Reflection Return Gap Period Scan - Nonlinear Dynamics and Stability Step by Step

Reflection rows separate fixed return from period-two return. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The midpoint returns but is not period two

One half maps to itself. The return bit is not enough without a non-fixed first step.

x=y=r=\frac{1}{2}

One third gives a true two-step return

The first update moves to two thirds, then returns to one third.

x=\frac{1}{3},\quad y=\frac{2}{3},\quad r=x

A larger first-step gap is still period two

Changing the starting point changes the gap but not the exact two-step return contract.

\begin{array}{c|c|c}x&|f(x)-x|&\text{period}\\\frac{1}{2}&0&0\\\frac{1}{3}&\frac{1}{3}&1\\\frac{1}{4}&\frac{1}{2}&1\\\end{array}