Swallowtail Bifurcation
Description
This model represents the swallowtail bifurcation from nonlinear dynamics. It represents the equilibrium points of the non-linear system: 1.25(x/10)^6+ - 6 (x/10)^4 + 6(x/10)^2 + r(x/10) + 10h == 0 The axis are oriented as follows: 1) Place the object so the largest opening is toward the table. 2) Rotate it so it is possible to look under the model 3) The x-axis ranges from -20 to 20 horizontal and parallel to you. The r-axis ranges from -10 to 10 horizontal and moving directly away from you. The h-axis ranges from -0.5 to 0.5 and runs upward from the table. Anyone familiar with dynamics, and especially non-linear dynamics knows how important the equilibrium points of a system are. The study of the locations of these points as the system parameters change is known as bifurcation theory. This surface is meant to provide a visual aid for students of nonlinear dynamics. The 3D surface can be created in Mathematica using ContourPlot3D[1.25*(x/10)^6 - 6 (x/10)^4 + 6*(x/10)^2 + r*(x/10) + h*10 == 0, {h, -1.5, .5}, {r, -10, 10}, {x, -25, 25}, MaxRecursion -> 15, PlotPoints -> 20, ContourStyle -> Opacity[.3], ViewPoint -> {0, 0, Infinity}, Boxed -> False]
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