Lorenz Attractor

The Lorenz attractor is one of the most iconic objects in the study of chaos theory. It arises from a simplified model of atmospheric convection proposed by meteorologist Edward Lorenz in 1963. Despite being governed by just three deterministic equations, the system exhibits wildly unpredictable behaviour — the famous "butterfly effect".

The Lorenz System

The system consists of three coupled ordinary differential equations:

$\dot{x} = \sigma(y - x) \qquad \dot{y} = x(\rho - z) - y \qquad \dot{z} = xy - \beta z$

Parameter	Symbol	Physical Meaning
Prandtl number	$\sigma$	Ratio of momentum diffusivity to thermal diffusivity
Rayleigh number	$\rho$	Driving force — temperature difference across the fluid layer
Geometric factor	$\beta$	Related to the aspect ratio of the convection cell

What Makes It Chaotic?

• The system is deterministic — given exact initial conditions, the future is completely determined.
• Yet it is extremely sensitive to initial conditions: two trajectories starting almost identically will diverge exponentially over time.
• The trajectory never repeats exactly and never settles to a fixed point, yet it stays confined to a strange attractor — that butterfly-shaped region in 3D space.

"Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?" — Edward Lorenz, 1972

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Things to Try

1. Classic chaos — Set $\sigma = 10$, $\rho = 28$, $\beta = 8/3$. The trajectory should trace two lobes, switching unpredictably between them.
2. Edge of chaos — Lower $\rho$ toward 21. The attractor collapses into a stable periodic orbit.
3. Increase $\rho$ beyond 28 — Watch the attractor expand and the switching pattern become even more irregular.
4. Vary $\sigma$ — A higher Prandtl number makes the trajectory "stickier" on each lobe before switching.

Why Does It Matter?

• Weather prediction: Lorenz's discovery showed that long-range weather forecasting has a fundamental limit — not because our models are bad, but because the atmosphere is inherently chaotic.
• Nature is full of chaos: Turbulence in fluids, population dynamics, the double pendulum and even cardiac rhythms can exhibit similar behaviour.
• Strange attractors & fractals: The Lorenz attractor has a fractal structure with a dimension of about 2.06. It occupies zero volume in 3-D space, yet has an infinite surface area.

A note on the numerics — This simulation uses the Euler method (simplest first-order integrator) with a fixed time step of 0.01. For chaotic systems, numerical errors grow exponentially, so the specific trajectory you see will diverge from the "true" solution after some time. However, the overall attractor shape and statistical properties are faithfully reproduced.