Fixed-Point Substitution Closes the Loop - Nonlinear Dynamics and Stability Step by Step

A fixed point is a checked finite substitution. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Fixed point means the update returns itself

The candidate one half is checked by substitution.

f\left(\frac{1}{2}\right)=\frac{1}{2}\cdot\frac{1}{2}+\frac{1}{4}=\frac{1}{2}

The fixed bit is computed, not chosen

Because the output is again one half, the fixed bit is 1.

f(x^*)=x^*\Rightarrow 1