z-4Ψ1(z) holds for z≠0. Hence, if z≠0 is a root of Ψ1(z;λ), then so is 1/z.Proof.
(a)–(c) Suppose thatΨ1(z)=0 and q1(z)=0 for some values of z. Observe that parameter λ exists in a linear fashion in all coefficients of both polynomials. By eliminating λ from the two polynomials, we obtain the relation: G(z)=z(z-1)(z2+z+1)=0. Hence, any root of G is a candidate for a common divisor of Ψ1(z) and q1(z). Substituting all the roots of G into Ψ1(z)=0 and q1(z)=0, we find required relations for λ and, solving them for λ, we find λ=-3,-1. The remaining part of the proof is straightforward. (d) If z is a divisor of q1(z), then q1(0)=9(1+λ)=0 yielding λ=-1, which is already handled in (b). If (z-1) is a divisor of q1(z), then q1(1)=2(27+17λ)=0, yielding λ=-27/17. Then remaining proof is trivial. (e) By direct substitution, we find Ψ1(1/z)=z-4Ψ1(z) without difficulty. Hence Ψ1(1/z)=0 if and only if Ψ1(z)=0 for z≠0. This completes the proof.Theorem 7.
Letm=2 in (15). Then the following hold.
(a) Ifλ=-4, then ϕ(z;λ)=-(-1+z)z(1+z)3/1+2z and the strange fixed points are given by z=0 and z=±1.
(b) Ifλ=-2, then ϕ(z;λ)=-(-1+z)z2+9z+15z2+9z3+2z4/2+7z+7z2+z3 and the strange fixed points are given by z=1,  z=-1.80198±0.880308i, and z=-0.448023±0.21887i.
(c) Ifλ=-988/409, then ϕ(z;λ)=z170+951z+1735z2+777z3-516z4+777z5+1735z6+951z7+170z8/-170-951z-1735z2-955z3+258z4+178z5 and the strange fixed points are given by z=-2.35462,  z=-0.424696,  z=-1.68698±0.790067i,  z=-0.486146±0.227678i, and z=0.765729±0.643163i.
(d) Letλ∉{-4,-2,-988/409}. Then Ψ2(1/z)=z-8Ψ2(z) holds for z≠0. Hence, if z≠0 is a root of Ψ2(z;λ), then so is 1/z.Proof.
The proofs immediately follow from the same argument as used in the proofs of Theorem6.As a result of Theorem5(a), we find the fixed points of J(z:λ), that is, the roots of ϕ(z;λ) explicitly as stated in the following corollary.Corollary 8.
Letz∉{0,1} be a root of ϕ(z;λ), that is, a root of Ψi(z;λ) for i=1,2 in (15). Suppose Ψi(z;λ) and qi(z) have no common factors for some suitable λ-values. Then the roots of ϕ(z;λ) for 1≤i≤2 are explicitly given by the following.
(a) The four roots ofΨ1(z;λ) are explicitly found to be (17)z11=11+7λ--59-126λ-51λ26+6λ,z21=1z11,z31=11+7λ+-59-126λ-51λ26+6λ,z41=1z31.
(b) The eight roots ofΨ2(z;λ) are explicitly found to be (18)z12=12-s1--4+s12,z22=1z12,z32=12-s2--4+s22,z42=1z32,z52=12-u1--4+u12,z62=1z52,z72=12-u2--4+u22,z82=1z72,where (19)s1=12c1-8+c12-4c2,s2=12c1+8+c12-4c2,u1=12d1-8+d12-4d2,u2=12d1+8+d12-4d2,c1=17+8λ+-7-λ6+λ22+λ,c2=39+4λ+-7-λ6+λ22+λ,d1=17+8λ--7-λ6+λ22+λ,d2=-3-9-4λ+-7-λ6+λ22+λ.Proof.
Sincez is a root of Ψi(z;λ) for 1≤i≤2, so is 1/z from the result of Theorem 5(a). For the proof of (a), thus Ψ1(z;λ) can be written as a product of two factors: (20)Ψ1z;λ=z2+az+1z2+bz+1. By expanding the right side of the above equation and comparing the coefficients of the first and second-order terms, we find two relations: (21)a+b=11+7λ31+λ,2+ab=54+34λ91+λ,which gives the desired values of c, d. Then the four roots can be found explicitly from z2+az+1=0 or z2+bz+1=0. Similarly for the proof of (b), Ψ2(z;λ) can be written as a product of four factors: (22)Ψ2z;λ=z2+s1z+1z2+s2z+1z2+u1z+1z2+u2z+1. By the same argument as used in the proof of (a), the desired result follows. This completes the proof.We find thatJ′(z;λ) can be reduced to a fraction of a common denominator as follows:(23)J′z;λ=Γz;λD2z;λ, where Γ(z) is a polynomial of degree at most 4m+1 defined by(24)Γz=D·H-z2D·4a2θ2+θ2z+θ3κλ+a1θ21+zτ1-2mθ1+a2θ2τ1-a1θ3τ1+H·θ11+z+4mz-2θ2a2m31+z1+κλ+mβ1+zβ21+zβ1β22+3m1+κλ+m-11+zβ2+β11+m+z+mz+2β2ω1+21+zβ1-2β2β1+zβ2κλ-β1+z-β2β2ω1+θ3a2m3+m-4+m2κλ+2a1ω2with a1=r2τ1+2(β1+zβ2)στ2,a2=β1β2(β1+zβ2),  θ1=(m+2)4m+1(1+z)4m-1κ2(1+κλ),θ2=(m+2)2m-1(1+z)2m-1(β1β2)m-2κ and θ3=(β1β2)2m-3(β1+zβ2);  D(z;λ) and H(z;λ) 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 ofJ from Theorem 19: (25)
J
′
z
;
λ
=
4
z
3
Q
1
z
w
1
z
2
,
if
m
=
1
,
2
z
3
Q
2
z
w
2
z
2
,
if
m
=
2
, where (26)Q1z=36z18+23λ+7λ2+36z318+23λ+7λ2+921+34λ+13λ2+9z421+34λ+13λ2+2z2459+546λ+175λ2,Q2z=840+38λ+9λ2+8z1240+38λ+9λ2+z4200+3850λ+878λ2+z114200+3850λ+878λ2+z224880+22056λ+4861λ2+z1024880+22056λ+4861λ2+2z343848+37745λ+8088λ2+2z943848+37745λ+8088λ2+6z555820+46235λ+9581λ2+6z755820+46235λ+9581λ2+2z4102448+86152λ+18081λ2+2z8102448+86152λ+18081λ2+z6393440+324176λ+66891λ2,w1z=91+λ+12z2+λ+z221+13λ,w2z=42+λ+4z15+7λ+z2188+81λ+6z352+21λ+z4280+111λ+10z512+5λ+z620+9λ.We first check the existence ofλ-values for common factors (divisors) of Qi(z) and wi(z). Besides, wi(z) will be checked if it has divisors z,z2 and z3. The following theorem best describes relevant properties of such existence as well as explicit strange fixed points.Theorem 9.
