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Extension, embedding and global stability in two dimensional monotone maps. (English) Zbl 07272958
Summary: 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.
MSC:
39A22 Growth, boundedness, comparison of solutions to difference equations
39A30 Stability theory for difference equations
39A10 Additive difference equations
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
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