Bulbs of Period Two in the Family of Chebyshev-Halley Iterative Methods on Quadratic Polynomials

Abstract

The parameter space associated to the parametric family of Chebyshev-Halley on quadratic polynomials shows a dynamical richness worthy of study. This analysis has been initiated by the authors in previous works. Every value of the parameter belonging to the same connected component of the parameter space gives rise to similar dynamical behavior. In this paper, we focus on the search of regions in the parameter space that gives rise to the appearance of attractive orbits of period two.

1. Introduction

The application of iterative methods for solving nonlinear equations 𝑓(𝑧)=0, 𝑓∶ℂ→ℂ gives rise to rational functions whose dynamical behavior provides us with important information about the stability and reliability of the corresponding iterative scheme. The best known iterative method, under the dynamical point of view, is Newton’s scheme (see, e.g., [1]).

This study has been extended by different authors to other point-to-point iterative methods for solving nonlinear equations (see, e.g., [2, 3] and more recently, [4–8]). In particular, the authors study the parametric family of Chebyshev-Halley, whose dynamical analysis has been started in [9].

The fixed point operator corresponding to the family of Chebyshev-Halley type methods is 𝐺()1𝑧,𝛼=𝑧−1+2𝐿_𝑓(𝑧)1−𝛼𝐿_𝑓(𝑧)𝑓(𝑧)𝑓^′(𝑧), (1) where 𝐿_𝑓(𝑧)=𝑓(𝑧)𝑓^′′(𝑧)𝑓^′(𝑧)² (2) and 𝛼 is a complex parameter. In [9], the authors have begun the study of the dynamics of this operator when it is applied on quadratic polynomial 𝑝(𝑧)=𝑧²+𝑐. For this polynomial, the operator (1) is the rational function: 𝐺_𝑝()=𝑧𝑧,𝛼⁴(−3+2𝛼)+6𝑐𝑧²+𝑐²(1−2𝛼)4𝑧𝑧²(),−2+𝛼+𝛼𝑐 (3) depending on parameters 𝛼 and 𝑐.

The parameter 𝑐 can be removed by applying the conjugacy map ℎ(𝑧)=𝑧−𝑖√𝑐𝑧+𝑖√𝑐, (4) with the properties ℎ(∞)=1, ℎ(𝑖√𝑐)=0, and ℎ(−𝑖√𝑐)=∞.

Then, the operator (3) becomes a one-parametric rational function: 𝑂_𝑝(𝑧,𝛼)=𝑧³𝑧−2(𝛼−1)()𝑧.1−2𝛼−1 (5)

As it is known (see [10]), for a rational function 𝑅∶ℂ→ℂ, on the Riemann sphere ℂ, the orbit of a point 𝑧₀∈ℂ is defined as 𝑧₀,𝑅𝑧₀,𝑅²𝑧₀,…,𝑅^𝑛z₀,…. (6)

Depending on the asymptotic behavior of their orbits, a point 𝑧₀∈ℂ is called a fixed point if it satisfies 𝑅(𝑧₀)=𝑧₀. A periodic point 𝑧₀ of period 𝑝>1 is a point such that 𝑅^𝑝(𝑧₀)=𝑧₀ and 𝑅^𝑘(𝑧₀)≠𝑧₀, 𝑘<𝑝. A preperiodic point is a point 𝑧₀ that is not periodic, but there exists a 𝑘>0 such that 𝑅^𝑘(𝑧₀) is periodic. A critical point 𝑧₀ is a point where the derivative of rational function vanishes, 𝑅^′(𝑧₀)=0.

On the other hand, a fixed point 𝑧₀ is called attractor if |𝑅^′(𝑧₀)|<1, superattractor if |𝑅^′(𝑧₀)|=0, repulsor if |𝑅^′(𝑧₀)|>1, and parabolic if |𝑅^′(𝑧₀)|=1. The stability of a periodic orbit is defined by the magnitude (lower than 1 or not) of |𝑅^′(𝑧₁)⋯𝑅^′(𝑧_𝑝)|, where {𝑧₁,…,𝑧_𝑝} are the points of the orbit of period 𝑝.

