Avoiding strange attractors in efficient parametric families of iterative methods for solving nonlinear problems

Searching zeros of nonlinear functions often employs iterative procedures. In this paper, we construct several families of iterative methods with memory from one without memory, that is, we have increased the order of convergence without adding new functional evaluations. The main aim of this manuscript yields in the advantage that the use of real multidimensional dynamics gives us to decide among the different classes designed and, afterwards, to select its most stable members. Moreover, we have found some elements of the family whose behavior includes strange attractors of different kinds that must be avoided in practice. In this sense, Feigenbaum diagrams have resulted an extremely useful tool. Finally, some of the designed classes with memory have been directly extended for solving nonlinear systems, getting an improvement in the efficiency in relation to other schemes with the same computational cost. These numerical tests confirm the theoretical results and show the good performance of the methods.