Dynamics and integrability of polynomial vector fields on the $n$-dimensional sphere
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915045797003264 |
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| author | Jana, Supriyo Sarkar, Soumen |
| author_facet | Jana, Supriyo Sarkar, Soumen |
| contents | In this paper, we characterize arbitrary polynomial vector fields on $S^n$. We establish a necessary and sufficient condition for a degree one vector field on the odd-dimensional sphere $S^{2n-1}$ to be Hamiltonian. Additionally, we classify polynomial vector fields on $S^n$ up to degree two that possess an invariant great $(n-1)$-sphere. We present a class of completely integrable vector fields on $S^n$. We found a sharp bound for the number of invariant meridian hyperplanes for a polynomial vector field on $S^2$. Furthermore, we compute the sharp bound for the number of invariant parallel hyperplanes for any polynomial vector field on $S^n$. Finally, we study homogeneous polynomial vector fields on $S^n$, providing a characterization of their invariant $(n-1)$-spheres. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02190 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dynamics and integrability of polynomial vector fields on the $n$-dimensional sphere Jana, Supriyo Sarkar, Soumen Dynamical Systems 34C08, 34A34, 34C14, 34C40, 34C45 In this paper, we characterize arbitrary polynomial vector fields on $S^n$. We establish a necessary and sufficient condition for a degree one vector field on the odd-dimensional sphere $S^{2n-1}$ to be Hamiltonian. Additionally, we classify polynomial vector fields on $S^n$ up to degree two that possess an invariant great $(n-1)$-sphere. We present a class of completely integrable vector fields on $S^n$. We found a sharp bound for the number of invariant meridian hyperplanes for a polynomial vector field on $S^2$. Furthermore, we compute the sharp bound for the number of invariant parallel hyperplanes for any polynomial vector field on $S^n$. Finally, we study homogeneous polynomial vector fields on $S^n$, providing a characterization of their invariant $(n-1)$-spheres. |
| title | Dynamics and integrability of polynomial vector fields on the $n$-dimensional sphere |
| topic | Dynamical Systems 34C08, 34A34, 34C14, 34C40, 34C45 |
| url | https://arxiv.org/abs/2412.02190 |