Quadratic Map Stops Before Denominators Grow - Elasticity and Nonlinear Dynamics Step by Step

A quadratic map bends the recurrence, x -> x^2 + c, so values can climb fast and fractions get ugly; the ledger halts at a chosen step before that happens. It is a toy model of nonlinear feedback, not a real trajectory of a physical system. The exact numbers apply only to the stated c, seed, and stop rule.

A nonlinear update stays exact by keeping the ledger short. 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 nonlinear update can still be exact

With seed one quarter, the training map gives three eighths.

x_{1}=2\cdot\frac{1}{4}\left(1-\frac{1}{4}\right)=\frac{3}{8}

The example stops before fractions become unreadable

The next state is fifteen thirty-seconds; this lesson does not claim a long-run orbit.

x_{2}=2\cdot\frac{3}{8}\left(1-\frac{3}{8}\right)=\frac{15}{32}