Computing Hopf Bifurcations in Chemical Reaction Networks Using Reaction Coordinates Hassan Errami 1 , Werner M. Seiler 1 , Markus Eiswirth 2 , and Andreas Weber 3 1 Institut f¨ ur Mathematik, Universit¨ at Kassel, Germany; seiler@mathematik.uni-kassel.de 2 Fritz-Haber Institut der Max-Planck-Gesellschaft, Berlin, Germany and Ertl Center for Electrochemisty and Catalysis, Gwangju Institute of Science and Technology (GIST), South Korea; eiswirth@fhi-berlin.mpg.de 3 Institut f¨ ur Informatik II, Universit¨ at Bonn, Germany; weber@cs.uni-bonn.de 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 oscil- lations. For low dimensional reaction systems without additional con- straints 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 con- centration coordinates, a technique that extends the range of networks, which can be analyzed in practice, considerably. 1 Introduction In chemical and biochemical systems, reactions networks can be represented as a set of reactions. If it is assumed they follow mass action kinetics then the dynamics of these reactions can be represented by ordinary differential equa- tions(ODE) for systems without additional constraints or algebraic differential equations (DAE) for systems with constraints. Particularly, in complex systems it is sometimes difficult to estimate the values of the parameters of these equa- tions, hence the simulation studies involving the kinetics is a daunting task. Nevertheless, quite a few things about the dynamics can be concluded from the structure of the reaction network itself. In this context there has been a surge of algebraic methods, which are based on the structure of network and the as- sociated Stoichiometry of the chemical species. These methods provide a way to understand the qualitative behaviour (e.g. steady states, stability, bifurcations, oscillations, etc) of the network. The analysis of chemical reaction networks by detecting of the Occurrence of Hopf bifurcation attracts especially more and more interests in chemical and biological field due to its linkage to the oscil- latory behaviour. During the last decade many theoretical advances have been made for computing Hopf bifurcation in low dimensional systems [?]. However