Past exam of the topics in applied mathematics course of the University of Oxford / 2007 / A / 1 / v / Solution

A fixed point of $F$ is also a fixed point of $F\circ F$. By the chain rule,If the original fixed-point multiplier has modulus greater than one, its square also exceeds one, so the same fixed point is unstable for the twice-iterated map. New stable fixed points of the twice-iterated map, if present, must be different points; they can represent a genuine two-cycle of the original map.