Swallowtail Bifurcation

thingiverse

This model illustrates 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 axes are oriented as follows: 1) Place the object so that the largest opening faces toward the table. 2) Rotate it to allow a view 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 crucial equilibrium points of a system are. Studying the locations of these points as system parameters change is known as bifurcation theory. This surface is intended 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].