Method of investigating the stability of the oscillations of a linear system subject to the action of a parametric load with continuous spectrum

On the asymptotics for solutions of system of two linear oscillators with slowly decreasing coupling

“…Some of these problems arise in connection with the theory of dynamic stability of elastic systems (see, e.g., [2]). Samokhin and Fomin [10] were the first who studied system (1) in resonance case ω = ω 1 + ω 2 , if α = β = 1. We will get the asymptotic formulas for the solutions of system (1), if α + β > 1.…”

Krylov-Bogolyubov averaging of asymptotically autonomous differential equations

“…We notice that Cassell pointed out the possibility of an extension of the methods in [6,5] to linear systems with almost periodic coefficients. In this respect, it is important to mention the method of asymptotic integration of the linear systems x = (A 0 + k j=1 A j (t)t −jα )x, where αk ∈ (0, 1] and the A j (t) are finite trigonometric polynomials, proposed by Samokhin and Fomin [14] for k = α = 1 and by Burd and Karakulin [4] in the general case. In fact, they used some appropriate expansions in powers of the small parameter ε.…”

On the asymptotic integration of systems of linear differential equations with oscillatory decreasing coefficients

“…The mean of a matrix from Σ is a constant matrix that consists of the constant terms (λ j = 0) of the elements of the matrix. Using the Bogoliubov averaged method, Shtokalo transformed system (1) into a system with constant coefficients depending on the parameter ε up to terms of any order in smallnes in ε.We also mention that the method of averaging in the first approximation was utilized in [14,16] for studying the asymptotic behavior of solutions of a particular class of systems of equations with oscillatory decreasing coefficients.…”

“…We also mention that the method of averaging in the first approximation was utilized in [14,16] for studying the asymptotic behavior of solutions of a particular class of systems of equations with oscillatory decreasing coefficients.…”

“…Finally, let α = 1 3 , λ = ±1, and λ = ±2. Then, it turns out that A 1 is zero, and A 2 is defined by (14). Matrix A 3 differs from zero only if λ = ± 2 3 .…”