Fixed Candidate Substitution Cross-Scan - Elasticity and Nonlinear Dynamics Step by Step

A substitution cross-scan tests several fixed-point candidates by plugging each into the map and reading off the residual x - f(x), so a true fixed point leaves zero. The arithmetic is exact only for the stated candidates, map rule, and rows in the ideal discrete-map model. Real systems add continuous dynamics, noise, unmodeled feedback, finite sampling, parameter uncertainty, and model/source limits.

Fixed-point substitution is scanned across candidate values. 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 fixed point is found by substitution

The checked source row is the middle row: one half returns one half.

f(x)=\frac{x}{2}+\frac{1}{4}

Neighbor candidates fail the same substitution

The table keeps the map fixed and changes only the candidate. The fixed bit turns on only when output and candidate match.

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