Hysteresis, bistability, and catastrophic transitions in ẋ = rx + x³ − x⁵
Normal form: ẋ = rx + x³ − x⁵. The cubic term makes the pitchfork subcritical: unstable branches appear for r < 0. Saddle-node (fold) bifurcations at r = r* ≈ −3/16 create and destroy equilibria. The system is bistable in a r-range, giving hysteresis: increasing vs decreasing r follows different paths, with sudden catastrophic jumps. Potential V(x) = −rx²/2 − x⁴/4 + x⁶/6 shows the wells and barriers.