On the cyclicity of Kolmogorov polycycles
In this paper we study planar polynomial Kolmogorov's differential systems Xμ{x˙=f(x,y;μ),y˙=g(x,y;μ), with the parameter μ varying in an open subset Λ⊂RN. Compactifying Xμ to the Poincaré disc, the boundary of the first quadrant is an invariant triangle Γ, that we assume to be a hyperbolic polycycl...
| Autores: | ; |
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| Tipo de documento: | Artículo |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Lenguaje: | Inglés |
| OAI Identifier: | oai:ddd.uab.cat:265184 |
| Acceso en línea: | https://ddd.uab.cat/record/265184 https://dx.doi.org/urn:doi:10.14232/ejqtde.2022.1.35 |
| Nivel de acceso: | Acceso abierto |
| Palabra clave: | Limit cycle Polycycle Cyclicity Asymptotic expansion |
| _version_ | 1878729646825013248 |
|---|---|
| author | Marín, David|||0000-0003-4422-6418 |
| author2 | Villadelprat Yagüe, Jordi|||0000-0002-1168-9750 |
| author2_role | author |
| author_browse | Marín, David|||0000-0003-4422-6418 Villadelprat Yagüe, Jordi|||0000-0002-1168-9750 |
| author_facet | Marín, David|||0000-0003-4422-6418 Villadelprat Yagüe, Jordi|||0000-0002-1168-9750 |
| author_role | author |
| collection | Dipòsit Digital de Documents de la UAB |
| description | In this paper we study planar polynomial Kolmogorov's differential systems Xμ{x˙=f(x,y;μ),y˙=g(x,y;μ), with the parameter μ varying in an open subset Λ⊂RN. Compactifying Xμ to the Poincaré disc, the boundary of the first quadrant is an invariant triangle Γ, that we assume to be a hyperbolic polycycle with exactly three saddle points at its vertices for all μ∈Λ. We are interested in the cyclicity of Γ inside the family {Xμ}μ∈Λ, i.e., the number of limit cycles that bifurcate from Γ as we perturb μ. In our main result we define three functions that play the same role for the cyclicity of the polycycle as the first three Lyapunov quantities for the cyclicity of a focus. As an application we study two cubic Kolmogorov families, with N=3 and N=5, and in both cases we are able to determine the cyclicity of the polycycle for all μ∈Λ, including those parameters for which the return map along Γ is the identity. |
| eu_rights_str_mv | openAccess |
| format | article |
| id | ES_0c1cd7a2a46479ecd2a12f7f2ad3bae2 |
| institution | Universitat Autònoma de Barcelona |
| instname_str | Universitat Autònoma de Barcelona |
| language | eng |
| network_acronym_str | ES |
| network_name_str | España |
| oai_identifier_str | oai:ddd.uab.cat:265184 |
| publishDate | 2022 |
| publishDateSort | 2022 |
| reponame_str | Dipòsit Digital de Documents de la UAB |
| spelling | On the cyclicity of Kolmogorov polycycles Marín, David|||0000-0003-4422-6418 Villadelprat Yagüe, Jordi|||0000-0002-1168-9750 Limit cycle Polycycle Cyclicity Asymptotic expansion In this paper we study planar polynomial Kolmogorov's differential systems Xμ{x˙=f(x,y;μ),y˙=g(x,y;μ), with the parameter μ varying in an open subset Λ⊂RN. Compactifying Xμ to the Poincaré disc, the boundary of the first quadrant is an invariant triangle Γ, that we assume to be a hyperbolic polycycle with exactly three saddle points at its vertices for all μ∈Λ. We are interested in the cyclicity of Γ inside the family {Xμ}μ∈Λ, i.e., the number of limit cycles that bifurcate from Γ as we perturb μ. In our main result we define three functions that play the same role for the cyclicity of the polycycle as the first three Lyapunov quantities for the cyclicity of a focus. As an application we study two cubic Kolmogorov families, with N=3 and N=5, and in both cases we are able to determine the cyclicity of the polycycle for all μ∈Λ, including those parameters for which the return map along Γ is the identity. https://ddd.uab.cat/record/265184 https://dx.doi.org/urn:doi:10.14232/ejqtde.2022.1.35 |
| spellingShingle | On the cyclicity of Kolmogorov polycycles Marín, David|||0000-0003-4422-6418 Limit cycle Polycycle Cyclicity Asymptotic expansion |
| title | On the cyclicity of Kolmogorov polycycles |
| title_full | On the cyclicity of Kolmogorov polycycles |
| title_fullStr | On the cyclicity of Kolmogorov polycycles |
| title_full_unstemmed | On the cyclicity of Kolmogorov polycycles |
| title_short | On the cyclicity of Kolmogorov polycycles |
| title_sort | On the cyclicity of Kolmogorov polycycles |
| topic | Limit cycle Polycycle Cyclicity Asymptotic expansion |
| topic_facet | Limit cycle Polycycle Cyclicity Asymptotic expansion |
| url | https://ddd.uab.cat/record/265184 https://dx.doi.org/urn:doi:10.14232/ejqtde.2022.1.35 |