# Dynamics and Stability on a Family of Optimal Fourth-Order Iterative Methods
## 4 Dynamical Analysis
### Stability of the Fixed Points  
In this section, we analyze the stability of fixed points. We give some important results on the stability of strange fixed points, and we analyze the behavior of the rational operator on free critical points. The strange fixed points are \(z=1\), and the roots of the polynomial are  
\[q(z,\beta)=6z^{6}+42z^{5}+126z^{4}+(180+\beta)z^{3}+126z^{2}+42z+6, \tag{18}\]  
which is denoted by \(s_{j}(\beta)\), \(j=1,2,\ldots,6\). The stability of the strange fixed points is analyzed in the following result. **Theorem 4**. : _The fixed points \(z=0\) and \(z=\infty\), which are associated with the roots \(a\) and \(b\), are superattractors regardless of the value of \(\beta\). \(z=1\) is a strange fixed point when \(\beta\
eq 240\), and its stability is determined as follows:__(a) If \(|\beta-240|>768\), then the fixed point \(z=1\) is an attractor, (b) If \(|\beta-240|<768\), then the fixed point \(z=1\) is repulsive, (c) If \(|\beta-240|=768\), then the fixed point \(z=1\) is neutral._  
Proof. : By evaluating the rational operator \(\mathcal{R}_{p}(z,\beta)\), as well as its derivative at \(z=0\), we conclude that \(z=0\) is a superattractor fixed point. For the punto \(z=\infty\), we derive the inverse of the operator \(\mathcal{R}_{p}(z,\beta)\), i.e., \(\left(\frac{1}{\mathcal{R}_{p}\left(\frac{1}{2}\right)}\right)^{\prime}\), and we evaluate it in \(z=0\). We can affirm that \(z=\infty\) is a superattractor fixed point. Now, we evaluate \(\mathcal{R}_{p}^{\prime}(z,\beta)\) on the fixed point \(z=1\) in order to determine its stability. (a) \(\mathcal{R}_{p}^{\prime}(1,\beta)=\left|\frac{768}{\beta-240}\right|<1\), which implies that \(768<|\beta-240|\), and therefore \(\left|\mathcal{R}_{p}^{\prime}(1,\beta)\right|<1\) if and only if \(|\beta-240|>768\) and \(\beta\
eq 240\). (b) \(\mathcal{R}_{p}^{\prime}(1,\beta)=\left|\frac{768}{\beta-240}\right|>1\), which implies that \(768<|\beta-240|\), and, therefore, \(\left|\mathcal{R}_{p}^{\prime}(1,\beta)\right|>1\) if and only if \(|\beta-240|<768\) and \(\beta\
eq 240\). This completes the proof. In Figure 1, the stability of the strange fixed point \(z=1\) is presented. Theorem 4 shows that we have two fixed points that are superattractors, \(z=0\) and \(z=\infty\), and the strange fixed point \(z=1\), for which we find different regions where this behaves as an attractor, repulsor, and parabolic for \(\beta\
eq 240\). However, when we solve the equation \(\mathcal{R}_{p}(z,\beta)=z\), six other solutions appear; these are strange fixed points that are solutions of the polynomial \(q(z,\beta)\) (18), which is denoted by \(s_{j}(\beta)\), \(j=1,2,\ldots,6\). In the following result, we analyze the stability of these strange fixed points. **Theorem 5**. : _The strange fixed points of the rational operator \(\mathcal{R}_{p}(z,\beta)\), \(s_{j}(\beta)\), \(j=1,2\), are all repulsors in a complex plane._  
_On the other hand, the strange fixed points \(s_{j}(\beta)\), \(j=3,4\) are:_  
1. _attractors in the open interval_ \(\frac{32}{9}<\beta<3.55728\)_. In particular, they are superattractors for_ \(\beta\approx 3.55728\mp 4.9603i\)_._
2. _parabolics for_ \(\beta=\frac{4}{9}(8\mp 5i\sqrt{5})\)_._
3. _repulsors in the rest of the complex plane._  
_Finally, the strange fixed points \(s_{j}(\beta)\), \(j=5,6\) are:_  
1. _attractors in_ \(-524.182<\beta<-289.865\) _and_ \(-289.865<\beta<-189.865)\)_. In particular, they are superattractors for_ \(\beta\approx-289.865\)_._
2. _parabolic at all points on the boundary of the circumference of the cone--in particular, for the complex numbers_ \(\beta=\frac{4}{9}(8\mp 5i\sqrt{5})\)_._
3. _repulsors in the rest of the complex plane._  
Figure 1: Strange fixed point stability region where \(z=1\). Theorem 5 cannot be analytically proven, since the substitution of these strange fixed points into the stability function results in roots of higher-order polynomials.