A trichotomy for generic sectional-hyperbolic chain-recurrent classes
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866911485580541952 |
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| author | Rego, Elias Vivas, Kendry |
| author_facet | Rego, Elias Vivas, Kendry |
| contents | The notion of sectional-hyperbolicity is a weakened form of hyperbolicity introduced for vector fields in order to understand the dynamical behavior of certain higher-dimensional systems such as the multidimensional Lorenz attractor. In this paper we address the questions proposed in [\emph{Math. Z.}, \textbf{298} (2021), 469-488] and we provide a partial answer by proving that a $C^1$-generic non-trivial sectional-hyperbolic chain-recurrent class, not necessarily Lyapunov stable, satisfies a trichotomy: it is either a homoclinic loop, a union of saddle connections between singularities, or it is robustly a homoclinic class. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01318 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A trichotomy for generic sectional-hyperbolic chain-recurrent classes Rego, Elias Vivas, Kendry Dynamical Systems Primary: 37C70, 37C10, 37C20, 37D45 The notion of sectional-hyperbolicity is a weakened form of hyperbolicity introduced for vector fields in order to understand the dynamical behavior of certain higher-dimensional systems such as the multidimensional Lorenz attractor. In this paper we address the questions proposed in [\emph{Math. Z.}, \textbf{298} (2021), 469-488] and we provide a partial answer by proving that a $C^1$-generic non-trivial sectional-hyperbolic chain-recurrent class, not necessarily Lyapunov stable, satisfies a trichotomy: it is either a homoclinic loop, a union of saddle connections between singularities, or it is robustly a homoclinic class. |
| title | A trichotomy for generic sectional-hyperbolic chain-recurrent classes |
| topic | Dynamical Systems Primary: 37C70, 37C10, 37C20, 37D45 |
| url | https://arxiv.org/abs/2601.01318 |