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

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}