The Logistic Map: How One Equation Produces Chaos

One line, infinite behaviour

The logistic map is the simplest possible model of a population that grows and is limited by its own crowding. It has one variable x, scaled to sit between 0 and 1 (a fraction of some carrying capacity), and one knob r, the growth rate:

x(n+1) = r . x(n) . (1 - x(n))          0 <= x <= 1,  0 <= r <= 4

The r . x term is ordinary exponential growth; the (1 - x) term is a brake that gets stronger as x approaches 1 — plenty of food and space when x is small, starvation and competition when x is near the carrying capacity. That single multiplicative brake is enough to produce every qualitative behaviour a dynamical system can have: fixed points, periodic cycles, and full-blown chaos, depending only on the value of r.

The road to chaos: period-doubling

Iterate the map for a fixed r, throw away the first few hundred steps as transient, and plot every x value that follows — that is one vertical slice of the bifurcation diagram. For 0 < r < 1 the population dies out to x = 0. For 1 < r < 3 it settles to a single stable value, x* = 1 - 1/r. At r = 3 that fixed point loses stability and the orbit starts alternating between two values — a period-2 cycle. Push r further and each cycle itself becomes unstable and splits again: period 2 to 4 at r ≈ 3.449, 4 to 8 at r ≈ 3.544, and so on, with the intervals between successive splits shrinking by a constant factor.

That factor, δ ≈ 4.6692016, is the Feigenbaum constant. Mitchell Feigenbaum discovered in the 1970s that it is universal — the same number governs the period-doubling route to chaos in almost any smooth one-hump map, and in real physical systems (dripping taps, convecting fluids, some electronic circuits) that have nothing to do with population biology. It was one of the first hard pieces of evidence that chaos has its own quantitative laws, not just qualitative ones.

Chaos, windows, and the cobweb

The cascade accumulates at r∞ ≈ 3.56995, beyond which the orbit is aperiodic for most values of r — this is the chaotic region, and it fills most of the interval up to r = 4. It is not chaos everywhere, though: the diagram has clear white gaps, periodic windows, where a stable cycle re-emerges from the chaos, most visibly a period-3 window opening near r ≈ 3.8284. By a theorem popularised as 'period three implies chaos', the existence of any period-3 orbit guarantees that cycles of every other period exist somewhere in the same map.

The easiest way to watch a single trajectory rather than the statistical diagram is the cobweb plot: draw y = r x (1-x) and the diagonal y = x, start at some x0 on the diagonal, step up to the curve, across to the diagonal, up to the curve again. In the stable region the cobweb spirals tightly into the fixed point; in the chaotic region it never settles, tracing a different tangled path from almost every starting value.

Measuring the chaos: the Lyapunov exponent

Chaos means sensitive dependence on initial conditions — two starting points a tiny distance apart drift apart exponentially fast. The rate of that drift is the Lyapunov exponent λ, computed by averaging the log of the local stretching factor |f'(x)| = |r(1 - 2x)| along the orbit:

lambda(r) = lim (1/N) . sum_{n=0}^{N-1} ln | r (1 - 2 x(n)) |

λ < 0 means nearby trajectories converge — a stable fixed point or cycle, read directly from the pre block above as text, not rendered math. λ = 0 marks a bifurcation. λ > 0 means they diverge exponentially — chaos — and 1/λ sets the Lyapunov time, the number of iterations before a rounding-error-sized uncertainty swamps the whole prediction. Plotting λ(r) alongside the bifurcation diagram makes the periodic windows obvious: λ dips back below zero exactly where a periodic orbit briefly wins back the interval.

Frequently asked questions

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