The basin of attraction of an attractor 𝑧 is defined as the set of pre images of any order: 𝒜(𝑧)=𝑧₀∈ℂ∶𝑅^𝑛𝑧₀⟶𝑧,𝑛⟶∞. (7)

The set of points 𝑧∈ℂ such that their families {𝑅^𝑛(𝑧)}_𝑛∈𝑁 are normal in some neighborhood 𝑈(𝑧) is the Fatou set, ℱ(𝑅), that is, the Fatou set is composed by the set of points whose orbits tend to an attractor (fixed point, periodic orbit, or infinity). Its complement in ℂ is the Julia set, 𝒥(𝑅); therefore, the Julia set includes all repelling fixed points, periodic orbits, and their pre images. That means that the basin of attraction of any fixed point belongs to the Fatou set. On the contrary, the boundaries of the basins of attraction belong to the Julia set.

The invariant Julia set for Newton’s method on quadratic polynomials is the unit circle 𝑆¹, and the Fatou set is defined by the two basins of attraction of the superattractor fixed points: 0 and ∞. However, as it can be seen in [11], the Julia set for Chebyshev’s method applied to quadratic polynomials is more complicated than for Newton’s method. These methods are two elements of the Chebyshev-Halley family. The dynamical study of operator of the Chebyshev-Halley family on quadratic polynomials (3) has been started for the authors in [9, 12].

The rest of the paper is organized as follows. in Section 2, we recall some results about the stability of the strange fixed points of operator 𝑂_𝑝(𝑧,𝛼). In Sections 3, 4, and 5, we analyze the black regions of the parameter space involving attractive cycles of period two. We finish the work with some conclusions.

2. Previous Results on Chebyshev-Halley Family

Fixed points of the operator 𝑂_𝑝(𝑧,𝛼) are 𝑧=0, 𝑧=∞, which correspond to the roots of the polynomial and the strange fixed points 𝑧=1 and 𝑧=(−3+2𝛼±√5−12𝛼+4𝛼²)/2, denoted by 𝑠₁(𝛼) and 𝑠₂(𝛼), respectively.

Moreover, 𝑧=0 and 𝑧=∞ are superattractors, and the stability of the other fixed points is established in the following results.

Proposition 1 (see [9, Proposition 1]). The fixed point 𝑧=1 satisfies the following statements.(1)If |𝛼−(13/6)|<1/3, then 𝑧=1 is an attractor, and, in particular, it is a superattractor for 𝛼=2.(2)If |𝛼−(13/6)|=1/3, then 𝑧=1 is a parabolic point.(3)If |𝛼−(13/6)|>1/3, then 𝑧=1 is a repulsive fixed point.

Proposition 2 (see [9, Proposition 2]). The fixed points 𝑧=𝑠_𝑖(𝛼), 𝑖=1,2, satisfy the following statements.(i)If |𝛼−3|<1/2, then 𝑠₁(𝛼) and 𝑠₂(𝛼) are two different attractive fixed points. In particular, 𝑠₁(3) and 𝑠₂(3) are superattractors.(ii)If |𝛼−3|=1/2, then 𝑠₁(𝛼) and 𝑠₂(𝛼) are parabolic points. In particular, 𝑠₁(5/2)=𝑠₂(5/2)=1.(iii)If |𝛼−3|>1/2, then 𝑠₁(5/2) and 𝑠₂(5/2) are repulsive fixed points.

On the other hand, the critical points of 𝑂_𝑝(𝑧,𝛼) are 𝑧=0, 𝑧=∞ and 𝑧=3−4𝛼+2𝛼²±√−6𝛼+19𝛼²−16𝛼³+4𝛼⁴3(),𝛼−1 (8) which are denoted by 𝑐₁(𝛼) and 𝑐₂(𝛼), respectively.

It is known that there is at least one critical point associated with each invariant Fatou component (see [13]). It is also shown in [9] that the critical points 𝑐_𝑖, 𝑖=1,2, are inside the basin of attraction of 𝑧=1 when it is attractive (11/6<𝛼<5/2) and coincide with 𝑧=1 for 𝛼=2. Then, they move to the basins of attraction of 𝑠₁(𝛼) and 𝑠₂(𝛼) when these fixed points become attractive (5/2<𝛼<7/2). Critical and fixed points coincide for 𝛼=3, and 𝑠₁(3) and 𝑠₂(3) become superattractors.

