Newton's Method For A Non-Analytical Perturbation of Z^2-1

Abstract

This paper aims to analyze the discrete dynamical system arising from New-ton’s method applied to the two-dimensional real quadratic polynomial given by the analogue of the complex function H_A(z) =z^2−1 +A ̄z with complex parameter A. This equation derives from a perturbation of the well known system of Newton’s method for F(z) =z^2−1. The complex dynamics for Newton’s method applied to F have many of the same properties as the perturbation. We use these properties to analyze and describe the orbits for our system. Then by restricting our parameter value A=a+bi we were able to fully describe the dynamics for when |a| < 2/(3√3), b= 0 and partially describe the dynamics for when a= 0, |b|<2. A key result is the existence of an invariant line on which the Newton map is topologically conjugate to the doubling map. The doubling map is D: [0,1)→[0,1), with D(x) = 2x(mod 1). We use results from this map to describe our system. We present conjectures to describe the behavior of orbits in both cases. Finally we present some preliminary results for the dynamics of the system for parameter values not in those two cases.