Method for deriving Hopf and saddle-node bifurcation hypersurfaces and application to a model of the Belousov–Zhabotinskii system

Abstract

Chemical mechanisms with oscillations or bistability undergo Hopf or saddle-node bifurcations on parameter space hypersurfaces, which intersect in codimension-2 Takens–Bogdanov bifurcation hypersurfaces. This paper develops a general method for deriving equations for these hypersurfaces in terms of rate constants and other experimentally controllable parameters. These equations may be used to obtain better rate constant values and confirm mechanisms from experimental data. The method is an extension of stoichiometric network analysis, which can obtain bifurcation hypersurface equations in special (h,j) parameters for small networks. This paper simplifies the approach using Orlando’s theorem and takes into consideration Wegscheider’s thermodynamic constraints on the rate constants. Large realistic mechanisms can be handled by a systematic method for approximating networks near bifurcation points using essential extreme currents. The algebraic problem of converting the bifurcation equations to rate constants is much more tractable for the simplified networks, and agreement is obtained with numerical calculations. The method is illustrated using a seven-species model of the Belousov–Zhabotinskii system, for which the emergence of Takens–Bogdanov bifurcation points is explained by the presence of certain positive and negative feedback cycles.