The butterfly that started chaos theory - sensitivity to initial conditions
In 1963, meteorologist Edward Lorenz made a startling discovery while running weather simulations. When he rounded a number from 0.506127 to 0.506, the entire simulation diverged completely - proving that long-term weather prediction is fundamentally impossible.
This "sensitive dependence on initial conditions" became known as the butterfly effect: the idea that a butterfly flapping its wings in Brazil could set off a tornado in Texas. The Lorenz attractor is the visual representation of this chaotic system.
Watch the trajectory trace out the iconic butterfly shape. Enable "Diverging Paths" to see how two particles starting 0.001 apart eventually end up in completely different places:
Drag to rotate, scroll to zoom. The classic Lorenz parameters are σ=10, ρ=28, β=8/3. Enable "Diverging Paths" to see how tiny differences lead to completely different trajectories - the butterfly effect!
The system is defined by three coupled differential equations:
Where σ (sigma), ρ (rho), and β (beta) are parameters. The classic chaotic values are σ=10, ρ=28, β=8/3.
A set that the system evolves toward. No matter where you start, you'll eventually end up near the attractor.
The attractor has fractal structure - it's neither a point nor a smooth curve, but something in between with non-integer dimension.
The Lorenz attractor has a fractal dimension of approximately 2.06. It's more than a surface but less than a volume - the trajectory fills space in a peculiar fractal way.
The attractor has two "wings" - the system spirals around one wing for a while, then unpredictably switches to the other. The number of times it circles each wing before switching is chaotic and impossible to predict long-term.
This is similar to weather: the system may be in one "mode" (sunny) for a while, then switch to another (rainy). When it switches and how long it stays is fundamentally unpredictable beyond a few days.
Before Lorenz, scientists believed that more precise measurements and more powerful computers would eventually allow perfect prediction. Chaos theory revealed a fundamental limit: some systems are inherently unpredictable, not because we lack information, but because tiny errors grow exponentially.
This has profound implications for weather forecasting, economics, ecology, and anywhere complex nonlinear systems appear. It's not that these systems are random - they follow precise rules - but they're deterministically unpredictable.