Letm=1 in (25). Then the following hold.
(a) Ifλ=-3, then J′(z;λ)=4z3.
(b) Ifλ=-1, then J′(z;λ)=2z29+22z+9z2/(3+2z)2.
(c) Ifλ=-3/5, then J′(z;λ)=12z3(11z2+34z+11)/(11z+3)2.
(d) Ifλ=-1/3, then J′(z;λ)=4z3(3z+5)(15z2+26z+15)/(5z+3)3.
(e) Ifλ=-5/3, then J′(z;λ)=4z3(3z-1)(3z2-14z+3)/(z-3)3.
(f) Ifλ=-27/17, then J′(z;λ)=-12z3(5+94z+5z2)/(z+15)2.
(g) Letλ∉{-3,-1,-3/5,-1/3,-5/3,-27/17}. Let z be a fixed point of J(z;λ) satisfying ϕ(z;λ)=0. Then J′(z;λ)=J′(1/z;λ) holds for z≠0.Proof.
The proofs of (a)–(f) immediately follow from the same argument as used in the proofs of Theorem19. Eliminating λ from the two polynomials Qi(z) and wi(z) plays a key role in obtaining the relation: G(z)=(-3+z)(-1+z)z(1+z)(3+5z)(3+2z+3z2)=0, whose roots enable us to deduce some desired ρ-values. Additional requirement that z,z2,z3 are candidates for common divisors of Qi(z) and wi(z) gives only λ=-1. For the proof of (g), we use Theorem 19 to find J′(z;λ)-J′(1/z;λ)=-4Q1(z)(z-1)Δ1/q1(z)q1(1/z)2z8ϕ(z;λ)=0, where Δ1=9(1+λ)+9z4(1+λ)+3z(11+7λ)+3z3(11+7λ)+z2(54+34λ). This completes the proof.Theorem 10.
Letm=2 in (25). Then the following hold.
(a) Ifλ=-2, then J′(z;λ)=2z2(3+35z+112z2+159z3+112z4+35z5+3z6)/(2+7z+7z2+z3)2.
(b) Ifλ=-4, then J′(z;λ)=z3(8+17z+8z2)/(1+2z)2.
(c) Ifλ=-988/409, then J′(z;λ)=z3σ1/(170+951z+1735z2+955z3-258z4-178z5)2, where σ1=121040+727134z+624992z2-5092795z3-17328190z4-24412987z5-17328190z6-5092795z7+
624992 z 8 + 727134 z 9 + 121040 z 10.
(d) Letλ∉{-2,-4,-988/409}. Let z be a fixed point of J(z;λ) satisfying ϕ(z;λ)=0. Then J′(z;λ)=J′(1/z;λ) holds for z≠0.Proof.
The proofs of (a)–(c) immediately follow from the same argument as used in the proofs of Theorem19. (d) We use Theorem 19 to find J′(z;λ)-J′(1/z;λ)=-2Q2(z)(z-1)Δ2/q2(z)q2(1/z)2z16ϕ(z;λ)=0, where Δ2=4(2+λ)+4z8(2+λ)+4z(17+8λ)+4z7(17+8λ)+3z4(236+97λ)+z2(256+113λ)+z6(256+113λ) + z3(548+230λ)+z5(548+230λ). This completes the proof.Table1 summarizes the stability results for the strange fixed points ζ of J for special λ-values with m∈{1,2}.Table 1
Stability check fromJ′ζ;λ of strange fixed points ζ for special λ-values with 1≤m≤2.
m λ ζ no. ofζ | J ′ ( ζ ; ρ ) |:t∗ 1 - 3 5 - 3.89179 - 0.256951 0.241037 ± 0.970516 i    4 20.212:r 20.212:r 3.2324:r −3 1 ± 1 + 3 2     3 4:r 4:r −1 1 - 0.833333 ± 0.552771 i     3 3.2:r 4.66667:r - 1 3 −1 - 2.46956 - 0.40493 0.270579 ± 0.962698 i   5 4:r 7.50978:r 7.50978:r 3.24022:r - 5 3 −1 0.89538 ± 0.445302 i - 0.0620469 ± 0.998073 i    5 5:r 7.45743:r 5.54257:r - 27 17 - 1 1 - 0.0333333 ± 0.999444 i    4 5.14286:r 4.875:r 4.99556:r  2 - 2 - 1 −2.23964 - 0.4465 0.0930703 ± 0.99566 i   5 1:r 9.33031:r 15.5842:r 3.44158:r - 4 - 2.29663 −0.435421 0.366025 ± 0.930605 i    4 10.4641:p 10.4641:r 3.5359:p - 988 409 - 1 1 1 5 ( - 7 ± 2 6 ) - 2.19404 -0.455779 - 0.0721469 ± 0.997394 i 8 3.04938:r 5.82133:r 49.2133:r 8.72642:r 8.72642:r 3.70561:r ∗ | J ′ ( ζ ; ρ ) |:t implies that ζ is attractive, parabolic, and repulsive, if t=a (|J′|<1), t=p (|J′|=1), and t=r (|J′|>1), respectively.We are ready to discuss the stability of the fixed points described in Theorems6 and 7 in terms of parameter λ.Theorem 11.
Letm=1 and λ∉{-3,-1,-3/5,-1/3,-5/3,-27/17}. Then the following hold.
(a) The strange fixed pointz=1 becomes an attractor, parabolic (indifferent, neutral) point, and a repulser, respectively, when 323+2λ/27+17λ<1,  323+2λ/27+17λ=1, and 323+2λ/27+17λ>1.
(b) The strange fixed pointz=1 is a superattractor if λ=-3/2.Proof.
(a) From the case ofm=1 in (25), we find J′(1;λ)=32(3+2λ)/(27+17λ). Solving |J′(1;λ)|=323+2λ/27+17λ=1 for λ, we obtain circle (x+5685/3807)2+y2=32/12692 in the cross-sectional λ-parameter plane for z=1 to be a parabolic point, where x=Re(λ) and y=Im(λ). (b) Solving |J′(1;λ)|=0 easily yields λ=-3/2.Theorem 12.
Letm=2 and λ∉{-2,-4,-988/409}. Then the following hold.
