2010. november 13. –11:02 1 BIFURCATIONS IN THE DIFFERENTIAL EQUATION MODEL OF A CHEMICAL REACTION By G ´ ABOR CS ¨ ORG ˝ O and P ´ ETER L. SIMON Abstract. In this paper a two-dimensional system of non-linear ordinary differential equations describing the oxygen reduction reaction on platinum surface is studied. The investigation is motivated by the fact that this reaction plays an important role in fuel cells. The mechanism of this reaction has been known for years, however, the detailed study of the mathematical model has not been carried out. The purpose of this paper is to reveal the dynamical behaviour of the ODE system, with emphasis on the number and type of the stationary points, the existence of periodic orbits and bifurcations. We point out that bistability occurs in the system, i.e. for certain values of the parameters two stable equilibria coexist that was not known before and is significant also from the chemical point of view. We also prove that the sys- tem has no periodic orbit. The saddle-node bifurcation curve is determined by using the Parametric Representation Method, and this enables us to determine numerically the parameter domain where bistability occurs. 1. Introduction We will consider the following system of non-linear ordinary differential equations. ˙ 1 =3 2 2 2 3 2 3 1 31 + 3 (1) ˙ 2 = 1 1 2 22 2 + 2 3 1 (2) where =1 1 2 and are positive parameters for = 1 2 3. AMS Subject Classification (2000): 52A40