Characterization and dynamics of certain classes of polynomial vector fields on the torus
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916443669397504 |
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| author | Jana, Supriyo |
| author_facet | Jana, Supriyo |
| contents | In this paper, we classify all polynomial vector fields in $\mathbb{R}^3$ of degree up to three such that their flow makes the torus $$\mathbb{T}^2=\{(x,y,z)\in \mathbb{R}^3:(x^2+y^2-a^2)^2+z^2-1=0\}~\mbox{with}~a\in (1,\infty)$$ invariant. We also classify cubic Kolmogorov vector fields on $\mathbb{T}^2$ and prove that they exhibit a rational first integral. We study `pseudo-type-$n$' vector fields on $\mathbb{T}^2$ and show that any such vector field is completely integrable. We prove that the Lie bracket of any two quadratic vector fields on $\mathbb{T}^2$ is completely integrable. We explicitly find all cubic vector fields on $\mathbb{T}^2$ which achieve the sharp bounds for the number of invariant meridians and parallels. We present necessary and sufficient conditions when invariant meridians and parallels of cubic vector fields on $\mathbb{T}^2$ are periodic orbits or limit cycles. We discuss invariant meridians and parallels of pseudo-type-$n$ vector fields as well. Moreover, we characterize the singular points of a class of polynomial vector fields on $\mathbb{T}^2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_07843 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterization and dynamics of certain classes of polynomial vector fields on the torus Jana, Supriyo Dynamical Systems Classical Analysis and ODEs 34A34, 34C14, 34C40, 34C45 In this paper, we classify all polynomial vector fields in $\mathbb{R}^3$ of degree up to three such that their flow makes the torus $$\mathbb{T}^2=\{(x,y,z)\in \mathbb{R}^3:(x^2+y^2-a^2)^2+z^2-1=0\}~\mbox{with}~a\in (1,\infty)$$ invariant. We also classify cubic Kolmogorov vector fields on $\mathbb{T}^2$ and prove that they exhibit a rational first integral. We study `pseudo-type-$n$' vector fields on $\mathbb{T}^2$ and show that any such vector field is completely integrable. We prove that the Lie bracket of any two quadratic vector fields on $\mathbb{T}^2$ is completely integrable. We explicitly find all cubic vector fields on $\mathbb{T}^2$ which achieve the sharp bounds for the number of invariant meridians and parallels. We present necessary and sufficient conditions when invariant meridians and parallels of cubic vector fields on $\mathbb{T}^2$ are periodic orbits or limit cycles. We discuss invariant meridians and parallels of pseudo-type-$n$ vector fields as well. Moreover, we characterize the singular points of a class of polynomial vector fields on $\mathbb{T}^2$. |
| title | Characterization and dynamics of certain classes of polynomial vector fields on the torus |
| topic | Dynamical Systems Classical Analysis and ODEs 34A34, 34C14, 34C40, 34C45 |
| url | https://arxiv.org/abs/2401.07843 |