The logistic map xₙ₊₁ = r·xₙ·(1−xₙ) is a one-line population model that encodes the entire path from stability to chaos. The bifurcation diagram is its DNA: it shows exactly where period doubling cascades begin and chaos takes over.
For r < 3 the map settles to a fixed point. At r ≈ 3 the period doubles to 2, at r ≈ 3.449 to 4, then 8, 16 … accumulating at the Feigenbaum point r ≈ 3.56995 where chaos begins. The ratio of successive bifurcation widths converges to δ ≈ 4.669 — a universal constant.
Drag the zoom window to magnify any region. The diagram is self-similar — every chaotic window contains a miniature copy of the entire diagram. Toggle Orbit paths to trace individual trajectories alongside the full bifurcation structure.
Mitchell Feigenbaum discovered the constant δ in 1975 on an HP-65 pocket calculator. Remarkably, the same constant appears in any smooth one-dimensional map with a single quadratic maximum — from the Hénon map to the driven pendulum. It is a universal law of chaos.
This simulation plots the bifurcation diagram of the logistic map, the iterated equation xn+1 = r·xn·(1−xn). For each growth parameter r along the horizontal axis (here 0 to 4), the map is iterated from x = 0.5, early transient steps are discarded, and the remaining values of x are plotted vertically. The resulting cloud of points reveals the long-term attractor: fixed points, periodic cycles, or chaotic bands.
The Iterations slider sets how many attractor points are collected per column (100–1000), while the Transient slider sets how many warm-up steps are skipped (50–500) so only the settled behaviour is shown. Preset buttons jump to the full range, the chaos onset near r = 3.57, or the period-2 region, and you can click and drag to zoom into any window. The diagram underpins chaos theory, modelling population dynamics, electronics and many nonlinear systems.
What is a bifurcation diagram?
It is a plot showing how the long-term behaviour of a dynamical system changes as a parameter is varied. Here the horizontal axis is the growth rate r and the vertical axis shows the values that x eventually settles onto. Where one line splits into two, the system has bifurcated from a period-1 cycle to a period-2 cycle.
What equation does this simulation use?
It iterates the logistic map, xn+1 = r·xn·(1−xn). This single quadratic recurrence is a simplified model of constrained population growth, where r is the reproduction rate and the (1−x) term represents resource limits that cap the population.
How is each column of the image computed?
For each pixel column a value of r is chosen, then the map is iterated starting from x = 0.5. The first Transient steps are discarded so the orbit reaches its attractor, after which the next Iterations values of x are plotted as points. Stable cycles appear as a few sharp dots; chaos appears as a dense vertical smear.
Transient (50–500) sets how many warm-up steps are run and thrown away so the orbit settles before plotting. Iterations (100–1000) sets how many attractor points are then drawn per column. More iterations reveal finer structure in chaotic regions, while a larger transient gives cleaner periodic bands.
For r below 3 the map settles to a single fixed point. At r ≈ 3 it doubles to a period-2 cycle, at r ≈ 3.449 to period-4, then 8, 16 and so on. These doublings accumulate at the Feigenbaum point r ≈ 3.56995, beyond which the behaviour becomes chaotic, interrupted by periodic windows.
It is the universal ratio δ ≈ 4.669 to which the spacing between successive period-doubling bifurcations converges. Remarkably, the same constant appears in any smooth one-dimensional map with a single quadratic maximum, making it one of the deep universal numbers of chaos theory, alongside the related scaling constant α ≈ 2.5029.
The bifurcation diagram is fractal and self-similar. Clicking and dragging a window restricts r and x to that range and re-renders at higher resolution. Inside the chaotic region you can find periodic windows, such as the prominent period-3 window near r ≈ 3.83, each containing a miniature copy of the whole diagram.
Yes, it iterates the exact logistic recurrence in double-precision floating point, so the bifurcation thresholds and the period-doubling cascade it draws match the textbook values. The main limits are pixel resolution and the finite number of iterations, which can blur very fine periodic windows or thin chaotic bands.
The logistic map keeps any x in the interval (0,1) within that interval for r up to 4, so x represents a normalised population fraction. Starting at x = 0.5 is a neutral mid-range seed; because the attractor is independent of the initial value (outside the unstable fixed points), the precise seed does not change the long-term picture.
The logistic map was popularised by ecologist Robert May to model insect populations with non-overlapping generations. The same period-doubling route to chaos appears in fluid convection, nonlinear electronic circuits, laser dynamics, cardiac rhythms and chemical oscillators, which is why the diagram is a touchstone of nonlinear dynamics.