Stability of the bromate–cerium–malonic acid network. II. Steady state formic acid case

Abstract

The stability of the nonequilibrium steady states of the reduced network for the complete mechanism of the bromate–cerium–malonic acid (Belousov–Zhabotinskii) system is examined for the case where formic acid does not appear in the overall reaction (i.e., is not added to the system initially). The stability analysis involves constructing, by computer, the third Hurwitz determinant, which is a 2575 term polynomial. The system is unstable when the polynomial is negative, and the asymptotes of the negative region are found by computer analysis of the terms in the polynomial. The asymptotic unstable region determined by this polynomial is the union of 19 convex polyhedral cones which are defined by 152 linear inequalities in eight parameters. The network is also unstable in the region where a second polynomial is negative. The asymptotes of this region are described by eleven inequalities. The terms in these polynomials are interpreted diagrammatically as products of feedback cycles of the reaction network. The terms which destabilize arise from three essential feedback cycles: (1) a positive feedback cycle of mutual inhibition between Br− and HBrO2; (2) a negative feedback cycle in which HBrO2 activates HOBr, HOBr activates Br−, and Br− inhibits HBrO2; and (3) a negative feedback cycle in which HBrO2 activates Ce(IV), Ce(IV) activates Br−, and Br− inhibits HBrO2. The network will stabilize if [Br−] becomes large; however, the boundary of stability for large bromide ion is made very irregular by the second and third feedback cycles above. About eighteen important inequalities characterize the irregularities. The stability is found to be very sensitive to small currents in the nondominant flux loops. The reduced network may in certain circumstances be taken as a model of the detailed network. The model is not valid for low concentrations of bromate, cerium, or malonic acid because the reverse reactions which are missing in the reduced network then stabilize the detailed mechanism and the stability inequalities obtained here must be supplemented with additional inequalities. The reduced network is the core of the stability problem for the full network. All simplifications made in the reduced network should have a sufficiently predictable effect on the stability analysis to be removable theoretically. This paper extracts new insights into reaction network stability from the computer results. A detailed theory of destabilization by positive feedback cycles is developed. A general principle for selecting all the destabilizing cycles of a network is discussed with reference to this example.