Period-doubling bifurcations in the family of Chebyshev–Halley-type methods

The choice of a member of a parametric family of iterative methods is not always easy. The family of Chebyshev–Halley schemes is a good example of it. The analysis of bifurcation points of this family allows us to define a real interval in which there exist several problematic behaviours: attracting points that become doubled, other ones that become periodic orbits, etc. These aspects should be avoided in an iterative procedure, so it is important to determine the regions where this conduct takes place. In this paper, we obtain that this family admits attractive 2-cycles in two different intervals, for real values of the parameter.


Introduction
The application of iterative methods for solving nonlinear equations f (z) = 0, with f : C → C, gives rise to rational functions whose dynamics are not well known. The simplest model is obtained when f (z) is a quadratic polynomial and the iterative process is Newton's method. This dynamical study has been extended to other point-to-point iterative methods used for solving nonlinear equations, with higher order of convergence (see e.g. [1,3] and, more recently [7,12]).
Most of the well-known point-to-point cubically convergent methods belong to the oneparameter family called Chebyshev-Halley. This set of iterative schemes has been widely analysed under different points of view. In this work, we focus our attention on the dynamical behaviour of the rational function associated with this family. From the numerical point of view, this dynamical behaviour gives us important information about its stability and reliability. In this line, Varona [13] described the dynamical behaviour of several well-known iterative methods. The dynamics of some third-order iterative methods is also studied in [1,2] and, more recently, in [7,8].
In this paper, we are interested in the study of period-doubling bifurcations in the Chebyshev-Halley family when it is applied on quadratic polynomials. In a previous paper by Cordero et al. [5], we have obtained that the corresponding rational function has different fixed points depending on *Corresponding author. Email: vindel@uji.es the values of the parameter.As we have proved, two of these fixed points are always superattractive, but the stability of the others (called strange fixed points) depends on the parameter values.
In fact, we check in Section 4 that there are, at least, two real intervals for which attractive 2cycles appear and therefore three period-doubling and one pitchfork bifurcations occur changing the stability of the strange fixed points.

Basic dynamical concepts
Now, let us recall some basic concepts on complex dynamics (see [4]). Given a rational function R :Ĉ →Ĉ, whereĈ is the Riemann sphere, the orbit of a point z 0 ∈Ĉ is defined as We are interested in the study of the asymptotic behaviour of the orbits depending on the initial condition z 0 , that is, we are going to analyse the phase plane of the map R defined by the different iterative methods.
To obtain this phase space, we classify the starting points from the asymptotic behaviour of their orbits. A On the other hand, a fixed point The basin of attraction of an attractor α is defined as the set of pre-images of any order: The set of points z ∈Ĉ such that their families {R n (z)} n∈N are normal in some neighbourhood 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 inĈ is the Julia set, J (R); therefore, the Julia set includes all repelling fixed points, periodic orbits and their preimages. 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 Chebyshev's method applied to quadratic polynomials is more complicated than that for Newton's method, and it has been studied in [10].
The rest of this paper is organized as follows. In Section 2, we introduce the previous results needed to develop the present study. In Section 3, we show the parametric space associated with the parametric family. Section 4 is devoted to characterize the bifurcation points and to classify them. Finally, we provide some remarks and conclusions in Section 5.

Previous results
The family of Chebyshev-Halley-type methods can be written as the iterative method Our interest is focused on the study of the dynamics of the corresponding fixed-point operator when it is applied on the quadratic polynomial p(z) = z 2 + c. For this polynomial, the operator corresponds to the rational function depending on two parameters α and c.
The parameter c can be obviated by considering the conjugacy map with the following properties: Accordingly, we are interested in the dynamics of the operator (1)

Fixed and critical points
We have began the study of the dynamics of this operator in function of the parameter α in [5].
As we have proved, the number and stability of fixed points depend on the parameter α: The last two fixed points are the roots of z 2 + (3 − 2α)z + 1 = 0, denoted by s 1 and s 2 . Furthermore, these two points are not independent as s 1 = 1/s 2 . We need the derivative of the operator (1) to study the stability of fixed points: From Equation (3), we obtain that the origin and ∞ are always superattractive fixed points, but the stability of the other fixed points change depending on the values of the parameter α. These points are called strange fixed points. Their stability satisfies the following statements.
Proposition 1 (see [5]) The stability of the fixed point z = 1 satisfies the following statements:

is a repulsive fixed point for any other value of α.
Proposition 2 (see [5]) The stability of the fixed points z = s i , i = 1, 2, satisfies the following statements: • If |α − 3| < 1 2 , then s 1 and s 2 are two different attractive fixed points. In particular, for α = 3, s 1 and s 2 are superattractors.
The critical points are those points where the first derivative of the rational operator vanishes, that is, the last two critical points are denoted by c 1 and c 2 . In addition, they are not independent as c 1 = 1/c 2 .

