Tangential homoclinic points for Lozi maps
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917396749484032 |
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| author | Kvaternik, Kristijan Kilassa |
| author_facet | Kvaternik, Kristijan Kilassa |
| contents | For the family of Lozi maps, we study homoclinic points for the saddle fixed point $X$ in the first quadrant. Specifically, in the parameter space, we examine the boundary of the region in which homoclinic points for $X$ exist. For all parameters on that boundary, all intersections of the stable and unstable manifold of $X$, apart from $X$, are tangential, or these manifolds intersect along a segment. We ultimately prove that for such parameters, all possible homoclinic points for $X$ are iterates of two special points $Z$ and $V$, or iterates of points on a segment joining $V$ with an iterate of $Z$. Additionally, we describe the parameter curves that form the boundary and provide explicit equations for several of them. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_12536 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tangential homoclinic points for Lozi maps Kvaternik, Kristijan Kilassa Dynamical Systems 37E30 (Primary) 37D05 (Secondary) For the family of Lozi maps, we study homoclinic points for the saddle fixed point $X$ in the first quadrant. Specifically, in the parameter space, we examine the boundary of the region in which homoclinic points for $X$ exist. For all parameters on that boundary, all intersections of the stable and unstable manifold of $X$, apart from $X$, are tangential, or these manifolds intersect along a segment. We ultimately prove that for such parameters, all possible homoclinic points for $X$ are iterates of two special points $Z$ and $V$, or iterates of points on a segment joining $V$ with an iterate of $Z$. Additionally, we describe the parameter curves that form the boundary and provide explicit equations for several of them. |
| title | Tangential homoclinic points for Lozi maps |
| topic | Dynamical Systems 37E30 (Primary) 37D05 (Secondary) |
| url | https://arxiv.org/abs/2412.12536 |