Logistic Fixed Residual Zero Scan - Elasticity and Nonlinear Dynamics Step by Step

A logistic fixed-residual-zero scan checks candidate fixed points across rows by plugging each into the map and reading off x - f(x), accepting only those that leave zero residual. The verdict is exact only for the stated candidates, parameters, and rows in the ideal logistic-map model. Real systems add continuous-time dynamics, noise, parameter drift, finite precision, model mismatch, sampling choices, and measurement uncertainty.

Fixed-point status is scanned by exact residual rows. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Zero is a true logistic fixed point

The first checked row has output zero and residual zero.

2\left(0\right)\left(1\mathbin{-}0\right)=0

One half is the second exact fixed point

The middle row returns one half, so its residual is also zero.

2\cdot\frac{1}{2}\left(1\mathbin{-}\frac{1}{2}\right)=\frac{1}{2}

A nearby candidate exposes the residual

At three quarters the output is three eighths. The residual table keeps the fixed claim numeric.

\begin{array}{c|c|c|c}x&f(x)&|f(x)-x|&\text{fixed}\\0&0&0&1\\\frac{1}{2}&\frac{1}{2}&0&1\\\frac{3}{4}&\frac{3}{8}&\frac{3}{8}&0\\\end{array}