Slope Less Than One Is a Local Contract - Elasticity and Nonlinear Dynamics Step by Step

A fixed point is locally stable when the map's slope at that point has magnitude below one, |f'(x*)| < 1, so nearby points are pulled in rather than pushed away. This is a linear test: it uses the local tangent of the map, not the whole curve. The clean verdict holds for the stated map and its linearization; real data add finite sampling, parameter uncertainty, and noise that blur the slope.

A local stability label must cite the fixed-point source and exact slope. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Stability cites the fixed-point source

The fixed point is reused, then the local slope one half is checked at that same map.

|f'(x^*)|=\left|\frac{1}{2}\right|=\frac{1}{2}

Less than one is the accepted local contract

Since one half is less than 1, the accepted bit is 1.

\frac{1}{2}<1\Rightarrow 1