The Logistic Map
The logistic map is the simplest one-dimensional dynamical system that exhibits chaos. Originally introduced by ecologist Robert May (1976) to model population dynamics, it became the paradigm for the period-doubling route to chaos:
Despite its seeming simplicity — a single quadratic equation — the long-run behaviour depends sensitively on r and fills a bifurcation diagram of remarkable complexity.
Period-Doubling Cascade
As r increases from 0 to 4, the logistic map undergoes successive bifurcations where a stable periodic orbit loses stability and splits into an orbit of twice the period:
| Period | Bifurcation at r ≈ | Width Δr | Ratio Δrₙ/Δrₙ₊₁ |
|---|---|---|---|
| 1 → 2 | 3.000 000 | 0.449 490 | — |
| 2 → 4 | 3.449 490 | 0.096 257 | 4.669 |
| 4 → 8 | 3.544 090 | 0.020 616 | 4.669 |
| 8 → 16 | 3.564 407 | 0.004 413 | 4.669 |
| 16 → 32 | 3.568 759 | 0.000 945 | 4.669 |
| ∞ (chaos onset) | 3.569 945 7… | 0 | δ ≈ 4.6692… |
The ratio of successive interval widths converges to Feigenbaum's constant δ = 4.6692040890097185…
Feigenbaum's Constants
In 1978 Mitchell Feigenbaum discovered that both constants are universal: they appear in any smooth one-dimensional map with a quadratic maximum — sin(πx), x(1−x²), and countless others all exhibit the same ratio. This universality is explained by a fixed-point equation in the space of maps, analogous to the renormalisation group in statistical physics.
Universality
δ and α appear in every smooth unimodal map. Period-doubling cascades have been observed in fluid turbulence, electronic circuits, and optical lasers.
Self-Similarity
Zoom into any chaotic band and you see miniature copies of the whole diagram, scaled by δ horizontally and α vertically — a fractal structure.
Periodic Windows
Within the chaotic regime, narrow windows of stable periodicity appear (period 3, 5, 7, …) before bifurcating into their own period-doubling cascades.
Sharkovskii Ordering
Period 3 implies all periods: 3 ▷ 5 ▷ 7 ▷ … ▷ 6 ▷ 10 ▷ … ▷ 2 ▷ 4 ▷ 8 ▷ … The period-3 window around r≈3.831 is the most prominent.
The Logistic Map in Detail
Fixed Points
x* = r·x*(1−x*) gives fixed points x* = 0 and x* = 1 − 1/r. The non-trivial fixed point is stable (|df/dx|<1) for r ∈ (1, 3). At r=3 the stability eigenvalue equals −1, triggering the first bifurcation.
Lyapunov Exponent
λ < 0 indicates a stable periodic orbit; λ > 0 indicates chaos. At the period-doubling accumulation point λ = 0. In the chaotic regime λ fluctuates around +0.69.
Chaos Onset r∞ ≈ 3.5699456718…
The accumulation point of period doublings is r∞ ≈ 3.5699456718789691. Beyond this value, for most r, the orbit densely fills one or more intervals (chaotic bands). The number of bands halves at each reverse-bifurcation as r increases further.
Zooming the Diagram
The simulation supports three zoom methods:
- Drag a rectangle with your mouse to zoom into any region
- Scroll the mouse wheel to zoom in/out centred on the cursor
- Double-click or use ← Back to undo the last zoom step
Moving your mouse over the diagram displays the live orbit (time series x_n vs n) for the r value under the cursor in the strip below the diagram.
Summary Table
| Quantity | Value / Formula | Significance |
|---|---|---|
| Feigenbaum δ | 4.6692040890097185… | Horizontal period-doubling ratio |
| Feigenbaum α | 2.5029078750958928… | Vertical attractor scaling |
| First bifurcation | r₁ ≈ 3.000 | Period 1→2 |
| Chaos onset r∞ | 3.5699456718… | Accumulation of period doublings |
| Period-3 window | r ≈ 3.8284 – 3.8571 | Sharkovskii: period 3 ⟹ all periods |
| Lyapunov λ (chaotic) | ≈ +0.69 | Exponential sensitivity to initial conditions |
| Hausdorff dimension | ≈ 0.538 | Fractal dimension of attractor at r∞ |
| Universality class | All unimodal maps | δ, α identical for all quadratic maxima |
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