Routes to Chaos

Period-doubling, Feigenbaum constants, intermittency, and quasiperiodicity

🎯 Hook

💡 Intuition

📚 Theory

🎮 Playground

💻 Code

🔧 Debug

✏️ Practice

The Universal Number of Chaos

Is there a mathematical constant hiding in the transition to chaos?

Watch a simple system transform from order to chaos - and discover the surprising pattern in how it happens.

The Period-Doubling Mystery

The logistic map transitions from stable behavior to chaos as we increase parameter r. But the transitions happen at specific values: r_1 = 3.0, r_2 = 3.449, r_3 = 3.544... Watch it unfold!

Long-term Behavior

Bifurcation Diagram

The Key Question:

Notice that the gaps between bifurcations get smaller: (3.449 - 3.0) = 0.449, then (3.544 - 3.449) = 0.095, then even smaller. What is the ratio of consecutive gaps?

0.449 / 0.095 = ?

What do you think this ratio converges to?

Try it! Click "Watch the Transition" to see chaos emerge step by step!

The Wild

The Wild

The Wild

The Wild

Routes to Chaos

Period-doubling, Feigenbaum constants, intermittency, and quasiperiodicity

The Universal Number of Chaos

Is there a mathematical constant hiding in the transition to chaos?

The Period-Doubling Mystery

Long-term Behavior

Bifurcation Diagram

The Key Question:

What do you think this ratio converges to?

Routes to Chaos

Period-doubling, Feigenbaum constants, intermittency, and quasiperiodicity

The Universal Number of Chaos

Is there a mathematical constant hiding in the transition to chaos?

The Period-Doubling Mystery

Long-term Behavior

Bifurcation Diagram

The Key Question:

What do you think this ratio converges to?