The Anishchenko-Ashtakhov Chaotic Attractor

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Jared Bergan 1 November 2021 George Mason University Math 401: Mathematics Through 3D Printing This week’s project was based around chaotic attractors. In general, an attractor is a mathematical model of a physical system that evolves in a closed subset through time. These are generated through systems of differential equations with continuous changes to t. A chaotic attractor, or a strange attractor, is a type of attractor that approaches its final point chaotically, or without any specific order, and can be very unstable based around its initial conditions. For this project, I decided to recreate the Anishchenko-Ashtakhov attractor. This chaotic attractor is made with the following system of differential equations: F1(x,y,z) = ax + y – xz F2(x,y,z) = -x F3(x,y,z) = -bz+b*H(x)*x^2 Where H(x) is the Heaviside function and a = 1.1 and b = 0.6. Initial values for x, y and z in the system were -0.1, 0.5, and -0.6 respectively. Using Mathematica, I used the NDSolve function to solve the system with a time range of 0 to 25 and then plotted it resulting in a system that indeed looks chaotic. Now I continued to solve the system to get the final product. Using NDSolve again picking up where the last calculation left off, I got a set containing the completed chaotic attractor and from there it was ready to be plotted using ParametricPlot3D and then saved to an STL for printing. There were a few issues that I encountered along the way that I was luckily able to solve. First off, the Heaviside function is a step function, implying that it is not continuous and therefore not differentiable. However, there was a rather simple fix to this. Dr. Sander suggested and came up with a way to write the function in a way that it is continuous, as the Heaviside function simply outputs 1 if x is positive and 0 if it is zero or negative. The rewritten function came to be as follows: H(x) = 0.5 + 0.5*(|x|/x) It is easily verifiable that this replacement yields the same results as the original step function. The second problem that I ran into was at the time of print. I printed the object on a MakerBot Replicator and when it came time to slice, the software kept failing to generate the file necessary to print. After messing around with a few settings, we found that it did not like having only 10% support density, where it defaulted to 20%. After pushing this value back up, we were able to get it to complete the slice and print the attractor! Unfortunately, the Ultimaker with dual extruders is out for repair so I was unable to print the supports with dissolvable filament. This led to a rough look from all the supports that were removed (and there were a lot). The print is nice and thick so there is no risk of breaking it and it looks incredible! If you are interested in learning more about chaotic attractors, feel free to visit these sites that I used! http://www.scholarpedia.org/article/Attractor https://www.stsci.edu/~lbradley/seminar/attractors.html