Infinite Turing Bifurcations in Chains of Van der Pol Systems
A chain of coupled systems of Van der Pol equations is considered. We study the local dynamics of this chain in the vicinity of the zero equilibrium state. We make a transition to the system with a continuous spatial variable assuming that the number of elements in the chain is large enough. The cri...
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| Format: | Article |
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MDPI AG
2022-10-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/20/3769 |
| _version_ | 1827649157096013824 |
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| author | Sergey Kashchenko |
| author_facet | Sergey Kashchenko |
| author_sort | Sergey Kashchenko |
| collection | DOAJ |
| description | A chain of coupled systems of Van der Pol equations is considered. We study the local dynamics of this chain in the vicinity of the zero equilibrium state. We make a transition to the system with a continuous spatial variable assuming that the number of elements in the chain is large enough. The critical cases corresponding to the Turing bifurcations are identified. It is shown that they have infinite dimension. Special nonlinear parabolic equations are proposed on the basis of the asymptotic algorithm. Their nonlocal dynamics describes the local behavior of solutions to the original system. In a number of cases, normalized parabolic equations with two spatial variables arise while considering the most important diffusion type couplings. It has been established, for example, that for the considered systems with a large number of elements, the dynamics change significantly with a slight change in the number of such elements. |
| first_indexed | 2024-03-09T19:52:00Z |
| format | Article |
| id | doaj.art-34ebf9dee96e461d9fd2ce520d94c5f1 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T19:52:00Z |
| publishDate | 2022-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-34ebf9dee96e461d9fd2ce520d94c5f12023-11-24T01:06:45ZengMDPI AGMathematics2227-73902022-10-011020376910.3390/math10203769Infinite Turing Bifurcations in Chains of Van der Pol SystemsSergey Kashchenko0Regional Scientific and Educational Mathematical Center «Centre of Integrable Systems», P. G. Demidov Yaroslavl State University, 150003 Yaroslavl, RussiaA chain of coupled systems of Van der Pol equations is considered. We study the local dynamics of this chain in the vicinity of the zero equilibrium state. We make a transition to the system with a continuous spatial variable assuming that the number of elements in the chain is large enough. The critical cases corresponding to the Turing bifurcations are identified. It is shown that they have infinite dimension. Special nonlinear parabolic equations are proposed on the basis of the asymptotic algorithm. Their nonlocal dynamics describes the local behavior of solutions to the original system. In a number of cases, normalized parabolic equations with two spatial variables arise while considering the most important diffusion type couplings. It has been established, for example, that for the considered systems with a large number of elements, the dynamics change significantly with a slight change in the number of such elements.https://www.mdpi.com/2227-7390/10/20/3769Van der Pol equationasymptoticsstabilitynormal formbifurcationcritical cases |
| spellingShingle | Sergey Kashchenko Infinite Turing Bifurcations in Chains of Van der Pol Systems Mathematics Van der Pol equation asymptotics stability normal form bifurcation critical cases |
| title | Infinite Turing Bifurcations in Chains of Van der Pol Systems |
| title_full | Infinite Turing Bifurcations in Chains of Van der Pol Systems |
| title_fullStr | Infinite Turing Bifurcations in Chains of Van der Pol Systems |
| title_full_unstemmed | Infinite Turing Bifurcations in Chains of Van der Pol Systems |
| title_short | Infinite Turing Bifurcations in Chains of Van der Pol Systems |
| title_sort | infinite turing bifurcations in chains of van der pol systems |
| topic | Van der Pol equation asymptotics stability normal form bifurcation critical cases |
| url | https://www.mdpi.com/2227-7390/10/20/3769 |
| work_keys_str_mv | AT sergeykashchenko infiniteturingbifurcationsinchainsofvanderpolsystems |