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Open AccessArticle

A Dynamical System with Random Parameters as a Mathematical Model of Real Phenomena

1
Faculty of Civil Engineering, Brno University of Technology, Veveří 331/95, 602 00 Brno, Czech Republic
2
Department of Computer Mathematics and Information Security, National University of Economics, 03068 Kyiv, Peremogy 54/1, Ukraine
3
Faculty of Mathematics, University of Białystok, K. Ciołkowskiego 1M, 15-245 Białystok, Poland
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(11), 1338; https://doi.org/10.3390/sym11111338
Received: 30 September 2019 / Revised: 27 October 2019 / Accepted: 28 October 2019 / Published: 30 October 2019
(This article belongs to the Special Issue Differential and Difference Equations and Symmetry)
In many cases, it is difficult to find a solution to a system of difference equations with random structure in a closed form. Thus, a random process, which is the solution to such a system, can be described in another way, for example, by its moments. In this paper, we consider systems of linear difference equations whose coefficients depend on a random Markov or semi-Markov chain with jumps. The moment equations are derived for such a system when the random structure is determined by a Markov chain with jumps. As an example, three processes: Threats to security in cyberspace, radiocarbon dating, and stability of the foreign currency exchange market are modelled by systems of difference equations with random parameters that depend on a semi-Markov or Markov process. The moment equations are used to obtain the conditions under which the processes are stable. View Full-Text
Keywords: Markov and semi-Markov chain; random transformation of solutions; L2-stability; jumps of solutions; moment equations Markov and semi-Markov chain; random transformation of solutions; L2-stability; jumps of solutions; moment equations
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Diblík, J.; Dzhalladova, I.; Růžičková, M. A Dynamical System with Random Parameters as a Mathematical Model of Real Phenomena. Symmetry 2019, 11, 1338.

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