Hopf thresholds in a two-pool calcium model ($f(\mu ,v)=0,f_v-\gamma f_u=ϵ$)

The unique equilibrium has $u=\mu$ and $f=0$. Its determinant is $-f_v/ϵ>0$ and trace is $-1+(f_v-\gamma f_u)/ϵ$. Thresholds are obtained from the two displayed equations, and a positive-trace interval exists only if the fast normal derivative rises above $ϵ$. At a nondegenerate trace crossing, the linear oscillation frequency is $\sqrt{-f_v/ϵ}$. The sign of a nonlinear Hopf coefficient is needed to classify the local periodic branch; in a bounded positive planar flow, a unique repelling equilibrium also allows a Poincaré-Bendixson existence argument for a periodic orbit.