Exploring deterministic systems that exhibit unpredictable, chaotic behavior due to extreme sensitivity to initial conditions.
In classical mechanics, if we know the exact initial state of a deterministic system (like a pendulum), we can perfectly predict its future behavior forever. However, many systems in nature are nonlinear and exhibit what is known as deterministic chaos.
A chaotic system is highly sensitive to its starting conditions. Even the smallest measurement error or numerical rounding will exponentially compound over time, making long-term prediction fundamentally impossible, despite the system obeying strict deterministic laws.
Lyapunov exponent $\lambda$ defines the rate of separation.
The atmosphere is a highly chaotic fluid system. Because we cannot measure the temperature and pressure of every cubic inch of air on Earth perfectly, our weather forecasts lose accuracy beyond a week. The tiny unknown errors rapidly multiply and change the predicted outcome.
In 1963, meteorologist Edward Lorenz simplified the Navier-Stokes equations for atmospheric convection into three coupled nonlinear differential equations. When plotted in 3D space, the solutions trace out a beautifully structured, infinite, non-repeating path known as a strange attractor.
The system never perfectly repeats its past path, yet it is confined to the bounds of the "butterfly" shape. In this simulation, you can control the Rayleigh number ($\rho$) which drives the convection instability.
An attractor is a set of numerical values toward which a system tends to evolve. A "strange" attractor has fractal structure and chaotic dynamics on it.
The phrase "Butterfly Effect" stems from Lorenz's realization that a butterfly flapping its wings in Brazil could set off a cascade of atmospheric events leading to a tornado in Texas weeks later.
In the simulation, we track two distinct particles. Their starting positions are identical except for a microscopic difference of $1 \times 10^{-5}$ in the x-coordinate. Watch how their paths overlap perfectly at first, but eventually diverge exponentially, ending up on entirely different wings of the attractor.