Cubic population iteration ($u_{n+1}=\lambda u_n(1-u_n^2)$)

The fixed points of this cubic iteration of a map satisfy $u=0$ or $u^2=1-1/\lambda$. The nonzero real pair exists for $\lambda <0$ or $\lambda >1$. Its derivative multipliers are $\lambda$ at zero and $3-2\lambda$ at either nonzero point. The fixed point stability for an iteration criterion therefore proves local asymptotic stability of zero for $-1<\lambda <1$ and of the nonzero pair for $1<\lambda <2$. A multiplier of modulus one requires a nonlinear argument rather than this strict derivative test.