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## Archive of Issues

Russia Izhevsk
Year
2016
Volume
26
Issue
1
Pages
79-86
 Section Mathematics Title About asymptotical properties of solutions of difference equations with random parameters Author(-s) Rodina L.I.a, Tyuteev I.I.a Affiliations Udmurt State Universitya Abstract We investigate the asymptotic behavior of solutions of difference equations. Their right-hand sides at given time depend not only on the value of state at the previous moment, but also on a random value from a given set $\Omega$. We obtain conditions of Lyapunov stability and asymptotic stability of the equilibrium for all values of random parameters and with probability one. We show that the problem of coexistence of stochastic cycles of different periods has a solution, which strongly differs from a known Sharkovsky result for a determined difference equation. Under some conditions, the existence of a stochastic cycle of length $k$ implies the existence of a cycle of any length $\ell>k$. Keywords difference equations with random parameters, Lyapunov stability, asymptotical stability, cyclic solution UDC 517.935, 517.938 MSC 34A60, 37N35, 49J15, 93B03 DOI 10.20537/vm160107 Received 20 January 2016 Language Russian Citation Rodina L.I., Tyuteev I.I. About asymptotical properties of solutions of difference equations with random parameters, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 1, pp. 79-86. References Riznichenko G.Yu. Lektsii po matematicheskim modelyam v biologii. Chast' 1 (Lectures on mathematical models in biology, Part 1), Izhevsk: Regular and Chaotic Dynamics, 2002, 232 p. Ten V.V. Modelling and tool support of the bank's financial stability, Dr. Sci. (Econom.) Dissertation, Tambov, 2006, 350 p. (In Russian). Sharkovskii A.N. The coexistence of cycles for a continuous mapping of the line in itself, Ukr. Mat. Zh., 1964, vol. 16, no. 1, pp. 61-71 (in Russian). Li T.-Y., Yorke J.A. Period three implies chaos, The American Mathematical Monthly, 1975, vol. 82, no. 10, pp. 985-992. Svirezhev Yu.M., Logofet D.O. Ustoichivost' biologicheskikh soobshchestv (Stability of biological communities), Moscow: Nauka, 1978, 352 p. Shapiro A.P., Luppov S.P. Rekurrentnye uravneniya v teorii populyatsionnoi biologii (The recurrent equations in the theory of population biology), Moscow: Nauka, 1983, 133 p. Sharkovskii A.N., Kolyada S.F., Sivak A.G., Fedorenko V.V. Dinamika odnomernykh otobrazhenii (Dynamics of one-dimensional mappings), Kiev: Naukova dumka, 1989, 216 p. Bobrovski D. Vvedenie v teoriyu dinamicheskikh sistem s diskretnym vremenem (Introduction to the theory of discrete-time dynamical systems), Izhevsk: Regular and Chaotic Dynamics, 2006, 360 p. Shiryaev A.N. Veroyatnost’ (Probability), Moscow: Nauka, 1989, 580 p. Masterkov Yu.V., Rodina L.I. Sufficient conditions for the local controllability of systems with random parameters for an arbitrary number of system states, Russian Mathematics, 2008, vol. 52, no. 3, pp. 34-44. Rodina L.I. On some probability models of dynamics of population growth, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2013, no. 4, pp. 109-124 (in Russian). Rodina L.I. About invariant sets of control systems with random coefficients, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2014, no. 4, pp. 109-121 (in Russian). Khas'minskii R.Z. Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems, Theory Probab. Appl., 1967, vol. 12, no. 1, pp. 144-147. Feller W. An introduction to probability theory and its applications, Vol. 2, Wiley, 1971. Translated under the title Vvedenie v teoriyu veroyatnostei i ee prilozheniya, vol. 2, Moscow: Mir, 1984, 738 p. Full text