# Convergence and Stability of a New Parametric Class of Iterative Processes for Nonlinear Systems
## 4 Bifurcation Analysis of Free Critical Points
### Parameter Line and Bifurcation Plane  
Now, we use a graphical tool that helps us to identify for which values of parameters there might be convergence to roots, divergence or any other performance. Real parametric lines, for \(n=2\), are presented in Figures 4 and 5 (see Theorem 5). In these figures, a different free critical point is employed as a seed of each member of the class, using \(-77<\alpha<-5\) and \(-5<\alpha<-1\) to ensure the existence of real critical points.  
To generate them, a mesh of \(1000\times 1000\) points is made in \([0,1]\times]-77,-5\) [ for the first figure and \([0,1]\times]-5,-1\) [ for the next. We use \([0,15]\) in Figure 4a to increase the interval where \(\alpha\) is defined and \([0,1]\) in Figures 4a and 5, allowing a better visualization. So, the color corresponding to each value of \(\alpha\) is red if the corresponding critical point converges to one of the roots of the polynomial system, blue in the case of divergence, and black in other cases (chaos or periodic orbits). In addition, we use \(500\) as the limit of iterations and tolerance \(10^{-3}\).  
The global performance of each pair of free critical points is similar, so only \((cri_{1}(\alpha),cri_{1}(\alpha))\) is shown in Figure 4. In Figure 4a, only a small black region shows non-convergence to the roots (red color). Now we show the parameter line for \(\alpha\in(-5,-1)\).  
In the line shown in Figure 5 it is observed that the zone shows global convergence to the roots. Therefore, it is a good area for choosing \(\alpha\).  
The concept of bifurcation is important in nonlinear systems since it allows us to study the behavior of the solutions of a family of iterative methods. In reference to dynamical systems, a bifurcation occurs when a small variation in the values of the system parameters (bifurcation parameters) causes a qualitative or topological change in the behavior of the system. Feigenbaum or bifurcation diagrams appear to analyze the changes of each class of methods on \(q(x)\) by using each critical point of the function as a seed and observing its performance for different ranges of \(\alpha\). By using a mesh with \(4000\) subintervals in each axis and after \(1000\) iterations, different b  
Figure 4: Parameter line of \(ACTV^{6}(x,\alpha)\) in \(\alpha\in(-77,-5)\). (**a**) \(\alpha\in(-77,-5)\). (**b**) \(\alpha\in(-77,-72)\).  
Figure 5: Parameter line of \(ACTV^{6}(x,\alpha)\) in \(\alpha\in(-5,-1)\).  
Figure 6 shows the bifurcation diagrams in the black area of the parameter line Figure 4b, specifically when \(\alpha\in(-73.5,-72.5)\). In Figure 6a, a general convergence to one of the roots appears. However, a quadruple-period orbit can be found in a small interval around \(\alpha=-73\). It includes not only periodic but also chaotic behavior (strange attractors, blue regions).  
To obtain the strange attractors, we plot in Figures 7 and 8 the orbit of 1000 initial guesses close to point \(x=(0.36,0.36)\) in the \((x_{1},x_{2})\)-space by iterating \(ACTV^{6}((x_{1},x_{2}),\alpha)\). The value of the parameter used is \(\alpha=-73.25\), laying in the blue region. For each seed, the first 500 iterations are ignored; meanwhile, the following 400 appear in blue and the last 100 in magenta color. We see in Figures 7 and 8 that a parabolic fixed point bifurcates in periodic orbits with increasing periods, and therefore falls in a dense orbit (chaotic behavior) in a small area of \((x_{1},x_{2})\) space.  
Figure 8: Details Strange attractors of \(ACTV6(x,\alpha)\). (**a**) \(\alpha=-73.25\), a detail. (**b**) \(\alpha=-73.25\), a detail.  
Figure 9: Feigenbaum diagrams of \(ACTV^{6}(x,\alpha)\), for \(-76.9<\alpha<-76.6\), from different critical points. (**a**) \((cri_{1}(\alpha),cri_{1}(\alpha))\) and \((cri_{2}(\alpha),cri_{2}(\alpha))\). (**b**) \((cri_{1}(\alpha),cri_{1}(\alpha))\) a detail. (**c**) \((cri_{2}(\alpha),cri_{2}(\alpha))\) a detail.