Dynamics of Newton's method in quadratic and fifth-degree polynomials

Abstract

This article studies the complex dynamics induced by Newton's method applied to polynomials, with an emphasis on the existence and classification of periodic orbits. Three main results are presented: (i) it is demonstrated that super-attracting orbits do not exist in the case of quadratic polynomials under Newton's iteration, a result previously established in a master's thesis; (ii) repulsive orbits associated with quadratic polynomials are characterized and exhibited; and (iii) the analysis is extended to fifth-degree polynomials, showing the presence of repulsive orbits and describing their dynamic properties. The methodology combines theoretical analysis with computational experimentation for the localization and classification of periodic cycles. The results contribute to a broader understanding of the dynamic behavior of Newton's method according to the degree of the polynomial and provide comparative evidence between low- and high-degree cases.