^{1, 2}

^{1, 2}

^{1, 2}

^{1}

^{2}

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

This study has been extended by different authors to other point-to-point iterative methods for solving nonlinear equations (see, e.g., [

The fixed point operator corresponding to the family of Chebyshev-Halley type methods is

The parameter

Then, the operator (

As it is known (see [

Depending on the asymptotic behavior of their orbits, a point

On the other hand, a fixed point

The set of points

The invariant Julia set for Newton’s method on quadratic polynomials is the unit circle

The rest of the paper is organized as follows. in Section

Fixed points of the operator

Moreover,

The fixed point

If

If

If

The fixed points

If

If

If

On the other hand, the critical points of

It is known that there is at least one critical point associated with each invariant Fatou component (see [

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

Parameter plane.

Briefly summarizing the results of [

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 (

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:

As we have seen in [

In addition, the authors showed in [

Two 2-cycles in the dynamical plane for

Two 2-cycles in the dynamical plane for

Dynamical plane for

Two 2-cycles in the dynamical plane for

Let us observe in Figure

For

For

The roots of

We can draw numerically the boundaries where these 2-cycles are parabolic, (see Figure

The first two drawings of Figure

Moreover, we find two disks that embed this bulb.

Let

The boundary of the bulb satisfies

As we see in Figures

Corona delimiting the bulb of period two in the head.

Stability function of the 2-cycle

The dynamical planes for values of

A similar study can be made on

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.

The four solutions of

It is easy to see that

The dynamical planes for values of the parameter

Let

The 2-cycles

The stability function on these two circles

Corona delimited by the circles

Corona delimited by the circles

Stability function of the 2-cycle

Stability function of the 2-cycle

The four solutions of

The study of the stability functions of these roots

Let

The 2-cycle

The stability function of these two circles

Stability function of the 2-cycle

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

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

Furthermore, by comparing Figures

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.