Exploring Chaotic Attractors via 3D printing Dylan Evans November 23 2024 George Mason University Math 401: Mathematics Through 3D printing

Chaotic attractors are mathematical structures that exhibit unpredictable behavior while remaining bounded within a region of space. The study of such structures connects fields of nonlinear dynamics, differential equations, and geometry offering a theoretical and applied point of view. For this project, I created an attractor that starts off as two separate spirals that, while the parameters increase, combines the spirals into one larger spiral much like two colliding galaxies. The attractor is defined by three differential equations with three parameters controlling the dynamics of the system. Using Mathematica, I numerically solved the system and visualized the data converting it into a 3D printable model in the end.

First I created the differential equations, F1(x,y,z):= a(y-x), F2(x,y,z):= x(b-z)-y and F3(x,y,z):= xy-cz where the coupled terms xy and xz are essential in creating the chaos aspect. Then, I set the parameters a,b,c to whatever I wanted to get the desired look of the object. These can be played around with freely and should be done iteratively to understand the effect it has on the object. We get three differential equations with three variables. Then, once these are set up I set the initial conditions, simply using 1 for each function's initial condition, that is, x0=1, y0=1, and z0=1. Then starting at the initial condition, I solve the differential equations using Mathematica's NDSolve function, but we want to see it converge to the attractor, so I start a new computation at the place where the last computation ended solving the differential equation at that place again. Finally, I plot the attractor using the ParametricPlot3D function with the solutions to the solved differential equations with the PlotStyle->Tube option in Mathematica which gives us a 3D plotted graph of the function that's thickened thanks to the PlotStyle used.

The chaotic nature of the attractor is driven by the combination of the nonlinear terms such as xy and xz as well as the sensitivity it has to the initial conditions. This creates a sort of feedback look that amplifies small changes resulting in the chaotic nature of these systems. While these equations appear relatively simple, you can obtain geometrically rich patterns that embody the unpredictable nature and as this shows, you can get wildly different objects from the same set of equations by simply changing the initial conditions. One challenge with such a system is creating something that looks nice and is visually compelling. Due to the sensitivity of the functions it's easy to create systems which don't show any compelling results or just don't output anything using the structure of the code shown. This chaotic attractor is reminiscent of two galaxies colliding. There are two separate spiraling structures that slowly come together and form one much larger spiral. When printing, it's recommended you use supports and to print the supports using dissolvable filament as the 3D print may be fragile and break when taking the supports off.