The study of the parameter space enables us to analyze the dynamics of the rational function associated to an iterative method: each point of the parameter plane is associated to a complex value of 𝛼, that is, to an element of the family. Moreover, every value of 𝛼 belonging to the same connected component of the parameter space gives rise to subsets of schemes of family with similar dynamical behavior. The parameter space of (3) is shown in Figure 1.

Figure 1 

Parameter plane.

Briefly summarizing the results of [9], we observe a black figure (the cat set) with two big disks corresponding to the 𝛼 values for those fixed points 𝑧=1 (the head, |𝛼−(13/6)|<1/3), and 𝑠₁(𝛼) and 𝑠₂(𝛼) (the body, |𝛼−3|<1/2) become attractive and have their own basin of attraction, and one critical point is in each basin. The intersection point of the head and the body of the cat is in their common boundary and corresponds to 𝛼=5/2. The parameter space inside the necklace (the curve similar to a circle that passes through the cat’s neck) is topologically equivalent to a disk. The boundary of the cat set is exactly the set of parameters for which the dynamics changes abruptly under small changes of 𝛼, that is, the bifurcation loci of the family of Chebyshev-Halley acting on quadratic polynomial.

Let us stress that the head and the body are surrounded by bulbs, of different sizes, that yield to the appearance of attractive cycles of different periods. In this paper, we focus on the study of all bulbs involving attractive cycles of period 2. As we see in the following sections, these attractive 2-cycles also appear in the small black figures passing through the necklace (little cats).

3. Bulbs Involving Attractive Cycles of Period 2

The 2-bulbs consist of values of the parameter which have been associated with an attracting periodic cycle of period two in their respective dynamical planes. Cycles of period 2 satisfy the equation: 𝑂²_𝑝(𝑧,𝛼)=𝑧. (9) The relation 𝑂²_𝑝(𝑧,𝛼)−𝑧=0 can be factorized as 𝑧(−1+𝑧)1+3𝑧−2𝛼𝑧+𝑧²𝑓(𝑧,𝛼)𝑔(𝑧,𝛼)=0, (10) where 𝑓(𝑧,𝛼)=1+(3−2𝛼)𝑧+(3−2𝛼)𝑧²+(3−2𝛼)𝑧³+𝑧⁴,𝑔(𝑧,𝛼)=1+(3−4𝛼)𝑧+2−6𝛼+4𝛼²𝑧²+3−6𝛼+4𝛼²𝑧³+9−22𝛼+20𝛼²−8𝛼³𝑧⁴+3−6𝛼+4𝛼²𝑧⁵+2−6𝛼+4𝛼²𝑧⁶+(3−4𝛼)𝑧⁷+𝑧⁸. (11)(12)

As we have seen in [9], the product 𝑧(−1+𝑧)(1+3𝑧−2𝛼𝑧+𝑧²) yields to the fixed points. So, 2 periodic points come from the roots of 𝑓(𝑧,𝛼)=0 or 𝑔(𝑧,𝛼)=0. In the following, we study the bulbs where 2-cycles become attractive, and, by imitating the notation of the Mandelbrot set (see [14]), we call them 2-bulbs.

In addition, the authors showed in [12] that the strange fixed points 𝑧=1 and 𝑠₁(𝛼),𝑠₂(𝛼) move from attractors to repulsor in some bifurcation points, and one attractive 2-cycle appears. If we study the dynamical plane for a value of 𝛼 inside these 2-bulbs, we observe that the Fatou set has a periodic component with two connected components containing the attracting 2-cycle. The two connected components touch at a common point that it is the fixed point from which the attractive cycle comes. Examples of these dynamical planes can be seen in Figures 2, 3, 4, and 5, where the fixed points are identified with little white stars. In these figures we also observe three different Fatou components: the orange one is the attraction basin of 𝑧=0, the blue one is the attraction basin of 𝑧=∞, and the black one corresponds to the attractive 2-cycle. Let us notice that the strange fixed points are repulsive and they are located on the Julia set.

(a)

(b)

Figure 2 

Two 2-cycles in the dynamical plane for 𝛼=0.376.

(a)

(b)

Figure 3 

Two 2-cycles in the dynamical plane for 𝛼=1.687616.

Figure 4 

Dynamical plane for 𝛼=(11/6)−0.05.

(a)

(b)

Figure 5 

Two 2-cycles in the dynamical plane for 𝛼=3.55.

