Stability Slope Threshold Cross-Scan - Elasticity and Nonlinear Dynamics Step by Step

A stability-slope cross-scan checks each fixed point's local slope |f'(x*)| against the threshold of one, marking it stable when below and unstable when above. The verdict is exact only for the stated slopes, fixed points, and rows in the ideal linearized discrete-map model. Real systems add continuous dynamics, noise, unmodeled feedback, finite sampling, parameter uncertainty, and model/source limits.

Local stability is scanned across exact slope magnitudes. 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 small slope is an accepted local contract

The first checked source has slope one half at the fixed point.

|m|=\left|\frac{1}{2}\right|<1

The threshold uses magnitude, not sign

The neutral reflection row has slope negative one. Its magnitude is one, so it is not accepted by the less-than-one contract.

\begin{array}{c|c|c|c}\text{map}&m&|m|&\text{accepted}\\\text{contract}&\frac{1}{2}&\frac{1}{2}&1\\\text{neutral}&-1&1&0\\\text{repel}&\frac{3}{2}&\frac{3}{2}&0\\\end{array}