= First Lyapunov coefficient
{title2=$l_1$}
The <first Lyapunov coefficient> is the <cubic polynomial>[cubic] radial coefficient in the center-manifold normal form of a smooth <Hopf bifurcation>, after an appropriate normalization. In $\dot r=\beta r+l_1r^3+\cdots$, its nonzero sign distinguishes <Lyapunov stability>[stable] and <Lyapunov stability>[unstable] bifurcating small <periodic orbit>[cycles] relative to the eigenvalue-crossing direction. If $l_1=0$, higher-order terms are required and the generic <cubic polynomial>[cubic] <Hopf bifurcation>[Hopf theorem] does not apply.
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