# Chaos and Stability in a New Iterative Family for Solving Nonlinear Equations
## 3 Complex Dynamics Behavior
### Analysis and Stability of Fixed Points  
We calculate the fixed points of the rational operator \(R_{\alpha}(x)\) given in (30) and analyze their stability. **Proposition 2**. : _The fixed points of \(R_{\alpha}(x)\) are the roots of the equation \(R_{\alpha}(x)=x\). That is, \(x=0\), \(x=\infty\) and the following strange fixed points:_  
* \(ex_{1}=1\) _(if_ \(\alpha\
eq-77\)_), and_* \(ex_{i}(\alpha)\) _that correspond to the 10 roots of polynomial_ \(x^{10}+6x^{9}+18x^{8}+37x^{7}+58x^{6}-(\alpha-67)x^{5}+58x^{4}+37x^{3}+18x^{2}+6x+1\)_, where_ \(i=2,...,11\)_. The total number of different fixed points varies with the value of_ \(\alpha\)_:_
* _If_ \(\alpha\in\mathbb{C}\) _and_ \(\alpha\
otin\{-77,-1,1,5,307\}\)_, then_ \(R_{\alpha}(x)\) _has 13 fixed points._
* _If_ \(\alpha=-77\)_, then_ \(ex_{1}=1\) _is not a fixed point and_ \(R_{\alpha}(x)\) _has 12 fixed points._
* _If_ \(\alpha\in\{-1,1,5\}\)_, then_ \(R_{\alpha}(x)\) _has 11 fixed points._
* _If_ \(\alpha=307\)_, then_ \(ex_{1}=ex_{2}=ex_{3}=1\) _and_ \(R_{\alpha}(x)\) _has 11 fixed points._ _The pairs of conjugated strange fixed points, satisfying_ \(ex_{i}=\dfrac{1}{ex_{j}}\) _for_ \(i\
eq j\)_, are_ \(ex_{2}\) _and_ \(ex_{3}\)_,_ \(ex_{4}\) _and_ \(ex_{5}\)_,_ \(ex_{6}\) _and_ \(ex_{9}\)_,_ \(ex_{7}\) _and_ \(ex_{8}\)_, and_ \(ex_{10}\) _and_ \(ex_{11}\)_._  
From Proposition 2, we establish there is a minimum of 11 and a maximum of 13 fixed points. Of these, 0 and \(\infty\) correspond to the roots of the original quadratic polynomial \(p(x)\), and the strange fixed point \(ex_{1}=1\) (if \(\alpha\
eq-77\)) corresponds to the divergence of the original method, before Mobius transformation. Proposition 3: _The stability of the strange fixed point \(ex_{1}=1\), \(\forall\alpha\in\mathbb{C}\smallsetminus\{-77\}\), verifies:_  
* _If_ \(\left|\dfrac{384}{77+\alpha}\right|<1\)_, then_ \(ex_{1}\) _is an attractor._
* _If_ \(\left|\dfrac{384}{77+\alpha}\right|>1\)_, then_ \(ex_{1}\) _is a repulsor._
* _If_ \(\left|\dfrac{384}{77+\alpha}\right|=1\)_, then_ \(ex_{1}\) _is parabolic._ \(ex_{1}\) _is never a superattractor because_ \(\left|\dfrac{384}{77+\alpha}\right|\
eq 0\)_. The superattracting fixed points that satisfy_ \(|R^{\prime}_{\alpha}(x)|=0\) _are_ \(x=0\)_,_ \(x=\infty\) _and the following strange fixed points:_
* \(ex_{4}\)_,_ \(ex_{5}\) _for_ \(\alpha=-0.949874\pm 0.16946i\)_,_
* \(ex_{6}\)_,_ \(ex_{9}\) _for_ \(\alpha=2.40285\pm 1.11088i\)_, and_
* \(ex_{10}\)_,_ \(ex_{11}\) _for_ \(\alpha=178.653\)_._  
_The repulsive fixed points, which always satisfy \(|R^{\prime}_{\alpha}(x)|>1\), are the strange fixed points \(ex_{2}\) and \(ex_{3}\)._  
It is clear that 0 and \(\infty\) are always superattracting fixed points, but the stability of the rest of fixed points depends on the values of parameter \(\alpha\). From Proposition 3, there are 6 strange fixed points that can become superattractors for certain values of \(\alpha\). This means that there would be a basin of attraction of the strange fixed point and it could cause the method not to converge to the solution. Figure 1 shows the stability surface of the strange fixed point \(ex_{1}\). In this figure, the zones of attraction (yellow surface) and repulsion (gray surface) are observed, being the first one much greater than the second one. Note that for values of \(\alpha\) inside disk, \(ex_{1}\) is a repulsor; and, for off-disk values of \(\alpha\), \(ex_{1}\) is an attractor. Therefore, it is in our interest to always work inside the disk because the strange fixed point \(ex_{1}=1\) comes from the divergence of the original method and, therefore, it is better for the performance of the iterative method that the divergence is repulsive. From Proposition 2, the study of the stability of strange fixed points is reduced by a half. This is because each pair of conjugated strange fixed points exhibits the same stability characteristics. Furthermore, due to Proposition 3, \(ex_{2}\) and \(ex_{3}\) are always repulsors regardless of the value of \(\alpha\).