Example of trajectories converging to an underlying two-dimensional SM in a three-dimensional phase-space. The EM for this specific kinetic scheme (calculations are performed with k 1 = 2, k 2 = 1, k 3 = 0.6, k 4 = 3 in proper units, see text) is shown in red.
Two-dimensional projections of several trajectories for the three kinetic schemes. Calculations have been performed with k 1 = 2, k 2 = 1, and k 3 = 0.6 for both (S1) and (S2) , and γ = 3 for (S4) . These values are employed for all calculations in this work. For (S1) and (S2) we use the chemist's notation where [ … ] stands for volumetric concentration. For (S1) (panel B), the directions of the projected fast (blue line) and slow (red line) eigenvectors of the kinetic matrix are shown.
Two-dimensional projections of randomly selected trajectories for the model hydrogen combustion mechanism (see (S5) and values of the rate constants therein given) under the mass-conservation constraints 2[H2] + 2[H2O] + [H] + [OH] = 2.0 and 2[O2] + [H2O] + [O] + [OH] = 1.0. The blue circle indicates the equilibrium point.
Pictorial representation of the sub-region (taken as the SM) and its “junction” with the embedded EM. In the inset, the strategy adopted to identify is sketched (see the text for details).
(Left panels) Pair of trajectories “from above” (solid black lines) and “from below” (dashed lines) with respect to the perceived SM for (S1) and (S2) . The trajectories are coloured in red where the conditions in Eq. (9) are satisfied for n max = 11 (see the text for details). (Right panels) Profiles of time-derivatives versus the displacement Δ as moving along the direction transverse to the local velocity vector (see Figure 3 ) at the points x marked on the trajectories “from above” in the left panels.
Individuation of the two-dimensional projection of the SM for (S1) and (S2) according to the fulfillment of conditions in Eq. (9) . Red marks are the outcome of the search with n max = 11, black lines are pairs of trajectories and the dashed lines delimit the region into which the scan has been performed (see the text for details).
Individuation of the SM for (S4) according to the fulfillment of conditions in Eq. (9) but excluding Q = 1 (see text). Red marks are the outcome of the search with n max = 11; dashed lines delimit the scanned region. The detail in the lower panel offers the same view as in Figure 6 of Ref. 14 ; the underlying dashed line is the exact SM (analytical solution).
Two-dimensional projections of 20 trajectories randomly selected (black points) for the hydrogen combustion mechanism (see (S5) and values of the rate constants therein given) under the mass-conservation constraints 2[H2] + 2[H2O] + [H] + [OH] = 2.0 and 2[O2] + [H2O] + [O] + [OH] = 1.0. The red spots are points that passed the checks of Eq. (10) with n max = 20 and n c = 5 (other parameters are given in the text); the magenta square corresponds to an anomalous point. The blue circle indicates the equilibrium point.
Three-dimensional representation, on the concentration space of the radical species, of the data presented in Figure 8 .
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