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arXiv · 2402.14127

Attractors in $k$-dimensional discrete systems of mixed monotonicity

Ziyad AlSharawi·Jose S. Cánovas·Sadok Kallel
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Abstract

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.

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Ziyad AlSharawi, Jose S. Cánovas, Sadok Kallel. 2024-02-21. Attractors in $k$-dimensional discrete systems of mixed monotonicity. https://arxiv.org/abs/2402.14127

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