Overshoot-undershoot kinetics

A more natural phenomenon seems to be the oligo-oscillation, or the overshoot-undershoot phenomenon. These expressions denote the case when there is only a finite number of local extrema on the concentration versus time functions. Natural as it is, it has rarely been studied in a well-controlled experiment (see, however, Rabai et a/., 1979). It has also rarely been studied from the theoretical point of view. This situation can be explained by the fact that the qualitative theory of differential equations usually makes statements on long-range behaviour and much less on transient behaviour. The only exception seems to be that Pota (1981) has given a complete proof of the statement called Jost s theorem which says in a closed reversible compart-mental system of M components none of the concentrations can have more than M - 2 strict extrema. The methods used by Pota makes it possible to extend this result (see Problem 6 below). Another result of this type, relating nonlinear kinetic differential equations, can also be found among the Problems. 

Show that the induced kinetic differential equation of the simple consecutive reaction 

Applying the last statement of Subsection 4.5.2 show that, if the number of atoms is one less than that of the components in an atomic reaction obeying the law of conservation of atomic numbers, and if the stoichio metric matrix is of full rank, then the induced kinetic differential equation of the reaction has no periodic solution. 

Show that the differential equation (4.9) (the induced kinetic differential equation of the Field-Koros-Noyes mechanism model of the Belousov-Zhabotinskii reaction) does have periodic solutions at certain values of the parameters. 

---

Pota, Gy. (1981). On a theorem of overshoot-undershoot kinetics. React. Kinet. Catal. Lett., 17, 35-9. 

Rabai, Gy., Bazsa, G. Beck, M. T. (1979). Design of reaction systems exhibiting overshoot-undershoot kinetics. J. Am. Chem. Soc., 131, 6746-8. 

---