# Chaos and Stability in a New Iterative Family for Solving Nonlinear Equations
## 1 Introduction  
Many problems in Computational Sciences and other disciplines can be stated in the form of a nonlinear equation or nonlinear systems using mathematical modeling. In particular, a large number of problems in Applied Mathematics and Engineering are solved by finding the solutions of these equations.  
In the literature, there are many methods and families of iterative schemes that have been designed by using different procedures to approximate the simple roots of a nonlinear equation \(f(x)=0\), where \(f:I\subseteq\mathbb{R}\rightarrow\mathbb{R}\) is a real function defined in an open interval \(I\). We can find in [1; 2; 3] several surveys and overviews of the iterative schemes published in the last years. Each method has a different behavior. This behavior is characterized with the efficiency criteria and the complex dynamics tools.  
In this paper, we introduce a new family of multistep iterative schemes to solve nonlinear equations, which contains as an element of this family, a particular method presented in [4]. This family is built from the [PERSON]'s scheme, adding a Newton step with a \"frozen\" derivative and using a divided difference operator. Therefore, the family has a three-step iterative expression. Furthermore, it has three arbitrary parameters named \(\alpha\), \(\beta\), and \(\gamma\), which can take real or complex values, and an order of convergence of at least four. The order of convergence will be discussed in Section 2.  
From the error equation we observe by fixing two parameters in function of the third one, an uniparametric family of sixth-order iterative methods is obtained. We analyze the dynamical behavior of this family in terms of values of the parameter, in order to detect its elements with good stability properties and others with chaotic behavior. The concept of chaos has been widely discussed (see, for example, in [5]) and it is commonly understood as the presence of complex orbit structure and extreme sensitivity of orbits to small perturbations. Moreover, the presence of unstable periodic orbits of all periodsis also included in the concept of chaotic system. For this study, we use tools of discrete complex dynamics that we introduce in Section 3.  
In Section 4, we present the performance of the presented schemes on several test functions. These numerical tests allow us to confirm the results obtained in the dynamical section and to compare our schemes with other known ones. The manuscript finishes with some conclusions and the references used in it.  
The parametric family object of study in this manuscript has the following iterative expression:  
\[\left\{\begin{aligned} y_{k}&=x_{k}-\frac{f(x_{k})}{f^{ \prime}(x_{k})},\\ z_{k}&=y_{k}-\frac{f(y_{k})}{2f[x_{k},y_{k}]-f^{ \prime}(x_{k})},\\ x_{k+1}&=z_{k}-(\alpha+\beta u_{k}+\gamma v_{k}) \frac{f(z_{k})}{f^{\prime}(x_{k})},\end{aligned}\right. \tag{1}\]  
where \(u_{k}=1-\frac{f[x_{k},y_{k}]}{f^{\prime}(x_{k})}\); \(v_{k}=\frac{f^{\prime}(x_{k})}{f[x_{k},y_{k}]}\); \(k=0,1,2,...\); and \(\alpha\), \(\beta\), and \(\gamma\) are arbitrary parameters.  
The divided difference operator \(f[\cdot,\cdot]:I\times I\subset\mathbb{R}\times\mathbb{R}\rightarrow\mathcal{ L}(\mathbb{R})\) defined by [PERSON] in [PERSON] (2016), satisfies  
\[f[x,y](x-y)=f(x)-f(y),\quad\forall x,y\in I. \tag{2}\]