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Russia Izhevsk
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 University^{a} 
Abstract  We investigate the asymptotic behavior of solutions of difference equations. Their righthand 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. 7986. 
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