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

and Applied Analysis 3 Figure 1: Parameter plane. 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: O p (z, α) = z. (9) The relation O p(z, α) − z = 0 can be factorized as z (−1 + z) (1 + 3z − 2αz + z ) f (z, α) g (z, α) = 0, (10) where f (z, α) = 1 + (3 − 2α) z + (3 − 2α) z 2 + (3 − 2α) z 3 + z, (11) g (z, α) = 1 + (3 − 4α) z + (2 − 6α + 4α ) z + (3 − 6α + 4α) z + (9 − 22α + 20α − 8α) z + (3 − 6α + 4α) z + (2 − 6α + 4α) z + (3 − 4α) z 7 + z. (12) As we have seen in [9], the product z(−1 + z)(1 + 3z − 2αz+z) 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 theMandelbrot 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 2cycle 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. 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 + α 2 + 1 4 √5 − 4α + 4α2 − 1 4 √−2−16α +8α2+(−6 +4α)√5−4α+4α2, z2 (α) = − 3 4 + α 2 + 1 4 √5 − 4α + 4α2 + 1 4 √−2 − 16α+8α2+(−6+4α)√5−4α+4α2, z3 (α) = − 3 4 + α 2 − 1 4 √5 − 4α + 4α2 − 1 4 √−2−16α+8α2−(−6+4α)√5−4α+4α2, z4 (α) = − 3 4 + α 2 − 1 4 √5 − 4α + 4α2 + 1 4 √−2 − 16α + 8α2 − (−6 + 4α)√5 − 4α + 4α2, (13) 4 Abstract and Applied Analysis