Owen Lewis

Harvey Mudd College Mathematics 2005

Thesis Proposal: Numeric Solution Methods and Fractals
Completed Thesis (12/20/2004): Generalized Julia sets: An Extension of Cayley's Problem
Thesis Advisor: Prof. Jon Jacobsen
Second Reader: Prof. Lesley A. Ward
E-Mail: olewis@math.hmc.edu

Generalized Julia sets: An extension of Cayley's Problem

Can you find a value, x, such that f(x) = 0? Despite the rather innocuous problem statement, for certain choices of the function in question, this can be a rather difficult question to answer in the affirmative. In light of this, various itterative methods for finding such values, x, have been proposed. One of the oldest and simplest of these is Newton's Method. Newton's Method gives rise to a discrete dynamical system in the domain of f, who's stable fixed points are precisely the zeros of f (i.e. the x, such that f(x) = 0). If there exists multiple such values, x, one might wonder which initial values are carried to each individual zero under itteration of Newton's Method. Put another way, what are the basins of attraction for the various zeros of f. In the late nineteenth century, A. Cayley investigated these basins of attraction for Newton's Method applied to complex polynomials. Cayley succeeded in completely characterizing such basins in the case of a quadratic polynomial, but made little progress for the case of a cubic. We now know that Cayley's difficulties are due to the fact that for polynomials of higher order, the basins of attraction are almost always bounded by an intricate set of fractal dimension (named the Julia set in honor of a pioneer of this field). The point of view taken in my thesis is that Newton's Method can be regarded as a first order approximation of a particular differential equation known as Continuous Newton's Method. I investigate the same basins of attraction mentioned above, while approximating Continous Newton's Method with more sophisticated numerical techniques, specifically a fourth order Runge-Kutta approximation. Remarkably I find that fractal sets associated with the basins of attraction persist, in both the complex, and Euclidean settings.