Abstract

We consider the probability model defined by the difference equation
$$x_{n+1}=f(\omega_n,x_n), \quad (\omega_n,x_n)\in \Omega\times [a,b], \quad n=0,1,\dots, \qquad\qquad (1)$$
where $\Omega$ is a given set with sigmaalgebra of subsets $\widetilde{\mathfrak A},$ on which a probability measure $\widetilde \mu$ is defined. Let $\mu $ be a continuation of the measure $\widetilde \mu $ on the sigmaalgebra generated by cylindrical sets. We study invariant sets and attractors of the equation with random parameters $(1).$ We receive conditions under which a given set is the maximal attractor. It is shown that, in invariant set $A\subseteq [a,b]$, there can be solutions, which are chaotic with probability one. It is observed in the case when exist an $m_i\in\mathbb N $ and sets $\Omega_i\subset\Omega $ such that $ \mu (\Omega_i)> 0,$ $i=1,2,$ and ${\rm cl}\, f^{m_1}(\Omega_1,A)\cap \,{\rm cl}\, f^{m_2}(\Omega_2,A)=\varnothing.$ It is shown, that solutions, chaotic with probability one, exist also in the case when the equation $(1)$ either has no any cycle, or all cycles are unstable with probability one. The results of the paper are illustrated by the example of a continuousdiscrete probabilistic model of the dynamics of an isolated population; for this model we investigate different modes of dynamic development, which have certain differences from the modes of determined models and describe the processes in real physical systems more exhaustively.

References

 Li T.Y., Yorke J.A. Period three implies chaos, The American Mathematical Monthly, 1975, vol. 82, no. 10, pp. 985992. DOI: 10.2307/2318254
 Svirezhev Yu.M., Logofet D.O. Ustoichivost' biologicheskikh soobshchestv (Stability of biological communities), Moscow: Nauka, 1978, 352 p.
 Sharkovskii A.N., Kolyada S.F., Sivak A.G., Fedorenko V.V. Dinamika odnomernykh otobrazhenii (Dynamics of onedimensional mappings), Kiev: Naukova dumka, 1989, 216 p.
 Bobrovski D. Vvedenie v teoriyu dinamicheskikh sistem s diskretnym vremenem (Introduction to the theory of discretetime dynamical systems), Izhevsk: Regular and Chaotic Dynamics, 2006, 360 p.
 Sharkovskii A.N. Attraktory traektorii i ikh basseiny (Attractors of trajectories and their basins), Kiev: Naukova dumka, 2013, 320 p.
 Shiryaev A.N. Veroyatnost' (Probability), Moscow: Nauka, 1989, 580 p.
 Rodina L.I., Tyuteev I.I. About asymptotical properties of solutions of difference equations with random parameters, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2016, vol. 26, issue 1, pp. 7986 (in Russian). DOI: 10.20537/vm160107
 Rodina L.I. On repelling cycles and chaotic solutions of difference equations with random parameters, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2016, vol. 22, no. 2, pp. 227235 (in Russian). DOI: 10.21538/013448892016222227235
 Bratus' A.S., Novozhilov A.S., Rodina E.V. Diskretnye dinamicheskie sistemy i matematicheskie modeli v ekologii (Discrete dynamic systems and mathematical models in ecology), Moscow: Moscow State University of Railway Engineering, 2005, 139 p.
 Feller W. An introduction to probability theory and its applications, Vol. 1, Wiley, 1971. Translated under the title Vvedenie v teoriyu veroyatnostei i ee prilozheniya, vol. 1, Moscow: Mir, 1984, 528 p.
 Nedorezov L.V., Nazarov I.N. Continuousdiscrete models of dynamics of an isolated population and two competing species, Mat. Strukt. Model., 1998, issue 2, pp. 7791 (in Russian).
 Nedorezov L.V., Nedorezova B.N. Modification of MoranRicker models for dynamics of number of the isolated population, Zhurnal Obshchei Biologii, 1994, vol. 55, no. 45, pp. 514521 (in Russian).
