Dynamical analysis of iterative methods for nonlinear systems or how to deal with the dimension?

This paper deals with the real dynamical analysis of iterative methods for solving nonlinear systems on vectorial quadratic polynomials. We use the extended concept of critical point and propose an easy test to determine the stability of fixed points to multivariate rational functions. Moreover, an Scaling Theorem for different known methods is satisfied. We use these tools to analyze the dynamics of the operator associated to known iterative methods on vectorial quadratic polynomials of two variables. The dynamical behavior of Newton's method is very similar to the obtained in the scalar case, but this is not the case for other schemes. Some procedures of different orders of convergence have been analyzed under this point of view and some “dangerous” numerical behavior have been found, as attracting strange fixed points or periodic orbits.