= Hopf thresholds in a two-pool calcium model
{c}
{title2=$f(\mu,v)=0,\quad f_v-\gamma f_u=\epsilon$}
The unique <equilibrium> has $u=\mu$ and $f=0$. Its <determinant> is $-f_v/\epsilon>0$ and <matrix trace>[trace] is $-1+(f_v-\gamma f_u)/\epsilon$. Thresholds are obtained from the two displayed <equations>, and a positive-trace interval exists only if the fast normal <derivative> rises above $\epsilon$. At a nondegenerate <matrix trace>[trace] crossing, the linear <oscillation> <frequency> is $\sqrt{-f_v/\epsilon}$. 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>.
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