Chains with Connections of Diffusion and Advective Types
The local dynamics of a system of oscillators with a large number of elements and with diffusive- and advective-type couplings containing a large delay are studied. Critical cases in the problem of the stability of the zero equilibrium state are singled out, and it is shown that all of them have inf...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/6/790 |
| _version_ | 1797240126479269888 |
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| author | Sergey Kashchenko |
| author_facet | Sergey Kashchenko |
| author_sort | Sergey Kashchenko |
| collection | DOAJ |
| description | The local dynamics of a system of oscillators with a large number of elements and with diffusive- and advective-type couplings containing a large delay are studied. Critical cases in the problem of the stability of the zero equilibrium state are singled out, and it is shown that all of them have infinite dimensions. Applying special methods of infinite normalization, we construct quasinormal forms, namely, nonlinear boundary value problems of the parabolic type, whose nonlocal dynamics determine the behavior of the solutions of the initial system in a small neighborhood of the equilibrium state. These quasinormal forms contain either two or three spatial variables, which emphasizes the complexity of the dynamical properties of the original problem. |
| first_indexed | 2024-04-24T18:02:29Z |
| format | Article |
| id | doaj.art-518bfeef824546b7b2962444953d4ad2 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-04-24T18:02:29Z |
| publishDate | 2024-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-518bfeef824546b7b2962444953d4ad22024-03-27T13:52:55ZengMDPI AGMathematics2227-73902024-03-0112679010.3390/math12060790Chains with Connections of Diffusion and Advective TypesSergey Kashchenko0Regional Scientific and Educational Mathematical Center “Centre of Integrable Systems”, P. G. Demidov Yaroslavl State University, 150003 Yaroslavl, RussiaThe local dynamics of a system of oscillators with a large number of elements and with diffusive- and advective-type couplings containing a large delay are studied. Critical cases in the problem of the stability of the zero equilibrium state are singled out, and it is shown that all of them have infinite dimensions. Applying special methods of infinite normalization, we construct quasinormal forms, namely, nonlinear boundary value problems of the parabolic type, whose nonlocal dynamics determine the behavior of the solutions of the initial system in a small neighborhood of the equilibrium state. These quasinormal forms contain either two or three spatial variables, which emphasizes the complexity of the dynamical properties of the original problem.https://www.mdpi.com/2227-7390/12/6/790boundary value problemdelaystabilitynormal formdynamicsasymptotics of solutions |
| spellingShingle | Sergey Kashchenko Chains with Connections of Diffusion and Advective Types Mathematics boundary value problem delay stability normal form dynamics asymptotics of solutions |
| title | Chains with Connections of Diffusion and Advective Types |
| title_full | Chains with Connections of Diffusion and Advective Types |
| title_fullStr | Chains with Connections of Diffusion and Advective Types |
| title_full_unstemmed | Chains with Connections of Diffusion and Advective Types |
| title_short | Chains with Connections of Diffusion and Advective Types |
| title_sort | chains with connections of diffusion and advective types |
| topic | boundary value problem delay stability normal form dynamics asymptotics of solutions |
| url | https://www.mdpi.com/2227-7390/12/6/790 |
| work_keys_str_mv | AT sergeykashchenko chainswithconnectionsofdiffusionandadvectivetypes |