Dubovsky’s Class of Mathematical Models for Describing Economic Cycles with Heredity Effects
The article is devoted to the study of economic cycles and crises, which are studied within the framework of the theory of N.D. Kondratiev long waves (K-waves). The object of the study is the fractional mathematical models of S. V. Dubovsky, consisting of two nonlinear differential equations of frac...
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| Language: | English |
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MDPI AG
2024-12-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/9/1/19 |
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| author | Danil Makarov Roman Parovik Zafar Rakhmonov |
| author_facet | Danil Makarov Roman Parovik Zafar Rakhmonov |
| author_sort | Danil Makarov |
| collection | DOAJ |
| description | The article is devoted to the study of economic cycles and crises, which are studied within the framework of the theory of N.D. Kondratiev long waves (K-waves). The object of the study is the fractional mathematical models of S. V. Dubovsky, consisting of two nonlinear differential equations of fractional order and describing the dynamics of the efficiency of new technologies and the efficiency of capital productivity, taking into account constant and variable heredity. Fractional mathematical models also take into account the dependence of the rate of accumulation on capital productivity and the influx of external investment and new technological solutions. The effects of heredity lead to a delayed effect of the response of the system in question to the impact. The property of heredity in mathematical models is taken into account using fractional derivatives of constant and variable orders in the sense of Gerasimov–Caputo. The fractional mathematical models of S. V. Dubovsky are further studied numerically using the Adams–Bashforth–Moulton algorithm. Using a numerical algorithm, oscillograms and phase trajectories were constructed for various values and model parameters. It is shown that the fractional mathematical models of S. V. Dubovsky may have limit cycles, which are not always stable. |
| format | Article |
| id | doaj-art-70b0c829b6484dae9fd4b8b0531557a8 |
| institution | Kabale University |
| issn | 2504-3110 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-70b0c829b6484dae9fd4b8b0531557a82025-01-24T13:33:23ZengMDPI AGFractal and Fractional2504-31102024-12-01911910.3390/fractalfract9010019Dubovsky’s Class of Mathematical Models for Describing Economic Cycles with Heredity EffectsDanil Makarov0Roman Parovik1Zafar Rakhmonov2Department of Applied Mathematics and Computer Analysis, National University of Uzbekistan Named After Mirzo Ulugbek, Tashkent 100174, UzbekistanDepartment of Applied Mathematics and Computer Analysis, National University of Uzbekistan Named After Mirzo Ulugbek, Tashkent 100174, UzbekistanDepartment of Applied Mathematics and Computer Analysis, National University of Uzbekistan Named After Mirzo Ulugbek, Tashkent 100174, UzbekistanThe article is devoted to the study of economic cycles and crises, which are studied within the framework of the theory of N.D. Kondratiev long waves (K-waves). The object of the study is the fractional mathematical models of S. V. Dubovsky, consisting of two nonlinear differential equations of fractional order and describing the dynamics of the efficiency of new technologies and the efficiency of capital productivity, taking into account constant and variable heredity. Fractional mathematical models also take into account the dependence of the rate of accumulation on capital productivity and the influx of external investment and new technological solutions. The effects of heredity lead to a delayed effect of the response of the system in question to the impact. The property of heredity in mathematical models is taken into account using fractional derivatives of constant and variable orders in the sense of Gerasimov–Caputo. The fractional mathematical models of S. V. Dubovsky are further studied numerically using the Adams–Bashforth–Moulton algorithm. Using a numerical algorithm, oscillograms and phase trajectories were constructed for various values and model parameters. It is shown that the fractional mathematical models of S. V. Dubovsky may have limit cycles, which are not always stable.https://www.mdpi.com/2504-3110/9/1/19mathematical modelphase trajectoryoscillogramlimit cyclefractional derivativeAdams–Bashforth–Moulton method |
| spellingShingle | Danil Makarov Roman Parovik Zafar Rakhmonov Dubovsky’s Class of Mathematical Models for Describing Economic Cycles with Heredity Effects Fractal and Fractional mathematical model phase trajectory oscillogram limit cycle fractional derivative Adams–Bashforth–Moulton method |
| title | Dubovsky’s Class of Mathematical Models for Describing Economic Cycles with Heredity Effects |
| title_full | Dubovsky’s Class of Mathematical Models for Describing Economic Cycles with Heredity Effects |
| title_fullStr | Dubovsky’s Class of Mathematical Models for Describing Economic Cycles with Heredity Effects |
| title_full_unstemmed | Dubovsky’s Class of Mathematical Models for Describing Economic Cycles with Heredity Effects |
| title_short | Dubovsky’s Class of Mathematical Models for Describing Economic Cycles with Heredity Effects |
| title_sort | dubovsky s class of mathematical models for describing economic cycles with heredity effects |
| topic | mathematical model phase trajectory oscillogram limit cycle fractional derivative Adams–Bashforth–Moulton method |
| url | https://www.mdpi.com/2504-3110/9/1/19 |
| work_keys_str_mv | AT danilmakarov dubovskysclassofmathematicalmodelsfordescribingeconomiccycleswithheredityeffects AT romanparovik dubovskysclassofmathematicalmodelsfordescribingeconomiccycleswithheredityeffects AT zafarrakhmonov dubovskysclassofmathematicalmodelsfordescribingeconomiccycleswithheredityeffects |