Numerical Methods: Euler to Runge–Kutta · Superchallenge problem

Euler's method on the logistic equation $y' = y(1 - y)$ gives the map $y_{n+1} = y_n + h\,y_n(1 - y_n)$. For $0 < h < 2$ iterates starting near $1$ converge to the equilibrium $y = 1$. When $h$ passes $2$, they instead settle onto a stable 2-cycle, a spurious oscillation that the differential equation does not have. Find the largest $h$ for which this 2-cycle is still stable.