Chapter 45 - A tutorial on the visualization of forward orbits associated with Siegel disks in the quadratic Julia Sets

Abstract

The quadratic Julia sets arising from the familiar dynamical system zn+1 = f(zn) = zn2+ c where z, c are complex, have one of five basic morphologies: (1) a Jordan curve–– homeomorphic to the unit circle; (2) a periodic basin of attraction; (3) a parabolic basin of attraction in which the Julia set reaches inward to the fixed point; (4) a dendrite where there is one basin of attraction; and (5) the Siegel disk.. The chapter discusses the usage of computer graphics to gain a better understanding of the dynamics occurring in the Siegel disk Julia Sets. Traditional images of the Julia Sets usually display only the final behavior of the forward orbits of each point in the discretized complex plane. For each pixel in the image, a forward orbit must be computed—usually the information contained in the orbit is thrown away. This approach examines the orbit as well as its final behavior.