Dynamics of a system of nonlinear differential equations

This is a qualitative analysis of a system of two nonlinear ordinary differential equations which arises in modeling the self-oscillations of the rate of heterogeneous catalytic reaction. The kinetic model under study accounts for the influence of the reaction environment on the catalyst; namely, we consider the reaction rate constant to be an exponential function of the surface concentration of oxygen with an exponent $\mu$. We study the necessary and sufficient conditions for the existence of periodic solutions of differential equations as depending on $\mu$. We formulate some sufficient conditions for all trajectories to converge to a steady state and study global behavior of the stable manifolds of singular saddle points.