| Abstract: |
| Inspired by the fractal images produced by the Electric Sheep project started in 1999 by Scott Draves, we seek to analyze the underlying iterated function systems (IFSs) that are used to produce these fractal images. Since the IFSs involve multiple nonlinear functions selected using a random process, some of the traditional analytic techniques from difference equations run into difficulties.
In this work stemming from a recent undergraduate student research collaboration, we will give some background on the Electric Sheep project and how the underlying IFSs are utilized to produce images. The Electric Sheep project focuses on the aesthetics of the resulting images. Here, we seek to develop several visual heuristics (some that are interactive) in order to analyze the mathematical behavior of orbits with various initial conditions for the IFSs that produce these images. We visualize the nullclines using approximations to account for the nonlinear and random aspects of the IFSs. We will develop several heuristics that incorporate ideas involving domain coloring from complex variables in order to construct the visualization for the heuristics. Finally, we apply these techniques to discrete analogs of certain DEs such as the Clairaut equation, the Bernoulli equation, and the Lotka-Volterra system of equations. |
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