E.V. Lisovsky1, O.V. Druzhinina2

1 Bauman Moscow State Technical University (Moscow, Russia)

2 FRC «Computer Science and Control» of RAS (Moscow, Russia)

2 V.A. Trapeznikov Institute of Control Sciences of RAS (Moscow, Russia)

1 levgenijv@gmail.com, 2 ovdruzh@mail.ru

Solving the problems of synthesizing and analyzing multidimensional nonlinear dynamic models of organizational and economic systems is one of the relevant areas of research. Analyzing the stability of systems described by differential equations taking into account uncertainties of various types, is of both theoretical and applied interest. One of the effective methods for studying stability is the method of Lyapunov functions which has received significant development in the works of domestic and foreign researchers. The approach to describing processes in models of enterprise complex development based on multidimensional differential inclusions, fuzzy and stochastic differential equations, aimed at identifying new qualitative effects, makes it possible to assess the impact of multivaluedness and stochastic perturbations in complex organizational and economic systems. The aim of the work is to develop an approach to studying multidimensional nonlinear dynamic models of organizational and economic systems based on the application of methods from the qualitative theory of differential equations and control theory, and to derive stability conditions for these models, taking into account uncertainties of various types. A nine‑dimensional dynamic model of enterprise interaction was constructed taking into account the use and growth of the total capital share. The case is considered in which the development and interaction of several enterprises in the industry including enterprises with distributed capacities are modeled. Using the Lyapunov function method sufficient conditions for stability in the deterministic case are obtained. A nonstationary model is constructed taking into account time‑dependent parameters. The transition to models of organizational and economic systems of enterprise interaction and development based on vector differential inclusions, fuzzy and stochastic differential equations is described. The problem of trajectory stability has been studied using the principle of reducing the problem of stability of differential inclusions to the problems of stability of fuzzy and stochastic differential equations. The possibilities of applying the considered approach to describe the development of railway car‑repair enterprises have been considered. The results can be used in problems related to studying stability and computer modeling of complex organizational and economic systems, in problems of improving technologies used in designing interaction schemes for enterprises in the transport industry, and for describing processes in complex production systems with distributed capacities.

Lisovsky E.V., Druzhinina O.V. Construction and analysis of the stability of generalized models of organizational and economic systems // Science Intensive Technologies. 2026. V. 27. № 5. P. 22−30. DOI: https://doi.org/10.18127/j19998465-202605-03

