Small parasitic parameters and chemical oscillations

Abstract

In treating chemical oscillators, it is common practice to use no more than two concentrations as dependent variables due to the mathematical difficulties with more than two. Other variables, necessarily involved to make the system nontrivial—the “pool” chemicals—are treated as parameters which are independent of time. We investigate here under what conditions the “pool” chemicals can be treated as constants without qualitatively effecting the behaviour of the system. Mathematical methods developed by Tikhonov are used to study the effect of a small parameter on the roots of the characteristic equation. This small parameter may be chosen as the ratio of the initial concentrations of the reactants to the “pool” chemicals or for dilute solutions, the reciprocal of the heat capacity. A number of interesting results are obtained in which the slow variation of the “pool” chemicals can either produce a limit cycle where there was none previously or place severe restrictions on the rate constants so as to exclude regions where interesting instabilities have been found in the two variable case. Multistability in the two variable system lends itself very well to the production of limit cycles of the relaxation type. Finally it is shown that it is possible to devise thermokinetic oscillators with very small temperature oscillations provided the energy equation is highly nonlinear.