Invariant circles and phase portraits of cubic vector fields on the sphere
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929261848297472 |
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| author | Benny, Joji Jana, Supriyo Sarkar, Soumen |
| author_facet | Benny, Joji Jana, Supriyo Sarkar, Soumen |
| contents | In this paper, we characterize and study dynamical properties of cubic vector fields on the sphere $\mathbb{S}^2 = \{(x, y, z) \in \mathbb{R}^3 ~|~ x^2+y^2+z^2 = 1\}$. We start by classifying all degree three polynomial vector fields on $\mathbb{S}^2$ and determine which of them form Kolmogorov systems. Then, we show that there exist completely integrable cubic vector fields on $\mathbb{S}^2$ and also study the maximum number of various types of invariant circles for homogeneous cubic vector fields on $\mathbb{S}^2$. We find a tight bound in each case. Further, we also discuss phase portraits of certain cubic Kolmogorov vector fields on $\mathbb{S}^2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_14238 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Invariant circles and phase portraits of cubic vector fields on the sphere Benny, Joji Jana, Supriyo Sarkar, Soumen Dynamical Systems Classical Analysis and ODEs 34A34, 34C14, 34C40, 34C45, 58J90 In this paper, we characterize and study dynamical properties of cubic vector fields on the sphere $\mathbb{S}^2 = \{(x, y, z) \in \mathbb{R}^3 ~|~ x^2+y^2+z^2 = 1\}$. We start by classifying all degree three polynomial vector fields on $\mathbb{S}^2$ and determine which of them form Kolmogorov systems. Then, we show that there exist completely integrable cubic vector fields on $\mathbb{S}^2$ and also study the maximum number of various types of invariant circles for homogeneous cubic vector fields on $\mathbb{S}^2$. We find a tight bound in each case. Further, we also discuss phase portraits of certain cubic Kolmogorov vector fields on $\mathbb{S}^2$. |
| title | Invariant circles and phase portraits of cubic vector fields on the sphere |
| topic | Dynamical Systems Classical Analysis and ODEs 34A34, 34C14, 34C40, 34C45, 58J90 |
| url | https://arxiv.org/abs/2310.14238 |