(a) The strange fixed pointz=1 is a parabolic (neutral, indifferent) point, respectively, when 5464+27λ/988+409λ<1,  5464+27λ/988+409λ=1, and 5464+27λ/988+409λ>1.
(b) The strange fixed pointz=1 is a superattractor if λ=-64/27.Proof.
From the case ofm=2 in (25), we find J′(1;λ)=54(64+27λ)/(988+409λ). Solving |J′(1;λ)|=5464+27λ/988+409λ=1 for λ, we obtain an ellipse (x+4634756/1958483)2/(27000/1958483)2+y2/(54005/898177930147)2=1 in the cross-sectional λ-parameter plane for z=1 to be a parabolic point, where x=Re(λ) and y=Im(λ). (b) Solving |J′(1;λ)|=0 easily yields λ=-64/27.We now proceed to discuss the stability of the strange fixed pointsζ for conjugate map J(z;ρ) with m∈{1,2} using J′(ζ;ρ). 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 z=1. 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 z=1. The stability surfaces for the remaining fixed points zi(1≤i≤8) are displayed in order from top to bottom and from left to right in each case of m=1 and m=2. The underlying theory is clearly verified via cross-sectional views of the stability surfaces with λ-parameter domains.Figure 1
Stability surfaces of the strange fixed pointsJ(z;ρ) for m=1.Figure 2
Stability surfaces of the strange fixed pointsJ(z;ρ) for m=2.The critical points of the iterative method are given by the roots ofJ′(z,λ)=0. Clearly, z=0 and z=∞ are critical points associated with the roots a and b of the polynomial (z-a)(z-b). The critical points that are not related to any roots of the polynomial (z-a)(z-b) as free critical points.Remark 13.
In view of (25), the following hold.
(i) Whenm=1 and (1+λ)(3+λ)(1+3λ)(5+3λ)(3+5λ)(27+17λ)≠0, four roots ξ of Q1(ξ) are the free critical points given by(27)ξ=12-a±-4+a2, where a=254+69λ+21λ2±(3+λ)27+105λ+160λ2+64λ3/3(1+λ)(21+13λ).
(ii) Whenm=2 and (λ+2)(λ+4)(409λ+988)(9λ+20)≠0,  12 roots ξ of Q2(ξ) can be found numerically for a given λ.For Remark13(i), we further find that since z is a root of Q1(z), so is 1/z from Theorem 19. Hence, there exist two constants a1(x) and a2(x) such that Q1(z)=3618+23λ+7λ2(z2+a1z+1)(z2+a2z+1). For convenience, we define a function ψ(t)=-t+t2-4/2. Then for λ satisfying (1+λ)(3+λ)(1+3λ)(5+3λ)(3+5λ)(27+17λ)(21+13λ)≠0, 4 roots (ξ1,ξ2,ξ3,ξ4) of θ1(ξ)=0 are the free critical points of J(ξ,λ),  ξ1=ψ(a1),  ξ2=ψ(a2),  ξ3=1/ξ1,  ξ4=1/ξ2, where (28)a1=254+69λ+21λ2-3+λ27+105λ+160λ2+64λ331+λ21+13λ,a2=254+69λ+21λ2+3+λ27+105λ+160λ2+64λ331+λ21+13λ.For Remark13(ii), we also find that if z is a root of Q2(z), then so is 1/z from Theorem 19. Hence there exist six constants a1(λ),a2(λ),…,a6(λ) such that (29)Q2z=∏i=161+aiz+z2. With the use of function ψ(t) introduced earlier, all 12 roots of Q2(ξ)=0 can be written as (30)ξi=ψai,ξi+6=1ξi,1≤i≤6.Then desired among six constants a1,a2,…,a6 can be found by comparing coefficients of six terms up to order 6 with those of Q2(z) in (25); eliminating a1,a2,…,a6 from these relations, we find (31)32768+23072λ+4145λ2+8a1640+38λ+9λ2-2a152100+1925λ+439λ2+a1422960+20232λ+4429λ2-2a1333348+28120λ+5893λ2-2a146416+35095λ+6674λ2+2a1254128+43408λ+8683λ2=0, which constitutes a sextic equation in a1. For simplicity, we denote the left side of (31) by Y(a1;λ). Then the six roots a1,a2,…,a6 of Y(a;λ)=0 are found to the desired six constants. By solving Y(a;λ)=0 numerically for a given λ, we can determine the desired six constants a1,a2,…,a6. Having found such a1,a2,…,a6, the six critical points ξi can be given by ξi=ψ(ai) for 1≤i≤6.


## 3. Parameter Spaces and Basins of Attraction

It is interesting to study the relevant complex dynamics from the viewpoint of parameter spaces and dynamical planes.


### 3.1. Parameter Spaces

It is our further interest to investigate the relevant complex dynamics from the viewpoint of parameter spaces and dynamical planes. The following lemma will be used for us to claim the favorable properties of symmetry on both parameter spaces and dynamical planes.Lemma 14.
LetF:C2→C be defined by F(λ,z)=P0(z)+P1(z)·λ+P2(z)·λ2, where Pi(z)(0≤i≤2) are complex polynomials with real coefficients. Suppose F(λ,z)=0. Let z¯ denote a complex conjugate of z. Then the following hold.
(a)λ(z¯)=λ(z)¯.
(b) Ifz(λ) is a root of F, then so is z¯(λ¯).Proof.
(a) By directly solvingF(λ,z)=0 for λ by means of z, we find(32)λ=λz=-P1z±P1z2-4P1zP2z2P1z.
SincePiz0≤i≤2 are complex polynomials with real coefficients, we get Pi(z¯)=Pi(z)¯.
In view of (32), (33)λz¯=-P1z¯±P1z¯2-4P1z¯P2z¯2P1z¯=-P1z¯±P1z¯2-4P1z¯·P2z¯2P1z¯. On the other hand, we also have (34)Fλ,z¯=0=P0z¯+P1z¯·λ¯+P2z¯·λ¯2, implying λ(z)¯=λ(z¯).
(b) Letz(λ) be a root of F(λ,z). Then (35)Fλ,z=0=Fλ,z¯=P0z+P1z·λ+P2z·λ2¯=P0z¯+P1z¯·λ¯+P2z¯·λ¯2=P0z¯+P1z¯·λ¯+P2z¯·λ¯2=Fλ¯,z¯, stating that z¯(λ¯) is also a root of F.Theorem 15.
Letz(λ) be a free critical point of J(z;λ) dependent upon parameter λ given by a root of Q(z)=0 described in (15). Then the corresponding parameter space is symmetric with respect to its horizontal axis.Proof.
