Tent Branch Boundary Scan - Elasticity and Nonlinear Dynamics Step by Step

A tent-branch-boundary scan varies the input across rows and records which side of the tent map each point lands on, making the branch switch visible at the boundary. The arithmetic is exact only for the stated inputs, parameters, branches, and rows in the ideal discrete tent-map model. Real systems add continuous-time dynamics, noise, parameter drift, finite precision, model mismatch, sampling choices, and measurement uncertainty.

The tent map branch boundary is scanned around the equality row. 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 split the tent map uses the low branch

One quarter doubles to one half on the low-or-equal branch.

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

The equality row still belongs to low-or-equal

At one half the output is one, and the branch name records the equality convention.

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

Past the split the high branch takes over

The right row returns to one half, so the table shows continuity without hiding the branch switch.

\begin{array}{c|c|c}x&f(x)&\text{branch}\\\frac{1}{4}&\frac{1}{2}&\text{low-or-equal}\\\frac{1}{2}&1&\text{low-or-equal}\\\frac{3}{4}&\frac{1}{2}&\text{high}\\\end{array}