| Abstract: |
| In this talk, we consider the scalar difference equation $x_{n+1}=f(x_n,x_{n-1})$ in which $f$ is sufficiently differentiable and has a fixed point $x_0.$ We say that $x_0$ is strongly locally stable (SLAS for short) whenever the sum of the absolute values of $f_x(x_0, x_0)$ and $f_y(x_0, x_0)$ is smaller than $1$. We show the advantages of the SLAS on developing the notion of dominance condition as introduced in [1] and the similar enveloping notion for one-dimensional difference equations given in [2]. We give some geometric results that make finding an enveloping one-dimensional map more practical, which makes proving global stability a manageable task.
When the map $f$ is of mixed monotonicity, we establish the connection between the enveloping and the embedding techniques. In particular, we prove that embedding is enough to give the existence of an enveloping function and provide ideas for finding enveloping functions. Some interesting questions will be posed.
References.
[1] H. A. El-Morshedy and V. Jimenez Lopez. Global attractors for difference equations dominated by one-dimensional maps. J. Difference Equ. Appl., 14(4):391-410, 2008
[2] Paul Cull. Enveloping implies global stability. In Difference equations and discrete dynamical systems, pages 71-85. World Sci. Publ., Hackensack, NJ, 2005.
[3] Ziyad AlSharawi, Jose S. Canovas. Integrating the expansion strategy with the enveloping technique to establish stability 2024. |
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