It suffices to show the claim form=1. We find that such z(λ) is a root of F(λ,z)=Q1(z)=27(1+z)27+10z+7z2+651+138z+182z2+138z3+51z4λ+3+2z+3z239+58z+39z2λ2 as shown in (15) for a given λ. Then z¯(λ¯) is also a root of Q (i.e., a free critical point of J) at λ¯ from Lemma 14. For such a free critical point z(λ), consider conjugated map J(z;λ) from (9): (36)Jz;λ=z437+8z+3z2+13+12z+9z2λ33+8z+7z2+9+12z+13z2λ. This expression allows us to get (37)Jz;λ=Jzλ;λ=Jzλ;λ¯=Jzλ¯;λ¯=Jz¯λ¯;λ¯, which states that the magnitude of the orbit of free critical point z at λ is that same as that of the orbit of free critical point z¯ at λ¯ and hence implies that the parameter space associated with map J(z;λ) is symmetric with respect to its horizontal axis. The proofs for m≥2 are similar.Theorem 16.
Given a parameterλ∈R, let z(λ) be a starting point of iterative map J(z;λ) described in (15). Then the corresponding dynamical plane is symmetric with respect to its horizontal axis.Proof.
We first note thatλ¯=λ and similarly follow the proof Theorem 15. We now consider J(z;λ): (38)Jz;λ=Jz;λ¯=Jz¯;λ¯=Jz¯;λ, which states that the magnitude of the orbit of z(λ) is the same as that of the orbit of z(λ)¯ and hence implies that the dynamical plane associated with map J(z;λ) is symmetric with respect to its horizontal axis if λ is real.We shall describe further properties on parameter spaces and dynamical planes associated with conjugated mapJ(z;λ). By direct computation from J(z;λ) in (9), we obtain the following lemma.Lemma 17.
J ( z ; λ ) = 1 / J 1 / z ; λ for any λ∈C and z≠0.Corollary 18.
Letq∈N be given. If z≠0 is a q-periodic point of J, then so is 1/z.Proof.
As a result of Lemma17 and via induction on q, we find that Jq(z;λ)=1/Jq(1/z;λ) holds for any integer q≥1 and for any λ∈C and z≠0. Hence Jq(1/z;λ)=1/Jq(z;λ)=1/z, stating that 1/z is also a q-periodic point of J.Theorem 19.
Letz be a critical point. Then the following holds.
(i)J′(z;λ)=J′(1/z;λ).
(ii) Ifz≠0 is a critical point, then so is 1/z.Proof.
(i) SinceJ′(1/z;λ)=Γ(z)/H2(z;λ) and z is a critical point satisfying Γ(z)=0, we find that (39)J′1z;λ-J′z;λ=Γz1H2z;λ-1D2z;λ=0,using (23).
(ii) HenceJ′(z;λ)=0 implies J′(1/z;λ)=0. This completes the proof.From (25), we first conveniently let k=4m-2(1≤m≤2) and denote the 2k branches of the free critical points ξ1,ξ2,…,ξk,1/ξ1,1/ξ2,…,1/ξk of J(z;λ) by cp1(λ),cp2(λ),…,cp2k(λ) in order. Indeed, we find that, for λ∉{-3,-1,-3/5,-1/3,-5/3,-27/17} if m=1 and for λ∉{-2,-4,-988/409} if m=2, (40)cpjλ=ψaj,forj=1,2,…,kcpjλ=1cpj-kλ,forj=k+1,…,2k, where ψ(t)=-t-t2-4/2; aj(1≤j≤k) are described after Remark 13. For instance, when m=1, we consider the orbit behavior of two branches cp1(λ)=ξ1 and cp2(λ)=ξ2 of the free critical points under the action of J(z;λ). The orbit behavior of other two branches cp3(λ)=1/cp1(λ) and cp4(λ)=1/cp2(λ) can be similarly described and hence its investigation will be omitted here.The following remark is useful in understanding the orbit behavior of two branches that are reciprocals of each other among the four free critical points ofJ(z;λ).Remark 20.
In view of Lemma17 and Corollary 18, we find that Jn(cpj;λ)=1/Jn(cpj+2;λ) for j=1,2 holds for any integer n≥1 and λ∉{-3,-1,-3/5,-1/3,-5/3,-27/17}. Hence, the critical orbit of one branch behaves in quite the same way as the other branch does in the following sense.
(1) If the critical orbit of one branch converges to aq-cycle with an integer q≥1, then so does the other branch.
(2) If the critical orbit of one branch is divergent but bounded, then so is the other branch.
(3) If the critical orbit of one branch converges to∞, then the other branch converges to 0, and vice versa.
Therefore, we find that theλ-parameter space associated with one branch has same components [16] and boundaries in a neighborhood of which different anomalies occur as the other branch does. It is interesting to observe that the component associated with fixed point ∞ shares its boundary with the component associated with fixed point 0, in each branch.The case whenm=2 can be similarly treated.Without loss of generality, we limit ourselves to considering the case whenm=1, since similar treatment applies to the case when m=2. In view of the above remark, it suffices to consider two branches cp1 and cp2 for their orbit behavior. Now, we are going to look for the best members of the family by means of the parameter space associated with the free critical points cp1(λ) or cp2(λ).LetP ={λ∈C: a critical orbit of z under Jz;λ tends to a number σp∈C¯}. Similarly, we let D = {z∈C: an orbit ofzunderJz;λtends to a numberσd∈C¯}. We call P and D, respectively, as the parameter space and the dynamical plane (showing long-term orbit behavior) associated with J(z;λ). If the number σp or σd is a finite constant, then there exist finite periods in the orbit. Otherwise, the orbit is nonperiodic but bounded or goes to infinity. One should find that z=∞ can be treated as a fixed point in the dynamics on the Riemann sphere.We now conveniently introduce a systematic scheme coloring a pointλ∈P or point z∈D based on the period of the orbit of z under the action of J(z;λ) as follows. Let t be a point in P or D. Then t is painted in specified color Cq if t induces a q-periodic orbit with q∈N∪0 under the action of J(z;λ). We accept the desired q-periodic convergence of an orbit associated with P or D after a maximum of 2000 iterations and with a tolerance of 10-6. We further identify color Cq according to the scheme shown in Table 2.Table 2
Stability check from|J′(ζ;λ)| of strange fixed points ζ for special λ-values with 1≤m≤2.
