Literature cited
N. N. Bogolyubov, Some Statistical Methods in Mathematical Physics [in Russian], Izdatel'stvo AN USSR, Kiev (1945).
A. Einstein and M. Smoluchowski, “Brownian motion” [in German], Ann. d. Physik,17, 549 (1905);21, 756 (1906).
A. N. Kolmogorov, “Analytic methods in probability theory,” UMN,5 (1933).
L. S. Pontryagin, A. A. Andronov, and A. A. Vitt, “On the statistical treatment of dynamic systems,” in: Collected Works of A. A. Andronov [in Russian], Izd. AN SSSR, Moscow (1956).
I. I. Gikhman
I. I. Gikhman, “On some differential equations with random functions,” Ukr. Matem. Zh., 2, No. 3 (1950).
K. Ito, “On stochastic differential equations” [Russian translation], Matematika,1, No. 1 (1957).
I. I. Gikhman, “On the theory of differential equations of random processes,” Ukr. Matem. Zh.,2, No. 4 (1950).
A. V. Skorokhod, Investigations in the Theory of Random Processes [in Russian], Izd. KGU, Kiev (1961).
I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations [in Russian], Naukova Dumka, Kiev (1968).
I. L. Bershtein, “Fluctuations of amplitude and phase of a vacuum-tube generator,” Izv. AN SSSR, Ser. Fiz.,14, No. 2 (1950).
S. M. Rytov, Introduction to Statistical Radio Engineering [in Russian], Nauka, Moscow (1966).
R. L. Stratonovich, Selected Questions in the Theory of Fluctuations in Radio Engineering [in Russian], Sovetskoe Radio, Moscow (1961).
V. I. Tikhonov, Statistical Radio Engineering [in Russian], Sovetskoe Radio, Moscow (1966).
R. Z. Khas'minskii, “On the averaging principle for parabolic and elliptic differential equations and Markov processes with low diffusion,” TVIP,8, No. 1 (1963).
I. I. Gikhman, “Differential Equations with Random Functions [in Russian] (Winter School on Probability Theory and Mathematical Statistics, Uzhgorod, 1964), Izd. Inst. Matem. AN UkrSSR, Kiev (1964).
I. Vrkoc, “Extension of the averaging method to stochastic equations,” Czech. Math. J., m.16 (91), No. 4 (1966).
Yu. A. Mitropol'skii and V. G. Kolomiets, “Application of the averaging principle to the investigation of the influence of random stimuli on oscillatory systems,” in: Mathematical Physics [in Russian], Vol. 3, Naukova Dumka, Kiev (1967).
V. G. Kolomiets, “Random oscillations of quasilinear systems” [in Russian], Abhandl. Deutsch. Akad. Wissensch. Berlin, Kl. Math. Phys. Tech.,1 (1965).
N. N. Bogolyubov and Yu. A. Mitropol'skii, Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Nauka, Moscow (1963).
V. G. Kolomiets, “Action of random forces on nonlinear oscillatory systems,” Dissertation Abstract [in Russian], Kiev (1963).
V. G. Kolomiets, “Random oscillations of nonautonomous quasilinear stochastic systems,” Ukr. Matem. Zh.,20, No. 3 (1968).
R. Z. Khas'minskii, “On random processes defined by differential equations with small parameters,” TVIP,2 (1966).
R. Z. Khas'minskii, “Limit theorem for the solution of differential equations with a random right side,” TVIP,3 (1966).
R. Z. Khas'minskii, “On the averaging principle for Ito stochastic differential equations,” Kubernetika,3, No. 4 (1968).
V. G. Kolomiets, “On the averaging principle for stochastic systems with after-effects,” in: Proceedings of the Fifth International Conference on Nonlinear Oscillations [in Russian], Vol. 1, Izd. Inst. Matem. AN UkrSSR (1970).
V. I. Kerzyuk and V. G. Kolomiets, “Investigation of oscillations of nonlinear systems with slowly varying random parameters and random functions,” in: Mathematical Physics [in Russian], Vol. 8, Naukova Dumka, Kiev (1970).
V. I. Kerzyuk and V. G. Kolomiets, “On the investigation of certain oscillatory systems with slowly varying random parameters,” in: Proceedings of the Seminar on Mathematical Physics and the Theory of Nonlinear Oscillations [in Russian], Vol. 3, Izd. Inst. Matem. AN UkrSSR (1969).
S. A. Vasilishin and V. G. Kolomiets, “Investigation of oscillations of nonlinear systems with distributed parameters under random perturbations,” Ukr. Matem. Zh.,19, No. 3 (1967).
Yu. A. Mitropol'skii and B. I. Moseenkov, Lectures on the Application of Asymptotic Methods to the Solution of Partial Differential Equations [in Russian], Izd. Inst. Matem. AN UkrSSR (1968).
V. G. Kolomiets and L. M. Porkhun, “Random oscillations of elastic nonlinear systems with distributed parameters,” in: Mathematical Physics [in Russian], Vol. 5, Naukova Dumka, Kiev (1968).
A. V. Voznyuk and V. G. Kolomiets, “Application of asymptotic methods of nonlinear mechanics for the investigation of single-frequency oscillations of rods of variable cross section under random perturbations,” in: Mathematical Physics [inRussian], Vol. 6, Naukova Dumka, Kiev (1969).
S. A. Vasilishin and V. G. Kolomiets, “Investigation of single-frequency oscillations of nonlinear systems with distributed parameters and lags under random stimuli,” in: Mathematical Physics [in Russian], Vol. 6, Naukova Dumka, Kiev (1969).
V. G. Kolomiets and D. G. Korenevskii, “Certain questions in the theory of nonlinear oscillations of quasilinear systems with random delays,” in: Mathematical Physics [in Russian], Vol. 3, Naukova Dumka, Kiev (1967).
Author information
Authors and Affiliations
Additional information
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 23, No. 3, pp. 318–345, May–June, 1971.
Rights and permissions
About this article
Cite this article
Mitropol'skii, Y.A., Kolomiets, V.G. A veraging in stochastic systems. Ukr Math J 23, 268–290 (1971). https://doi.org/10.1007/BF01085350
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01085350