If α = ν, (11) has nonzero roots b = ± 1 2 4α 2 − 8αν + 2ω 2 1 + 2ω 2 2 . When this b is substituted into (10), we obtain four potential roots for α at the Hopf point. Two are complex, which is unphysical and can be ignored. The one corresponding to the Hopf bifurcation is α c = ν − 1 2 4ν 2 − (ω 1 − ω 2 ) 2 . (The fourth root is larger and corresponds to loss of stability of the second pair of eigenvalues.) This α c is real and thus gives the Hopf point location when |ω 1 − ω 2 | ≤ 2ν. When ω 1 − ω 2 = 0 and ν > 0, α c is positive, indicating stabilization due to the asymmetry and coupling.