ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

9. Ordinary Differential Equations and Dynamical Systems

1991 MS Classification: 34 Samoilenko, Anatolij Investigation of m-frequency oscillatory systems The perturbed system of equations in ℝ^{2n} of the form

\frac{dx}{dt} = λHx + εX(x)

is considered, where λH = diag{λ_{1}H_{1}, …, λ_{n}H_{n}}, λ = (λ_{1}, …, λ_{n}) are the frequencies of the unperturbed system with the frequency basis ω = (ω_{1}, …, ω_{m}), H_{ν}^{2} = −E, ν = \overline{1, n}, and X ∈ 𝕂[x] is the ring of polynomials, 0 < ε ≪ 1. For the system under consideration the algorithm of an asymptotic method [1] is presented, the analysis of the averaged equations is carried out, and the conditions of existence of p-dimensional invariant tori with arbitrary 1 ≤ p ≤ m are obtained.

Reference: [1] Yu. A. Mitropolsky and A. M. Samoilenko, To the problem of asymptotic expansions in nonlinear mechanics // Ukr. Mat. Zh. 31 (1979), pp. 42–53.