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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 536910, 10 pages
Bulbs of Period Two in the Family of Chebyshev-Halley Iterative Methods on Quadratic Polynomials
1Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Cno. de Vera s/n, 46022 Valencia, Spain
2Instituto de Matemáticas y Aplicaciones de Castellón, Universitat Jaume I, Avenida de Vicent Sos Baynat s/n, 12071 Castelló de la Plana, Spain
Received 1 March 2013; Accepted 18 April 2013
Academic Editor: Jan Andres
Copyright © 2013 Alicia Cordero et al. 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.
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.
The application of iterative methods for solving nonlinear equations , 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., ).
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 .
The fixed point operator corresponding to the family of Chebyshev-Halley type methods is where and is a complex parameter. In , the authors have begun the study of the dynamics of this operator when it is applied on quadratic polynomial . For this polynomial, the operator (1) is the rational function: depending on parameters and .
The parameter can be removed by applying the conjugacy map with the properties , , and .
Then, the operator (3) becomes a one-parametric rational function:
As it is known (see ), for a rational function , on the Riemann sphere , the orbit of a point is defined as
Depending on the asymptotic behavior of their orbits, a point is called a fixed point if it satisfies . A periodic point of period is a point such that and , . A preperiodic point is a point that is not periodic, but there exists a such that is periodic. A critical point is a point where the derivative of rational function vanishes, .
On the other hand, a fixed point is called attractor if , superattractor if , repulsor if , and parabolic if . The stability of a periodic orbit is defined by the magnitude (lower than 1 or not) of , where are the points of the orbit of period .
The basin of attraction of an attractor is defined as the set of pre images of any order:
The set of points such that their families are normal in some neighborhood is the Fatou set, , 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 is the Julia set, ; 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 , and the Fatou set is defined by the two basins of attraction of the superattractor fixed points: and . However, as it can be seen in , 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 . 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 are , , which correspond to the roots of the polynomial and the strange fixed points and , denoted by and , respectively.
Moreover, and 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 satisfies the following statements.(1)If , then is an attractor, and, in particular, it is a superattractor for .(2)If , then is a parabolic point.(3)If , then is a repulsive fixed point.
Proposition 2 (see [9, Proposition 2]). The fixed points , , satisfy the following statements.(i)If , then and are two different attractive fixed points. In particular, and are superattractors.(ii)If , then and are parabolic points. In particular, .(iii)If , then and are repulsive fixed points.
On the other hand, the critical points of are , and which are denoted by and , respectively.
It is known that there is at least one critical point associated with each invariant Fatou component (see ). It is also shown in  that the critical points , , are inside the basin of attraction of when it is attractive () and coincide with for . Then, they move to the basins of attraction of and when these fixed points become attractive (). Critical and fixed points coincide for , and and 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 , that is, to an element of the family. Moreover, every value of 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.
Briefly summarizing the results of , we observe a black figure (the cat set) with two big disks corresponding to the values for those fixed points (the head, ), and and (the body, ) 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 . 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 , 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: The relation can be factorized as where
As we have seen in , the product yields to the fixed points. So, 2 periodic points come from the roots of or . In the following, we study the bulbs where 2-cycles become attractive, and, by imitating the notation of the Mandelbrot set (see ), we call them 2-bulbs.
In addition, the authors showed in  that the strange fixed points and 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 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 , the blue one is the attraction basin of , 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.
Let us observe in Figure 4 that the two components, where the 2-cycle is included, touch each other in the strange fixed point .
For (Figure 5), two attractive 2-cycles appear, one comes from the strange fixed point , and the other comes from .
For (Figure 3), two attractive 2-cycles appear from the bifurcation of the 2-cycle coming from the fixed point . 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
The roots of are where and form two cycles of period two: The stability functions of these 2-cycles depends on :
We can draw numerically the boundaries where these 2-cycles are parabolic, (see Figure 6). In it, blue points correspond to and red points represent the stability function .
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 . As we see in the following result, these real values permit to obtain the “radius” and the center of this bulb (see ).
Moreover, we find two disks that embed this bulb.
Proposition 3. Let be the stability function of the 2-cycle . Then, , for , such that where , , and .
Proof. The boundary of the bulb satisfies , and it is not a circle, but there exists a corona delimited by two circles and . These circles are centered in the middle point of and , , and have radii and , respectively, see Figure 7.
As we see in Figures 7 and 8, the value of the stability function in these circles is such that if and if . In Figure 8 the red curve corresponds to when and the blue one corresponds to . The value corresponds to the real values and .
The dynamical planes for values of inside this bulb are similar to those obtained in Figure 4.
5. Bulbs Coming from
A similar study can be made on in order to obtain the other bulbs of period 2 belonging to the cat set. To simplify the study we factorize , where are polynomials of degree two, , . The relationships that the coefficients must satisfy are whose solution is
We define the following functions, in terms of the behavior of the cycles
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
The four solutions of are
It is easy to see that and , so there are two 2-cycles coming from this function, and . We know (see ) that for the strange fixed points, and become parabolic , and for , these strange fixed points are repulsive. As we prove in  these two 2-cycles become attractive for real , . In this paper, we are interested in locating the boundaries of the bulb. In the stability study of the cycles and , we obtain two 2-bulbs where these 2-cycles are attractive.
Proposition 4. Let and be the stability functions of the 2-cycles and , respectively. Then, and , for , such that
Proof. The 2-cycles and derive from the strange points and for and , being attractive for the same values of the parameter. There are two different bulbs where these 2-cycles are attractive, and . The boundaries of these bulbs satisfy , and they are not circles. Nevertheless, there exist two coronas delimited by the circles and , see Figures 9 and 10, surrounding the boundaries of these bulbs.
The stability function on these two circles is drawn for all of them: and , and , , 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 for in the second picture; this point corresponds to , and it is precisely the bifurcation point.
5.2. Cycles of Period 2 Coming from
The four solutions of are
The study of the stability functions of these roots and their boundaries gives three different regions where the 2-cycles and are attractive (see Figure 13).
Proposition 5. Let be the stability function of the 2-cycle . Then, , for , such that
Proof. The 2-cycle is attractive in three different bulbs. One of them cuts the real line, and its boundary satisfies ; as in the previous cases, it is not circle but there exists one corona delimited by the circles , (Figure 14).
The stability function of these two circles is drawn for and , , 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 inside this bulb are similar to those obtained in Figure 2.
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 where the fixed points (left one, ), and (right one, ), respectively, are attractive. The other figures correspond to the different bulbs where the 2-cycles studied in the previous sections are attractive.
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 , whereas there are two 2-cycles in the regions and . This is because the attractive 2-cycle for comes from the bifurcation of the strange fixed point , and the two attractive 2-cycles for come from the bifurcation of the strange fixed points , and the two attractive 2-cycles for come from the bifurcation of the 2-cycle .
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.
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