the pitchfork bifurcation
/ PITCH-fork /
Press down slowly on the top of a thin vertical ruler. For light pressure it stays straight — there is one obvious equilibrium, dead centre. Past a critical load it cannot stay straight: it buckles, and it must choose to bow left or right, two new equilibria appearing symmetrically while the straight state goes unstable. The plot of equilibria looks like a three-pronged garden fork. That symmetric splitting of one fixed point into three is the pitchfork bifurcation, the signature of symmetry breaking.
Its normal form is x' = r x - x^3 (the supercritical case). The cubic is odd, so the system has the symmetry x -> -x: if x(t) is a solution so is -x(t), and the central fixed point x = 0 always exists. Solve r x - x^3 = 0: factor as x(r - x^2) = 0, giving x = 0 always, plus x = +/- sqrt(r) when r > 0. For r < 0 only the origin exists and it is stable. As r passes 0 the origin loses stability and two new stable equilibria branch off symmetrically at +/- sqrt(r) — one fixed point became three. The system must pick one of the two new outer states, breaking the left-right symmetry even though the equations keep it.
There are two flavours. The supercritical pitchfork (above) is gentle: the new branches are stable and the system slides smoothly onto a small nearby state — a soft, reversible transition. The subcritical pitchfork, normal form x' = r x + x^3, is dangerous: the branching states are unstable and exist on the 'wrong' side, so when the central state loses stability the system jumps abruptly to a distant state, often with hysteresis. Pitchforks are the language of buckling beams, convection rolls choosing a direction, and spontaneous pattern formation, and they appear precisely when a problem has a built-in symmetry that the solution is forced to break.
A bead on a rotating hoop: at low spin rate the bead's only equilibrium is at the bottom (x = 0). As the rotation speed crosses a critical value, the bottom becomes unstable and two new symmetric resting angles appear on either side, where centrifugal effect balances gravity. The bead must settle to one side or the other — a supercritical pitchfork.
One central equilibrium splits into two symmetric ones as a parameter crosses threshold — symmetry of the law, broken by the solution.
The pitchfork's perfect three-pronged shape relies on exact symmetry. Add any small asymmetry (a slight tilt, an imperfection) and the fork breaks apart into a smooth branch plus a disconnected saddle-node — pure pitchforks are idealizations that real, imperfect systems only approximate.