Current and eigenvector analyses of chemical reaction networks at Hopf bifurcations

Abstract

We show how quenching data can be combined with extreme current representation of stationary states for systematic analysis and optimization of models of oscillatory chemical reactions at a Hopf bifurcation. The method is global: it considers all possible models based on a given reaction network, and does not rely on continuation techniques; all possible stationary states can be directly and systematically generated, and Hopf bifurcations can be compared with experimental (quenching) data at one or more operating points. It avoids any integration of kinetic equations. Therefore it is fast and a dense subset of all Hopf bifurcation points can be considered in a search. In this sense the method is complete. Experimentally known stationary concentrations can be fixed and reduce the dimension of the space to be searched. A scaling principle lets all Hopf bifurcation points of the current cone be treated in terms of corresponding points of the current polytope. It admits of a simple, direct calculation of points that match a given (experimental) oscillation frequency. A graphical method of testing the compatibility of a reaction network with a set of experimental quenching data may help selecting possible networks for systematic optimization. The method exploits the eigenvector property of quenching amplitudes. The paper develops the two methods; an accompanying paper applies them to two specific networks for the Belousov–Zhabotinsky reaction.