Nonlinear stability at the Ricker flip threshold

At the upper multiplier-$-1$ threshold of $x\mapsto sx+kx{e}^{-\gamma x}$, set $u=\gamma (x-x_{\ast })$. The second iterate is $u-2(s^2-s/2+1/6)u^3+O(u^4)$. Since its cubic damping coefficient is positive for $0\le s<1$, this endpoint is locally asymptotically stable despite the inconclusive linear multiplier.