Sketch · 2026-10-02
Logistic map, cobweb and bifurcation.
A one-line rule, x next equals r times x times one minus x, repeated. Below r = 3 the sequence settles on a single value. Push r past 3 and it splits in two. Past 3.449 it splits in four, then eight, and the doublings pile up on each other faster and faster until, around 3.5699, order gives out and the orbit is chaotic. But not everywhere: slide further and clean three-cycle, five-cycle and six-cycle windows open inside the noise.
Click or drag anywhere on the diagram to pick r.
Cobweb
Trajectory, 120 steps
Note
Robert May put this map in a 1976 Nature paper as a model for insect populations with no overlapping generations. He wanted to make a point about teaching: a very simple equation can have very complicated solutions, and biologists should know this before fitting curves to field data. The paper ends with a plea to put chaos on the undergraduate syllabus.
The bifurcation diagram here is drawn column by column. For each value of r it iterates the map 400 times from a fixed seed to let the orbit settle, then plots the next 100 points. In a period-doubling regime those 100 points land on 2, 4, 8 distinct values and the column shows that many dots. In the chaotic regime they land anywhere in the attractor and the column fills in.
The ratio of successive bifurcation widths tends to a constant 4.6692..., discovered by Mitchell Feigenbaum in 1975. It also shows up for the Mandelbrot set along the real axis and for a long list of other smooth maps with one hump. The constant is a property of the shape of the hump, not of the equation.
The Lyapunov exponent λ is the long-run average of log |r (1 - 2x)|. Negative means neighbours contract and the orbit is stable; positive means neighbours diverge exponentially and the orbit is chaotic. It crosses zero at each bifurcation and dives to minus infinity at the superstable points inside each window.
Sources
- May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature, 261, pages 459-467.
- Feigenbaum, M. J. (1978). Quantitative universality for a class of nonlinear transformations. Journal of Statistical Physics, 19, pages 25-52.
- Strogatz, S. H. (2015). Nonlinear Dynamics and Chaos, 2nd ed. Westview, chapter 10 on one-dimensional maps.