Attractors in $k$-dimensional discrete systems of mixed monotonicity
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913238698950656 |
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| author | AlSharawi, Ziyad Cánovas, Jose S. Kallel, Sadok |
| author_facet | AlSharawi, Ziyad Cánovas, Jose S. Kallel, Sadok |
| contents | We consider $k$-dimensional discrete-time systems of the form $x_{n+1}=F(x_n,\ldots,x_{n-k+1})$ in which the map $F$ is continuous and monotonic in each one of its arguments. We define a partial order on $\mathbb{R}^{2k}_+$, compatible with the monotonicity of $F$, and then use it to embed the $k$-dimensional system into a $2k$-dimensional system that is monotonic with respect to this poset structure. An analogous construction is given for periodic systems. Using the characteristics of the higher-dimensional monotonic system, global stability results are obtained for the original system. Our results apply to a large class of difference equations that are pertinent in a variety of contexts. As an application of the developed theory, we provide two examples that cover a wide class of difference equations, and in a subsequent paper, we provide additional applications of general interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14127 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Attractors in $k$-dimensional discrete systems of mixed monotonicity AlSharawi, Ziyad Cánovas, Jose S. Kallel, Sadok Dynamical Systems 39A30, 39A60, 37N25 We consider $k$-dimensional discrete-time systems of the form $x_{n+1}=F(x_n,\ldots,x_{n-k+1})$ in which the map $F$ is continuous and monotonic in each one of its arguments. We define a partial order on $\mathbb{R}^{2k}_+$, compatible with the monotonicity of $F$, and then use it to embed the $k$-dimensional system into a $2k$-dimensional system that is monotonic with respect to this poset structure. An analogous construction is given for periodic systems. Using the characteristics of the higher-dimensional monotonic system, global stability results are obtained for the original system. Our results apply to a large class of difference equations that are pertinent in a variety of contexts. As an application of the developed theory, we provide two examples that cover a wide class of difference equations, and in a subsequent paper, we provide additional applications of general interest. |
| title | Attractors in $k$-dimensional discrete systems of mixed monotonicity |
| topic | Dynamical Systems 39A30, 39A60, 37N25 |
| url | https://arxiv.org/abs/2402.14127 |