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At every equilibrium the store balance gives $f=0$, and adding $\gamma$ times that balance to the cytoplasmic one gives $u=\mu$. The unique-root result therefore gives a single positive equilibrium $(\mu ,V(\mu ))$. Its Jacobian iswith derivatives evaluated at that equilibrium. At small $\mu$, $f_u>0$ and the trace is negative; sufficiently large $\mu$ also has $f_u>0$, since $J^{\prime }$ is of order $u^{-3}$ and $K^{\prime }$ of order $u^{-5}$. Thus both extreme regimes have a linearly stable unique equilibrium.

The fast variable is not $v$ alone. Using total-calcium fast fibres, introduce total scaled calcium $w=u+\gamma v$. Then $\dot{w}=\mu -u$, while on fast time $\sigma =\tau /ϵ$ the leading equations give $du/d\sigma =-\gamma f$, $dv/d\sigma =f$ and $w$ constant. Fast paths are therefore straight lines of slope $-1/\gamma$ in the $(u,v)$ plane. Attracting portions of the critical manifold satisfy $f_v-\gamma f_u<0$, rather than merely $f_v<0$. Along $v=V(u)$, slow motion obeysStarting with large $u$ and $v=0$, $f>0$ drives a rapid move upward and left, loading the store until the trajectory approaches the right attracting branch. For large $\mu$ in the right stable regime, slow motion on that branch carries it toward $u=\mu$; the direction reverses appropriately if the starting value lies below that equilibrium. For sufficiently small $\mu$, slow motion travels left along the right branch until its fold, then a fast approximately constant-$w$ jump carries the trajectory to the left attracting branch. Subsequent slow motion reaches the small-$u$ equilibrium. At a fold the slow approximation is singular, so the full equations resolve the transition. At very small $u$ an additional loss of time-scale separation is possible because the rates themselves vanish; the fast/slow drawing describes the separated portions, not a uniform estimate arbitrarily near the origin.

In a relaxation cycle, cytoplasmic calcium rises slowly on the left attracting branch while total calcium is accumulated, then rises rapidly during store discharge along a constant-$w$ fast path to the right branch. It subsequently decreases on the right slow branch as extrusion dominates input, and finally drops rapidly during store uptake back to the left branch. The result is a repeated pulse with slow sections joined by sharp transitions, rather than a sinusoid except near a small-amplitude Hopf bifurcation.

For explicit time labelling, the compatible scaling $u=c/K_1$, $v=c_s/K_2$ gives $\tau =kt$, $\gamma =K_2/K_1$, $ϵ=k{K}_2/V_2$, $\mu =r/(k{K}_1)$, $\beta =V_1/V_2$ and $\delta =k_s{K}_2/V_2$. Hence an interval $\Delta \tau$ corresponds to physical duration $\Delta t=\Delta \tau /k$. Slow-branch times can be computed by integratingin the direction of motion; fast-jump durations are omitted at leading order. The plot uses that quadrature, with vertical lines representing the limiting rapid transitions. Finite $ϵ$ rounds those lines and changes the precise amplitude and period, especially near folds.