Bifurcation Diagrams: Period-Doubling & the Feigenbaum Constant

The logistic map and its fixed points

The logistic map is the simplest nonlinear difference equation that exhibits the full route from order to chaos, introduced by Robert May in 1976 as a population-dynamics model:

xₙ₊₁ = r · xₙ · (1 − xₙ)     x ∈ [0,1],  r ∈ [0,4]

Fixed points: x* = 0  or  x* = 1 − 1/r
Stability: |f'(x*)| = |r(1 − 2x*)|  — loses stability at r = 3

At r = 3, the derivative at the fixed point hits exactly −1: this is a period-doubling (flip) bifurcation, and the system settles into a stable 2-cycle instead. The bifurcation diagram is built by iterating thousands of times to let transients die out, then plotting the next 200 iterates as dots at each horizontal position r — repeated across thousands of r values to produce the famous "period-doubling tree."

The Feigenbaum constant: universal, not coincidental

The period-2 cycle itself loses stability at r₂ ≈ 3.449, spawning period-4, then period-8, and so on — the sequence of bifurcation points {rₙ} converges to r∞ ≈ 3.5699456, the onset of chaos. Mitchell Feigenbaum discovered in 1975–1978 that the ratio between successive gaps converges to a universal constant:

δ = lim (rₙ − rₙ₋₁) / (rₙ₊₁ − rₙ)  ≈ 4.669201609...

Appears in EVERY smooth unimodal map with a single quadratic maximum —
logistic map, sine map, tent map, and physical systems alike.

This is not numerology: δ has been measured directly in Rayleigh-Bénard convection experiments, nonlinear electrical circuits, and modulated laser oscillations. The explanation is a renormalisation-group fixed point in function space — every smooth unimodal map is attracted to the same universal function, then travels along one unstable direction at rate δ, exactly analogous to universality classes near critical points in statistical mechanics.

Beyond chaos onset: periodic windows

Past r∞, the system is predominantly chaotic but not uniformly so — the diagram is interspersed with narrow periodic windows where a stable cycle re-emerges. The most prominent, near r ≈ 3.828, is the period-3 window, guaranteed by the Li-Yorke theorem: "period 3 implies chaos" — if a continuous map has a period-3 orbit, it has orbits of every period. Each window undergoes its own sub-cascade of period-doublings, producing a fractal structure repeated at every scale of parameter space.

Frequently asked questions

Try it live

Everything above runs in your browser — open Bifurcation Diagram and drag to zoom into any region of the logistic map, revealing the period-doubling cascade and self-similar structure at every scale.