To Integration of the Damped Mathieu Equation in the Monograph of N. N. Bogoliubov and Y. A. Mitropolsky "Asymptotic Methods in the Theory of Nonlinear Oscillations"

Using the asymptotic method described in the monograph referred to in the title, expressions are obtained that determine the boundaries of three regions of parametric resonance of the damped homogeneous Mathieu equation. The formulas for the boundaries of the second and third regions, validated by solving the equation numerically, differ significantly from the known ones obtained in the monograph. It is shown that the very existence of resonance regions depends on the choice of orders of smallness of the three small parameters of the problem.