What do the sigma, rho and beta sliders change?

They are the three system parameters: sigma (the Prandtl number, 1 to 30), rho (the Rayleigh number, 10 to 60) and beta (a geometric factor, 0.5 to 6). The classic chaotic regime uses sigma=10, rho=28 and beta=8/3, about 2.67. Lowering rho below about 24.74 makes trajectories spiral into a stable point instead of behaving chaotically.

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🦋 Lorenz Attractor Dynamics Simulation

3D Lorenz attractor: up to 80 trajectories forming butterfly wings in real time. Adjust σ, ρ, β parameters. RK4 integration, chaos theory visualised.

Chaos & Dynamics3DModerate60 FPS

The Lorenz attractor is the icon of chaos theory: a fully deterministic system that is impossible to predict long-term. Tiny differences in starting conditions diverge exponentially — the famous butterfly effect.

🔬 What It Demonstrates

Three coupled differential equations, derived by Edward Lorenz in 1963 to model atmospheric convection, produce trajectories that never repeat yet stay confined to a fractal butterfly-shaped attractor with dimension ≈ 2.06.

🎮 How to Use

Drag to rotate the 3D attractor. Launch multiple particles with slightly different starting positions to visualise exponential divergence. Adjust ρ (rho), σ (sigma), and β (beta) to explore different attractor shapes.

💡 Did You Know?

Lorenz discovered chaos accidentally in 1961 by re-running a simulation with rounded values (0.506 instead of 0.506127). The forecast diverged completely, leading to his 1972 lecture asking whether a butterfly's wing-flap could trigger a tornado.

About this simulation

This visualiser traces the Lorenz attractor, the set of three coupled ordinary differential equations Edward Lorenz published in 1963 to model heat-driven atmospheric convection. Up to 80 particles are stepped through 3D state space, each leaving a colour-coded trail that settles onto the famous butterfly-shaped strange attractor. Because nearby trajectories separate exponentially, the system is the textbook illustration of deterministic chaos: completely rule-bound, yet impossible to forecast far ahead.

🔬 What it shows

Each curve obeys dx/dt = σ(y − x), dy/dt = x(ρ − z) − y and dz/dt = xy − βz. The animation advances every particle in tiny time steps (five sub-steps per frame) so you can watch initially neighbouring paths fan apart while remaining bound to the two-lobed attractor, whose fractal dimension is roughly 2.06.

🎮 How to use

Drag to rotate the scene and scroll to zoom. The σ, ρ and β sliders reshape the attractor; Particles sets how many trajectories run (1–80); Trail length controls how many recent points each trail keeps (100–3000); Speed × scales simulation time (0.2–6). Press ↺ Restart to re-seed every particle from fresh starting points.

💡 Did you know?

Lorenz met chaos by accident in 1961, when restarting a forecast from a rounded value (0.506 instead of 0.506127) produced a wildly different result. That insight led to his 1972 talk asking whether a butterfly's wing-flap in Brazil might trigger a tornado in Texas — the origin of the term butterfly effect.

Frequently asked questions

What is the Lorenz attractor?

It is the long-term behaviour of a three-variable system of differential equations introduced by Edward Lorenz in 1963 as a simplified model of convection. Instead of settling to a point or a loop, the solution wanders forever along a fixed butterfly-shaped surface in 3D space, called a strange attractor.

How does the simulation compute each trajectory?

For every particle it evaluates the three Lorenz equations using the current σ, ρ and β values, then advances x, y and z by a small time increment. The frame loop performs five sub-steps per frame for stability, and recent positions are stored in a ring buffer and drawn as a fading 3D trail.

What do the σ, ρ and β sliders change?

They are the three system parameters: σ (the Prandtl number, 1–30), ρ (the Rayleigh number, 10–60) and β (a geometric factor, 0.5–6). The classic chaotic regime uses σ=10, ρ=28 and β=8/3 ≈ 2.67. Lowering ρ below about 24.74 makes trajectories spiral into a stable point instead of behaving chaotically.

Why do particles that start together drift apart?

This is sensitive dependence on initial conditions, the defining feature of chaos. The attractor has a positive Lyapunov exponent, so the gap between two nearby trajectories grows roughly exponentially with time. Even identical equations and a microscopic difference in start point soon yield completely different paths.

Is the visualisation physically accurate?

The equations and the attractor shape are faithful, and the divergence you see is a genuine property of the system. It is, however, a heavily idealised toy model of convection rather than real weather, and the numerical integration uses a finite time step, so individual trajectories are approximations whose detailed paths depend on the chosen step size.

⚙ Under the hood

Observe the formation of intricate butterfly wing patterns as you manipulate parameters within this chaotic system, exploring the core principles of the Lorenz attractor.

Three.jsStrange AttractorRK4Chaos

3D · Three.js / WebGL renderer · 60 FPS target · runs fully client-side, no install