q C q 1 Magenta, for fixed point∞   Cyan, for fixed point 0Yellow, for fixed point 1Red, for other strange fixed point  2 ≤ q ≤ 68 C 2  orange C 3  light green C 4  dark red C 5  dark blue C 6  dark green C 7  dark yellow C 8  floral white C 9  light pink C 10  khaki C 11  dark orange C 12  turquoise C 13  lavender C 14  thistle C 15  plum C 16  orchid C 17  medium orchid C 18 blue violet C 19  dark orchid C 20  purple C 21  powder blue C 22  sky blue C 23  deep sky blue C 24  dodger blue C 25  royal blue C 26  medium spring green C 27  spring green C 28  medium sea green C 29  sea green C 30  forest green C 31  olive drab C 32  bisque C 33  moccasin C 34  light salmon C 35  salmon C 36  light coral C 37  Indian red C 38  brown C 39  fire brick C 40  peach puff C 41  wheat C 42  sandy brown C 43  tomato C 44  orange red C 45  chocolate C 46  pink C 47  pale violet red C 48   deep pink C 49  violet red C 50  gainsboro C 51  light gray C 52  dark gray C 53  gray C 54  charteruse C 55  electric indigo C 56  electric lime C 57  lime C 58  silver C 59  teal C 60  pale turquoise C 61  sandy brown C 62  honeydew C 63  misty rose C 64  lemon chiffon C 65  lavender blush C 66  gold C 67  crimson C 68  tan   q = 0 ∗ or q>68 black q   ∗ = 0 implies that the orbit is nonperiodic but bounded.In Figures3-4, parameter spaces P associated with cpj(λ),(1≤j≤4) are shown. A point λ in P is painted according to the coloring scheme defined in Table 2. One should note that every point of the parameter space P whose color is not cyan (root z=a), magenta (root z=b), yellow, and red is not a good choice of λ in terms of relevant numerical behavior.Figure 3
Parameter spaces associated with free critical pointcp1 for m=1.
(a)
- 2000 ≤ R ( λ ) ≤ 2000,  |I(λ)|≤2000 (b)
1.1875 ≤ R ( λ ) ≤ 1.1925,  |I(λ)|≤0.0025 (c)
1.335 ≤ R ( λ ) ≤ 1.355,  |I(λ)|≤0.01 (d)
5.858 ≤ R ( λ ) ≤ 5.86,  |I(λ)|≤0.01Figure 4
Parameter spaces associated with free critical pointcp2 for m=1.
(a)
- 2000 ≤ R ( λ ) ≤ 2000 ,   | I ( λ ) | ≤ 2000 (b)
1.15 ≤ R ( λ ) ≤ 1.25 ,   | I ( λ ) | ≤ 0.05 (c)
- 1.55 ≤ R ( λ ) ≤ - 1.4 ,   | I ( λ ) | ≤ 0.075For convenience of analysis, we now letPi denote the parameter space associated with branch cpi for 1≤i≤4.Focusing the attention on the regions shown in Figures3–14, it is evident that there exist members of the family with complicated behavior. We clearly see from Figure 11 that there exist components where q-periodic orbits are generated with q∈{2,3,…,11} budding from period-1 component (in red color) and 4-periodic orbits budding from period-2 component (in orange color). We further observe that q-periodic components with q∈{5,7,9,11,…} can be evidently seen as indicated by arrow lines. It is interesting for us to watch these components budded along the boundary of the period-1 component in the manner of Farey sequence [17], according to the Schleicher’s Algorithm [18].Figure 5
Parameter spaces associated with free critical pointcp1 for m=2.
(a)
| R ( λ ) | ≤ 2000,  |I(λ)|≤2000 (b)
- 1.1 ≤ R ( λ ) ≤ - 1.05 ,   | I ( λ ) | ≤ 0.025 (c)
1.525 ≤ R ( λ ) ≤ 1.545 ,   | I ( λ ) | ≤ 0.01Figure 6
Parameter spaces associated with free critical pointcp2 for m=2.
(a)
- 2000 ≤ R ( λ ) ≤ 2000 ,   | I ( λ ) | ≤ 2000 (b)
30 ≤ R ( λ ) ≤ 70 ,   | I ( λ ) | ≤ 20 (c)
- 20.5 ≤ R ( λ ) ≤ - 18.5 ,   | I ( λ ) | ≤ 1Figure 7
Parameter spaces associated with free critical pointcp3 for m=2.
(a)
- 5 ≤ R ( λ ) ≤ 1 ,   | I ( λ ) | ≤ 3 (b)
- 4.4 ≤ R ( λ ) ≤ - 3.9 ,   | I ( λ ) | ≤ 0.25 (c)
- 2.4 ≤ R ( λ ) ≤ - 2.35 ,   | I ( λ ) | ≤ 0.025 (d)
- 2.35 ≤ R ( λ ) ≤ - 2.25 ,   0.175 ≤ I ( λ ) ≤ 0.275Figure 8
Parameter spaces associated with free critical pointcp4 for m=2.
(a)
- 5 ≤ R ( λ ) ≤ 6 ,   | I ( λ ) | ≤ 5.5 (b)
- 4.4 ≤ R ( λ ) ≤ - 3.8 ,   | I ( λ ) | ≤ 0.3 (c)
- 3.7 ≤ R ( λ ) ≤ - 3.5 ,   | I ( λ ) | ≤ 0.1Figure 9
Parameter spaces associated with free critical pointcp5 for m=2.
(a)
- 5 ≤ R ( λ ) ≤ 7 ,   | I ( λ ) | ≤ 6 (b)
- 4.525 ≤ R ( λ ) ≤ - 3.65 ,   | I ( λ ) | ≤ 0.4375 (c)
- 4.445 ≤ R ( λ ) ≤ - 4.345 ,   | I ( λ ) | ≤ 0.05Figure 10
Parameter spaces associated with free critical pointcp6 for m=2.Figure 11
Farey-sequential components along the boundary of a red component form=1.Figure 12
Farey-sequential components along the boundary of a red component form=2.Figure 13
Dynamical planes associated withJ(z;λ) for m=1.
(a)
λ = 1.8875 (b)
λ = 1.35 (c)
λ = 1.215 (d)
λ = - 1.525Figure 14
Dynamical planes associated withJz;λ for m=2.
