Stability Margin Boundary Scan - Nonlinear Dynamics and Stability Step by Step

The strict stability gate is scanned through positive, zero, and negative margins. 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 zero slope sits inside the stability gate

The logistic fixed point at one half has slope zero, leaving a full margin below one.

m=0,\quad 1-|m|=1

Magnitude one is the rejected edge

The reflection row returns to itself but its slope magnitude is exactly one, so the strict gate rejects it.

m=\mathbin{-}1,\quad 1-|m|=0

A larger slope gives a negative margin

The repeller row shows the outside side of the same gate.

\begin{array}{c|c|c|c}x^*&m&\text{margin}&\text{accepted}\\\frac{1}{2}&0&1&1\\\frac{1}{2}&\mathbin{-}1&0&0\\\frac{1}{2}&\frac{3}{2}&\mathbin{-}\frac{1}{2}&0\\\end{array}