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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.02226 |
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| _version_ | 1866909216603635712 |
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| author | Zhuang, Ziwei Liu, Changjian |
| author_facet | Zhuang, Ziwei Liu, Changjian |
| contents | Consider a family of planar polynomial systems $\dot x = y^{2l-1} - x^{2k+1}, \dot y =-x +m y^{2s+1},$ where $l,k,s\in\mathbb{N^*},$ $2\le l \le 2s$ and $m\in\mathbb{R}.$ We study the center-focus problem on its origin which is a monodromic nilpotent critical point. By directly calculating the generalized Lyapunov constants, we find that the origin is always a focus and we complete the classification of its stability. This includes the most difficult case: $s=kl$ and $m=(2k+1)!!/(2kl+1)!_{(2l)}.$ In this case, we prove that the origin is always unstable. Our result extends and completes a previous one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_02226 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stability for a family of planar systems with nilpotent critical points Zhuang, Ziwei Liu, Changjian Dynamical Systems Consider a family of planar polynomial systems $\dot x = y^{2l-1} - x^{2k+1}, \dot y =-x +m y^{2s+1},$ where $l,k,s\in\mathbb{N^*},$ $2\le l \le 2s$ and $m\in\mathbb{R}.$ We study the center-focus problem on its origin which is a monodromic nilpotent critical point. By directly calculating the generalized Lyapunov constants, we find that the origin is always a focus and we complete the classification of its stability. This includes the most difficult case: $s=kl$ and $m=(2k+1)!!/(2kl+1)!_{(2l)}.$ In this case, we prove that the origin is always unstable. Our result extends and completes a previous one. |
| title | Stability for a family of planar systems with nilpotent critical points |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2406.02226 |