Qualitative Theory of Two-Dimensional Polynomial Dynamical Systems
A qualitative theory of two-dimensional quadratic-polynomial integrable dynamical systems (DSs) is constructed on the basis of a discriminant criterion elaborated in the paper. This criterion enables one to pick up a single parameter that makes it possible to identify all feasible solution classes a...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-10-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/10/1884 |
| _version_ | 1797513100699631616 |
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| author | Yury Shestopalov Azizaga Shakhverdiev |
| author_facet | Yury Shestopalov Azizaga Shakhverdiev |
| author_sort | Yury Shestopalov |
| collection | DOAJ |
| description | A qualitative theory of two-dimensional quadratic-polynomial integrable dynamical systems (DSs) is constructed on the basis of a discriminant criterion elaborated in the paper. This criterion enables one to pick up a single parameter that makes it possible to identify all feasible solution classes as well as the DS critical and singular points and solutions. The integrability of the considered DS family is established. Nine specific solution classes are identified. In each class, clear types of symmetry are determined and visualized and it is discussed how transformations between the solution classes create new types of symmetries. Visualization is performed as series of phase portraits revealing all possible catastrophic scenarios that result from the transition between the solution classes. |
| first_indexed | 2024-03-10T06:10:55Z |
| format | Article |
| id | doaj.art-e962526253904065be377d4be7cc770f |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T06:10:55Z |
| publishDate | 2021-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-e962526253904065be377d4be7cc770f2023-11-22T20:10:33ZengMDPI AGSymmetry2073-89942021-10-011310188410.3390/sym13101884Qualitative Theory of Two-Dimensional Polynomial Dynamical SystemsYury Shestopalov0Azizaga Shakhverdiev1Faculty of Engineering and Sustainable Development, Academy of Technology and Environment, University of Gävle, 801 76 Gävle, SwedenDepartment of Geology and Exploration of Hydrocarbon Deposits, Russian State Geological Prospecting University, 117997 Moscow, RussiaA qualitative theory of two-dimensional quadratic-polynomial integrable dynamical systems (DSs) is constructed on the basis of a discriminant criterion elaborated in the paper. This criterion enables one to pick up a single parameter that makes it possible to identify all feasible solution classes as well as the DS critical and singular points and solutions. The integrability of the considered DS family is established. Nine specific solution classes are identified. In each class, clear types of symmetry are determined and visualized and it is discussed how transformations between the solution classes create new types of symmetries. Visualization is performed as series of phase portraits revealing all possible catastrophic scenarios that result from the transition between the solution classes.https://www.mdpi.com/2073-8994/13/10/1884polynomial integrable dynamical systemsdiscriminant criterioncritical pointsvisualizationqualitative theoryphase portraits |
| spellingShingle | Yury Shestopalov Azizaga Shakhverdiev Qualitative Theory of Two-Dimensional Polynomial Dynamical Systems Symmetry polynomial integrable dynamical systems discriminant criterion critical points visualization qualitative theory phase portraits |
| title | Qualitative Theory of Two-Dimensional Polynomial Dynamical Systems |
| title_full | Qualitative Theory of Two-Dimensional Polynomial Dynamical Systems |
| title_fullStr | Qualitative Theory of Two-Dimensional Polynomial Dynamical Systems |
| title_full_unstemmed | Qualitative Theory of Two-Dimensional Polynomial Dynamical Systems |
| title_short | Qualitative Theory of Two-Dimensional Polynomial Dynamical Systems |
| title_sort | qualitative theory of two dimensional polynomial dynamical systems |
| topic | polynomial integrable dynamical systems discriminant criterion critical points visualization qualitative theory phase portraits |
| url | https://www.mdpi.com/2073-8994/13/10/1884 |
| work_keys_str_mv | AT yuryshestopalov qualitativetheoryoftwodimensionalpolynomialdynamicalsystems AT azizagashakhverdiev qualitativetheoryoftwodimensionalpolynomialdynamicalsystems |