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| We say that a map $h:I\to I$ belongs to the class $S$ if , roughly speaking, it is monotone or unimodal, has a unique fixed point $u$ and the Schwarzian derivative of $h$ is negative. These maps are important in dynamics because i if $u$ is a local attractor for (1): $x_{n+1}=h(x_n)$ (which is equivalent to $|h'(u)|\leq 1$), then it is a global attractor of (1) as well.
In this work we address the question of whether the local stability of $u$ for the higher order difference equation (2): $x_{n+1}=\alpha x_{n-k} + (1-\alpha) h(x_n)$ may imply global attraction whenever $h$ belongs to the class $S$. This equation has been widely used in the literature in population dynamics and control of chaos models.
Our main results:
a) If $k$ is odd, then $u$ is locally attracting for (1) if and only if it is locally attracting for (2); moreover, in this case $u$ is globally attracting for (2).
b) If $k$ is even, local attraction need not imply global attraction for (2); a counterexample for $k=2$ is the Ricker function $h(x)= p x ^{q x}$. Yet we give some arguments supporting the validity of the statement when $k$ is large enough. |
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