Reflection Parameter Period Cross-Scan - Nonlinear Dynamics and Stability Step by Step

A reflection map family is scanned with finite period-row tests. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A reflection-family row is a finite two-step test

The candidate is fixed at one quarter. Each parameter row checks the first update and the return update.

f_p(x)=p-x

Period labels follow the two checked updates

The first parameter row is period one. The next two rows return after two steps without stopping after one step.

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