For solving a nonlinear scalar equation by a third-order Newton-type method, the authors discuss the behaviour of the dynamics of iterations for this method, when it is applied to polynomial equations of degree two or three. It is well known that the classical Newton method converges quadratically in a neighbourhood of a simple root of the equation, and if the left hand side of the equation is a polynomial, then the corresponding Newton iterative function is a rational map (a quotient of two polynomials without common factors). The third-order Newton-type method, obtained by a composition of two Newton iterative function, requires more computational cost and this type of methods is applied only in some cases. Moreover, each iteration consists in two steps of the Newton method having the same derivative.
The main interest of the authors is the study of the dynamics of the discrete dynamical system defined by this third-order Newton-type method. The analysis is based on the so called “scaling theorem”, the main theorem proved in this paper. For several simple quadratic polynomials, the analysis shows that the dynamics of the method is chaotic. For cubic polynomials, the results of analysis show that bifurcations and chaos appear. From the numerical point of view, this represents a great difficulty to determine the region of convergence of the method to the solution of the given equation.