Planar Kolmogorov systems with infinitely many singular points at infinity
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916581803556864 |
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| author | Diz-Pita, Érika Llibre, Jaume Otero-Espinar, M. Victoria |
| author_facet | Diz-Pita, Érika Llibre, Jaume Otero-Espinar, M. Victoria |
| contents | We classify the global dynamics of the five-parameter family of planar Kolmogorov systems \begin{equation*}
\begin{split}
\dot{y}&=y \left( b_0+ b_1 y z + b_2 y + b_3 z\right),
\dot{z}&=z\left( c_0 + b_1 y z + b_2 y + b_3 z\right),
\end{split} \end{equation*} which is obtained from the Lotka-Volterra systems of dimension three. These systems have infinitely many singular points at inifnity. We give the topological classification of their phase portraits in the Poincaré disc, so we can describe the dynamics of these systems near infinity. We prove that these systems have 13 topologically distinct global phase portraits. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_14416 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Planar Kolmogorov systems with infinitely many singular points at infinity Diz-Pita, Érika Llibre, Jaume Otero-Espinar, M. Victoria Dynamical Systems We classify the global dynamics of the five-parameter family of planar Kolmogorov systems \begin{equation*} \begin{split} \dot{y}&=y \left( b_0+ b_1 y z + b_2 y + b_3 z\right), \dot{z}&=z\left( c_0 + b_1 y z + b_2 y + b_3 z\right), \end{split} \end{equation*} which is obtained from the Lotka-Volterra systems of dimension three. These systems have infinitely many singular points at inifnity. We give the topological classification of their phase portraits in the Poincaré disc, so we can describe the dynamics of these systems near infinity. We prove that these systems have 13 topologically distinct global phase portraits. |
| title | Planar Kolmogorov systems with infinitely many singular points at infinity |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2501.14416 |