Solution
= Solution
The positive fixed point satisfies $x_*=x_*e^{r(1-x_*)}$, hence $x_*=1$. Put $x_n=1+u_n$ and retain first-order terms:
$$
1+u_{n+1}=(1+u_n)e^{-ru_{n-1}}
=1+u_n-ru_{n-1}+O(u^2).
$$
Thus
$$
\boxed{u_{n+1}=u_n-ru_{n-1}},
$$
with characteristic polynomial $\lambda^2-\lambda+r$.