Oscillatory period

Potential differences between the nitrobenzene and aqueous phases at the interfaces in the presence [Fig. 2(B)] and absence of surfactant (C) were measured simultaneously. KCl salt bridges were inserted into the octanol phase to monitor potential. Oscillation measurement data across the nitrobenzene membrane are given in Fig. 2(A) for comparison. The oscillation mode in Fig. 2(C) is virtually the same as that in (A) with respect to oscillatory period and amplitude but quite different with that in (B). Although the potential across the nitrobenzene membrane (A) was not recorded simultaneously with that between nitrobenzene-water phases (B) and (C) but successively, it was noted that the algebraic sum of (B) and (C) should be essentially the same as (A). This is an indication that potential oscillation across the nitrobenzene membrane is likely generated at the interface between the nitrobenzene phase and aqueous phase initially containing no surfactant. 

In this study, potential oscillation was measured in the presence of lOOmM sodium salts of barbital, allobarbital, phenobarbital, and amobarbital in phase wl [19]. Their chemical structures are shown in Fig. 15. Amplitude and the oscillatory and induction periods were noted to depend on the particular hypnotic used. Amplitude decreased in the order, barbital > allobarbital > phenobarbital > amobarbital. The oscillatory period increased in the order, barbital < allobarbital < phenobarbital < amobarbital. Induction period increased in the order, barbital < allobarbital < phenobarbital < amobarbital. These parameters changed depending on drug concentration. Hypnotics at less than 5 mM had virtually no effect on the oscillation mode. 

Fig. 1,10. Actual time-dependent concentration of intermediate A for consecutive first-order reactions with cubic autocatalysis showing pseudo-steady-state behaviour, pre-oscillatory evolution, an oscillatory period, and then the return to pseudo-steady-state behaviour.

Fig. 2.4. Computed concentration.histories for autocatalytic model with rate constants given exactly as in Table 2.1 (a) exponential decay of precursor (b) intermediate concentrations a(t) and 6(r), showing initial pseudo-stationary-state behaviour but subsequent development of an oscillatory period of finite duration, 1752 s < t < 3940 s.

In a similar way, the system will reach the end of the oscillatory period when the reactant concentration falls to pf at time t given by, ... 

Fig. 2.7. Variation in (a) absolute amplitude and (b) oscillatory period across region of instability for the pool chemical model with rate data from Table 2.1. (The qualitative form is appropriate for all combinations of rate constants giving oscillatory behaviour with this model.)...

Though reduced to the barest of essentials, the scheme shows many features observed in real examples of oscillatory reactions a pre-oscillatory period, a period of oscillatory behaviour, and then a final monotonic decay of reactant and intermediate concentrations to their equilibrium values. We can identify from the model such features as the dependence of the length of the pre-oscillatory period on the initial reactant concentration and the rate constants, an estimate for the number of oscillations, and the length of the oscillatory phase. By tuning the parameters we can obtain as many oscillations as we wish. 

Note that the dimensionless time rf, which gives the length of the pre-oscillatory period, will only be positive if the initial concentration /i0 exceeds the upper Hopf bifurcation value /if. If we start with a lower initial reactant concentration, so that /i0 < /if (but still with /i0 > /if), there will be no pre-oscillatory period the system will jump straight into oscillations which will persist until time rf. 

These general quantitative predictions concerning the existence and length of pre-oscillatory and subsequent oscillatory periods depending on /i0, ku, and e are borne out by numerical computation of the exact equations with... 

Because oscillatory behaviour persists only for a finite length of time, only a finite number of excursions can occur. We can estimate this number by obtaining an approximate value for the mean oscillatory period, im. For this we take a geometric mean of the periods at the two Hopf bifurcation points. These latter quantities can be evaluated from the frequency co0 defined by... 

The pre-oscillatory period, during which the temperature excess decreases and the concentration of A increases, will last at least until i has fallen from H0 to the value n, i.e. until the time tf given by... 

A would then be close to its maximum value, so the post-oscillatory period should see both 6 and a decreasing, eventually to zero. The actual computed behaviour is shown in Fig. 4.10 which is the dimensionless analogue of Fig. 4.1. 

At the point of Hopf bifurcation, the emerging limit cycle has zero amplitude and an oscillatory period given by 2n/a>0. As we begin to move away from the bifurcation point the amplitude A and period T grow in a form we can calculate according to the formulae... 

In these equations th term (p — p ) represents how far we have moved away from the bifurcation point, in terms of the dimensionless concentration of reactant. There are two new quantities p2 and t2 which tell us a number of things. The amplitude A grows as the square root of the distance from the bifurcation point (p — p ), and so the term (p — p )/p2 must be positive. If lx2 turns out to be positive, then the limit cycle must grow as p is increased beyond p if fx2 is negative, the limit cycle grows as p decreases below fx. The growth (or decrease) in oscillatory period is linear in (p — p ) and depends on the ratio t2jp2. 

The slowest motion of all, and hence that which dominates the total oscillatory period, is that along DA. On this section the temperature rise is relatively low, so y0 1 and we may make the exponential approximation to the Arrhenius term. The resulting integral to be evaluated is... 

Fig. 9.7. Non-stationary behaviour in the diffusive autocatalysis model showing sustained temporal and spatial oscillations with D = 5.2 x 10 3, / = 0.08, and k2 = 0.05 (a) z = 0 or 465 (the oscillatory period) (b) z = 115 (c) z = 140 (d) z = 160 (e) z = 235. The limit cycle obtained by plotting the concentrations at the centre of the reaction zone, a,s(0) and /J (0), versus each other is shown in (f). The broken curve in (a) is the unstable stationary-state profile about...

As the motion around the limit cycle is periodic, we can only talk of a perturbation decaying or growing if we compare successive measurements made at the same point on the cycle. Thus we impose an initial perturbation Ax0 and observe its temporal evolution at the end of each successive circuit of the limit cycle. For a system with n independent variables, the perturbation Ax is an n-component vector. If the oscillatory period is given by Tp, then the perturbation at the end of the first cycle can be represented in the form... 

Plot of the oscillatory period TQ versus the partial pressure of O2 in N2. 

Fig. 5 A control target including a highly oscillating pattern. The control target (dashed) and the achieved current (solid) are displayed in the middle panel while in the top panel an average over five oscillatory periods is shown. The control field is given in the lowest part of the figure. (Reproduced from Ref. [73]. Copyright 2008 by the American Physical Society.)...

Fig. 4.6. Comparison of temperature traces from the 22- and 16-step reduced models with the full 47-step non-isothermal model at = 790 K. The removal of reactions 14, 27, 29, 35, 37 and 45 causes only a small decrease in the oscillatory period.

Fig. 5.23. Variation of oscillatory period in vicinity of ignition limit (saddle-node bifurcation point) showing extreme lengthening as limit is approached. (Reprinted with permission from reference [32], Manchester University Press.)...

With different mixture compositions, however, the range of behaviour supported becomes richer. The variation of the oscillatory period with the ambient temperature for a stoichiometric mixture (2H.2 -i- O2) at an operating pressure of 16 Torr and a residence time of 4 s is compared with... 

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