Auto-oscillating regime

The correlation functions for the Lotka model in the auto-oscillating regime are presented in Figs 8.14 and 8.15. The value of the parameter k = 0.02 corresponds to the curves plotted in Figs 8.7(c) and 8.8. The correlation functions motion is completely periodic, the results shown here correspond to the the minimum and maximum of K(t). 

Fig. 27. Boundaries separating the regions of steady-state and auto-oscillation regimes of binary copolymerization in CSTR. Curve 2 is calculated for kinetic parameters resulting in the best approximation of experimental data reported in Ref. [345] curves 1 and 3 are calculated at the limits of confidence interval of the values of these parameters [17, 6]...

The values of its dispersion under an auto-oscillating regime of copolymerization of styrene with butyl acrylate in CSTR have been presented [17]. 

The solution of the first kind is stable and arises as the limit, t —> oo, of the non-stationary kinetic equations. Contrary, the solution of the second kind is unstable, i.e., the solution of non-stationary kinetic equations oscillates periodically in time. The joint density of similar particles remains monotonously increasing with coordinate r, unlike that for dissimilar particles. The autowave motion observed could be classified as the non-linear standing waves. Note however, that by nature these waves are not standing waves of concentrations in a real 3d space, but these are more the waves of the joint correlation functions, whose oscillation period does not coincide with that for concentrations. Speaking of the auto-oscillatory regime, we mean first of all the asymptotic solution, as t —> oo. For small t the transient regime holds depending on the initial conditions. 

More complicated case of standing waves emerges in the regime of chaotic oscillations. Here the equations for the correlation dynamics are able to describe auto-oscillations (for d — 3). However, a noise in concentrations changes stochastically the amplitude and period of the standing waves. It results finally in the correlation functions with non-monotonous behaviour. Despite the fact that the motion of both concentrations and of the correlation functions is aperiodic, the time evolution of the correlation functions reveals several distinctive distributions shown in Fig. 8.6. 

Fig. 4.19. Bifurcation diagram for the reduced system (4.4b,c) as a function of parameter a. On the ordinate, the steady-state concentration of /3 in that system is shown, as well as the maximum value reached by jS in the course of oscillations. The diagrcuns are obtained numerically by means of the program AUTO (Doedel, 1981), for decreasing values of parameter (in s" ) (a) 10 (b) 7.78 (c) 4.5 (d) 2.7 (e) 1.7 (f) 1.2. Solid or dashed lines denote, respectively, stable and unstable (steady or periodic) regimes. The arrowed trajectories represent, schematically, the dynamic behaviour of the full, three-variable system (4.1a-c). The particular values of a relate to the limit points of the hysteresis curve (au, L2), the Hopf bifurcation points ( hi, h2)> and the points corresponding to the appearance of homoclinic orbits (an, a ) (Decroly Goldbeter, 1987).

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