pitchfork bifurcation
A thin vertical column bears a growing load; while the load is light it stays straight, but past a critical weight it suddenly buckles, and it is equally likely to buckle left or right. A single symmetric state has given way to two mirror-image states. Draw the outcome against the load and you get a shape like a pitchfork, three tines branching from one handle.
The pitchfork bifurcation is the generic bifurcation of systems with a symmetry (typically x -> -x). Its supercritical normal form is dx/dt = r x - x^3. For r < 0 the origin x* = 0 is the only fixed point and it is stable; at r = 0 it loses stability; for r > 0 the origin becomes unstable and two new stable fixed points appear at x* = +sqrt(r) and x* = -sqrt(r). One symmetric solution has smoothly branched into a symmetric-unstable state flanked by two stable, symmetry-broken states. There is also a subcritical version, dx/dt = r x + x^3, in which the branching pair is unstable and appears for r < 0, producing an abrupt, hysteretic jump rather than a gentle split.
This is the mechanical picture behind spontaneous symmetry breaking: the equations keep the left-right symmetry, but any actual realized state must choose one side, breaking it. You meet the supercritical pitchfork in buckling beams, in convection rolls that can turn either way, in the onset of magnetization in the Ising model near its critical point, and throughout Landau's theory of continuous phase transitions. The pitchfork's very existence depends on the symmetry; add a small asymmetric term and the clean fork breaks apart into an imperfect bifurcation.
A vertical elastic strut under axial load stays straight until the load reaches the Euler buckling threshold, then bows either left or right; the straight state persists mathematically but is now unstable, exactly as x* = 0 does for r > 0 in dx/dt = r x - x^3.
Past the threshold the symmetric state survives on paper but not in practice: reality must pick a side.
A pitchfork is not generic without a symmetry to protect it: perturb dx/dt = r x - x^3 by a constant term h and the fork unfolds into a smooth branch plus a disconnected saddle-node, so a perfect pitchfork in a real system is a sign of an underlying symmetry.