Ricker population map two-cycle ($u_{\pm }=L\pm d,\tanh (d/2)=d/L$)

For $H(u)=ru{e}^{-u}$ and $L=\log r$, a nonzero second-iterate fixed point satisfies $u+H(u)=2L$. A genuine pair has the displayed form, and its periodic-orbit multiplier is $(1-L{)}^2-d^2$. At $r=9$, the pair is exactly $\log 3,3\log 3$, with periodic-orbit multiplier $(1-\log 3)(1-3\log 3)\simeq 0.2264$. For large $r$ its upper point is asymptotic to $2\log r$ and the pair periodic-orbit multiplier to $1-2\log r$, so eventual instability follows despite the small derivative at the upper point.