Bifurcation Diagrams: Transcritical & Pitchfork

Explore steady-state solutions and their stability as a control parameter varies. Watch symmetry-breaking transitions emerge in real time.

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ẋ = rx − x²
Transcritical: exchange of stability between two fixed points. Appears in population dynamics (SIR threshold).

Pitchfork: spontaneous symmetry breaking. Supercritical: soft transition. Subcritical: hard (hysteresis).

Cusp catastrophe: fold lines in 2D parameter space. Imperfection h breaks the Z₂ symmetry.