Delayed Loss of Stability in Systems with Degenerate Linear Parts

Abstract

We study singularly perturbed scalar and planar differential equations with linear parts independent of time. The associated autonomous equations undergo a bifurcation of equilibria in the scalar case and the Hopf bifurcation in the case of planar systems at a bifurcation point where the zero equilibrium loses stability. We suggest natural sufficient conditions for the phenomenon of delayed loss of stability for the singularly perturbed equations and estimate the asymptotic delay. Bifurcation points, stability of the zero equilibrium, and the asymptotic delay are determined by superlinear terms in the expansions of the right-hand sides of the associated and singularly perturbed equations.