Saturating Clamp Boundary Scan - Nonlinear Dynamics and Stability Step by Step

A saturating recurrence is scanned below, at, and above the cap. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Below the clamp, doubling is still linear

One quarter doubles to one half with no clipping.

2\cdot\frac{1}{4}=\frac{1}{2}

At the cap, the linear branch reaches one exactly

The equality row is not clipped yet because the raw output is one.

2\cdot\frac{1}{2}=1

Above the cap, different inputs share the capped output

Three rows separate linear growth from the clamped plateau.

\begin{array}{c|c|c}x&f(x)&\text{branch}\\\frac{1}{4}&\frac{1}{2}&\text{linear}\\\frac{1}{2}&1&\text{linear}\\\frac{3}{4}&1&\text{clamped}\\\end{array}