Cusp Catastrophe — Codimension-Two Bifurcation
ẋ = −x³ + bx + a: fold catastrophe unfolding, hysteresis, and the cusp surface
Cusp catastrophe (Thom 1972): Normal form ẋ = −x³ + bx + a. Equilibria: −x³ + bx + a = 0.
Catastrophe set (fold curve): 4b³ = 27a² (meets at cusp b=a=0). Inside this curve: three equilibria (two stable, one unstable).
Hysteresis: sweeping a at fixed b>0 → system jumps discontinuously at fold edges. State depends on history.
Applications: stock market crashes, heartbeat rhythm, opinion formation, optics (caustics), cell differentiation (Waddington landscape).
Codimension 2: Requires 2 parameters (a,b) to fully unfold — generic 1D ODE can't have a cusp without tuning 2 params.