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 pol...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | 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 |
| Access Level: | acceso abierto |
| Palabra clave: | Limit cycle Polycycle Cyclicity Asymptotic expansion |
| Sumario: | 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. |
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