Lorenz Attractor — The Butterfly Effect Visualized (Python Simulation)

Two paths start just 0.00000001 apart. By the end, they are in completely different places. This is the butterfly effect -- made visible.

The Lorenz Attractor was discovered in 1963 by meteorologist Edward Lorenz while running a simplified weather simulation. He noticed that rounding a number from 0.506127 to 0.506 caused the entire forecast to diverge. That tiny difference changed everything. Chaos theory was born.

WHAT YOU ARE SEEING:
- Red path (A) and blue path (B) start offset by just 10^-8
- They follow the same butterfly-shaped attractor at first
- Watch the separation graph -- it grows exponentially until the paths are completely independent
- The 3D view slowly rotates so you can see the full attractor structure

THE LORENZ SYSTEM (3 equations):
dX/dt = sigma * (Y - X)
dY/dt = X * (rho - Z) - Y
dZ/dt = X*Y - beta*Z

Parameters used: sigma=10, rho=28, beta=8/3 (the classic chaotic regime)

WHY IT NEVER REPEATS:
The attractor has infinite complexity -- the path spirals around two lobes forever, switching between them unpredictably, but NEVER crossing itself and NEVER repeating. This is called a strange attractor.

WHAT STUDENTS LEARN:
- Nonlinear differential equations and dynamical systems
- Sensitive dependence on initial conditions (Lyapunov exponents)
- Why long-range weather forecasting is fundamentally impossible
- 3D visualization and RK4 numerical integration in Python
- The difference between chaos and randomness

Integrator: RK4 -- 4th order Runge-Kutta, dt=0.005

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