Attractors in k-dimensional discrete systems of mixed monotonicity

2024/02/21 by AlSharawi, Ziyad, Cánovas, Jose S., Kallel, Sadok · 1 citation
#37N25 #39A30 #39A60 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2402.14127

Abstract

We consider k-dimensional discrete-time systems of the form x_n+1=F(x_n,…,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 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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