Trajectories of vector fields asymptotic to formal invariant curves
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914452262092800 |
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| author | Gal, Olivier Le Sánchez, Fernando Sanz |
| author_facet | Gal, Olivier Le Sánchez, Fernando Sanz |
| contents | We prove that a formal curve $Γ$ that is invariant by a $C^\infty$ vector field $ξ$ of $\mathbb{R}^m$ has a geometrical realization, as soon as the Taylor expansion of $ξ$ is not identically zero along $Γ$. This means that there is a trajectory $γ$ of $ξ$ which is asymptotic to $Γ$. This result solves a natural question proposed by Bonckaert nearly forty years ago. We also construct an invariant $C^0$ manifold $S$ in some open horn around $Γ$ which is composed entirely of trajectories asymptotic to $Γ$, and contains the germ of any such trajectory. If $ξ$ is analytic, we prove that there exists a trajectory asymptotic to $Γ$ which is, moreover, non-oscillating with respect to subanalytic sets. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_06821 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Trajectories of vector fields asymptotic to formal invariant curves Gal, Olivier Le Sánchez, Fernando Sanz Dynamical Systems Classical Analysis and ODEs 34E05, 34C20, 34C08, 34A25, 37D10, 37C25 We prove that a formal curve $Γ$ that is invariant by a $C^\infty$ vector field $ξ$ of $\mathbb{R}^m$ has a geometrical realization, as soon as the Taylor expansion of $ξ$ is not identically zero along $Γ$. This means that there is a trajectory $γ$ of $ξ$ which is asymptotic to $Γ$. This result solves a natural question proposed by Bonckaert nearly forty years ago. We also construct an invariant $C^0$ manifold $S$ in some open horn around $Γ$ which is composed entirely of trajectories asymptotic to $Γ$, and contains the germ of any such trajectory. If $ξ$ is analytic, we prove that there exists a trajectory asymptotic to $Γ$ which is, moreover, non-oscillating with respect to subanalytic sets. |
| title | Trajectories of vector fields asymptotic to formal invariant curves |
| topic | Dynamical Systems Classical Analysis and ODEs 34E05, 34C20, 34C08, 34A25, 37D10, 37C25 |
| url | https://arxiv.org/abs/2311.06821 |