The Logistic Map
One equation, one parameter, infinite complexity. The logistic map is the canonical example of how a completely deterministic system can produce unpredictable, chaotic behaviour — and how order and chaos are separated by a razor-thin boundary.
1. Introduction
In 1976, biologist Robert May published a landmark paper in Nature titled "Simple mathematical models with very complicated dynamics." He urged scientists and students to spend time exploring the behaviour of a single difference equation — the logistic map — because it contained lessons that textbooks had completely missed.
The equation models how a population changes from one generation to the next, balancing two forces:
- Growth — the more individuals, the more offspring
- Competition — the more individuals, the more they crowd each other out
What May found shocked the scientific community: depending on a single parameter, this tiny equation produces stable equilibrium, regular oscillations, and completely chaotic behaviour — with no noise or randomness involved.
Like Kauffman NK networks, the logistic map shows a phase transition from order to chaos as a single parameter increases. Both systems sit at the edge between predictability and complexity — and both have been proposed as models of how life harnesses the power of that boundary.
2. The Equation
Definition
This is a recurrence relation: you plug in the current value xn to get the next value xn+1.
| Symbol | Meaning | Range |
|---|---|---|
xn | Population at generation n, as a fraction of the maximum possible | [0, 1] |
r | Growth rate — the single control parameter | [0, 4] |
x0 | Initial condition — where the iteration starts | (0, 1) |
Why (1 − x)?
The term r·x represents exponential growth — the more animals, the
more offspring. The term (1 − x) represents a ceiling: as x approaches
1 (maximum capacity), the braking force approaches 1 and growth stops.
Together they create a parabola — the defining shape of the map.
Why r ≤ 4?
At r = 4 the parabola just touches the top of the unit square at x = 0.5 (giving f(0.5) = 1). For r > 4, iterates escape outside [0, 1] and the population-biology interpretation breaks down. The interesting dynamics all happen within 0 ≤ r ≤ 4.
The Phase Landscape
3. Fixed Points and Stability
What Is a Fixed Point?
A fixed point x* satisfies f(x*) = x*
— the map leaves it unchanged. Starting at a fixed point, the
population never changes.
Finding Fixed Points
Solve r·x·(1 − x) = x:
r·x·(1 − x) = x x · [r(1 − x) − 1] = 0 Solution 1: x* = 0 (trivial — extinction) Solution 2: x* = 1 − 1/r (non-trivial — for r > 1)
Stability of the Non-Trivial Fixed Point
A fixed point is stable (attracting) if a small push away from it shrinks over time. The test is the magnitude of the derivative of f at the fixed point:
At x* = 1 − 1/r: f'(x*) = 2 − r. This has magnitude less than 1 when 1 < r < 3. At r = 3, |f'| = 1 exactly — the fixed point becomes unstable and the first bifurcation occurs.
In the cobweb diagram below, a stable fixed point is where the parabola crosses the y = x line at a shallow enough angle to trap the orbit. When the crossing is too steep (|slope| > 1) the orbit bounces away to a cycle or chaos.
4. Period Doubling — The Route to Chaos
The First Bifurcation (r = 3)
At r = 3 the stable fixed point loses stability and splits into a period-2 cycle: the population alternates between two values every other generation. This is called a pitchfork bifurcation.
The Cascade
As r increases further, each stable cycle undergoes its own pitchfork bifurcation — period 2 becomes 4, then 8, then 16… Each successive bifurcation happens in a shorter interval of r:
The intervals shrink by a constant ratio each time. That ratio is the Feigenbaum constant δ ≈ 4.669 — one of the most surprising universal numbers in mathematics (see §8).
At r ≈ 3.5699… the cascade of period doublings has completed infinitely many times. Beyond this point the system is mostly chaotic — but with scattered windows of order where stable cycles reappear, most famously a period-3 window near r ≈ 3.83.
5. Demo: Orbits, Cobweb & Time Series
Drag the r slider to explore different regimes. The cobweb diagram (left) shows how the iteration bounces between the parabola y = rx(1−x) and the diagonal y = x. The time series (right) shows xn vs n. Click a preset to jump to an interesting region.
6. The Bifurcation Diagram
The bifurcation diagram shows the long-run attractor values of x for every r simultaneously. For each r, the first 400 iterates are discarded (transient), then the next 300 are plotted as dots. The orange vertical line tracks the r value from the sliders above.
Drag the diagram to set r and see the cobweb and time series update instantly.
7. Chaos and Sensitive Dependence
What Is Deterministic Chaos?
The logistic map is perfectly deterministic — given r and x0, every subsequent xn is uniquely determined. Yet for r above the accumulation point (≈ 3.5699), two initial conditions that differ by as little as 10-15 will eventually produce completely different trajectories. This is sensitive dependence on initial conditions — the defining feature of chaos.
Set r = 3.9 and press the Compare x₀±ε button in the demo above. The red orbit starts at x₀ + 0.001. Watch how the two nearly-identical orbits quickly diverge.
The Lyapunov Exponent
The Lyapunov exponent λ quantifies how fast nearby orbits diverge (or converge). It is the average log-rate of stretching:
| λ value | Meaning | Regime |
|---|---|---|
λ < 0 | Nearby orbits converge — attractor is stable | Ordered / periodic |
λ = 0 | Neutral — bifurcation or onset of chaos | Critical boundary |
λ > 0 | Nearby orbits diverge exponentially | Chaotic |
The Lyapunov exponent is displayed live in the demo above as you move the r slider. Watch it cross zero at r ≈ 3.5699 and spike at r = 4 where λ = ln(2) ≈ 0.693.
