Affine Map Keeps a Finite Ledger - Elasticity and Nonlinear Dynamics Step by Step

An affine map updates the state by scaling and then shifting it, x -> a*x + b, so each step has the same shape and the ledger stays small and easy to audit. It is a discrete state-update rule, not a continuous physical law of motion. The exact figures below hold only for the stated a, b, and starting value.

A finite map lesson begins with a seed, a rule, and checked states. 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 map is a rule plus a seed

The seed is 0 and the checked rule sends it to one quarter.

x_{1}=\frac{x_{0}}{2}+\frac{1}{4}=\frac{1}{4}

The ledger stops at the pinned depth

The four displayed states are the seed, one quarter, three eighths, and seven sixteenths.

0\to\frac{1}{4}\to\frac{3}{8}\to\frac{7}{16}