Delay in Solving Autonomous Singularly Perturbed Equations Near an Unstable Equilibrium Position

Abstract

This paper considers an autonomous system of singularly perturbed equations of fast variables, consisting of \(2n\) first-order equations and one equation of a slow variable. The first approximation matrix of singularly perturbed equations has pairwise complex conjugate eigenvalues. The system has an equilibrium position, and the stability of the equilibrium position is lost by all eigenvalues at some value of the slow variable. It is proven that the solution of a singularly perturbed equation remains near an unstable equilibrium position during a finite time. Thus, the solution is delayed near the unstable equilibrium position. Early works considered cases when the stability of the equilibrium position is lost by one pair of complex conjugate eigenvalues.