Homoclinic tangencies in $\mathbb{R}^n$
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916363809849344 |
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| author | Rayskin, Victoria |
| author_facet | Rayskin, Victoria |
| contents | Let $f: M \to M$ denote a diffeomorphism of a smooth manifold $M$. Let $p$ in $M$ be its hyperbolic fixed point with stable and unstable manifolds $W_S$ and $W_U$, respectively. Assume that $W_S$ is a curve. Suppose that $W_U$ and $W_S$ have a degenerate homoclinic crossing at a point $B\ne p$, i.e., they cross at $B$ tangentially with a finite order of contact.
It is shown that, subject to $C^1$-linearizability and certain conditions on the invariant manifolds, a transverse homoclinic crossing will arise arbitrarily close to $B$. This proves the existence of a horseshoe structure arbitrarily close to $B$, and extends a similar planar result of Homburg and Weiss. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_11314 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homoclinic tangencies in $\mathbb{R}^n$ Rayskin, Victoria Dynamical Systems 37B10, 37C05, 37C15, 37D10 Let $f: M \to M$ denote a diffeomorphism of a smooth manifold $M$. Let $p$ in $M$ be its hyperbolic fixed point with stable and unstable manifolds $W_S$ and $W_U$, respectively. Assume that $W_S$ is a curve. Suppose that $W_U$ and $W_S$ have a degenerate homoclinic crossing at a point $B\ne p$, i.e., they cross at $B$ tangentially with a finite order of contact. It is shown that, subject to $C^1$-linearizability and certain conditions on the invariant manifolds, a transverse homoclinic crossing will arise arbitrarily close to $B$. This proves the existence of a horseshoe structure arbitrarily close to $B$, and extends a similar planar result of Homburg and Weiss. |
| title | Homoclinic tangencies in $\mathbb{R}^n$ |
| topic | Dynamical Systems 37B10, 37C05, 37C15, 37D10 |
| url | https://arxiv.org/abs/2408.11314 |