Critical stability of solutions to linear ordinary differential equations with large high-frequency terms

The stability problem is considered for certain classes of systems of linear ordinary differential equations with almost periodic coefficients. These systems are characterized by the presence of rapidly oscillating terms with large amplitudes. For each class of equations, a procedure for analyzing the critical stability of solutions is constructed on the basis of the Shtokalo–Kolesov method. A verification scheme is described. The theory proposed is illustrated by using a linearized stability problem for the upper equilibrium of a pendulum with a vibrating suspension point.