Let us observe in Figure 4 that the two components, where the 2-cycle is included, touch each other in the strange fixed point 𝑧=1.

For 𝛼=3.55 (Figure 5), two attractive 2-cycles appear, one comes from the strange fixed point 𝑠₁(3.55), and the other comes from 𝑠₂(3.55).

For 𝛼=1.687616 (Figure 3), two attractive 2-cycles appear from the bifurcation of the 2-cycle coming from the fixed point 𝑧=1. So, this fixed point is in the boundary of its basin of attraction but not in the boundary of the two immediate basins.

4. Bulbs Coming from 𝑓(𝑧,𝛼)

The roots of 𝑓(𝑧,𝛼)=0 are 𝑧₁(𝛼)3=−4+𝛼2+14√5−4𝛼+4𝛼²−14−2−16𝛼+8𝛼²+(−6+4𝛼)√5−4𝛼+4𝛼²,𝑧₂(𝛼)3=−4+𝛼2+14√5−4𝛼+4𝛼²+14−2−16𝛼+8𝛼²+(−6+4𝛼)√5−4𝛼+4𝛼²,𝑧₃(𝛼)3=−4+𝛼2−14√5−4𝛼+4𝛼²−14−2−16𝛼+8𝛼²−(−6+4𝛼)√5−4𝛼+4𝛼²,𝑧₄(𝛼)3=−4+𝛼2−14√5−4𝛼+4𝛼²+14−2−16𝛼+8𝛼²−(−6+4𝛼)√5−4𝛼+4𝛼², (13) where {𝑧₁(𝛼),𝑧₂(𝛼)} and {𝑧₃(𝛼),𝑧₄(𝛼)} form two cycles of period two: 𝑂_𝑝𝑧₁(𝛼),𝛼=𝑧₂(𝛼),𝑂_𝑝𝑧₃(𝛼),𝛼=𝑧₄(𝛼). (14) The stability functions of these 2-cycles depends on 𝛼: 𝑆₁₂(𝛼)=𝑂^′_𝑝𝑧₁(𝛼),𝛼⋅𝑂^′_𝑝𝑧₂(𝛼)(),𝛼=2−1+𝛼)(−3+2𝛼−9+13𝛼−12𝛼²+4𝛼³×√5−4𝛼+4𝛼²+263−210𝛼+363𝛼²−376𝛼³+248𝛼⁴−96𝛼⁵+16𝛼⁶,𝑆₃₄(𝛼)=𝑂^′_𝑝𝑧₃(𝛼),𝛼⋅𝑂^′_𝑝𝑧₄(𝛼)(),𝛼=−2−1+𝛼)(−3+2𝛼−9+13𝛼−12𝛼²+4𝛼³×√5−4𝛼+4𝛼²+263−210𝛼+363𝛼²−376𝛼³+248𝛼⁴−96𝛼⁵+16𝛼⁶. (15)

We can draw numerically the boundaries where these 2-cycles are parabolic, (see Figure 6). In it, blue points correspond to |𝑆₁₂(𝛼)|=|𝑂^′_𝑝(𝑧₁(𝛼),𝛼)⋅𝑂^′_𝑝(𝑧₂(𝛼),𝛼)|=1 and red points represent the stability function |𝑆₃₄(𝛼)|=|𝑂^′_𝑝(𝑧₃(𝛼),𝛼)⋅𝑂^′_𝑝(𝑧₄(𝛼),𝛼)|=1.

(a)

(b)

(c)

Figure 6 

|𝑆12(𝛼)|=1,|𝑆34(𝛼)|=1.

The first two drawings of Figure 6 are part of the boundary of the little cats and correspond to values of the parameter where the imaginary part is always different from zero. The third one admits real values of 𝛼. As we see in the following result, these real values permit to obtain the “radius” and the center of this bulb (see [12]).

Moreover, we find two disks that embed this bulb.

Proposition 3. Let 𝑆₁₂(𝛼) be the stability function of the 2-cycle {𝑧₁(𝛼),𝑧₂(𝛼)}. Then, |𝑆₁₂(𝛼)|<1, for 𝛼∈𝐷₁, such that 𝐶₂∶={𝛼∶|𝛼−𝑚|<0.064}⊂𝐷₁⊂𝐶₁∶={𝛼∶|𝛼−𝑚|<𝑟}, (16) where 𝑚=(1/2)((11/6)+𝛼^∗), 𝑟=(1/2)((11/6)−𝛼^∗)≈0.064617, and 𝛼^∗=(1/6)³(134+18√57)−(4/(3³(134+18√57)))+(5/6)≈1.7041.

