Qualitative analysis of a generalized system of saturated enzyme reactions

In this paper, we discuss a generalized system of saturated enzyme reactions. In order to see the tendency of concentrations over a large range, we discuss not only its finite equilibria but also its equilibria at infinity. The case of high degeneracy is dealt with by constructing generalized normal sectors. Furthermore, we calculate coefficients of a normal form of the system at the sole finite equilibrium and prove that a unique limit cycle of small amplitude bifurcates from the equilibrium. The existence of periodic solutions of large amplitude is given by Poincaré–Bendixson theorem. Nonexistence of periodic solutions is investigated by discussing the integral of divergence.