Logistic Map

5 min read#chaos

A one-line population model, x_{n+1}=r x_n(1-x_n), whose period-doubling cascade is the canonical road from order into chaos.

Contents

Logistic Map

The logistic map is the cleanest proof that chaos needs no complexity. It is a single line of arithmetic, iterated:

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

Think of x_n as a population, scaled to lie between 0 (extinct) and 1 (carrying capacity), measured generation by generation. The factor r x_n is growth; the factor (1-x_n) is the brake of overcrowding. One parameter, r, controls the whole story — and as you raise it from 0 to 4, this innocent map marches through fixed points, oscillations, a cascade of bifurcations, and finally full-blown chaos.

The cast of behaviors

For small r the population just dies out. Past r = 1 it settles to a steady Fixed Point. At r = 3 that fixed point loses stability and the population starts alternating between two values — a period-2 cycle. Raise r further and the period doubles to 4, then 8, then 16, the windows of stability shrinking geometrically, until near r \approx 3.5699 the period becomes infinite and the dynamics turn chaotic.

The bifurcation diagram

Here is the showpiece. For every value of r across the width of the canvas, we iterate the map, throw away the transient, and plot the values it settles onto down the height. Read it left to right and you watch order dissolve into chaos: one branch splits into two, two into four, four into eight — the period-doubling cascade — then shatters into a fractal mist, threaded by surprising white windows of restored periodicity (the widest is the period-3 window near r \approx 3.83).

Move your mouse across the diagram to pick an r and see the live orbit it produces.

Bifurcation diagram of x→r x(1-x) for r from 2.5 to 4.0. Computed once into an offscreen buffer; hover to probe the attracting orbit at any r. Density (darker) shows where the iterates spend their time.

Reading the chaos quantitatively

Whether a given r is periodic or chaotic is decided by the Lyapunov Exponent of the map,

\lambda(r) = \lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}\ln\big|\,r(1 - 2x_n)\,\big|.

Where \lambda < 0 neighboring states converge and the orbit is a stable cycle; where \lambda > 0 they diverge and the orbit is chaotic. The plot below sketches this growth-rate factor f'(x)=r(1-2x) — sensitivity per step lives in its magnitude.

Per-step stretch factor f'(x)=r(1-2x) at r=2.5 (stable) and r=4 (chaotic)
Where |f'| exceeds 1, a single step stretches small errors — the seed of a positive Lyapunov exponent.

In the bifurcation diagram, what is happening at the values of r where a single curve splits into two?

See also