Computations for the first Lyapunov coefficient

These notes are a supplementary file to the paper Hopf bifurcations for HANDY-type models (M. Badiale and I. Cravero, under submission), providing full details of the computations developed in Section 4.2. The purpose of this supplement is to derive explicitly the first Lyapunov coefficient associated with a Hopf bifurcation, following the framework of Yu. A. Kuznetsov (Elements of Applied Bifurcation Theory, Springer, 4th ed., 2023). We compute the multilinear forms $B$ and $C$, the right and left eigenvectors and their normalization, and the resolvents $A^{-1}$ and $(2i{\omega }_{0}I-A{)}^{-1}$. Using asymptotic expansions with respect to the small parameter $\epsilon$, we derive explicit formulas for $\mu (\epsilon )$, ${\omega }_0$, and the Lyapunov coefficient $a(\mu (\epsilon ),\epsilon )$, which characterize the criticality of the Hopf bifurcation in the main model.