| Abstract: |
| In this talk, we consider the scalar difference equation $x_{n+1}=F_0(x_n,\ldots,x_{n-k+1})$ in which $F_0$ is sufficiently differentiable and has a fixed point $\bar x.$ When $F_0(x_{n-1},\ldots,x_{n-k})$ replaces $x_n,$ we obtain a new system with higher delay, namely
$$y_{n+1}=F_1(y_{n-1},\ldots,y_{n-k})=F_0(F_0(y_{n-1},\ldots,y_{n-k}),y_{n-1},\ldots,y_{n-k+1}).$$
The authors define the expansion strategy as successively repeating the above process, i.e., repeating the process $j$-times gives the system
$$u_{n+1}=F_j(u_{n-j},u_{n-j-1},\ldots,u_{n-j-k+1}).$$
The fixed point $\bar x$ of $F_0$ is a fixed point of $F_j$ for all $j.$ In this talk, we discuss the relationship between the local stability of $F_0$ and $F_j$ at $\bar x.$ In particular, $\bar x$ is locally asymptotically stable (LAS) under $F_0,$ if and only if $\|\nabla F_j\|_1 |
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