Proof. The boundary of the bulb satisfies |𝑆₁₂(𝛼)|=1, and it is not a circle, but there exists a corona delimited by two circles 𝐶₁ and 𝐶₂. These circles are centered in the middle point of 𝛼^∗ and 11/6,  𝛼^∗₀=(1/2)((11/6)+𝛼^∗)≈1.76871, and have radii 𝑟=(1/2)((11/6)−𝛼^∗)≈0.0646 and 𝑟^′=0.064, respectively, see Figure 7.
As we see in Figures 7 and 8, the value of the stability function |𝑆₁₂(𝛼)| in these circles is such that |𝑆₁₂(𝛼)|≥1 if 𝛼∈𝐶₁ and |𝑆₁₂(𝛼)|<1 if 𝛼∈𝐶₂. In Figure 8 the red curve corresponds to |𝑆₁₂(𝛼)| when 𝛼∈𝐶₁ and the blue one corresponds to 𝛼∈𝐶₂. The value |𝑆₁₂(𝛼)|=1 corresponds to the real values 𝛼=11/6 and 𝛼=𝛼^∗.

Figure 7 

Corona delimiting the bulb of period two in the head.

Figure 8 

Stability function of the 2-cycle {𝑧1(𝛼),𝑧2(𝛼)} on 𝐶1 and 𝐶2.

The dynamical planes for values of 𝛼 inside this bulb 𝐷₁ are similar to those obtained in Figure 4.

5. Bulbs Coming from 𝑔(𝑧,𝛼)

A similar study can be made on 𝑔(𝑧,𝛼) in order to obtain the other bulbs of period 2 belonging to the cat set. To simplify the study we factorize 𝑔(𝑧,𝛼)=ℎ₁(𝑧,𝛼)ℎ₂(𝑧,𝛼)ℎ₃(𝑧,𝛼)ℎ₄(𝑧,𝛼), where ℎ_𝑖(𝑧,𝛼) are polynomials of degree two, ℎ_𝑖(𝑧,𝛼)=1+𝑏_𝑖𝑧+𝑧², 𝑖=1,2,3,4. The relationships that the coefficients must satisfy are 𝑏₁+𝑏₂+𝑏₃+𝑏₄𝑏=3−4𝛼,₁𝑏₂+𝑏₁𝑏₃+𝑏₁𝑏₄+𝑏₂𝑏₃+𝑏₂𝑏₄+𝑏₃𝑏₄=4𝛼²𝑏−6𝛼−2,₁𝑏₂𝑏₃+𝑏₁𝑏₂b₄+𝑏₁𝑏₃𝑏₄+𝑏₂𝑏₃𝑏₄=4𝛼²𝑏−6𝛼−6,₁𝑏₂𝑏₃𝑏₄=−8𝛼³+12𝛼²−10𝛼+7, (17) whose solution is 𝑏₁(𝛼)=143−4𝛼−√−√−3+8𝛼223−16𝛼+8𝛼²+−3+20𝛼−32𝛼²√,𝑏−3+8𝛼₂(𝛼)=143−4𝛼−√+√−3+8𝛼223−16𝛼+8𝛼²+−3+20𝛼−32𝛼²√,𝑏−3+8𝛼₃(𝛼)=143−4𝛼+√−√−3+8𝛼223−16𝛼+8𝛼²−−3+20𝛼−32𝛼²√,𝑏−3+8𝛼₄(𝛼)=143−4𝛼+√+√−3+8𝛼223−16𝛼+8𝛼²−−3+20𝛼−32𝛼²√.−3+8𝛼 (18)

We define the following functions, in terms of the behavior of the cycles 𝑔₁(𝑧,𝛼)=ℎ₁(𝑧,𝛼)ℎ₂()1𝑧,𝛼=1+23−4𝛼−√𝑧+1−3+8𝛼2(−1+2𝛼)1+√−3+8𝛼𝑧²+123−4𝛼−√−3+8𝛼𝑧³+𝑧⁴,𝑔₂(𝑧,𝛼)=ℎ₃(𝑧,𝛼)ℎ₄()1𝑧,𝛼=1+23−4𝛼+√𝑧+1−3+8𝛼2(−1+2𝛼)1−√−3+8𝛼𝑧²+123−4𝛼+√−3+8𝛼𝑧³+𝑧⁴. (19)

We see in the following sections that the roots of these functions yield to the appearance of attractive 2-cycles.

