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Iraq; Russia Al Diwaniyah; Izhevsk
Section Mathematics
Title On the invariant sets and chaotic solutions of difference equations with random parameters
Author(-s) Rodina L.I.a, Hammady A.H.ab
Affiliations Udmurt State Universitya, University of Al-Qadisiyahb
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 sigma-algebra 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 sigma-algebra 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 continuous-discrete 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.
Keywords difference equations with random parameters, stable and unstable cycles, chaotic solutions
UDC 517.962.24
MSC 37H10, 34F05, 60H25, 93E03
DOI 10.20537/vm170207
Received 12 April 2017
Language Russian
Citation Rodina L.I., Hammady A.H. On the invariant sets and chaotic solutions of difference equations with random parameters, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 2, pp. 238-247.
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