# Chaos and Stability in a New Iterative Family for Solving Nonlinear Equations
## 3 Complex Dynamics Behavior
### Dynamical Planes  
We begin this section by presenting how we generate a dynamical plane that will allow us to see the stability of a method for a specific value of \(\alpha\). This is defined as a mesh in the complex plane where each point of this mesh corresponds to a different value of the initial estimate \(x_{0}\). Its graphical representation shows the convergence of the method to any of the roots starting from \(x_{0}\) with a maximum of 50 iterations and a tolerance of \(10^{-3}\). Fixed points are illustrated with a white circle \"\(\Circ^{\prime}\), critical points with a white square \"\(\square\)\" and attractors with a white asterisk \"\(\ast\)\". Moreover, the basins of attraction are depicted in different colors. The resulting graphic is made in Matlab R2020a with a resolution of \(1000\times 1000\) pixels.  
Here, we study the stability of some CMT(\(\alpha\)) family methods through the use of dynamical planes. We will consider the methods proposed in Section 3.4 for values of \(\alpha\) inside and outside the stability regions of the parameter spaces.  
On the one hand, examples of methods inside the stability region are given for \(\alpha=-1,\;0,\;1\). Their dynamical planes with some convergence orbits in yellow are shown in Figure 4. Note that all three methods present only two basins of attraction associated with the roots: the basin of 0 colored in orange and the basin of \(\infty\) colored in blue. Furthermore, there are no black areas of non-convergence to the solution. Consequently, these methods show good dynamical behavior: they are very stable. Of these methods, the best member of CMT(\(\alpha\)) family is for \(\alpha=1\), as it has fewer strange fixed points and free critical points.  
Figure 3: Parameter spaces of free critical points (in red color, the complex area where the corresponding critical point converges to \(0\) or \(\infty\), that is, the stability region).  
On the other hand, examples of methods outside the stability region are given for \(\alpha=-300,\ 200,\ 400\). Their dynamical planes with some convergence orbits in yellow are shown in Figure 5. Note that all three methods present more than two basins of attraction, that is, there are other basins of attraction that do not correspond to the roots. The basins of \(0\) and \(\infty\) are colored in orange and blue, respectively, and the other basins are colored in black, red, and green. Figure 5a shows the convergence to an attracting periodic orbit of period \(2\). Figure 5b,c shows the convergence to an attracting strange fixed point. Furthermore, let us remark that in the three figures the basin of \(0\) is very small, due to the presence of the other basins of attraction, which reduces the chances of convergence to the solution. Likewise,  
Figure 4: Dynamical planes for methods inside the stability region (basin of attraction of \(0\) in orange color; in blue color, the basin of \(\infty\)).  
there are black areas of slow convergence of the methods. Consequently, these methods have poor dynamical behavior: they are unstable.  
Figure 5: Dynamical planes for methods outside the stability region (basin of attraction of 0 in orange color; in blue color, the basin of \(\infty\); in green or red color, the basin of attracting strange fixed points).