Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 32A / v / Solution

The exact second-iterate equation factors asThe positive nonfixed period-two points come from the final factor. If $y=x^2$, their squared values areThey are real and distinct for $\lambda >2$.

The multiplier of a periodic orbit of an iteration isFor $\lambda =2+\mu$ with $0<\mu \ll 1$,whose modulus is less than one. The new two-cycle is therefore locally asymptotically stable.