When α = ν, (11) is always satisfied. In that case, we must turn to (10) to determine b. Equation (10) has four roots, b = ± 1 2 (ω 1 + ω 2 ) ± 1 2 (ω 1 − ω 2 ) 2 − 4ν 2 . By our assumption, b must be real, and this occurs when |ω 1 − ω 2 | ≥ 2ν, which is precisely the case not covered by the above result. Thus when |ω 1 − ω 2 | ≥ 2ν, the Hopf point is at α c = ν.