📈 Population Chaos — Logistic Map & Bifurcation

The logistic map xn+1 = r·xₙ·(1−xₙ) models population growth with limited resources. For small r the population reaches a fixed point; as r increases it undergoes a period-doubling cascade — 1 → 2 → 4 → 8 → … fixed points — until around r ≈ 3.57 it enters chaos. The ratio of successive period-doublings converges to the Feigenbaum constant δ ≈ 4.669. The bifurcation diagram (top) reveals this entire structure; the time series (bottom) shows individual trajectories.

🇺🇦 Українська

View

Time series

Stats

r3.700
Lyapunov λ—
Behavior—
Period—
Logistic map:
xₙ₊₁ = r·xₙ·(1−xₙ)

Lyapunov exp.:
λ = (1/N)·Σ ln|r(1−2xₙ)|

λ < 0 → stable
λ = 0 → period-doubling
λ > 0 → chaos

δ = 4.6692… (Feigenbaum)

Route to Chaos

Population models were the first systems where deterministic chaos was clearly demonstrated. Robert May's 1976 paper showed that the simple logistic recurrence — meant to model insect population dynamics — produces complexity rivalling genuinely random processes. The Feigenbaum constants (δ ≈ 4.669 and α ≈ 2.502) are universal: they appear in every one-dimensional map with a quadratic maximum, from the logistic map to real experiments in fluid convection and laser physics. This universality is explained by the renormalization group theory of Feigenbaum (1978).

About this simulation

This simulation iterates the logistic map xn+1 = r·xn·(1−xn), the simplest equation that turns smooth population growth into deterministic chaos. A live bifurcation diagram plots the long-run values x settles into as the growth rate r sweeps from 2.8 to 4.0, while a time-series panel shows the actual sequence x0, x1, x2… for your chosen r, alongside a computed Lyapunov exponent that tells you whether the trajectory is stable, periodic, or chaotic.

🔬 What it shows

The bifurcation diagram plots hundreds of r values on the x-axis against the long-run x values the map settles into after discarding transients, revealing period-1 fixed points, then period-doubling into period-2, period-4, period-8, and finally chaotic bands after r≈3.57. The time-series panel shows the current r as a moving marker line and iterates xn directly so you can watch a fixed point, a repeating cycle, or an unpredictable chaotic sequence unfold in real time.

🎮 How to use

Drag the r slider (0–4) to move through the logistic map's entire behaviour range and watch the marker line sweep across the bifurcation diagram; drag x₀ (0.001–0.999) to change the starting population fraction and see that it does not affect the long-run periodic behaviour, only the transient. The Stats box reports the current r, the computed Lyapunov exponent λ, whether the behaviour is stable/periodic/chaotic, and the detected period length.

💡 Did you know?

Robert May's 1976 paper on the logistic map, originally intended to model insect populations with limited resources, became one of the most cited papers in chaos theory because it showed that a single-line equation can generate infinite complexity. The ratio between successive period-doubling intervals converges to the Feigenbaum constant δ ≈ 4.6692, a number that shows up universally in completely unrelated physical systems, from fluid convection to lasers, whenever they follow the same route to chaos.

Frequently asked questions

What is the logistic map and what does r control?

The logistic map is the recurrence xn+1 = r·xn·(1−xn), where x is a population fraction between 0 and 1 and r is the growth rate. For low r (below about 3.0) the population settles to a single stable value; as r increases past 3.0 the map begins period-doubling into cycles of 2, 4, 8 and more points, and beyond r≈3.57 it typically becomes chaotic, jumping unpredictably between values even though the rule generating them is completely deterministic.

How is the Lyapunov exponent calculated and what does it mean?

The simulation computes λ = (1/N)·Σ ln|r(1−2xn)| by averaging the log of the map's local slope magnitude over many iterations. A negative λ means nearby starting points converge together over time, indicating stable or periodic behaviour; λ = 0 marks a bifurcation point where periodic behaviour is about to split; and a positive λ means nearby trajectories diverge exponentially, the defining signature of chaos, where tiny differences in x₀ eventually produce completely different outcomes.

Why does changing x₀ not change the bifurcation diagram?

The bifurcation diagram is built by iterating the map many times for each r value and discarding an initial transient, then plotting only the values it settles into afterward. Because the logistic map's long-run behaviour (fixed point, cycle, or chaotic attractor) depends on r rather than on where you started, changing x₀ only alters the early transient path — the "warm-up" iterations thrown away — not the ultimate pattern the population reaches.

What is the period-doubling route to chaos?

As r increases, the logistic map's stable behaviour repeatedly splits in two: a single fixed point becomes an oscillation between two values, that oscillation splits into four values, then eight, then sixteen, with each doubling happening at ever-shrinking intervals of r. This cascade of period-doublings accumulates at a finite value r≈3.57, beyond which the period becomes effectively infinite and the system enters chaos, interspersed with narrow windows where periodic behaviour briefly reappears.

What is the Feigenbaum constant and why is it universal?

The Feigenbaum constant δ≈4.6692 is the limiting ratio of the r-intervals between successive period-doublings: each doubling happens roughly 4.6692 times closer to the previous one than the interval before it. Mitchell Feigenbaum discovered in 1975 that this exact number appears in every one-dimensional map with a single quadratic maximum, not just the logistic map, and later experiments confirmed it in fluid convection, electronic circuits and laser systems, a universality now explained by Feigenbaum's renormalization-group theory.