5.1. Cycles of Period 2 Coming from 𝑔₁(𝑧,𝛼)

The four solutions of 𝑔₁(𝑧,𝛼)=0 are 𝑤₁(𝛼)=18−3+4𝛼+√−3+8𝛼−√223−16𝛼+8𝛼²+(1−4𝛼)√−1−3+8𝛼4−3−12𝑎+8𝑎²−√√−3+8𝛼+223−4𝛼−√8𝛼−323−16𝛼+8𝛼²+(1−4𝛼)√𝑤−3+8𝛼,₂(𝛼)=18−3+4𝛼+√−3+8𝛼+√223−16𝛼+8𝛼²+(1−4𝛼)√−1−3+8𝛼4−3−12𝑎+8𝑎²−√√−3+8𝛼−223−4𝛼−√8𝛼−323−16𝛼+8𝛼²+(1−4𝛼)√𝑤−3+8𝛼,₃(𝛼)=18−3+4𝛼+√−3+8𝛼−√223−16𝛼+8𝛼²+(1−4𝛼)√+1−3+8𝛼4−3−12𝑎+8𝑎²−√√−3+8𝛼+223−4𝛼−√8𝛼−323−16𝛼+8𝛼²+(1−4𝛼)√𝑤−3+8𝛼,₄(𝛼)=18−3+4𝛼+√−3+8𝛼+√223−16𝛼+8𝛼²+(1−4𝛼)√+1−3+8𝛼4−3−12𝑎+8𝑎²−√√−3+8𝛼−223−4𝛼−√8𝛼−323−16𝛼+8𝛼²+(1−4𝛼)√−3+8𝛼. (20)

It is easy to see that 𝑤₁(7/2)=𝑤₂(7/2)=𝑠₁(7/2) and 𝑤₃(7/2)=𝑤₄(7/2)=𝑠₂(7/2), so there are two 2-cycles coming from this function, {𝑤₁(𝛼),𝑤₂(𝛼)} and {(𝑤₃(𝛼),𝑤₄(𝑎)}. We know (see [9]) that for 𝛼=7/2 the strange fixed points, 𝑠₁(7/2)=2−√3 and 𝑠₂(7/2)=2+√3 become parabolic |𝑂^′_𝑝(𝑠₁(7/2),𝛼)|=|𝑂^′_𝑝(𝑠₂(7/2),𝛼)|=1, and for 𝛼>7/2, these strange fixed points are repulsive. As we prove in [12] these two 2-cycles become attractive for real 𝛼∈(7/2,𝛼^∗∗),  𝛼^∗∗≈3.738271. In this paper, we are interested in locating the boundaries of the bulb. In the stability study of the cycles {𝑤₁(𝛼),𝑤₂(𝛼)} and {(𝑤₃(𝛼),𝑤₄(𝑎)}, we obtain two 2-bulbs where these 2-cycles are attractive.

The dynamical planes for values of the parameter 𝛼 inside these 2-bulbs are similar to those obtained in Figures 3 (region 𝐷₂) and 5 (region 𝐷₃).

Proposition 4. Let 𝑆𝑤₁₂(𝛼) and 𝑆𝑤₃₄(𝛼) be the stability functions of the 2-cycles {𝑤₁(𝛼),𝑤₂(𝛼)} and {𝑤₃(𝛼),𝑤₄(𝛼)}, respectively. Then, |𝑆𝑤₁₂(𝛼)|<1 and |𝑆𝑤₃₄(𝛼)|<1, for 𝛼∈𝐷₂∪𝐷₃, such that 𝐶₂∶={𝛼∶|𝛼−1.6876|<0.015}⊂𝐷₂⊂𝐶₁{||},𝐶∶=𝛼∶𝛼−1.6876<0.017₄∶={𝛼∶|𝛼−3.62|<0.117}⊂𝐷₃⊂𝐶₃∶={𝛼∶|𝛼−3.62|<0.12}. (21)

