The Dynamics of Newton's Method on Cubic Polynomials

Abstract

THE DYNAMICS OF NEWTON’S METHOD ON CUBIC POLYNOMIALS By Shannon N. Miller The field of dynamics is itself a huge part of many branches of science, including the motion of the planets and galaxies, changing weather patterns, and the growth and decline of populations. Consider a function f and pick x0 in the domain of f . If we iterate this function around the point x0, then we will have the sequence x0, f(x0), f(f(x0)), f(f(f(x0))), ..., which becomes our dynamical system. We are essentially interested in the end behavior of this system. Do we have convergence? Does it diverge? Could it do neither? We will focus on the functions obtained with Newton’s method on polynomials and will apply our knowledge of dynamics. We also will be studying the types of graphs one would get if they looked at these same functions in the complex plane.