Null dines

For the simplest three-step adsorption mechanism (4) at n = 2, m = 1 and p = q = 1, a retardation in the relaxation rate is observed in the region ("hole ) between two null dines... 

Fig. 14. Possible cases for mutual disposal of the null dines for system (9) in reaction simplex S.

We have already mentioned the sharp difference in the relaxation times outside the region between the null dines ( 1 s) and inside it (as high as hundreds of seconds). The motion outside this region depends on the "fastest reaction. Inside this region the relaxation rate is dependent on the complicated complex of rate constants, and in the general case we cannot suggest that the reaction rate is limited by some reaction. The common trajectory near which the relaxation is retarded is no more than a specific trajectory that is a separatrix going from the unstable into the stable steady... 

Localization of this steady state as a point of intercept for the null dines x = 0 and y = 0 as a function of the k x value is shown in Fig. 16. At low k x this point is localized sufficiently close to the region of probable initial conditions (at k x = 0 it becomes a boundary steady state). It is the proximity of the initial conditions to the steady state outside the reaction polyhedron that accounts for the slow transition regime. Note that, besides two real-valued steady states, the system also has two complex-valued steady states. At bifurcation values of the parameters, the latter become real and appear in the reaction simplex as an unrough internal steady state. The proximity of complex-valued roots of the system to the reaction simplex also accounts for the generation of slow relaxations. 

The transient characteristics, 0az(O and 0bz(O> of the adsorption model (Eq. (7.102)) show different relaxation times, which can differ by orders of magnitude. The trajectories are often characterized by fast initial motion and slow motion in the vicinity of a slow trajectory. This slow trajectory is a simple case of the slow manifolds known in the literature (Gorban and Karlin, 2003) and is located in the region between the two null dines dOpo ldt = 0 and dO-az/dt = 0. Both null dines are second-order curves with an axis of symmetry 6az = 0bz... 

Figure 4.2a shows a small perturbation to the steady state. The value of u is increased to point A. The system jumps back to the M-null dine at D because the 1/e term in eq. (4.1a) makes u change much more rapidly than v. Now, the system... 

Figure 4.1 The null dines for the two-variable model of eqs. (4.1). Arrows indicate the direction of the flow in phase space. The steady state occurring at the intersection of g = 0 and / = 0 is stable.

Figure 8.3 Schematic diagram of model A. System I undergoes a Hopf bifurcation to nearly sinusoidal periodic oscillation in A and B when fi falls below a critical value. System II has a slow manifold (the dU/dt = 0 null dine) and is excitable. Changes in B in system I perturb the steady-state point in system II, while changes in V in system II alter the effective value of p in system I. (Reprinted with permission from Barkley, D. 1988. Slow Manifolds and Mixed-Mode Oscillations in the Belousov-Zhabotinskii Reaction, J. Chem. Phys. 89, 5547-5559. 1988 American Institute of Physics.)...

In the figure, the dV/dt = 0 null dine in system II is located so that this subsystem is excitable rather than oscillatory. The steady state is stable, but if V is lowered beyond the turn in the s-shaped null dine, the system will undergo a large-scale excursion, as shown by the arrows, before returning to the steady state. 

If the timing parameter k is chosen appropriately, the s-shaped null dine in the two-dimensional subsystem II becomes the slow manifold in the fully coupled model A, as shown in Figure 8.4. 

Suppose the rate constants k and a " are infinite except the ones of step i for which k and /c" are finished, as the concentrations of the reactants and products are not null (where it is important to fix the concentrations of the reactants and products of the reaction dining the experiments), the rate factors / and are thns also infinite for p different from i. 

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