Is this just for rabbits?

No. This pattern (The Feigenbaum Constant) controls dripping faucets, heart fibrillations, and even the onset of turbulence in fluid dynamics.

What is that white vertical stripe?

That is an 'Island of Stability' ($r \approx 3.83$). Even deep in the chaotic zone, certain values force the system back into a predictable rhythm for a short time.

Why is it called the 'Logistic' map?

It comes from the 'Logistic Equation' used by Pierre François Verhulst in 1838 to model limited population growth.

The Chaos Microscope

Explore the fractal edge of chaos.

📜 The Origins

The Bifurcation Diagram of the Logistic Map is the most famous image in Chaos Theory. It shows how simple population growth formulas can fracture into infinite complexity.

🚀 Master the Tool

Click to zoom into the diagram. Use the 'Feigenbaum Ruler' to discover the universal constant δ ≈ 4.669 hidden in the spacing of the splits.

The Chaos Microscope

Click anywhere on the diagram to zoom in. Discover the fractal hidden within the chaos.

Feigenbaum Ruler

Detail (Iterations)1000 pts

Settle Time (Transient)200 skips

Viewport R: [2.80000 ... 4.00000]

Constant δ ≈ 4.6692016...

What am I looking at?

This is the Bifurcation Diagram of the Logistic Map. It shows how a population changes over time based on its growth rate ($r$).

r < 3.0: The population stabilizes at a single value (1 line).
r = 3.0: The first split! The population oscillates between two values (2 lines).
r ≈ 3.45: Second split (4 lines). This is "Period Doubling".
r ≈ 3.57: Chaos. The population becomes unpredictable/infinite.

The Feigenbaum Constant

Mitchell Feigenbaum discovered that the rate at which these splits happen is universal.

If you measure the distance between the first split and the second, then divide it by the distance between the second and the third, you get 4.669...

This number appears in fluid dynamics, heart beats, and electronic circuits. It is the universal fingerprint of chaos.

Zooming into Infinity

The image you are exploring is the Bifurcation Diagram of the Logistic Map. It is the "Roadmap to Chaos." It answers a simple question: *If a population grows and dies based on a fixed rate, will it stabilize?*

How to Read the Map

The X-Axis (Growth Rate $r$): Moves from a stagnant population (left) to an explosive one (right).
The Y-Axis (Population): Shows the stable population size(s) for that year.

The Journey

$r < 3.0$: One single line. The population is stable.
$r = 3.0$: The First Split. The population bounces between two numbers (Boom year, Bust year).
$r = 3.45$: The Second Split. Now a 4-year cycle.
$r = 3.5699...$: CHAOS. The lines dissolve into a cloud. The population effectively becomes random...
...Except for the Windows: Look at $r=3.83$. Suddenly, order returns! A stable 3-year cycle emerges from the noise. This "Island of Stability" contains a miniature copy of the entire diagram within it.