Computing Hopf Bifurcations in Chemical Reaction Networks Using Reaction Coordinates

Abstract

Abstract. The analysis of dynamic of chemical reaction networks by computing Hopf bifurcation is a method to understand the qualitative behavior of the network due to its relation to the existence of oscillations. For low dimensional reaction systems without additional constraints Hopf bifurcation can be computed by reducing the question of its occurrence to quantifier elimination problems on real closed fields. However deciding its occurrence in high dimensional system has proven to be difficult in practice. In this paper we present a fully algorithmic technique to compute Hopf bifurcation fixed point for reaction systems with linear conservation laws using reaction coordinates instead of concentration coordinates, a technique that extends the range of networks, which can be analyzed in practice, considerably.

A Survey of Some Methods for Real Quantifier Elimination, Decision, and Satisfiability and Their Applications

“…If y > 0, then we consider instead − f , which has the same zeros as f . Therefore we may assume in 22 For independent reasons one knows with the first Hopf bifurcation at n = 9 that there will be Hopf bifurcations for all n > 9. 23 http://www.ebi.ac.uk/biomodels-main/.…”

“…6a. 22 We use extended positive quantifier elimination, which assumes that all occurring variables are strictly positive, in combination with a case distinction on the sign of λ. In the satisfiable case we obtain exact symbolic sample solutions.…”

“…The subnetworks form a convex cone in flux space [68]. There are techniques for decomposing that cone and limiting the search for Hopf bifurcations to lower dimensional facets [22]. For our purposes here it is sufficient to understand that we are going to obtain a whole family of formulas for a single model corresponding to those facets.…”

Investigations and Analysis of Dynamical and Steady State Properties of Chemical Reaction Systems

“…Monotone systems are useful for analytic and computational purposes because there is no asymptotic behavior in which solutions grow to infinity or eventually vanish. While monotonicity is not common in practice according to references [12], [13], one can say more about the global stability of solutions if a dynamical system is strongly monotone. One wants all solutions to be bounded.…”

“…Errami, Eiswirth, Grigoriev, Seiler, Sturm, and Weber presented chemical reaction systems that have Hopf Bifurcations in the papers [12], [13].…”