Saddle-Node Bifurcation Ghost

Two fixed points collide and annihilate at r = 0. For r > 0, the trajectory slows dramatically near the "ghost" — the remnant of the fixed point. Transit time diverges as 1/√r.

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Flow: ẋ = r + x². For r < 0: two fixed points at x = ±√|r| (saddle + node).
At r = 0: bifurcation — both merge at x=0. For r > 0: no fixed points, ghost at x≈0.
Transit time through ghost: T = π/√r — diverges as r→0⁺.