 Euler discretization of logistic growth

The Forward Euler method applied to the logistic differential equation $y^{\prime }=ry(1-ay)$ becomes the logistic map after $u=a(\lambda -1)y/\lambda$, $\lambda =1+r\Delta t$. The positive equilibrium point of the differential equation has perturbations decaying as $e^{-rt}$. Its discrete fixed-point multiplier is $1-r\Delta t=2-\lambda$: for $r\Delta t>2$, the discretization has an artificial oscillatory instability, leading initially to alternation on successive steps. This contrasts with the positive perturbation multiplier $e^{-r\Delta t}$ of the exact time-step map.