AI authored to showcase ironpad capabilities.

Lorenz Attractor: Deterministic Chaos 🦋

In 1963, meteorologist Edward Lorenz discovered that a simple system of three coupled differential equations could produce wildly unpredictable, non-repeating motion. This was the birth of chaos theory and the famous butterfly effect.

The Lorenz System

The three equations describe convective fluid motion:

\frac{dx}{dt} = \sigma(y - x)

\frac{dy}{dt} = x(\rho - z) - y

\frac{dz}{dt} = xy - \beta z

where:

• \sigma (sigma): Prandtl number, ratio of momentum diffusivity to thermal diffusivity
• \rho (rho): Rayleigh number ratio, related to the temperature difference driving convection
• \beta (beta): geometric factor

The Butterfly Attractor

With the classic parameters \sigma = 10, \rho = 28, \beta = 8/3 \approx 2.667, the system never settles: the trajectory orbits two lobes forever, tracing the iconic butterfly-shaped strange attractor.

Sensitivity to initial conditions: Two trajectories starting 10^{-10} apart will diverge exponentially. This is the mathematical heart of the butterfly effect: a butterfly flapping its wings in Brazil could (in principle) set off a tornado in Texas.

This Notebook

• Cell 2: interactive sliders to tune \sigma, \rho, \beta in real time
• Cell 3: live 60 FPS simulation with RK4 integration and spectral trail rendering
• Cell 4: real-time dashboard showing current position and parameters