Dynamics of Newton method for symmetric quartic polynomial

Abstract

We consider the Newton method on symmetric quartic polynomials. The parameter space is divided into different regions with different dynamical behaviours. In this paper, we study the dynamics for values of the parameter on the bifurcations curves that separate those regions, checking its dynamical behaviour due to the presence of multiple roots. We also consider a modified Newton method and we prove that this method is always conjugate to the rational map m/n z2 when it is applied on polynomials with two roots of multiplicity m and n, respectively.