11.1 Sensitive dependence and the logistic map

The cleanest place to meet chaos is not a differential equation but a one-line rule you can iterate by hand. Take a number between 0 and 1, apply a fixed quadratic map, take the result, apply the same map again, and repeat. For some settings the sequence settles to a single value; for others it ends up alternating between two; and for others still it never settles into any pattern at all, despite the rule never changing and nothing random ever entering. That last case is deterministic chaos, and this lesson is about how a rule a child could follow produces it.

Iterated maps

A map (or difference equation) advances a state in discrete steps:

x_{n+1} \;=\; f(x_n), \qquad n = 0, 1, 2, \ldots

where

$x_n$: the state after n steps (here a number in [0,1])
$f$: the map: a fixed function applied once per step
$x_0$: the initial condition

Given $x_0$ , the orbit is the sequence $x_0, x_1, x_2, \ldots$ obtained by applying $f$ over and over. Maps are the discrete-time cousins of the flows $\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x})$ from the ODE chapter; everything qualitative about long-term behaviour — equilibria, stability, oscillation, chaos — has a version for both. Maps are simpler to study because there is no integration to do: iteration is the dynamics.

The example that carries this chapter is the logistic map,

\boxed{\; x_{n+1} \;=\; r\, x_n (1 - x_n). \;}

where

$r$: the growth parameter, the single knob we turn 0 ≤ r ≤ 4
$x_n$: a normalised population, as a fraction of the maximum the environment can support

It was introduced as a toy model of a population that grows when small but is held back when crowded. The factor $r x_n$ is unchecked reproduction — each generation is $r$ times the last — and the factor $(1 - x_n)$ is the brake: as $x_n$ approaches its ceiling of 1, the term $(1-x_n)$ approaches 0 and growth shuts off. Keeping $0 \le r \le 4$ guarantees that any $x_n$ in $[0,1]$ maps back into $[0,1]$ , so the population stays physical. The model is a caricature — real populations are not this clean — but as a dynamical system it is inexhaustible.

Fixed points and their stability

A fixed point $x^\*$ is a state the map leaves unchanged: $f(x^\*) = x^\*$ , so an orbit that lands on it stays forever. For the logistic map these solve $x = r x (1-x)$ :

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

The first is extinction; the second is a nonzero steady population, which is only physical (positive) once $r > 1$ . But existence is not the whole story. The decisive question for any equilibrium is stability: if the state is nudged slightly off $x^\*$ , does it return, or run away? The answer is set by the slope of the map at the fixed point.

▶ The stability criterion |f′(x*)| < 1 Derivation

Let the orbit sit a small distance $\varepsilon_n$ from the fixed point: $x_n = x^\* + \varepsilon_n$ . Apply the map and expand $f$ to first order (Taylor expansion) about $x^\*$ :

x_{n+1} \;=\; f(x^\* + \varepsilon_n) \;=\; f(x^\*) + f'(x^\*)\,\varepsilon_n + O(\varepsilon_n^2).

Since $f(x^\*) = x^\*$ and $x_{n+1} = x^\* + \varepsilon_{n+1}$ , the deviation obeys

\varepsilon_{n+1} \;=\; f'(x^\*)\,\varepsilon_n \quad\Longrightarrow\quad \varepsilon_n \;=\; \big[f'(x^\*)\big]^n\, \varepsilon_0.

The deviation is multiplied by the multiplier $\mu \equiv f'(x^\*)$ every step. So the geometric sequence $\varepsilon_n = \mu^n \varepsilon_0$ shrinks to zero — the fixed point is stable (attracting) — exactly when $|\mu| < 1$ , and grows — unstable (repelling) — when $|\mu| > 1$ . At $|\mu| = 1$ the linear term decides nothing and the fixed point is marginal; this is precisely where bifurcations happen.

For the logistic map $f'(x) = r(1 - 2x)$ . Evaluate at each fixed point:

At $x^\* = 0$ : $\;f'(0) = r$ . Stable for $r < 1$ , unstable for $r > 1$ . Below $r=1$ the population dies out; above it, extinction becomes a repeller and the nonzero state takes over.
At $x^\* = 1 - 1/r$ : $\;f'(x^\*) = r\big(1 - 2(1-1/r)\big) = 2 - r$ . This satisfies $|2 - r| < 1$ — i.e. stability — for $1 < r < 3$ . The steady population is attracting throughout this range.

So for $1 < r < 3$ the story is dull and reassuring: wherever you start (other than exactly 0), the orbit converges to the single value $1 - 1/r$ . The nonzero fixed point is globally attracting. At $r = 3$ its multiplier hits $f'(x^\*) = 2 - 3 = -1$ , the marginal case, and stability is lost. What happens next is the subject of the next lesson.

Reading the cobweb

The orbit of a 1-D map has a beautiful graphical construction. Draw the curve $y = f(x)$ and the diagonal $y = x$ on the same axes. Starting at $x_0$ on the horizontal axis: go vertically to the curve (that height is $x_1 = f(x_0)$ ), then horizontally to the diagonal (which copies $x_1$ onto the horizontal axis), then vertically to the curve again ( $x_2$ ), and so on. The resulting staircase or spiral — the cobweb — is the orbit made visible.

regime: period-2 cycle

growth r = 3.200 seed x₀ = 0.20

Drag r up through 3, 3.45, 3.54… and watch the cobweb settle to 1 point, then 2, then 4, then fill the interval.

