Asymptotics of the solution of a differential equation in a saddle–node bifurcation

A second-order semilinear differential equation with slowly varying parameters is considered. With frozen parameters, the corresponding autonomous equation has fixed points: a saddle point and stable nodes. Upon deformation of the parameters, the saddle–node pair merges. An asymptotic solution near such a dynamic bifurcation is constructed. It is found that, in a narrow transition layer, the principal terms of the asymptotics are described by the Riccati and Kolmogorov–Petrovsky–Piskunov equations. An important result is finding the dragging out of the stability: the moment of disruption significantly shifts from the moment of bifurcation. The exact assertions are illustrated by the results of numerical experiments.