Existence of at most two limit cycles for some non-autonomous differential equations
It is know that the non-autonomous differential equations dx/dt = a(t)+b(t)|x|, where a(t) and b(t) are 1-periodic maps of class C1, have no upper bound for their number of limit cycles (isolated solutions satisfying x(0) = x(1)). We prove that if either a(t) or b(t) does not change sign, then their...
| Autores: | ; |
|---|---|
| Tipo de documento: | Artículo |
| Estado: | Accepted version |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repositorio: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:2072/536856 |
| Acceso en línea: | http://hdl.handle.net/2072/536856 |
| Nivel de acceso: | Acceso abierto |
| Palabra clave: | limit cycle Non-autonomous differential equation periodic orbit |
| _version_ | 1878737695197364224 |
|---|---|
| author | Gasull, A. |
| author2 | Zhao, Y. |
| author2_role | author |
| author_browse | Gasull, A. Zhao, Y. |
| author_facet | Gasull, A. Zhao, Y. |
| author_role | author |
| collection | Recercat. Dipósit de la Recerca de Catalunya |
| description | It is know that the non-autonomous differential equations dx/dt = a(t)+b(t)|x|, where a(t) and b(t) are 1-periodic maps of class C1, have no upper bound for their number of limit cycles (isolated solutions satisfying x(0) = x(1)). We prove that if either a(t) or b(t) does not change sign, then their maximum number of limit cycles is two, taking into account their multiplicities, and that this upper bound is sharp. We also study all possible configurations of limit cycles. Our result is similar to other ones known for Abel type periodic differential equations although the proofs are quite different. © 2023 American Institute of Mathematical Sciences. All rights reserved. |
| eu_rights_str_mv | openAccess |
| format | article |
| id | ES_a38b8b2ac437e5d32f12b7632a6fabd0 |
| institution | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| instname_str | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| network_acronym_str | ES |
| network_name_str | España |
| oai_identifier_str | oai:recercat.cat:2072/536856 |
| publishDate | 2023 |
| publishDateSort | 2023 |
| publisher | American Institute of Mathematical Sciences |
| publisherStr | American Institute of Mathematical Sciences |
| reponame_str | Recercat. Dipósit de la Recerca de Catalunya |
| spelling | Existence of at most two limit cycles for some non-autonomous differential equations Gasull, A. Zhao, Y. limit cycle Non-autonomous differential equation periodic orbit It is know that the non-autonomous differential equations dx/dt = a(t)+b(t)|x|, where a(t) and b(t) are 1-periodic maps of class C1, have no upper bound for their number of limit cycles (isolated solutions satisfying x(0) = x(1)). We prove that if either a(t) or b(t) does not change sign, then their maximum number of limit cycles is two, taking into account their multiplicities, and that this upper bound is sharp. We also study all possible configurations of limit cycles. Our result is similar to other ones known for Abel type periodic differential equations although the proofs are quite different. © 2023 American Institute of Mathematical Sciences. All rights reserved. The first author is supported by Spanish State Research Agency, through the projects PID2019-104658GB-I00 grant and the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M) and by the grant 2017-SGR-1617 from AGAUR, Generalitat de Catalunya. The second author is supported by the NNSF of China (No. 11971495), Guangdong-Hong Kong-Macau Applied Math Center (No. 2020B1515310014) and Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515012105). ∗Corresponding author: Yulin Zhao. American Institute of Mathematical Sciences http://hdl.handle.net/2072/536856 |
| spellingShingle | Existence of at most two limit cycles for some non-autonomous differential equations Gasull, A. limit cycle Non-autonomous differential equation periodic orbit |
| status_str | acceptedVersion |
| title | Existence of at most two limit cycles for some non-autonomous differential equations |
| title_full | Existence of at most two limit cycles for some non-autonomous differential equations |
| title_fullStr | Existence of at most two limit cycles for some non-autonomous differential equations |
| title_full_unstemmed | Existence of at most two limit cycles for some non-autonomous differential equations |
| title_short | Existence of at most two limit cycles for some non-autonomous differential equations |
| title_sort | Existence of at most two limit cycles for some non-autonomous differential equations |
| topic | limit cycle Non-autonomous differential equation periodic orbit |
| topic_facet | limit cycle Non-autonomous differential equation periodic orbit |
| url | http://hdl.handle.net/2072/536856 |