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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.