Nonlinear stability at the Ricker flip threshold
= Nonlinear stability at the Ricker flip threshold
At the upper multiplier-$-1$ threshold of $x\mapsto sx+kxe^{-\gamma x}$, set $u=\gamma(x-x_*)$. 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 $0\leq s<1$, this endpoint is locally <asymptotically stable> despite the inconclusive linear multiplier.