Visualizing Chaos

Chaos is typically visualized on an infinite 2D plane. This summer, we utilized a third dimension to plot iterative root finding methods on a subset of the complex plane in which the initial starting point can be plotted on the z‐axis, creating 3D images of spheres. These spheres are then shaded in accordance to the speed in which the particular initial point converges, creating unique images that can be used to visualize what is happening at infinity on a finite 3D surface. The resulting images are used to explore efficiency of root finding methods as well as evaluating the choice of addition or subtraction in the denominator of the Hansen‐Patrick root finding method. There are many theories suggesting the sign choice for positive alpha values; however, in the case of a negative alpha value, these theories do not hold. Using programs based off of those developed by Andrew Nicklawsky and Dr. Robert Hesse, we developed rules to dictate this choice between addition and subtraction in order to maximize the speed of convergence for negative and imaginary alpha values