The cobweb makes the stability criterion geometric. Where the parabola crosses the diagonal is a fixed point; the steepness of the parabola at that crossing is the multiplier $f'(x^\*)$ . Three things to walk through with the slider:

$r < 3$ (e.g. $r = 2.7$ ): the cobweb spirals or steps straight into the single crossing point. Whatever $x_0$ you choose, the staircase collapses onto $1 - 1/r$ . The crossing is shallow ( $|f'| < 1$ ).
$3 < r < 3.45$ (e.g. $r = 3.2$ ): the crossing has become too steep ( $|f'| > 1$ ); the cobweb can no longer land on it and instead settles into a rectangle bouncing between two values — a period-2 cycle. The population oscillates between a boom year and a bust year forever.
$r$ near 4 (e.g. $r = 3.9$ ): the cobweb never closes. It wanders over the whole interval, never repeating. This is chaos.

The time-series panel on the right shows the same orbit as $x_n$ versus $n$ : flat line, then two-level oscillation, then an irregular jitter that no eye can predict.

Sensitive dependence

Set $r = 4$ , where the logistic map is fully chaotic, and run two orbits from almost-identical starts — say $x_0 = 0.4$ and $x_0' = 0.4001$ . For the first handful of steps they track each other; the difference is a ten-thousandth and stays small. But the gap grows, roughly multiplying each step, and within twenty-odd iterations the two orbits are as different as two unrelated sequences — one near 0.9 while the other sits near 0.1. This is sensitive dependence on initial conditions: the defining property of chaos.

▶ Why the error grows exponentially at r = 4 Derivation

At $r = 4$ the logistic map is conjugate to an exactly solvable one. Substitute $x_n = \sin^2(\pi \theta_n)$ . Then

x_{n+1} = 4 x_n (1 - x_n) = 4 \sin^2(\pi\theta_n)\cos^2(\pi\theta_n) = \sin^2(2\pi\theta_n),

using $\sin 2\alpha = 2\sin\alpha\cos\alpha$ . So in the $\theta$ coordinate the map is simply the doubling map

\theta_{n+1} = 2\,\theta_n \pmod 1.

Doubling mod 1 is transparent in binary: if $\theta_n = 0.b_1 b_2 b_3\ldots$ in base 2, then $\theta_{n+1} = 0.b_2 b_3 b_4\ldots$ — every step deletes the leading bit and shifts the rest up. Two starting values that agree to the first $k$ binary digits agree for about $k$ steps, then diverge once the shift exhausts the digits they shared. An initial error $\varepsilon_0$ is doubled each step, $\varepsilon_n \approx 2^n \varepsilon_0$ , so it grows as $e^{n\ln 2}$ . The rate $\ln 2$ per step is the map’s Lyapunov exponent, and it is positive — the unambiguous signature of chaos.

The doubling-map picture also explains why the orbit looks random while being fully deterministic: it is reading out the binary digits of $\theta_0$ one at a time. If $\theta_0$ is irrational, its digits never repeat, so the orbit never repeats — yet nothing is random. The “randomness” was sitting in the infinitely many digits of the initial condition all along; the chaotic map merely exposes them, one per step. No measurement pins down infinitely many digits, so prediction fails in finite time even though the rule is exact.

⏳ The history — May's 1976 plea to study the simple map

The logistic map as a population model goes back to Pierre-François Verhulst in the 1840s, but its chaotic side was not appreciated until the 1970s. In a 1976 review in Nature, the physicist-turned-ecologist Robert May laid out how this “very simple” first-order difference equation runs through stable points, stable cycles, and “an apparently chaotic regime in which… the trajectory looks like the sample function of a random process” — all by varying one parameter (May 1976). May’s closing was a manifesto: he urged that the model “be studied in the elementary mathematics courses,” arguing that the widespread intuition that simple equations have simple solutions was doing real harm in fields from ecology to economics. The deeper structure — that the rate of period-doubling is universal — was uncovered the same year by Mitchell Feigenbaum, the subject of the next lesson.

The conceptual roots run back further, to Henri Poincaré, who in his 1890 work on the three-body problem found orbits “so tangled that I cannot even begin to draw them,” and grasped that small differences in initial conditions could produce large differences later — sensitive dependence, four decades before anyone had a name for it (Poincaré 1890).

What this sets up

The logistic map is the whole of chaos in miniature. It gives us, in a setting with no calculus and one parameter:

Fixed points and linear stability — the $|f'(x^\*)| < 1$ test, which is the discrete-time analogue of the eigenvalue test for equilibria of flows.
Loss of stability and the onset of oscillation at $r = 3$ — a bifurcation, examined next.
Sensitive dependence and a positive growth rate of errors — made quantitative in 11.4.

The next lesson turns the parameter $r$ up past 3 and follows the cascade of period-doublings — period 2, then 4, then 8, … — that accumulates, at a universal rate, on the edge of chaos.