Langford Chaotic Attractor

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Langford Chaotic AttractorAlexis Ventura10/31/2023. Fall 2023.George Mason University 401: Mathematics Through 3D Printing.For the following print, the topic that was concentrated was “Chaotic Attractors”. The chaotic attractors (Lorenz, Rossler, Langford, Arneodo-Coullet-Tresser/Rucklidge, etc.) are functional with several variables to create a dynamical system implying certain periodic orbits of such chaotic attractors in the third dimension. Each attractor is comprised of its own ordinary differential equations (ODEs) and parameterization to create their chaotic model. In addition to this, the sensitive dependence and transitivity are the main ingredients for the model to be in a chaotic state. The Langford Chaotic Attractor is the focus of the printed model. The attractor was studied and discovered by William F. Langford. In his discovery, the model of the chaotic attractor “undergoes a supercritical Neimark Sacker bifurcation resulting in the appearance of a smooth attracting invariant torus”, mentioned in his work of Chaos In A Vector Field With A Neimark Sacker Bifurcation. He asserts parameterization being the general functional framework in studying the invariants of the dynamical system. The following is the ordinary differential equations (ODEs) used for this project:f (x, y, z) = {(z-b) x – dydx + (z-b) yc + az - (z^3/3) - (x^2+y^2) (1+ez) + fzx^3}.The parameter used for this project are as follows:a = 0.8, b = 0.7, c = 0.6, d = 3.5, e = 0.25, and f = 0.1.With these as the ODEs and parameters, I was able to design a Langford chaotic attractor shown in the following design code below. My Langford chaotic attractor object was printed on a Creality Ender-3 Pro 3-D printer and the 3-D object took about four hours to print. It is scaled at 50mm in height and length as an appropriate size to print the object. It uses tree supports touching the build-plate, skirt as the build-plate adhesion, the infill density at 100% using grid pattern, and the printing temperature at 215 degrees. While printing out the chaotic attractor the first time, there was a printing issue, thus aborting the print leading me to print on another 3D printer. The second print worked accordingly therefore creating a 3-D printed object of a Langford chaotic attractor. Meiss, James. "Differential Dynamical System." Society for Industrial and Applied Mathematics, 2007. Fluerantin, Emmanuel. Mireles James, Jason. Resonant Tori, Transport Barriers, and Chaos in a Vector Field with a Neimark-Sacker Bifurcation, 2020.