Iterated Maps and the Logistic Map

Not every system runs on continuous time. A population censused once a year, a bank balance compounded each month, an algorithm stepped one iteration at a time — these march in discrete ticks. The rule is not a differential equation but a map: the next state is a function of the current one,

x_{n+1} = f(x_n).

Starting from x_0 you generate the orbit x_0, x_1, x_2, \dots by feeding each output back in as the next input — iteration. It looks childishly simple next to continuous flows, yet this is where the deepest surprise in the whole subject lives: a single, smooth, one-line map can behave with the full ferocity of chaos. This page sets up the machinery; the fireworks come next.

Fixed points and the multiplier

A fixed point of a map is a value the map leaves alone: x^\* = f(x^\*). Once you land on it you stay forever. Stability is again a derivative question, but the test is different from the continuous case. Perturb by \eta_n = x_n - x^\* and linearize:

\eta_{n+1} = f(x^\* + \eta_n) - x^\* \approx f'(x^\*)\,\eta_n \;\Rightarrow\; \eta_n \approx \big(f'(x^\*)\big)^n \eta_0.

The disturbance is multiplied by f'(x^\*) every step, so it shrinks exactly when that multiplier is smaller than one in magnitude.

Let m = f'(x^\*) be the multiplier at a fixed point.

(Contrast the continuous rule, which compared f' to zero. For maps the dividing line is |f'| = 1, and the sign of m matters too: m < 0 makes the orbit alternate sides as it converges.)

The cobweb: how to iterate with a picture

There is a gorgeous graphical way to run a map. Draw the curve y = f(x) and the diagonal y = x together. From x_n go vertically to the curve — that height is x_{n+1} = f(x_n) — then horizontally to the diagonal to copy that value back onto the x-axis, ready to be the next input. Repeat, and the path draws a cobweb. Fixed points are where the curve meets the diagonal; a stable one sucks the web inward, an unstable one flings it out.

Explore it on the logistic map f(x) = r\,x(1-x), the fruit-fly of chaos theory. Slide the growth rate r and the start x_0, and watch the web settle, alternate, or wander:

For 1 < r < 3 the web spirals neatly into the nonzero fixed point — the population settles to a steady level. Push r past 3 and the web refuses to settle onto a point, closing instead into a rectangle that bounces between two values: a 2-cycle. That first split is the opening move of a story we finish next lesson.

Worked example: the logistic fixed points

Find and classify the fixed points of f(x) = r\,x(1-x).

Step 1 — solve x = r x(1-x). Either x = 0, or dividing by x gives 1 = r(1-x), so

x^\* = 0 \qquad\text{and}\qquad x^\* = 1 - \frac{1}{r}.

The second exists in (0,1) only for r > 1.

Step 2 — the multiplier. f'(x) = r(1 - 2x). At the origin f'(0) = r; at the interior point,

f'\!\left(1 - \tfrac1r\right) = r\left(1 - 2\left(1 - \tfrac1r\right)\right) = 2 - r.

Step 3 — classify. The origin is stable for r < 1 (|r| < 1) and loses stability at r = 1 — a transcritical bifurcation where the interior point takes over. The interior point is stable while |2 - r| < 1, i.e. for

1 < r < 3.

At r = 3 the multiplier hits 2 - 3 = -1 — the borderline |m| = 1, with the tell-tale negative sign that makes the orbit alternate. That is exactly the value where the cobweb above stops settling and the 2-cycle is born. The whole cascade to chaos begins at this one clean algebraic threshold.

The logistic map was popularised by the biologist Robert May in a famous 1976 Nature paper, "Simple mathematical models with very complicated dynamics." His point was a warning to his own field: you do not need a complicated model, random noise, or many species to get wildly irregular population counts. A single deterministic parabola, iterated, already produces booms, crashes, and apparent randomness. Ecologists staring at erratic field data had been reaching for complicated explanations; May showed the simplest possible model was fully capable of the mayhem on its own.

The stability rule for maps is not the rule for flows. For a continuous flow you compared f'(x^\*) to 0; for a map you compare it to \pm 1. A fixed point of a map with f'(x^\*) = -0.9 is stable even though the derivative is negative, because |-0.9| < 1 — the orbit just converges while flip-flopping from side to side. Reading the map's picture, remember too that x_{n+1} is a height on the curve, not a step along the axis: the horizontal move to the diagonal is what turns that height back into the next input. Skip the diagonal and the whole cobweb construction falls apart.