Hopf Bifurcation

Stable fixed point → limit cycle as bifurcation parameter μ crosses zero

-0.50
2.0
0.30
300
8
Supercritical: ṙ = μr - r³, θ̇ = ω. μ<0: stable node; μ>0: stable limit cycle of radius √μ.
Subcritical: ṙ = μr + r³ - r⁵. μ<0: bistable (fixed pt + large limit cycle); μ>0: only limit cycle.