Quadratic, Homogeneous and Kolmogorov vector fields on $S^1\times S^2$ and $S^2 \times S^1$
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929664120848384 |
|---|---|
| author | Jana, Supriyo Sarkar, Soumen |
| author_facet | Jana, Supriyo Sarkar, Soumen |
| contents | In this paper, we consider the following two algebraic hypersurfaces $$S^1\times S^2=\{(x_1,x_2,x_3,x_4)\in \mathbb{R}^4:(x_1^2+x_2^2-a^2)^2 + x_3^2 + x_4^2 -1=0;~ a>1\}$$ and $$S^2\times S^1=\{(x_1,x_2,x_3,x_4)\in \mathbb{R}^4:(x_1^2+x_2^2+x_3^2-b^2)^2+x_4^2-1=0;~ b>1\}$$ embedded in $\mathbb{R}^4$. We study polynomial vector fields in $\mathbb{R}^4$ separately, having $S^1\times S^2$ and $S^2\times S^1$ invariant by their flows. We characterize all linear, quadratic, cubic Kolmogorov and homogeneous vector fields on $S^1\times S^2$ and $S^2\times S^1$. We construct some first integrals of these vector fields and find which of the vector fields are Hamiltonian. We give upper bounds for the number of the invariant meridian and parallel hyperplanes of these vector fields. In addition, we have shown that the upper bounds are sharp in many cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_09439 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quadratic, Homogeneous and Kolmogorov vector fields on $S^1\times S^2$ and $S^2 \times S^1$ Jana, Supriyo Sarkar, Soumen Dynamical Systems Classical Analysis and ODEs 34A34, 34C14, 34C40, 37J35 In this paper, we consider the following two algebraic hypersurfaces $$S^1\times S^2=\{(x_1,x_2,x_3,x_4)\in \mathbb{R}^4:(x_1^2+x_2^2-a^2)^2 + x_3^2 + x_4^2 -1=0;~ a>1\}$$ and $$S^2\times S^1=\{(x_1,x_2,x_3,x_4)\in \mathbb{R}^4:(x_1^2+x_2^2+x_3^2-b^2)^2+x_4^2-1=0;~ b>1\}$$ embedded in $\mathbb{R}^4$. We study polynomial vector fields in $\mathbb{R}^4$ separately, having $S^1\times S^2$ and $S^2\times S^1$ invariant by their flows. We characterize all linear, quadratic, cubic Kolmogorov and homogeneous vector fields on $S^1\times S^2$ and $S^2\times S^1$. We construct some first integrals of these vector fields and find which of the vector fields are Hamiltonian. We give upper bounds for the number of the invariant meridian and parallel hyperplanes of these vector fields. In addition, we have shown that the upper bounds are sharp in many cases. |
| title | Quadratic, Homogeneous and Kolmogorov vector fields on $S^1\times S^2$ and $S^2 \times S^1$ |
| topic | Dynamical Systems Classical Analysis and ODEs 34A34, 34C14, 34C40, 37J35 |
| url | https://arxiv.org/abs/2307.09439 |