The parameter space
The dynamical behaviour of the operator (1) depends on the values of the parameter α. It can be seen in the parameter space, as shown in Figure 1.
In this parameter space, we observe a black figure (let us call it the cat set), with a certain similarity with the Mandelbrot set (see [6]): for values of α outside this cat set the Julia set is disconnected. The two disks in the main body of the cat set correspond to the α values for those the fixed points z = 1 (the head) and s 1 and s 2 become attractive (the body). We also observe a curve similar to a circle that passes through the cat's neck; we call it the necklace. As we have proved in [5], the parameter space inside this curve is topologically equivalent to a disk.
The head of the cat corresponds to |α − 13 6 | < 1 3 , for which the fixed point z = 1 is attractive. The body of the cat set corresponds to values of the parameter such that |α − 3| < 1 2 . In this case, s 1 and s 2 are attractors and have their own basin of attraction, as there exists one critical point in each basin. The intersection point of both disks is in their common boundary and corresponds to α = 5 2 . For α = 5 2 , the three strange fixed points coincide z = 1 = s 1 = s 2 and it is parabolic, |O p (1)| = 1, with multiplicity 3. We know, by the flower theorem of Latou (see e.g. [11]), that this parabolic point is in the common boundary of two attractive regions. The points of an orbit inside each region approach to z = 1 without leaving the region.
The boundary of the cat set is exactly the bifurcation locus of Chebyshev-Halley-type family acting on quadratic polynomial, that is, the set of parameters for which the dynamics changes abruptly under small changes of α. In this paper, we are interested in the study of some of these bifurcations, those that involve cycles of period 2.
In this paper, we are interested in the study of bifurcations for this family of iterative methods. As we will see in the follow-ing section, this family suffers various period-doubling and one pitchfork bifurcations for different values of the parameter. It is known that period-doubling bifurcation is characterized by one attractive (or repulsive) fixed point that becomes repulsive (attractive) after the bifurcation point, and simultaneously one attractive (repulsive) 2-cycle appears. For the pitchfork bifurcation, one attractive (or repulsive) fixed point becomes repulsive (attractive) after the bifurcation point, and simultaneously two attractive (repulsive) fixed points appear.

The bulb of period 2 of the head
It is easy to check that z = 1 is a hyperbolic point for all those values of α belonging to the circle |α − 13 6 | = 1 3 , as As we have seen in [5], if α > 11 6 then O p (1) < 1, if α = 11 6 then O p (1) = 1 and if α < 11 6 then O p (1) > 1.
We are going to show that there is a doubling period bifurcation for α = 11 6 . For α < 11 6 , the periodic point z = 1 becomes repulsive and one attractive cycle of period 2 appears. So, (z = 1, α = 11 6 ) will be a doubling period bifurcation point.
The points of the 2-cycle in the bulb of period 2 on the head are The function of stability is that is, To know the range where this cycle is an attractor, we demand For α ∈ R, this implies that The only real root inside the head of the cat (corresponding to the real root of the third-degree polynomial in the last equation) is Moreover, we observe that this cycle is attractive in the interval α * < α < 11 6 by drawing the function O p (z 1 ) · O p (z 2 ) in this interval (Figure 2). We observe that there is one value where this cycle is superattractive that coincides with the minimum of the function Let us point out that these points are complex for values of α in the interval (α * , 11 6 ).

The bulb of period 2 in the body of the cat
We know (see [5]) that for α = 7 2 , the strange fixed points, s 1 = 2 − √ 3 and s 2 = 2 + √ 3, become parabolic |O (s 1 )| = |O (s 2 )| = 1. For real α > 7 2 , these strange fixed points are repulsive. In this section, we show that for real α > 7 2 , there is a bulb where two attractive cycles of period 2 appear. As in the previous section, a doubling period bifurcation occurs for each strange fixed point.
Proof We can prove that, for α = 7 2 , s 1 and s 2 are two roots of O 2 Therefore, as happened in the previous section with f (z, α), the cycles of period 2 that collapse with s 1 and s 2 , for α = 7 2 , come from the roots of g(z, a). We can factorize g(z, α) = g 1 (z, α)g 2 (z, α)g 3 (z, α)g 4 (z, α), where g i (z, α) are polynomials of degree 2, g i (z, α) = 1 + b i z + z 2 . The relationships that the coefficients must satisfy are It easy to see that z 1 ( 7 2 ) = z 3 ( 7 2 ) = s 1 and z 2 ( 7 2 ) = z 4 ( 7 2 ) = s 2 . Therefore, the cycles of period 2 will be (z 1 , z 3 ) and (z 2 , z 4 ). It can be checked that Moreover, it can be checked graphically that So, we observe that there is one interval where these 2-cycles are attractive (Figures 3 and 4). We also obtain that In particular, it can be checked that is superattractive for α ≈ 3.6218839102; and the 2-cycle (z 2 , z 4 ) is superattractive for α ≈ 3.621883885696156.

Conclusions
We have characterized bifurcations of Chebyshev-Halley family for real values of parameter α. It has been stated that some doubling period and pitchfork bifurcations appear for values of the parameter in the interval (α * , α * * ). This is a very useful information from the numerical point of view, as the behaviour of the iterative methods out of this interval will be more reliable, since there are no attracting elements different from the roots.