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

A saturating-clamp-boundary scan varies the unclamped update across rows and marks when the clamp takes over, separating free evolution from the saturated branch. The arithmetic is exact only for the stated parameters, clamp limits, iterates, and rows in the ideal discrete-map model. Real systems add continuous-time dynamics, noise, parameter drift, finite precision, model mismatch, sampling choices, and measurement uncertainty.

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}