(a)
λ = - 1.07 (b)
λ = 1.53 (c)
λ = 1.5295 (d)
λ = 1.5383 (e)
λ = 45 (f)
λ = 19.75 (g)
λ = - 4.1 (h)
λ = - 4.2 (i)
λ = - 2.365 (j)
λ = - 4.0844 + 0.1134 i (k)
λ = - 3.601 + 0.0312 i (l)
λ = - 4.44055 + 0.0129 iWe have painted the points both in the parameter space and in the dynamical plane with the same coloring scheme as defined in Table2. In Figure 13, the dynamical planes of members of the family with regions of convergence to any of the strange fixed points are shown for the values of λ∈{1.8875,1.35,1.215,-1.525}, respectively. Some interesting cycles of finite period are clearly seen. In Figures 13, we observe the dynamical planes of a member of the family with regions of convergence to q-periodic cycles for q∈{4,3,8,2} in order from left to right and from top to bottom.In summary, whenm=1, we have thus far observed that there exist regions of finite period for stable q-cycles with q≥1. We also observe the fascinating fractal boundaries between the basins of attraction associated with different cycles.For the remaining case ofm=2, we have carried out similar analysis to explore the relevant dynamics described in subsequent paragraphs.Figures5–10 show the parameter spaces Pi associated with cpi(λ),(1≤i≤6) for m=2. We evidently observe period-doubling components from Figures 5–9. For instance, Figures 5(c) and 6(c) exhibit components with periodic-doubling sequences (4→8→16→32→⋯) and (2→4→8→16→⋯), respectively.It is interesting to note in Figure7(c) that q-periodic components (q≥1) arise along the boundary of a 1-periodic component in yellow where any λ-value can induce an orbit in the dynamical plane converging to a fixed point 1. The q-periodic components here are not likely to occur in the manner of Farey sequence. It is worthwhile to note that both Figures 7(d) and 8(c) contain Mandelbrot-like components. By magnifying Figure 8(b), we obtain Figure 12 leading to Farey-sequential components along the boundary of a red component for m=2.Figure14 displays dynamical planes associated with J(z;λ) when m=2, with various values of λ inducing q-periodic cycles for q∈{6,8,16,12,4,2,1,2,1(fixed point 1),3,10,5} in order from left to right and from top to bottom.


## 3.1. Parameter Spaces

It is our further interest to investigate the relevant complex dynamics from the viewpoint of parameter spaces and dynamical planes. The following lemma will be used for us to claim the favorable properties of symmetry on both parameter spaces and dynamical planes.Lemma 14.
LetF:C2→C be defined by F(λ,z)=P0(z)+P1(z)·λ+P2(z)·λ2, where Pi(z)(0≤i≤2) are complex polynomials with real coefficients. Suppose F(λ,z)=0. Let z¯ denote a complex conjugate of z. Then the following hold.
(a)λ(z¯)=λ(z)¯.
(b) Ifz(λ) is a root of F, then so is z¯(λ¯).Proof.
(a) By directly solvingF(λ,z)=0 for λ by means of z, we find(32)λ=λz=-P1z±P1z2-4P1zP2z2P1z.
SincePiz0≤i≤2 are complex polynomials with real coefficients, we get Pi(z¯)=Pi(z)¯.
In view of (32), (33)λz¯=-P1z¯±P1z¯2-4P1z¯P2z¯2P1z¯=-P1z¯±P1z¯2-4P1z¯·P2z¯2P1z¯. On the other hand, we also have (34)Fλ,z¯=0=P0z¯+P1z¯·λ¯+P2z¯·λ¯2, implying λ(z)¯=λ(z¯).
(b) Letz(λ) be a root of F(λ,z). Then (35)Fλ,z=0=Fλ,z¯=P0z+P1z·λ+P2z·λ2¯=P0z¯+P1z¯·λ¯+P2z¯·λ¯2=P0z¯+P1z¯·λ¯+P2z¯·λ¯2=Fλ¯,z¯, stating that z¯(λ¯) is also a root of F.Theorem 15.
Letz(λ) be a free critical point of J(z;λ) dependent upon parameter λ given by a root of Q(z)=0 described in (15). Then the corresponding parameter space is symmetric with respect to its horizontal axis.Proof.
It suffices to show the claim form=1. We find that such z(λ) is a root of F(λ,z)=Q1(z)=27(1+z)27+10z+7z2+651+138z+182z2+138z3+51z4λ+3+2z+3z239+58z+39z2λ2 as shown in (15) for a given λ. Then z¯(λ¯) is also a root of Q (i.e., a free critical point of J) at λ¯ from Lemma 14. For such a free critical point z(λ), consider conjugated map J(z;λ) from (9): (36)Jz;λ=z437+8z+3z2+13+12z+9z2λ33+8z+7z2+9+12z+13z2λ. This expression allows us to get (37)Jz;λ=Jzλ;λ=Jzλ;λ¯=Jzλ¯;λ¯=Jz¯λ¯;λ¯, which states that the magnitude of the orbit of free critical point z at λ is that same as that of the orbit of free critical point z¯ at λ¯ and hence implies that the parameter space associated with map J(z;λ) is symmetric with respect to its horizontal axis. The proofs for m≥2 are similar.Theorem 16.
Given a parameterλ∈R, let z(λ) be a starting point of iterative map J(z;λ) described in (15). Then the corresponding dynamical plane is symmetric with respect to its horizontal axis.Proof.
We first note thatλ¯=λ and similarly follow the proof Theorem 15. We now consider J(z;λ): (38)Jz;λ=Jz;λ¯=Jz¯;λ¯=Jz¯;λ, which states that the magnitude of the orbit of z(λ) is the same as that of the orbit of z(λ)¯ and hence implies that the dynamical plane associated with map J(z;λ) is symmetric with respect to its horizontal axis if λ is real.We shall describe further properties on parameter spaces and dynamical planes associated with conjugated mapJ(z;λ). By direct computation from J(z;λ) in (9), we obtain the following lemma.Lemma 17.
J ( z ; λ ) = 1 / J 1 / z ; λ for any λ∈C and z≠0.Corollary 18.
Letq∈N be given. If z≠0 is a q-periodic point of J, then so is 1/z.Proof.
As a result of Lemma17 and via induction on q, we find that Jq(z;λ)=1/Jq(1/z;λ) holds for any integer q≥1 and for any λ∈C and z≠0. Hence Jq(1/z;λ)=1/Jq(z;λ)=1/z, stating that 1/z is also a q-periodic point of J.Theorem 19.
Letz be a critical point. Then the following holds.
(i)J′(z;λ)=J′(1/z;λ).
(ii) Ifz≠0 is a critical point, then so is 1/z.Proof.
(i) SinceJ′(1/z;λ)=Γ(z)/H2(z;λ) and z is a critical point satisfying Γ(z)=0, we find that (39)J′1z;λ-J′z;λ=Γz1H2z;λ-1D2z;λ=0,using (23).