Windows of Order
Within the chaotic region, periodic windows reappear. The most prominent is the period-3 window near r ≈ 3.828. By the period-3 theorem (Li & Yorke, 1975): "Period three implies chaos" — any continuous map with a period-3 orbit also has orbits of every other period. These windows are also fractal — zoom in and you find smaller copies of the entire bifurcation diagram nested inside.
8. Feigenbaum Universality
In 1978, physicist Mitchell Feigenbaum made a stunning discovery. He computed the ratio of successive bifurcation intervals on a pocket calculator and found they converged to the same constant, regardless of the map being studied. For the logistic map:
A second constant α ≈ 2.502 describes how the width of the bifurcated branches scales. Both constants are universal: they appear identically in the period-doubling cascade of any smooth one-dimensional map with a single quadratic maximum — the sine map, the tent map, real physical experiments in fluid turbulence, electronic circuits, and chemical oscillators.
Universality means the fine details of the equation don't matter — only the topology (one hump, quadratic peak). The route to chaos is governed by the same two numbers across all of physics, chemistry and biology. Feigenbaum's constants have been measured experimentally and confirmed to several decimal places.
Renormalisation Explanation
Feigenbaum explained universality using renormalisation borrowed from quantum field theory: near the accumulation point, the map looks like a scaled copy of itself under a specific transformation. The Feigenbaum constants are the eigenvalues of that renormalisation operator — universal because they depend only on the local shape of the fixed point of an infinite-dimensional function space.
9. Guided Experiments
Use the demo above for all experiments. Drag the r slider or click the preset buttons to reach each starting point.
Experiment 1 — Watching a Fixed Point Appear and Disappear
r = 0.8 → 2.9 Start at r = 0.8 (extinction) and slowly drag toward r = 2.9. Watch the cobweb spiral inward to the fixed point at x* = 1 − 1/r. At r = 2.9 the fixed point is near x ≈ 0.655. As you approach r = 3, observe the cobweb spiral take longer and longer to converge — the system is slowing down near the bifurcation.
Experiment 2 — The First Bifurcation
r = 2.99 → 3.01 Cross r = 3 slowly. Before: one fixed point. After: a period-2 cycle. The cobweb changes from a spiral into a rectangle bouncing between two values. The time series switches from a flat line to an alternating signal. Note that the Lyapunov exponent crosses zero exactly here.
Experiment 3 — Period Doubling Cascade
r = 3.1, 3.45, 3.544, 3.567 Visit each preset in order: Period-2, Period-4, Period-8. Each time the cobweb rectangle splits into a more complex shape. Notice how rapidly the bifurcations compress — between period-4 and period-8 the r gap is already tiny (≈ 0.02).
Experiment 4 — Onset of Chaos
r = 3.57 Set r ≈ 3.570 and increase steps to 300. The time series becomes aperiodic — it never exactly repeats — but it is still confined to a strange-looking band. The cobweb fills a dense region of the parabola rather than tracing a closed loop. The Lyapunov exponent hovers near zero with large fluctuations.
Experiment 5 — Sensitive Dependence on Initial Conditions
r = 3.9 Set r = 3.9 and enable Compare x₀±ε. The blue and red orbits start only 0.001 apart. Watch them track each other for several steps, then suddenly diverge. Try r = 2.9 — the two orbits converge together instead, demonstrating that chaos is not universal to the equation but is a property of the parameter regime.
Experiment 6 — The Period-3 Window
r = 3.828 Drag to r ≈ 3.828 and observe a clean period-3 cycle emerge from the chaos. The cobweb traces a precise triangle. Drag slightly away from 3.828 in either direction and chaos resumes immediately. Notice in the bifurcation diagram the bright narrow band within the dark chaotic region.
Experiment 7 — Full Chaos at r = 4
r = 4.0 At r = 4 the map is fully chaotic and ergodic — xn visits every sub-interval of [0, 1] with a specific frequency. The Lyapunov exponent reaches its maximum: λ = ln 2 ≈ 0.693. The cobweb sweeps the entire unit square. No periodic structure remains anywhere.
Experiment 8 — Explore the Bifurcation Diagram
Click and drag across the bifurcation diagram directly. Notice the self-similar structure: the chaotic region contains mini-copies of the entire diagram. Every dark "bay" in the chaos is a periodic window with its own period-doubling cascade shrinking into it. This self-similarity is a fingerprint of the fractal structure underlying chaos.
Further Reading
- May, R.M. (1976). Simple mathematical models with very complicated dynamics. Nature, 261, 459–467.
- Feigenbaum, M.J. (1978). Quantitative universality for a class of nonlinear transformations. Journal of Statistical Physics, 19, 25–52.
- Li, T.Y. & Yorke, J.A. (1975). Period three implies chaos. The American Mathematical Monthly, 82(10), 985–992.
- Gleick, J. (1987). Chaos: Making a New Science. Viking Penguin. (accessible popular science)
- Strogatz, S.H. (1994). Nonlinear Dynamics and Chaos. Perseus Books. (rigorous undergraduate text)
- Devaney, R.L. (1989). An Introduction to Chaotic Dynamical Systems. Addison-Wesley.