Logistic Map & Bifurcation Diagram
One equation. Infinite complexity. The logistic map xn+1 = r xn(1−xn) reveals how simple rules generate chaos with a universal constant hidden in the branching points.
💡 In Bifurcation view, click anywhere on the diagram to set r at that point.
The Logistic Map
The logistic map was introduced as a population model by biologist Robert May in 1976, though its mathematical properties were studied earlier by Verhulst (1845). It models a population with a limited food supply: the term r x_n provides exponential growth, while −r x_n² provides density-dependent decay.
xn+1 = r · xn · (1 − xn)
x ∈ (0, 1) normalised population • r ∈ [0, 4] growth rate
r < 1: Extinction
Population collapses to zero regardless of initial condition. The only fixed point is x* = 0, which is globally stable.
1 < r < 3: Stable Fixed Point
Converges to x* = 1 − 1/r. At r = 2, x* = 0.5. The cobweb spirals inward to a single attractor.
3 < r < 3.449: Period 2
The fixed point becomes unstable. The orbit oscillates between two values a period-doubling bifurcation.
r > 3.569: Chaos
After infinite period doublings, the orbit becomes aperiodic. Sensitive dependence on initial conditions the hallmark of chaos.
The Bifurcation Diagram
The bifurcation diagram plots the long-run behaviour of x for each value of r. For each r, the simulation runs 300 iterations to discard the transient, then plots the next 200 points. Each column is a vertical slice of the attractor at that r. The result is the iconic "pitchfork" tree splitting forever into chaos.
🌐 The Feigenbaum Constant: δ ≈ 4.6692016
The period-doubling bifurcations occur at r values: 3.000, 3.4495, 3.5441, 3.5644, 3.5688… The ratio of successive gaps converges to δ = 4.6692… Feigenbaum’s constant. It is universal: every smooth unimodal (single-humped) map has the same δ, from the logistic map to the sine map to physical systems like dripping faucets and the Belousov-Zhabotinsky reaction. No one predicted this universality until Feigenbaum’s 1978 discovery.
Lyapunov Exponent
The Lyapunov exponent λ measures the average logarithmic divergence rate of nearby orbits:
λ = (1/N) Σn ln|f′(xn)| = (1/N) Σ ln|r(1−2xn)|
- λ < 0 stable orbit (fixed point or periodic cycle); perturbations decay
- λ = 0 bifurcation point; neutral stability at the transition
- λ > 0 chaos; nearby trajectories diverge exponentially
At the period-doubling bifurcations, λ = 0 exactly. The peaks of λ between the periodic windows indicate the most strongly chaotic parameter values.
⚠ Sensitivity & Predictability
For r = 3.9 (deep chaos), two starting conditions differing by 10−10 diverge to opposite sides of (0,1) after roughly 33 iterations: uncertainty doubles every ln(2)/λ ≈ 2 iterations. This is why weather forecasting beyond ~10 days is fundamentally limited not by measurement technology, but by the mathematics of chaotic systems.
Islands of Order: Periodic Windows
Look carefully at the bifurcation diagram: within the chaotic region there are bright periodic windows stripes of order amid the chaos. The largest is the period-3 window near r ≈ 3.828. The Li-Yorke theorem (1975) proves that period 3 implies chaos: any system with a period-3 orbit must also have orbits of every other period. Period-3 is the signature of full topological chaos.
Self-Similarity and Fractals
The bifurcation diagram is self-similar: zoom into any chaotic band and you find a smaller copy of the entire diagram infinite nested structure at every scale. The boundary at the onset of chaos at r ≈ 3.5688 has Hausdorff dimension d ≈ 0.538. This fractal structure is the fingerprint of the universal period-doubling route to chaos.
Connections to Other Simulations
Lorenz Attractor
The Lorenz system shows chaos in 3D continuous time. The logistic map shows the same universal features in 1D discrete time. Both are examples of deterministic chaos.
Fourier Epicycles
Chaotic time series have complex frequency spectra broad-band noise rather than discrete peaks. Fourier analysis can distinguish periodic from chaotic time series.
Double Pendulum
The double pendulum exhibits the same positive Lyapunov exponents and sensitive dependence on initial conditions as the logistic map in its chaotic regime.
Evolution Simulator
Population dynamics with competition and limited resources mirror the logistic model. Chaotic population cycles have been observed in real lynx/hare data (Hudson Bay Company records, 1845-1935).
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View Premium PlansLogistic Map Equation
The recurrence relation xn+1 = r xn(1−xn) models a population with limited resources. Despite its simplicity, it exhibits the full spectrum of behaviour from steady states to period-doubling cascades and deterministic chaos as the growth rate r increases from 0 to 4.
Feigenbaum Constant
The period-doubling bifurcations occur at r values whose successive gaps shrink by the universal ratio δ ≈ 4.6692016 Feigenbaum's constant. This constant is the same for every smooth unimodal (single-humped) map, from the sine map to physical dripping faucets, linking mathematics to real-world chaos via a universal number.
Lyapunov Exponent
The Lyapunov exponent λ = (1/N)Σn ln|r(1−2xn)| quantifies the average divergence rate of nearby trajectories. λ < 0 indicates stable periodicity; λ > 0 marks chaos nearby orbits diverge exponentially, rendering long-term prediction impossible despite the system being entirely deterministic.
Period-Doubling Route to Chaos
As r increases through 3.0, 3.45, 3.54… the orbit period doubles each time a period-doubling cascade. After infinitely many doublings (accumulating at r ≈ 3.5699), the orbit becomes aperiodic. Within the chaotic region, periodic windows appear most notably the period-3 window at r ≈ 3.828, which by the Li–Yorke theorem guarantees orbits of all periods.
Key Equations
| Concept | Formula | Notes |
|---|---|---|
| Logistic map | xn+1 = r xn(1−xn) | r ∈ [0, 4]; x ∈ (0, 1) |
| Non-trivial fixed point | x* = 1 − 1/r | Stable for 1 < r < 3 |
| Lyapunov exponent | λ = (1/N) Σ ln|r(1−2xn)| | λ > 0 ⇒ chaos |
| Feigenbaum constant | δ ≈ 4.6692016091… | Universal for all unimodal maps |
| Bifurcation onset | r1 = 3.0, r2 ≈ 3.449, r∞ ≈ 3.5699 | Period-doubling accumulation point |
| Hausdorff dimension | d ≈ 0.538 | Fractal boundary at onset of chaos |
| Period-3 window | r ≈ 3.828 | By Li–Yorke: period 3 implies all periods |
Curriculum Relevance
| Level | Topic | Relevance |
|---|---|---|
| GCSE | Sequences & iteration | Recurrence relations xn+1 = f(xn), cobweb diagrams |
| A-Level / IB | Numerical methods, limits | Fixed-point iteration, convergence & divergence criteria |
| AP Calculus | Sequences, stability | Stability of fixed points via derivative condition |f′(x*)| < 1 |
| Undergraduate | Dynamical systems | Bifurcation theory, Lyapunov exponents, symbolic dynamics |
| Postgraduate | Ergodic theory, topology | Measure-preserving maps, topological entropy, universality |