On Shilnikov's scenario in 3D: Topological chaos for vectorfields of class $C^1$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912176347807744 |
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| author | Walther, Hans-Otto |
| author_facet | Walther, Hans-Otto |
| contents | Shilnikov's scenario in 3D consists of a vectorfield $V$ so that the equation $$ x'(t)=V(x(t))\in\mathbb{R}^3 $$ with $V(0)=0$ has a solution homoclinic to the origin and the eigenvalues of $DV(0)$ are $u>0$ and $σ\pm iμ$, $σ<0<μ$, with $0<σ+u$. We give a detailed proof that close to the homoclinic loop complicated motion exists provided $V$ is just once continuously differentiable. The result requires working with flows instead of an ODE, which necessitates major modifications compared to the earlier approach for twice continuously differentiable vectorfields in arXiv:2406.18289 . |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_01878 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Shilnikov's scenario in 3D: Topological chaos for vectorfields of class $C^1$ Walther, Hans-Otto Dynamical Systems 37D45, 34C28 Shilnikov's scenario in 3D consists of a vectorfield $V$ so that the equation $$ x'(t)=V(x(t))\in\mathbb{R}^3 $$ with $V(0)=0$ has a solution homoclinic to the origin and the eigenvalues of $DV(0)$ are $u>0$ and $σ\pm iμ$, $σ<0<μ$, with $0<σ+u$. We give a detailed proof that close to the homoclinic loop complicated motion exists provided $V$ is just once continuously differentiable. The result requires working with flows instead of an ODE, which necessitates major modifications compared to the earlier approach for twice continuously differentiable vectorfields in arXiv:2406.18289 . |
| title | On Shilnikov's scenario in 3D: Topological chaos for vectorfields of class $C^1$ |
| topic | Dynamical Systems 37D45, 34C28 |
| url | https://arxiv.org/abs/2501.01878 |