METHOD FOR DERIVING HOPF AND SADDLE-NODE BIFURCATION HYPERSURFACES AND APPLICATION TO A MODEL OF THE BELOUSOV-ZHABOTINSKII SYSTEM

Chemical mechanisms with oscillations or bistability undergo Hopf or s addle-node bifurcations on parameter space hypersurfaces, which inters ect in codimension-2 Takens-Bogdanov bifurcation hypersurfaees. This p aper develops a general method for deriving equations for these hypers urfaces in terms of rate constants and other experimentally controllab le parameters. These equations may be used to obtain better rate const ant values and confirm mechanisms from experimental data. The method i s an extension of stoichiometric network analysis, which can obtain bi furcation hypersurface equations in special (hj) parameters for small networks. This paper simplifies the approach using Orlando's theorem a nd takes into consideration Wegscheider's thermodynamic constraints on the rate constants. Large realistic mechanisms can be handled by a sy stematic method for approximating networks near bifurcation points usi ng essential extreme currents. The algebraic problem of converting the bifurcation equations to rate constants is much more tractable for th e simplified networks, and agreement is obtained with numerical calcul ations. The method is illustrated using a seven-species model of the B elousov-Zhabotinskii system, for which the emergence of Takens-Bogdano v bifurcation points is explained by the presence of certain positive and negative feedback cycles.