The Bifurcation Diagram
This is the "fig tree" — a map of every long-term behavior of the logistic map at once. Each vertical slice shows the values the population settles onto for that growth rate. Where one line splits into two, a bifurcation has happened. Drag a box to zoom in and see what's hiding inside.
Jump to
Drag a box on the diagram to zoom into any region. Zoom into a fork and you'll find the whole tree again.
The tree inside the tree
Zoom into any branch tip in the doubling cascade. You'll see it splits, and those splits split, and so on — forever. The diagram contains tiny copies of itself. That's a fractal.
Now visit the period-3 window — a slice of calm inside the chaos — and zoom into its edge. The same branching tree grows there too.
Keep zooming and you'll eventually hit a wall: the computer only stores about 15 digits, so it runs out of room. The math goes on forever — the machine can't.
A ruler for the road to chaos
Turn on Mark the doubling points. Each split happens about 4.669× sooner than the one before — a number called Feigenbaum's constant.
The astonishing part: that same 4.669 shows up in dripping faucets, heartbeats, and electronic circuits. The route into chaos has a universal shape.
For teachers & grown-ups
Each column is one value of r: the logistic map xₙ₊₁ = r·xₙ·(1−xₙ) is iterated, an initial transient is discarded, and the remaining long-run values are plotted — so a single dot means a fixed point, two dots a period-2 cycle, a smear means chaos. Zooming actually recomputes the selected window at full resolution, which is why new detail keeps appearing: the diagram is self-similar (a fractal). The period-doubling bifurcations at r ≈ 3, 3.449, 3.544, 3.5644, … accumulate at r∞ ≈ 3.5699, and the ratios of successive intervals converge to the Feigenbaum constant δ ≈ 4.6692 — a universal constant shared by all maps with a smooth quadratic maximum, which is why the same number governs the onset of chaos in many real physical systems. The period-3 window (≈ 3.8284) is a good payoff: by Sharkovskii's theorem, "period 3 implies chaos," yet the window itself contains its own clean period-doubling cascade.