Immediate exchange of stabilities in singularly perturbed systems

Abstract

We study the initial value problem for singularly perturbed systems of ordinary differential equations whose associated systems have two transversally intersecting families of equilibria (transcritical bifurcation) which exchange their stabilities. By means of the method of upper and lower solutions we derive a sufficient condition for the solution of the initial value problem to exhibit an immediate exchange of stabilities. Concerning its asymptotic behavior with respect to $\varepsilon$ we prove that an immediate exchange of stabilities implies a change of the asymptotic behavior from $0(\varepsilon)$ to $0(\sqrt{\varepsilon})$ near the point of exchange of stabilities.