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For the fast layer dynamics of the (4F, 1SS) system, the critical manifold is our former superslow manifold, (mathscr {M}_{ss}), which lies within (mathscr {M}_{s}).
Eventually, the trajectory passes the fold of (mathscr {M}_{ss}) (blue triangle) where (mathscr {M}_{ss}) destabilizes, a transition to the fast layer problem occurs, and hence the solution of the (4F, 1SS) system follows the fast layer flow to another stable branch of (mathscr {M}_{ss}).
end{aligned} (8b) We define the critical manifold (mathscr {M}_{s}) to be the manifold of equilibrium points of the fast layer problem, i.e., mathscr {M}_{s}:=bigl{ (v, h, ca_{i}, c_{mathrm{tot}},l): hat{f}_{1}= hat{f}_{2}=0bigr}.
Introducing a fast time (t_{f}=t_{s}/varepsilon) and letting (varepsilon rightarrow0), we can derive a 2-dimensional fast layer problem that describes the dynamics of the fast variables v and (ca_{i}), for fixed values of the other variables: begin{aligned} frac{dv}{dt_{f}} =&hat{f}_{1} v,h,ca_{i}), end{aligned} (8a) begin{aligned} frac{d ca_{i}}{dt_{f}} =& hat{f}_{2} v,ca_{i},c_{mathrm{tot}},l).
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