Proof. The 2-cycles {𝑤₁(𝛼),𝑤₂(𝛼)} and {𝑤₃(𝛼),𝑤₄(𝛼)} derive from the strange points 𝑠₁(𝛼) and 𝑠₂(𝛼) for 𝛼=7/2 and 𝛼=𝛼^∗, being attractive for the same values of the parameter. There are two different bulbs where these 2-cycles are attractive, 𝐷₂ and 𝐷₃. The boundaries of these bulbs satisfy |𝑆𝑤₁₂(𝛼)|=|𝑆𝑤₃₄(𝛼)|=1, and they are not circles. Nevertheless, there exist two coronas delimited by the circles (𝐶₁,𝐶₂) and (𝐶₃,𝐶₄), see Figures 9 and 10, surrounding the boundaries of these bulbs.
The stability function on these two circles |𝑆𝑤₁₂(𝛼)|=|𝑆𝑤₃₄(𝛼)| is drawn for all of them: 𝜕𝐶₁∶𝛼=1.6876+0.017𝑒^{2𝜋𝑡𝑖} and 𝜕𝐶₂∶𝛼=1.6876+0.015𝑒^{2𝜋𝑡𝑖}, 𝜕𝐶₃∶𝛼=3.62+0.12𝑒^{2𝜋𝑡𝑖} and 𝜕𝐶₄∶𝛼=3.62+0.117𝑒^{2𝜋𝑡𝑖}, 0≤𝑡≤1, respectively, and they can be seen in Figures 11 and 12, where the color of the stability function is the same as the corresponding circle. Let us notice that the stability function is equal to 1 for 𝑡=𝜋 in the second picture; this point corresponds to 𝛼=7/2∈𝐶₄, and it is precisely the bifurcation point.

Figure 9 

Corona delimited by the circles (𝐶1,𝐶2).

Figure 10 

Corona delimited by the circles (𝐶3,𝐶4).

Figure 11 

Stability function of the 2-cycle {𝑤1(𝛼),𝑤2(𝛼)} for 𝐶1 and 𝐶2.

Figure 12 

Stability function of the 2-cycle {𝑤3(𝛼),𝑤4(𝛼)} for 𝐶3 and 𝐶4.

5.2. Cycles of Period 2 Coming from 𝑔₂(𝑧,𝛼)

The four solutions of 𝑔₂(𝑧,𝛼)=0 are𝑡₁(𝛼)=18−3+4𝛼−√−3+8𝛼−√223−16𝛼+8𝛼²−(1−4𝛼)√−1−3+8𝛼4−3−12𝛼+8𝛼²+√√−3+8𝛼+223−4𝛼+√8𝛼−323−16𝛼+8𝛼²−(1−4𝛼)√𝑡−3+8𝛼,₂(𝛼)=18−3+4𝛼−√−3+8𝛼−√223−16𝛼+8𝛼²−(1−4𝛼)√+1−3+8𝛼4−3−12𝛼+8𝛼²+√√−3+8𝛼+223−4𝛼+√8𝛼−323−16𝛼+8𝛼²−(1−4𝛼)√𝑡−3+8𝛼,₃(𝛼)=18−3+4𝛼−√−3+8𝛼+√223−16𝛼+8𝛼²−(1−4𝛼)√−1−3+8𝛼4−3−12𝛼+8𝛼²+√√−3+8𝛼−223−4𝛼+√8𝛼−323−16𝛼+8𝛼²−(1−4𝛼)√𝑡−3+8𝛼,₄(𝛼)=18−3+4𝛼−√−3+8𝛼+√223−16𝛼+8𝛼²−(1−4𝛼)√−1−3+8𝛼4−3−12𝛼+8𝛼²+√√−3+8𝛼−223−4𝛼+√8𝛼−323−16𝛼+8𝛼²−(1−4𝛼)√−3+8𝛼. (22)

The study of the stability functions of these roots 𝑆𝑡₁₄(𝛼)=𝑂^′_𝑝𝑡₁(𝛼),𝛼𝑂^′_𝑝𝑡₄(𝛼),𝛼,𝑆𝑡₂₃(𝛼)=𝑂^′_𝑝𝑡₂(𝛼),𝛼𝑂^′_𝑝𝑡₃(𝛼),𝛼, (23) and their boundaries |𝑆𝑡₁₄(𝛼)|=|𝑆𝑡₂₃(𝛼)|=1 gives three different regions where the 2-cycles {𝑡₁(𝛼),𝑡₄(𝛼)} and {t₂(𝛼),𝑡₃(𝛼)} are attractive (see Figure 13).

