Thomas Cyclically Symmetric Attractor by FormulaJockey

Last crawled date: 3 years, 1 month ago

This was created at George Mason University for Math 493: Mathematics Through 3D Printing, taught by Dr. Evelyn Sander.
Provided here is a model of the Thomas Cyclically Symmetric Attractor, created in Mathematica using the equations
dx/dt=sin⁡y-bx
dy/dt=sin⁡z-by
dz/dt=sin⁡x-bz
using the parameter b = 0.1998 with initial conditions (1,0,1).
Discovered by Rene Thomas, the Thomas attractor is an example of a strange or chaotic attractor. Attractors are values for which a system of ODEs tend towards in the limit, i.e. they are stable equilibria. Strange attractors differ from attractors in that they are chaotic, meaning that slight changes in the initial conditions cause huge changes in the solutions.
Slight changes in this parameter b have drastic effects on the stability of the solutions to this ODE. This particular .thing file uses b = 0.1998 because it creates a very interesting geometry. For b > 1 we see a stable equilibrium, and for b = 1 we see the first bifurcation. At b = 0.32899, the solution bifurcates and a periodic solution is created. The lower the b value, the more interesting the solution becomes. The system becomes chaotic around b = 0.2, which is why this particular b value was chosen.
The last part was guessing initial conditions that created decent-looking solutions. Since this b value is chaotic, small changes in the initial conditions created large changes in the solutions. The initial condition seen here is x(0) = z(0) = 1, y(0) = 0. The Mathematica code I used to generate this model is shown below. First I used NDSolve, which can solve a system of ODEs, and the next part was to plot it. In order to “3-dimensionalize” it, I plotted the style in tubes of thickness 0.2. However, this thickness might be a little too thin, as the first print created broke.