Universal Bifurcation Chaos Theory and Its New Applications
In this work, an analytical and numerical analysis of the transition to chaos in five nonlinear systems of ordinary and partial differential equations, which are models of autocatalytic chemical processes and interacting populations, is carried out. It is shown analytically and numerically that in a...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/11/2536 |
| _version_ | 1797597148870606848 |
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| author | Nikolai A. Magnitskii |
| author_facet | Nikolai A. Magnitskii |
| author_sort | Nikolai A. Magnitskii |
| collection | DOAJ |
| description | In this work, an analytical and numerical analysis of the transition to chaos in five nonlinear systems of ordinary and partial differential equations, which are models of autocatalytic chemical processes and interacting populations, is carried out. It is shown analytically and numerically that in all considered systems of equations, further complication of the dynamics of solutions and the transition to chemical and biological turbulence is carried out in full accordance with the universal Feigenbaum-Sharkovsky-Magnitskii bifurcation theory through subharmonic and homoclinic cascades of bifurcations of stable limit cycles. In this case, irregular (chaotic) attractors in all cases are exclusively singular attractors in the sense of the FShM theory. The obtained results once again indicate the wide applicability of the universal bifurcation FShM theory for describing laminar–turbulent transitions to chaotic dynamics in complex nonlinear systems of differential equations and that chaos in the system can be confirmed only by detection of some main cycles or tori in accordance with the universal bifurcation diagram presented in the article. |
| first_indexed | 2024-03-11T03:01:44Z |
| format | Article |
| id | doaj.art-7cdd48d5b38543ce9b190a008c2d5caa |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T03:01:44Z |
| publishDate | 2023-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-7cdd48d5b38543ce9b190a008c2d5caa2023-11-18T08:13:21ZengMDPI AGMathematics2227-73902023-05-011111253610.3390/math11112536Universal Bifurcation Chaos Theory and Its New ApplicationsNikolai A. Magnitskii0Federal Research Center “Computer Science and Control” of Russian Academy of Sciences, 119333 Moscow, RussiaIn this work, an analytical and numerical analysis of the transition to chaos in five nonlinear systems of ordinary and partial differential equations, which are models of autocatalytic chemical processes and interacting populations, is carried out. It is shown analytically and numerically that in all considered systems of equations, further complication of the dynamics of solutions and the transition to chemical and biological turbulence is carried out in full accordance with the universal Feigenbaum-Sharkovsky-Magnitskii bifurcation theory through subharmonic and homoclinic cascades of bifurcations of stable limit cycles. In this case, irregular (chaotic) attractors in all cases are exclusively singular attractors in the sense of the FShM theory. The obtained results once again indicate the wide applicability of the universal bifurcation FShM theory for describing laminar–turbulent transitions to chaotic dynamics in complex nonlinear systems of differential equations and that chaos in the system can be confirmed only by detection of some main cycles or tori in accordance with the universal bifurcation diagram presented in the article.https://www.mdpi.com/2227-7390/11/11/2536autocatalytic reactionspredator–prey modelshidden attractorsbifurcation cascadeschaossingular attractors |
| spellingShingle | Nikolai A. Magnitskii Universal Bifurcation Chaos Theory and Its New Applications Mathematics autocatalytic reactions predator–prey models hidden attractors bifurcation cascades chaos singular attractors |
| title | Universal Bifurcation Chaos Theory and Its New Applications |
| title_full | Universal Bifurcation Chaos Theory and Its New Applications |
| title_fullStr | Universal Bifurcation Chaos Theory and Its New Applications |
| title_full_unstemmed | Universal Bifurcation Chaos Theory and Its New Applications |
| title_short | Universal Bifurcation Chaos Theory and Its New Applications |
| title_sort | universal bifurcation chaos theory and its new applications |
| topic | autocatalytic reactions predator–prey models hidden attractors bifurcation cascades chaos singular attractors |
| url | https://www.mdpi.com/2227-7390/11/11/2536 |
| work_keys_str_mv | AT nikolaiamagnitskii universalbifurcationchaostheoryanditsnewapplications |