(ii) HenceJ′(z;λ)=0 implies J′(1/z;λ)=0. This completes the proof.From (25), we first conveniently let k=4m-2(1≤m≤2) and denote the 2k branches of the free critical points ξ1,ξ2,…,ξk,1/ξ1,1/ξ2,…,1/ξk of J(z;λ) by cp1(λ),cp2(λ),…,cp2k(λ) in order. Indeed, we find that, for λ∉{-3,-1,-3/5,-1/3,-5/3,-27/17} if m=1 and for λ∉{-2,-4,-988/409} if m=2, (40)cpjλ=ψaj,forj=1,2,…,kcpjλ=1cpj-kλ,forj=k+1,…,2k, where ψ(t)=-t-t2-4/2; aj(1≤j≤k) are described after Remark 13. For instance, when m=1, we consider the orbit behavior of two branches cp1(λ)=ξ1 and cp2(λ)=ξ2 of the free critical points under the action of J(z;λ). The orbit behavior of other two branches cp3(λ)=1/cp1(λ) and cp4(λ)=1/cp2(λ) can be similarly described and hence its investigation will be omitted here.The following remark is useful in understanding the orbit behavior of two branches that are reciprocals of each other among the four free critical points ofJ(z;λ).Remark 20.
In view of Lemma17 and Corollary 18, we find that Jn(cpj;λ)=1/Jn(cpj+2;λ) for j=1,2 holds for any integer n≥1 and λ∉{-3,-1,-3/5,-1/3,-5/3,-27/17}. Hence, the critical orbit of one branch behaves in quite the same way as the other branch does in the following sense.
(1) If the critical orbit of one branch converges to aq-cycle with an integer q≥1, then so does the other branch.
(2) If the critical orbit of one branch is divergent but bounded, then so is the other branch.
(3) If the critical orbit of one branch converges to∞, then the other branch converges to 0, and vice versa.
Therefore, we find that theλ-parameter space associated with one branch has same components [16] and boundaries in a neighborhood of which different anomalies occur as the other branch does. It is interesting to observe that the component associated with fixed point ∞ shares its boundary with the component associated with fixed point 0, in each branch.The case whenm=2 can be similarly treated.Without loss of generality, we limit ourselves to considering the case whenm=1, since similar treatment applies to the case when m=2. In view of the above remark, it suffices to consider two branches cp1 and cp2 for their orbit behavior. Now, we are going to look for the best members of the family by means of the parameter space associated with the free critical points cp1(λ) or cp2(λ).LetP ={λ∈C: a critical orbit of z under Jz;λ tends to a number σp∈C¯}. Similarly, we let D = {z∈C: an orbit ofzunderJz;λtends to a numberσd∈C¯}. We call P and D, respectively, as the parameter space and the dynamical plane (showing long-term orbit behavior) associated with J(z;λ). If the number σp or σd is a finite constant, then there exist finite periods in the orbit. Otherwise, the orbit is nonperiodic but bounded or goes to infinity. One should find that z=∞ can be treated as a fixed point in the dynamics on the Riemann sphere.We now conveniently introduce a systematic scheme coloring a pointλ∈P or point z∈D based on the period of the orbit of z under the action of J(z;λ) as follows. Let t be a point in P or D. Then t is painted in specified color Cq if t induces a q-periodic orbit with q∈N∪0 under the action of J(z;λ). We accept the desired q-periodic convergence of an orbit associated with P or D after a maximum of 2000 iterations and with a tolerance of 10-6. We further identify color Cq according to the scheme shown in Table 2.Table 2
Stability check from|J′(ζ;λ)| of strange fixed points ζ for special λ-values with 1≤m≤2.
q C q 1 Magenta, for fixed point∞   Cyan, for fixed point 0Yellow, for fixed point 1Red, for other strange fixed point  2 ≤ q ≤ 68 C 2  orange C 3  light green C 4  dark red C 5  dark blue C 6  dark green C 7  dark yellow C 8  floral white C 9  light pink C 10  khaki C 11  dark orange C 12  turquoise C 13  lavender C 14  thistle C 15  plum C 16  orchid C 17  medium orchid C 18 blue violet C 19  dark orchid C 20  purple C 21  powder blue C 22  sky blue C 23  deep sky blue C 24  dodger blue C 25  royal blue C 26  medium spring green C 27  spring green C 28  medium sea green C 29  sea green C 30  forest green C 31  olive drab C 32  bisque C 33  moccasin C 34  light salmon C 35  salmon C 36  light coral C 37  Indian red C 38  brown C 39  fire brick C 40  peach puff C 41  wheat C 42  sandy brown C 43  tomato C 44  orange red C 45  chocolate C 46  pink C 47  pale violet red C 48   deep pink C 49  violet red C 50  gainsboro C 51  light gray C 52  dark gray C 53  gray C 54  charteruse C 55  electric indigo C 56  electric lime C 57  lime C 58  silver C 59  teal C 60  pale turquoise C 61  sandy brown C 62  honeydew C 63  misty rose C 64  lemon chiffon C 65  lavender blush C 66  gold C 67  crimson C 68  tan   q = 0 ∗ or q>68 black q   ∗ = 0 implies that the orbit is nonperiodic but bounded.In Figures3-4, parameter spaces P associated with cpj(λ),(1≤j≤4) are shown. A point λ in P is painted according to the coloring scheme defined in Table 2. One should note that every point of the parameter space P whose color is not cyan (root z=a), magenta (root z=b), yellow, and red is not a good choice of λ in terms of relevant numerical behavior.Figure 3
Parameter spaces associated with free critical pointcp1 for m=1.
(a)
- 2000 ≤ R ( λ ) ≤ 2000,  |I(λ)|≤2000 (b)
1.1875 ≤ R ( λ ) ≤ 1.1925,  |I(λ)|≤0.0025 (c)
1.335 ≤ R ( λ ) ≤ 1.355,  |I(λ)|≤0.01 (d)
5.858 ≤ R ( λ ) ≤ 5.86,  |I(λ)|≤0.01Figure 4
Parameter spaces associated with free critical pointcp2 for m=1.
(a)
- 2000 ≤ R ( λ ) ≤ 2000 ,   | I ( λ ) | ≤ 2000 (b)
1.15 ≤ R ( λ ) ≤ 1.25 ,   | I ( λ ) | ≤ 0.05 (c)
- 1.55 ≤ R ( λ ) ≤ - 1.4 ,   | I ( λ ) | ≤ 0.075For convenience of analysis, we now letPi denote the parameter space associated with branch cpi for 1≤i≤4.Focusing the attention on the regions shown in Figures3–14, it is evident that there exist members of the family with complicated behavior. We clearly see from Figure 11 that there exist components where q-periodic orbits are generated with q∈{2,3,…,11} budding from period-1 component (in red color) and 4-periodic orbits budding from period-2 component (in orange color). We further observe that q-periodic components with q∈{5,7,9,11,…} can be evidently seen as indicated by arrow lines. It is interesting for us to watch these components budded along the boundary of the period-1 component in the manner of Farey sequence [17], according to the Schleicher’s Algorithm [18].Figure 5
Parameter spaces associated with free critical pointcp1 for m=2.
