Extension, embedding and global stability in two dimensional monotone maps

TL;DR

This paper develops a method to analyze the global stability of two-dimensional monotone maps with mixed monotonicity, using extension and embedding techniques to understand the asymptotic behavior of second order difference equations.

Contribution

It introduces an extension and embedding approach for second order difference equations with mixed monotonicity, enabling the study of global stability within invariant domains.

Findings

Extension preserves continuity and monotonicity of the original map.

Embedding into higher dimensions helps characterize asymptotic dynamics.

Illustrative examples demonstrate the method's application.

Abstract

We consider the general second order difference equation $x_{n+1}=F(x_n,x_{n-1})$ in which $F$ is continuous and of mixed monotonicity in its arguments. In equations with negative terms, a persistent set can be a proper subset of the positive orthant, which motivates studying global stability with respect to compact invariant domains. In this paper, we assume that $F$ has a semi-convex compact invariant domain, then make an extension of $F$ on a rectangular domain that contains the invariant domain. The extension preserves the continuity and monotonicity of $F.$ Then we use the embedding technique to embed the dynamical system generated by the extended map into a higher dimensional dynamical system, which we use to characterize the asymptotic dynamics of the original system. Some illustrative examples are given at the end.