Investigation of the problem on a parametric resonance in Lurie systems with weakly oscillating coefficients

Based on perturbation theory methods, criteria for the Lyapunov stability of Lurie systems with weakly oscillating parameters are proposed. The main attention is paid to obtaining the first approximation formulas for perturbations of multiple definite and indefinite multipliers of linear Hamiltonian systems and their applications to stability analysis. The formulas proposed lead to new criteria for the Lyapunov stability of Lurie systems in critical cases. Applications to the problem of a parametric resonance in fundamental resonances are considered. The results obtained are stated in terms of the original equations and brought to the stage of design formulas and algorithms. The efficiency of the formulas is illustrated by the example of the problem on the parametric resonance in a system of coupled oscillators.