Invariant graphs and dynamics of a family of continuous piecewise linear planar maps
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909490265194496 |
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| author | Cima, Anna Gasull, Armengol Mañosa, Víctor Mañosas, Francesc |
| author_facet | Cima, Anna Gasull, Armengol Mañosa, Víctor Mañosas, Francesc |
| contents | We consider the family of piecewise linear maps $$F_{a,b}(x,y)=\left(|x| - y + a, x - |y| + b\right),$$ where $(a,b)\in \mathbb{R}^2$. This family belongs to a wider one that has deserved some interest in the recent years as it provides a framework for generalized Lozi-type maps. Among our results, we prove that for $a\ge 0$ all the orbits are eventually periodic and moreover that there are at most three different periodic behaviors formed by at most seven points. For $a<0$ we prove that for each $b\in\mathbb{R}$ there exists a compact graph $Γ,$ which is invariant under the map $F$, such that for each $(x,y)\in \mathbb{R}^2$ there exists $n\in\mathbb{N}$ (that may depend on $x$) such that $F_{a,b}^n(x,y)\in Γ.$ We give explicitly all these invariant graphs and we characterize the dynamics of the map restricted to the corresponding graph for all $(a,b)\in\mathbb{R}^2$ obtaining, among other results, a full characterization of when $F_{a,b}|_Γ$ has positive or zero entropy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_01052 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Invariant graphs and dynamics of a family of continuous piecewise linear planar maps Cima, Anna Gasull, Armengol Mañosa, Víctor Mañosas, Francesc Dynamical Systems 37C05, 37E25, 37B40, 39A23 We consider the family of piecewise linear maps $$F_{a,b}(x,y)=\left(|x| - y + a, x - |y| + b\right),$$ where $(a,b)\in \mathbb{R}^2$. This family belongs to a wider one that has deserved some interest in the recent years as it provides a framework for generalized Lozi-type maps. Among our results, we prove that for $a\ge 0$ all the orbits are eventually periodic and moreover that there are at most three different periodic behaviors formed by at most seven points. For $a<0$ we prove that for each $b\in\mathbb{R}$ there exists a compact graph $Γ,$ which is invariant under the map $F$, such that for each $(x,y)\in \mathbb{R}^2$ there exists $n\in\mathbb{N}$ (that may depend on $x$) such that $F_{a,b}^n(x,y)\in Γ.$ We give explicitly all these invariant graphs and we characterize the dynamics of the map restricted to the corresponding graph for all $(a,b)\in\mathbb{R}^2$ obtaining, among other results, a full characterization of when $F_{a,b}|_Γ$ has positive or zero entropy. |
| title | Invariant graphs and dynamics of a family of continuous piecewise linear planar maps |
| topic | Dynamical Systems 37C05, 37E25, 37B40, 39A23 |
| url | https://arxiv.org/abs/2410.01052 |