Thomas Cyclically Symmetric Attractor

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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 equations that describe its behavior. The equations used are dx/dt = sin(y) - bx, dy/dt = sin(z) - by, and dz/dt = sin(x) - bz. These equations were solved with initial conditions (1, 0, 1). Discovered by Rene Thomas, the Thomas attractor is a strange or chaotic attractor that exhibits unstable behavior over time. Attractors are stable equilibria towards which a system of differential equations tends in the limit. Strange attractors differ from regular attractors because they exhibit chaos, meaning slight changes in initial conditions can cause huge changes in solutions. Small variations in parameter b have significant effects on solution stability. This model uses b = 0.1998 to create an interesting geometry that showcases chaotic behavior. For values of b greater than 1, we see a stable equilibrium; at b = 1, the first bifurcation occurs. When b reaches 0.32899, a periodic solution emerges, and as b decreases further, the solution becomes more complex and chaotic. The system becomes unpredictable around b = 0.2, which is why this value was chosen for this model. To find initial conditions that produced visually appealing solutions, I experimented with different values. Since the chosen b value produces chaotic behavior, even slight changes in initial conditions resulted in significantly different solutions. In this case, the initial conditions were x(0) = z(0) = 1 and y(0) = 0. To generate this model, I used Mathematica's NDSolve function to solve the system of differential equations. Then, I plotted the solution using ParametricPlot3D, with the tube style set to a thickness of 0.22. However, the first print broke due to this thin thickness. Print Settings: Printer Brand: MakerBot; Printer: MakerBot Replicator 2X; Rafts: Yes; Supports: Yes; Resolution: .23; Infill: diamond. Notes: Extra supports were used and might have been sufficient without a raft. Mathematica Code: b = .1998; Timing[soln = NDSolve[{\nx'[t] == Sin[y[t]] - bx[t], \y'[t] == Sin[z[t]] - by[t], \z'[t] == Sin[x[t]] - b*z[t], \x[0] == z[0] == 1, \y[0] == 0\n}, {x, y, z}, {t, 0, 400}, MaxSteps -> Infinity]]; plot = ParametricPlot3D[ Evaluate[{x[t], y[t], z[t]} /. soln], {t, 0, 400}, PlotStyle -> Tube[.22, PlotPoints -> 100], PlotRange -> All, Ticks -> None]