First Lyapunov coefficient ($l_1$)

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