One stable equilibrium splits symmetrically into two — symmetry breaking made visible, with the characteristic pitchfork shape
Controls
μ < 0: Single stable fixed point at x*=0
Supercritical (ẋ=μx−x³):
μ<0: One stable FP at x*=0
μ>0: x*=0 unstable, two stable FPs at x*=±√μ appear.
Subcritical (ẋ=μx+x³):
μ<0: x*=0 stable, two unstable FPs at x*=±√(-μ) exist.
μ>0: All three merge — x*=0 unstable, hysteresis!
Symmetry: The system has Z₂ symmetry (x→-x). The pitchfork shape reflects this — bifurcating branches are always symmetric. Real examples: buckling of a beam under load, magnetization below Curie temp, Bénard convection onset.