Cusp Catastrophe
z = ((x⁴)/4) +((a · x²)/8) +((y²)/2)
Open in the app The dials and keys named below are the app's.
What it draws
The height is a potential, and the two directions do quite different work. y²/2 is a plain trough that rises away from the line y = 0 and never changes. Everything else is in x: x⁴/4 climbs steeply at the edges, and a·x²/8 is the term the slider owns. The profile along y = 0 is the whole story, and at the opening value a = −8 it is two wells at x = ±1.41 with a bump of height 1 between them.
Where the wells come from
The slope along y = 0 is x³ + a·x/4, which factors as x·(x² + a/4). For a at zero or above the only root is x = 0, a single well. For a below zero two more roots open at x = ±√(−a/4) and the origin turns from the bottom of the well into the top of a barrier. The barrier is a²/64 high, so it starts at nothing and grows fast: 0.25 at a = −4 and 1 at a = −8. The wells spread as a square root instead, which is slow, reaching only ±1.58 at the end of the slider.
One minimum becoming two as a parameter crosses zero is a pitchfork bifurcation. Read the height as the energy of something free to slide in x. Below the threshold there is one place for it to rest. Above it the middle is a summit, so the thing must pick a side, and the barrier is what then keeps it there.
History
René Thom set out catastrophe theory through the 1960s, a classification of the few ways the minima of a smooth potential can appear and vanish as its parameters move. The cusp is the second on that list and the one drawn most. Christopher Zeeman took it up in the 1970s and pressed it on everything from heartbeats to prison riots, which is most of why the subject grew famous and then unfashionable. The cusp carries a second control parameter, a tilt, left at zero here, and that is what keeps the split symmetric.
Try
- Press Front to look along y: the profile in x alone, the two wells and the barrier seen edge on.
- Drag a up to 0 and watch the wells walk in and merge. What is left is x⁴/4, a single well with an unusually flat floor.
- Carry on to a = 4: one well again, and now a steep one, since the two terms in x no longer fight.
- Put the tilt back with (x⁴)/4 + (a·x²)/8 + x/2 + (y²)/2: at a = −8 the left well sinks to −1.736 and the right one only to −0.328, so one side is now clearly favoured.
- Raise Mesh to its top stop for the lit solid, which reads the depth of the wells better than the wireframe.