(a)

(b)

(c)

Figure 13 

|𝑆𝑡12(𝛼)|=1 and |𝑆𝑡34(𝛼)|=1.

Proposition 5. Let 𝑆𝑡₁₄(𝛼) be the stability function of the 2-cycle {𝑡₁(𝛼),𝑡₄(𝛼)}. Then, |𝑆𝑡₁₄(𝛼)|<1, for 𝛼∈𝐷₄, such that 𝐶₂∶={𝛼∶|𝛼−0.376|<0.001}⊂𝐷₄⊂𝐶₁∶={𝛼∶|𝛼−0.376|<0.002}. (24)

Proof. The 2-cycle {𝑡₁(𝛼),𝑡₄(𝛼)} is attractive in three different bulbs. One of them cuts the real line, and its boundary satisfies |𝑆𝑡₁₄(𝛼)|=1; as in the previous cases, it is not circle but there exists one corona delimited by the circles (𝐶₁,𝐶₂), (Figure 14).
The stability function of these two circles |𝑆𝑡₁₄(𝛼)| is drawn for 𝜕𝐶₁∶𝛼=0.376+0.02𝑒^{2𝜋𝑡𝑖} and 𝜕𝐶₂∶𝛼=0.376+0.01𝑒^{2𝜋𝑡𝑖}, 0≤𝑡≤1, and can be seen in Figure 15 where the colour of stability function is the same as the corresponding circle. The dynamical planes for values of 𝛼 inside this bulb are similar to those obtained in Figure 2.

Figure 14 

|𝑆𝑡14(𝛼)|=1 and the corona (𝐶1,𝐶2).

Figure 15 

Stability function of the 2-cycle {𝑡1(𝛼),𝑡4(𝛼)} for 𝐶1 and 𝐶2.

6. Conclusions

The cat set as a parameter space of the Chebyshev-Halley family on quadratic polynomials is dynamically very wealthy, as it happens with Mandelbrot set. The head and the body of the cat set are surrounded by bulbs of different sizes. In this paper, we study those which give rise to attractive cycles of period two. We observe that these attractive 2-cycles exist for many different parameter values, that is, for many different members of the family of iterative methods.

We can draw the stability functions of all these 2-cycles. Mathematica permits us to draw these stability functions for values between 0 and 1, Figure 16. In it, the big circles correspond to values of 𝛼 where the fixed points 𝑧=1 (left one, |𝛼−(13/6)|<1/3), and 𝑧=𝑠₁(𝛼),𝑠₂(𝛼) (right one, |𝛼−3|<1/2), respectively, are attractive. The other figures correspond to the different bulbs where the 2-cycles studied in the previous sections are attractive.

Figure 16 

Stability functions of cycles of period two.

Let us remark that the number of 2-cycles is different depending on the bulb considered. There is only one attractive 2-cycle in the region 𝐷₁, whereas there are two 2-cycles in the regions 𝐷₂ and 𝐷₃. This is because the attractive 2-cycle for 𝛼∈𝐷₁ comes from the bifurcation of the strange fixed point 𝑧=1, and the two attractive 2-cycles for 𝛼∈𝐷₂∪𝐷₃ come from the bifurcation of the strange fixed points 𝑧=𝑠₁(𝛼),𝑠₂(𝛼), and the two attractive 2-cycles for 𝛼∈𝐷₂ come from the bifurcation of the 2-cycle {𝑧₁(𝛼),𝑧₂(𝛼)}.

Furthermore, by comparing Figures 1 and 16, we can conclude that the little cats on the necklace also correspond to values of 𝛼 where some attractive 2-cycles appear in the Chebyshev-Halley family.

Acknowledgments

This research was supported by Ministerio de Ciencia y Tecnología MTM2011-28636-C02-02, by Vicerrectorado de Investigación, Universitat Politècnica de València PAID SP20120498 and by Vicerrectorado de Investigación, Universitat Jaume I P11B2011-30. The authors would like to thank Mr. Francisco Chicharro for his valuable help with the numerical and graphic tools for drawing the dynamical planes.