Fixed Points and Local Stability

Slope Greater Than One Is Rejected

A valid fixed point can have a rejected local stability bit. 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 can still fail the stability contract

The repeller map also has candidate one half as a fixed point.

g(12)=32⋅12−14=12g\left(\frac{1}{2}\right)=\frac{3}{2}\cdot\frac{1}{2}-\frac{1}{4}=\frac{1}{2}
Repeller fixed pointFixed-point validity and stability are separate.slope=3/2slopeMagnitude=3/2acceptedBit=0 bit

Slope greater than one is rejected

The exact local slope is three halves, so accepted bit is 0.

∣32∣>1⇒0\left|\frac{3}{2}\right|>1\Rightarrow 0
Rejected local stabilityA wrong accepted bit cannot render.slope=3/2slopeMagnitude=3/2acceptedBit=0 bit