Abstract
For a second-order quasilinear differential system whose coefficients have the form of Fourier series with slowly varying coefficients and frequency, we prove that, under certain conditions, there exists a particular solution with a similar structure in the case of purely imaginary roots of the characteristic equation for the matrix of coefficients of the linear part.
Similar content being viewed by others
References
N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Nauka, Moscow (1974).
A. M. Samoilenko, “Invariant toroidal manifolds of systems with slowly varying variables,” in: Problems of the Asymptotic Theory of Nonlinear Oscillations [in Russian], Naukova Dumka, Kiev (1977), pp. 181–191.
N. I. Shkil’, A. N. Voronoi, and V. N. Leifura, Asymptotic Methods in Differential and Integro-Differential Equations [in Russian], Vyshcha Shkola, Kiev (1985).
A. V. Kostin and S. A. Shchegolev, “On one class of solutions of differential systems with slowly varying parameters,” Ukr. Mat. Zh., 41, No. 1, 101–103 (1989).
S. A. Shchegolev, “On the existence of one class of solutions of a differential equation with oscillating coefficients in a Banach space,” Differents. Uravn., 32, No. 4, 570–571 (1996).
M. P. Merkin and V. M. Fridman, “Projective method for the solution of the problem of forced oscillations in nonlinear systems with slowly varying parameters,” Prikl. Mat. Mekh., 45, No. 1, 71–76 (1981).
N. G. Malkin, Some Problems of the Theory of Nonlinear Oscillations [in Russian], Gostekhizdat, Moscow (1956).
A. V. Kostin, Stability and Asymptotics of Quasilinear Nonautonomous Differential Systems [in Russian], Odessa University, Odessa (1984).
V. M. Starzhinskii, Applied Methods of Nonlinear Oscillations [in Russian], Nauka, Moscow (1977).
Author information
Authors and Affiliations
Additional information
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 5, pp. 654–664, May, 1998.
Rights and permissions
About this article
Cite this article
Kostin, A.V., Shchegolev, S.A. On solutions of a second-order quasilinear differential system representable by Fourier series with slowly varying parameters. Ukr Math J 50, 741–753 (1998). https://doi.org/10.1007/BF02514327
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02514327