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Extension, embedding and global stability in two dimensional monotone maps

  • * Corresponding author: Ziyad AlSharawi

    * Corresponding author: Ziyad AlSharawi 
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  • 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.

    Mathematics Subject Classification: Primary: 39A30, 39A10; Secondary: 37C25.


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  • Figure 2.1.  In part (a) of this figure, we illustrate our notation of projecting a point $(x, y)$ to the boundary $\partial \Omega.$ In part (b), we illustrate the notion of putting the invariant domain $\Omega$ inside an Origami domain

    Figure 2.2.  This figure shows the possible options for the boundary of a convex $\Omega$ and a possible ray extension. Part (a) is based on the assumption that $f$ is non-decreasing and Part (b) is based on the assumption that $f$ is non-increasing. Note that the missing quarter in Part (a) is due to the fact that we cannot have $f(\uparrow)$ and $F(\downarrow, \uparrow)$ at the same time. Similarly for the missing quarter of Part (b)

    Figure 2.3.  Grafting $\gamma_1$ along $\gamma_2$

    Figure 2.4.  The various cases for the boundary pieces of an origami domain

    Figure 2.5.  A semi-convex domain is shown in Part (a). Next is shown a piece $ C_1\cup C_2 $ of the boundary of $ \Omega $ oriented positively, with $ f(t) = F(r(t)) $ changing monotonicity from $ C_1 $ to $ C_2 $. Parts (b) and (c) show two different ray extensions, while (d) shows the adjustment to make to these extensions, before filling the sector. As explained in the text, (b) is not a valid extension if monotonicity is to be preserved in the extension

    Figure 2.6.  Extending from semi-convex domain to the origami domain reduces to extending over sectors, with cases labeled (a) through (d)

    Figure 2.7.  Filling a rectangle where $ F $ is known on three sides. Extending partially by $ \widetilde{F}_1 $ as shown in (b) then extending over the resulting sector

    Figure 4.1.  Part (a) of this figure shows the invariant region $\Omega, $ while part (b) shows $T(\Omega)$ together with the extension through horizontal projections. The scale is missing to indicate the general form of the region when $0 < h < \min\{p, \frac{1}{2}\}.$

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