Controls
μ < 0: Stable spiral
System (polar):
ṙ = μr − r³
θ̇ = ω = 1
μ < 0: Origin is a stable spiral — all trajectories spiral inward to (0,0).
μ = 0: Weakly stable — spirals in very slowly (as t⁻¹/²).
μ > 0: Origin becomes unstable. A stable limit cycle of radius r = √μ is born. All trajectories (except origin) spiral toward this cycle.
Bifurcation diagram (right): amplitude vs μ. The curve r = √μ branches at μ=0 — the characteristic square-root scaling.