Dynamical analysis of an iterative method with memory on a family of third-degree polynomials

Qualitative analysis of iterative methods with memory has been carried out a few years ago. Most of the papers published in this context analyze the behaviour of schemes on quadratic polynomials. In this paper, we accomplish a complete dynamical study of an iterative method with memory, the Kurchatov scheme, applied on a family of cubic polynomials. To reach this goal we transform the iterative scheme with memory into a discrete dynamical system defined on ${\mathbf{R}}^2$. We obtain a complete description of the dynamical planes for every value of parameter of the family considered. We also analyze the bifurcations that occur related with the number of fixed points. Finally, the dynamical results are summarized in a parameter line. As a conclusion, we obtain that this scheme is completely stable for cubic polynomials since the only attractors that appear for any value of the parameter, are the roots of the polynomial.