Mathematics and Statistics Vol. 8(4), pp. 424 - 429
DOI: 10.13189/ms.2020.080408
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Logistic Map on the Ring of Multisets and Its Application in Economic Models

Iryna Halushchak ¹, Zoriana Novosad ², Yurii Tsizhma ¹, Andriy Zagorodnyuk ^3,*
1 Faculty of Economics, Vasyl Stefanyk Precarpathian National University, 57 Shevchenka Str., Ivano-Frankivsk 76018, Ukraine
² Department of Higher Mathematics and Quantitative Methods, Lviv University of Trade and Economics, 10, Tuhan-Baranovsky Str., Lviv 79005, Ukraine
³ Faculty of Mathematics and Informatics, Vasyl Stefanyk Precarpathian National University, Shevchenka Str., Ivano-Frankivsk 76018, Ukraine

ABSTRACT

In this paper, we extend complex polynomial dynamics to a set of multisets endowed with some ring operations (the metric ring of multisets associated with supersymmetric polynomials of infinitely many variables). Some new properties of the ring of multisets are established and a homomorphism to a function ring is constructed. Using complex homomorphisms on the ring of multisets, we proposed a method of investigations of polynomial dynamics over this ring by reducing them to a finite number of scalarvalued polynomial dynamics. An estimation of the number of such scalar-valued polynomial dynamics is established. As an important example, we considered an analogue of the logistic map, defined on a subring of multisets consisting of positive numbers in the interval [0; 1]: Some possible application to study the natural market development process in a competitive environment is proposed. In particular, it is shown that using the multiset approach, we can have a model that takes into account credit debt and reinvestments. Some numerical examples of logistic maps for different growth rate multiset [r] are considered. Note that the growth rate [r] may contain both "positive" and "negative" components and the examples demonstrate the influences of these components on the dynamics.

KEYWORDS
Logistic Map, Multisets, Supersymmetric Polynomials, Chaos, Market Development Model

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Iryna Halushchak , Zoriana Novosad , Yurii Tsizhma , Andriy Zagorodnyuk , "Logistic Map on the Ring of Multisets and Its Application in Economic Models," Mathematics and Statistics, Vol. 8, No. 4, pp. 424 - 429, 2020. DOI: 10.13189/ms.2020.080408.

(b). APA Format:
Iryna Halushchak , Zoriana Novosad , Yurii Tsizhma , Andriy Zagorodnyuk (2020). Logistic Map on the Ring of Multisets and Its Application in Economic Models. Mathematics and Statistics, 8(4), 424 - 429. DOI: 10.13189/ms.2020.080408.