= Cubic population iteration
{title2=$u_{n+1}=\lambda u_n(1-u_n^2)$}
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.
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