= Spectral Hopf condition for a cubic characteristic polynomial
{title2=$p(\lambda)=(\lambda+A)(\lambda^2+B),\qquad B>0,\quad AB=C$}
A real <cubic polynomial>[cubic] $p(\lambda)=\lambda^3+A\lambda^2+B\lambda+C$ has <root of a polynomial>[roots] $-A,\pm i\sqrt B$ when $B>0$ and $AB=C$. For $A>0$ this gives a <Lyapunov stability>[stable] <transversality>[transverse] third direction. To conclude a generic <Hopf bifurcation> as a <mathematical model parameter>[parameter] varies, also require a <transversality>[transverse crossing] of the imaginary pair and a nonzero <first Lyapunov coefficient>. The algebraic imaginary-root condition alone is not sufficient.
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