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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2501.13714 |
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| _version_ | 1866916581362106368 |
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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 | Consider a general $3$-dimensional Lotka-Volterra system with a rational first integral of degree two of the form $H=x^i y^j z^k$. The restriction of this Lotka-Volterra system to each surface $H(x,y,z)=h$ varying $h\in \mathbb{R}$ provide Kolmogorov systems. With the additional assumption that they have a Darboux invariant of the form $x^\ell y^m e^{st}$ they reduce to the Kolmogorov systems \begin{equation*} \begin{split} \dot{x}&=x \left( a_0- μ(c_1 x + c_2 z^2 + c_3 z)\right),\\ \dot{z}&=z\left( c_0+ c_1 x + c_2 z^2 + c_3 z\right). \end{split} \end{equation*} In this paper we classify the phase portraits in the Poincaré disc of all these Kolmogorov systems which depend on six parameters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13714 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Phase portraits of a family of Kolmogorov systems depending on six parameters Diz-Pita, Érika Llibre, Jaume Otero-Espinar, M. Victoria Dynamical Systems Consider a general $3$-dimensional Lotka-Volterra system with a rational first integral of degree two of the form $H=x^i y^j z^k$. The restriction of this Lotka-Volterra system to each surface $H(x,y,z)=h$ varying $h\in \mathbb{R}$ provide Kolmogorov systems. With the additional assumption that they have a Darboux invariant of the form $x^\ell y^m e^{st}$ they reduce to the Kolmogorov systems \begin{equation*} \begin{split} \dot{x}&=x \left( a_0- μ(c_1 x + c_2 z^2 + c_3 z)\right),\\ \dot{z}&=z\left( c_0+ c_1 x + c_2 z^2 + c_3 z\right). \end{split} \end{equation*} In this paper we classify the phase portraits in the Poincaré disc of all these Kolmogorov systems which depend on six parameters. |
| title | Phase portraits of a family of Kolmogorov systems depending on six parameters |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2501.13714 |