# Dynamics and Stability on a Family of Optimal Fourth-Order Iterative Methods
## 4 Dynamical Analysis  
In this part, we use several tools of complex dynamics. Thus, we need the function \(f\) to be defined on the Riemann sphere \(\hat{\mathbb{C}}\). In the following, we recall some concepts of complex dynamics. Let us suppose a fixed-point function on an arbitrary polynomial \(p(z)=(z-a)(z-b)\); this gives a rational function, \(\mathcal{R}\). Indeed, if given a rational function \(\mathcal{R}:\hat{\mathbb{C}}\rightarrow\hat{\mathbb{C}}\), where \(\hat{\mathbb{C}}\) represents the Riemann sphere, the orbit at a point \(z_{0}\) can be defined as  
\[\{z_{0},\mathcal{R}(z_{0}),\mathcal{R}^{2}(z_{0}),\mathcal{R}^{3}(z_{0}), \ldots,\mathcal{R}^{n}(z_{0}),\ldots\}.\]  
It is necessary to study the phase plane of \(\mathcal{R}\) by classifying the fixed points according to the asymptotic behavior related to their orbits. A \(z_{0}\in\hat{\mathbb{C}}\) is said to be a fixed point if \(\mathcal{R}(z_{0})=z_{0}\). A periodic point \(z_{0}\), of period \(p>1\), is a point where \(\mathcal{R}^{p}(z_{0})=z_{0}\) and \(\mathcal{R}^{k}(z_{0})\
eq z_{0}\), where \(k<p\). A preperiodic point is a point \(z_{0}\) that is not periodic, but there exists a \(k>0\) such that \(\mathcal{R}^{k}(z_{0})\) is periodic. A point \(z_{0}\) is a critical point of the rational function \(\mathcal{R}\) if \(\mathcal{R}^{\prime}(z_{0})=0\). If a critical point is different from any of those related to the roots of polynomial \(p(z)\), it is called a free critical point. In a similar way, the fixed points that are different from the roots of \(p(z)\) are called strange fixed points. Moreover, a fixed point \(z_{0}\) is called an attractor if \(|\mathcal{R}^{\prime}(z_{0})|<1\), a superattractor if \(|\mathcal{R}^{\prime}(z_{0})|=0\), repulsive if \(|\mathcal{R}^{\prime}(z_{0})|>1\), and parabolic if \(|\mathcal{R}^{\prime}(z_{0})|=1\). Fixed points other than those associated with the roots of the polynomial \(p(z)\) are called strange fixed points. The basin of attraction of an attractor \(\alpha\) is defined as:  
\[\mathcal{A}(\alpha)=\Big{\{}z_{0}\in\hat{\mathbb{C}}:\mathcal{R}^{n}(z_{0}) \rightarrow\alpha,n\rightarrow\infty\Big{\}}.\]  
For the rational function \(\mathcal{R}\), the Fatou set of \(\mathcal{R}\), which is denoted by \(\mathcal{F}(\mathcal{R})\), is the set of points \(z\in\hat{\mathbb{C}}\) whose orbits tend to one attractor (a fixed point or periodic orbit). Its complement in \(\hat{\mathbb{C}}\) is the Julia set, \(\mathcal{J}(\mathcal{R})\). That is, the basin of attraction from every fixed or periodic point belongs to the Fatou set, while the boundaries of these basins of attraction are in the Julia set (for more explanation, see [12; 13; 14]). The following result justifies the interest in the parameter planes that we will introduce later. The connected component of the basin of attraction, which contains the point or periodic orbit of attraction, is called the immediate basin of attraction. **Theorem 3** (Fatou-Julia). : _Let \(\mathcal{R}\) be a rational function. An immediate basin of attraction of any attractor has at least one critical point._  
To any rational operator that satisfies the scaling theorem, we can apply a Mobius transformation, which is stated as follows. If \(p(z)=(z-a)(z-b)\), then \(\mathcal{R}_{p}\) depends on parameters \(a\) and \(b\). To eliminate this dependency, we apply a Mobius transformation \(h\) as follows:  
\[O_{p}(z)=\left(h\circ\mathcal{R}_{p}\circ h^{-1}\right)(z),\]  
where  
\[h(z)=\frac{z-a}{z-b},\]  
which satisfies  
* \(h(\infty)=1\),
* \(h(a)=0\)
* \(h(b)=\infty\). Operators \(O_{p}\) and \(\mathcal{R}_{p}\) have the same qualitative properties of stability. With Theorem 2, we may find all of the stable behaviors of a rational function \(\mathcal{R}\) by analyzing its performance on the set of critical points, as we see below.