(a)
| R ( λ ) | ≤ 2000,  |I(λ)|≤2000 (b)
- 1.1 ≤ R ( λ ) ≤ - 1.05 ,   | I ( λ ) | ≤ 0.025 (c)
1.525 ≤ R ( λ ) ≤ 1.545 ,   | I ( λ ) | ≤ 0.01Figure 6
Parameter spaces associated with free critical pointcp2 for m=2.
(a)
- 2000 ≤ R ( λ ) ≤ 2000 ,   | I ( λ ) | ≤ 2000 (b)
30 ≤ R ( λ ) ≤ 70 ,   | I ( λ ) | ≤ 20 (c)
- 20.5 ≤ R ( λ ) ≤ - 18.5 ,   | I ( λ ) | ≤ 1Figure 7
Parameter spaces associated with free critical pointcp3 for m=2.
(a)
- 5 ≤ R ( λ ) ≤ 1 ,   | I ( λ ) | ≤ 3 (b)
- 4.4 ≤ R ( λ ) ≤ - 3.9 ,   | I ( λ ) | ≤ 0.25 (c)
- 2.4 ≤ R ( λ ) ≤ - 2.35 ,   | I ( λ ) | ≤ 0.025 (d)
- 2.35 ≤ R ( λ ) ≤ - 2.25 ,   0.175 ≤ I ( λ ) ≤ 0.275Figure 8
Parameter spaces associated with free critical pointcp4 for m=2.
(a)
- 5 ≤ R ( λ ) ≤ 6 ,   | I ( λ ) | ≤ 5.5 (b)
- 4.4 ≤ R ( λ ) ≤ - 3.8 ,   | I ( λ ) | ≤ 0.3 (c)
- 3.7 ≤ R ( λ ) ≤ - 3.5 ,   | I ( λ ) | ≤ 0.1Figure 9
Parameter spaces associated with free critical pointcp5 for m=2.
(a)
- 5 ≤ R ( λ ) ≤ 7 ,   | I ( λ ) | ≤ 6 (b)
- 4.525 ≤ R ( λ ) ≤ - 3.65 ,   | I ( λ ) | ≤ 0.4375 (c)
- 4.445 ≤ R ( λ ) ≤ - 4.345 ,   | I ( λ ) | ≤ 0.05Figure 10
Parameter spaces associated with free critical pointcp6 for m=2.Figure 11
Farey-sequential components along the boundary of a red component form=1.Figure 12
Farey-sequential components along the boundary of a red component form=2.Figure 13
Dynamical planes associated withJ(z;λ) for m=1.
(a)
λ = 1.8875 (b)
λ = 1.35 (c)
λ = 1.215 (d)
λ = - 1.525Figure 14
Dynamical planes associated withJz;λ for m=2.
(a)
λ = - 1.07 (b)
λ = 1.53 (c)
λ = 1.5295 (d)
λ = 1.5383 (e)
λ = 45 (f)
λ = 19.75 (g)
λ = - 4.1 (h)
λ = - 4.2 (i)
λ = - 2.365 (j)
λ = - 4.0844 + 0.1134 i (k)
λ = - 3.601 + 0.0312 i (l)
λ = - 4.44055 + 0.0129 iWe have painted the points both in the parameter space and in the dynamical plane with the same coloring scheme as defined in Table2. In Figure 13, the dynamical planes of members of the family with regions of convergence to any of the strange fixed points are shown for the values of λ∈{1.8875,1.35,1.215,-1.525}, respectively. Some interesting cycles of finite period are clearly seen. In Figures 13, we observe the dynamical planes of a member of the family with regions of convergence to q-periodic cycles for q∈{4,3,8,2} in order from left to right and from top to bottom.In summary, whenm=1, we have thus far observed that there exist regions of finite period for stable q-cycles with q≥1. We also observe the fascinating fractal boundaries between the basins of attraction associated with different cycles.For the remaining case ofm=2, we have carried out similar analysis to explore the relevant dynamics described in subsequent paragraphs.Figures5–10 show the parameter spaces Pi associated with cpi(λ),(1≤i≤6) for m=2. We evidently observe period-doubling components from Figures 5–9. For instance, Figures 5(c) and 6(c) exhibit components with periodic-doubling sequences (4→8→16→32→⋯) and (2→4→8→16→⋯), respectively.It is interesting to note in Figure7(c) that q-periodic components (q≥1) arise along the boundary of a 1-periodic component in yellow where any λ-value can induce an orbit in the dynamical plane converging to a fixed point 1. The q-periodic components here are not likely to occur in the manner of Farey sequence. It is worthwhile to note that both Figures 7(d) and 8(c) contain Mandelbrot-like components. By magnifying Figure 8(b), we obtain Figure 12 leading to Farey-sequential components along the boundary of a red component for m=2.Figure14 displays dynamical planes associated with J(z;λ) when m=2, with various values of λ inducing q-periodic cycles for q∈{6,8,16,12,4,2,1,2,1(fixed point 1),3,10,5} in order from left to right and from top to bottom.


## 4. Conclusion

Given multiplicitym, we have developed a uniparametric family of optimal fourth-order multiple-zero solvers with rational weight functions and investigated their complex dynamics via Möbius conjugacy map applied to a polynomial of the form f(z)=[(z-A)(z-B)]m along with the stability analysis of strange fixed points. The relevant dynamical analysis has been carried out from the viewpoint of stability analysis and in terms of parameter spaces and dynamical planes associated with basins of attraction. In the current approach, although we have encountered technical difficulties in explicitly finding critical points from (25) when m=2 in terms of parameter λ, a numerical approach developed based on Remark 13(ii) has been employed to resolve such technical difficulties. With the use of continuous dependence of zeros of a polynomial on its coefficients, that is, on λ-parameter values, it has been possible to construct numerical parameter spaces associated with twelve roots of Q2(z) in (25). Nevertheless, such continuous dependence on λ-parameter is not globally guaranteed since a numerical solution along one solution branch may trace along another nearby solution branch. This unfavorable result would be inevitable unless each solution branch is exactly found.A future study dealing with the dynamics of other types of iterative methods will be primarily concerned with exact free critical points in order to construct favorable parameter spaces via global continuity ofλ-parameter.



---

*Source: 7486125-2018-03-29.xml*