1. Andrianov D.L., Arbuzov V.O., Ivliev S.V., Maksimov V.P., Simonov P.M. Dinamicheskie modeli ekonomiki: teoriya, prilozheniya, programmnaya realizaciya. Vestnik Permskogo universiteta. Seriya «Ekonomika». 2015. №4 (27). S. 8–32. (In Russian).
2. Batishcheva G.A., Zhuravleva M. I., Kuznecov M. V., Trofimenko E.A. Analiz neravnovesnyh modelej dinamicheskih ekonomicheskih sistem. Vestnik Rostovskogo gosudarstvennogo ekonomicheskogo universiteta (RINH). 2017. № 4(60). S. 129–135. (In Russian).
3. Druzhinina O.V., Voroncova V.L., Zajcev D.S., Kabanov M.A., Shmel'kova A.A. Postroenie i analiz mnogomernyh nelinejnyh dinamicheskih modelej social'no-ekonomicheskih processov. Naukoemkie tekhnologii. 2022. T. 23. № 6. S. 60–68. (In Russian).
4. Kochetkov S.A. Razrabotka mekhanizma upravleniya ekonomicheskoj ustojchivost'yu predpriyatiya. Izv. vuzov. Seriya: Ekonomika, finansy i upravlenie proizvodstvom. 2014. № 2(20). S. 118-125. (In Russian).
5. Ganina G.E. Upravlenie proizvodstvenno-tekhnologicheskoj ustojchivost'yu predpriyatiya pri vnedrenii novyh vidov produkcii. Kontrolling. 2016. № 60. S. 30–37. (In Russian).
6. Shapovalov V.I., Kablov V.F., Bashmakov V.A., Avakumov V.E. Sinergeticheskaya model' ustojchivosti srednej firmy. Sinergetika i problemy teorii upravleniya. M.: Fizmatlit. 2004. S. 454–464 (In Russian).
7. Lorenc E. Determinirovannoe neperiodicheskoe dvizhenie. Strannye attraktory. M.: Mir. 1981. S. 88–116 (In Russian).
8. Danilov Yu.A. Lekcii po nelinejnoj dinamike. M.: KomKniga. 2006 (In Russian).
9. Gurina T.A., Dorofeev I.A. Sushchestvovanie gomoklinicheskoj babochki v modeli ustojchivosti srednej firmy. Dinamicheskie sistemy. 2010. Vyp. 28. S. 63 –68 (In Russian).
10. Gladkih O.B., Lisovskij E.V., Lyudagovskaya M.A., Oborotov A.V., Petrova S.N. Postroenie i komp'yuternoe modelirovanie trekhmernyh dinamicheskih modelej s haoticheskimi rezhimami. Nelinejnyj mir. 2019. T. 17. № 5. S. 67–75. DOI: 10.18127/j20700970-201905-08. (In Russian).
11. Shcherbakov A.V., Gladkih O.B., Petrova S.N., Voroncova V.L. Kachestvennye svojstva konechnomernyh nelinejnyh dinamicheskih modelej. Nelinejnyj mir. 2020.T. 18. №5. S. 15–23. (In Russian).
12. Petrova S.N., Korzhavina N.V., Gevejler N. S., Duplinskij I.A., Medyaev I.N. Obobshchenie i analiz matematicheskoj modeli razvitiya i vzaimodejstviya predpriyatij. Sistemy upravleniya, slozhnye sistemy: modelirovanie, ustojchivost', stabilizaciya, intellektual'nye tekhnologii. Materialy VIII Mezhdunar. nauch.-prak. konf. (Elec, 21–22 aprelya 2022 g.). Elec: EGU im. I.A. Bunina, 2022. S. 57–62. (In Russian).
13. Voroncova V. L., Ashurov A. A., Shorgin V. S. Postroenie upravlyaemyh dinamicheskih modelej social'no-ekonomicheskih processov. Sistemy upravleniya, slozhnye sistemy: modelirovanie, ustojchivost', stabilizaciya, intellektual'nye tekhnologii. Materialy IX Mezhdunarodnoj nauchno-prakticheskoj konferencii (Elec, 24–25 aprelya 2023 g.). Elec: EGU im. I.A. Bunina. 2023. S. 180–184. (In Russian).
14. Chetaev N.G. Ustojchivost' dvizheniya. M.: Nauka, 1992 (In Russian).
15. Demidovich B.P. Lekcii po matematicheskoj teorii ustojchivosti. M.: MGU. 1998 (In Russian).
16. Filatov A.N. Teoriya ustojchivosti. M.: IVM RAN, 2002 (In Russian).
17. Shestakov A.A. Obobshchennyj pryamoj metod Lyapunova dlya sistem s raspredelennymi parametrami. M.: Nauka. 1990 (2-e dopoln. izd. M.: URSS, 2007) (In Russian).
18. Merenkov Yu.N. Ustojchivopodobnye svojstva differencial'nyh vklyuchenij, nechetkih i stohasticheskih differencial'nyh uravnenij. M.: RUDN. 2000 (In Russian).
19. Druzhinina O.V., Masina O.N. Metody issledovaniya ustojchivosti i upravlyaemosti nechetkih i stohasticheskih dinamicheskih sistem. M.: VC RAN. 2009 (In Russian).
20. Druzhinina O.V., Masina O.N. Sistemnyj podhod k issledovaniyu ustojchivosti modelej, opisyvaemyh differencial'nymi uravneniyami razlichnyh tipov. Vestnik Rossijskoj akademii estestvennyh nauk. Differencial'nye uravneniya. 2015. T. 15. № 3. S. 24–30. https://raen.info/upload/000/vestnik/2015/3/24-30.pdf (In Russian)
21. Price K.., Storn R., Lampinen J. Differential Evolution: A Practical Approach to Global Optimization. Springer, 2005.
22. Druzhinina O.V., Masina O.N., Vasil'eva I.I. Differencial'naya evolyuciya v zadachah poiska optimal'nyh parametrov populyacionno-migracionnyh modelej. Sovremennye informacionnye tekhnologii i IT-obrazovanie. 2024. T. 20. № 1. S. 58–69 (In Russian).
23. Druzhinina O.V., Lisovskij E.V. Issledovanie ustojchivosti po Zhukovskomu traektorij dinamicheskih sistem, modeliruemyh nelinejnymi vektorno-matrichnymi uravneniyami. Nelinejnyj mir. 2019. T. 17. № 2. S. 40–47. DOI: 10.18127/j20700970-201902-05. (In Russian).