Halvorsen Chaotic Attractors

Halvorsen Chaotic Attractors by ckhai
November 20, 2024
George Mason University Math 401: Mathematics Through 3D Printing

An attractor is a set of states, such as points, curves, or more complex objects, toward which a system tends to evolve over time. In a chaotic system, small changes in the initial conditions lead to exponentially different outcomes. A chaotic attractor is typically a complex, fractal-like structure that the system approaches in the long term, but it never settles into a single point or periodic orbit. They emerge from nonlinear differential equations when the system’s behavior is sensitive to initial conditions and exhibits complex.
These chaotic attractors are specifically called Halvorsen attractors. It has one parameter and is generated by plotting the trajectory of the points (x,y,z) with initial values (0.1,0,0). The system of differential equations for it is dx/dt=-ax-4y-4z-y^2, dy/dt=-ay-4z-4x-z^2, dz/dt=-az-4x-4y-x^2. The parameters for the green and silver are a=1.4 and a=1.7 respectively. The time values used are timelengtha=40 and timelengthb=65 so that the attractor achieves the desired accuracy.
Both attractors were printed using the Ultimaker S5 and S7 printers. They were printed with PLA filament and normal supports. The supports were made with PVA dissolvable filament. It took 3-6 hours to print each attractor. A stand was created in OpenSCAD for both attractors and was printed using a Creality Ender-3 Pro printer. It takes approximately 2 hours to print one stand.