Darboux first integrals of Kolmogorov systems with invariant $n$-sphere
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911437678444544 |
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| author | Jana, Supriyo Sarkar, Soumen |
| author_facet | Jana, Supriyo Sarkar, Soumen |
| contents | In this paper, we characterize all polynomial Kolmogorov vector fields for which the standard $n$-sphere is invariant. We exhibit completely integrable Kolmogorov vector fields of degree $m$ on $\mathbb{S}^n$ for any $m >2$. Then, we show that there is no cubic Hamiltonian Kolmogorov vector field that makes an odd-dimensional sphere invariant. We examine the conditions under which a cubic Kolmogorov vector field has a Darboux first integral. In many cases, we determine whether they constitute necessary and sufficient conditions. Moreover, we study the complete integrability of cubic Kolmogorov vector fields having an invariant $n$-sphere. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_09747 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Darboux first integrals of Kolmogorov systems with invariant $n$-sphere Jana, Supriyo Sarkar, Soumen Dynamical Systems 34A34, 34C40, 34C45, 34C14, 34A05 In this paper, we characterize all polynomial Kolmogorov vector fields for which the standard $n$-sphere is invariant. We exhibit completely integrable Kolmogorov vector fields of degree $m$ on $\mathbb{S}^n$ for any $m >2$. Then, we show that there is no cubic Hamiltonian Kolmogorov vector field that makes an odd-dimensional sphere invariant. We examine the conditions under which a cubic Kolmogorov vector field has a Darboux first integral. In many cases, we determine whether they constitute necessary and sufficient conditions. Moreover, we study the complete integrability of cubic Kolmogorov vector fields having an invariant $n$-sphere. |
| title | Darboux first integrals of Kolmogorov systems with invariant $n$-sphere |
| topic | Dynamical Systems 34A34, 34C40, 34C45, 34C14, 34A05 |
| url | https://arxiv.org/abs/2602.09747 |