Asymptotics of Dynamical Saddle-node Bifurcations

Dynamical bifurcations occur in one-parameter families of dynamical systems, when the parameter is slow time. In this paper we consider a system of two nonlinear differential equations with slowly varying right-hand sides. We study the dynamical saddle-node bifurcations that occur at a critical instant. In a neighborhood of this instant the solution has a narrow transition layer, which looks like a smooth jump from one equilibrium to another. The main result is asymptotics for a solution with respect to the small parameter in the transition layer. The asymptotics is constructed by the matching method with three time scales. The matching of the asymptotics allows us to find the delay of the loss of stability near the critical instant.