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Lorenz Attractor
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Lorenz Attractor
MATHStudio
A 3D Lorenz attractor: pick sigma, rho, and beta, then watch the iconic butterfly-wing trajectory unfold.
UpdatedMar 2026
ExportsPNG, JPG
Runs onAny browser
PriceFree
Overview
A dynamic math playground.
Lorenz Attractor integrates the Lorenz system — three coupled equations that exhibit sensitive dependence on initial conditions — and renders the path as a fading trail.
The default parameters (σ=10, ρ=28, β=8/3) produce the famous butterfly shape. Small changes in the parameters or the initial state produce wildly different trajectories within seconds.
Features
Built for math-scale experimentation.
01 / Spotlight
Real-time integration of
The three Lorenz equations with adjustable σ, ρ, β.
02
Fading trail; rotate
The 3D view by dragging and zoom with scroll.
03
Export PNG or
JPG of the current frame from the toolbar.
How to use
Workflow
Steps take about a minute.
Use the sidebar to dial in σ, ρ, β, and the initial (x, y, z). Drag to rotate the 3D view, scroll to zoom. Press Reset to clear the trail, and Export PNG/JPG to save a frame.
Use cases
What people use it for
Demonstrating chaos theory and the butterfly effectTeaching dynamical systems and strange attractorsGenerating butterfly-wing trail art for posters or coversExploring sensitive dependence on initial conditions interactively
FAQ
Common questions
What are the 'classic' parameters?+
σ = 10, ρ = 28, β = 8/3. With these values the system is chaotic and produces the iconic butterfly-wing shape.
Why does a small change in the parameters produce a totally different shape?+
That's sensitive dependence on initial conditions — the defining property of chaos. The Lorenz system has a positive Lyapunov exponent, so two trajectories that start arbitrarily close eventually diverge exponentially.
Related
More math apps
Lorenz Attractor
MATHStudio
A 3D Lorenz attractor: pick sigma, rho, and beta, then watch the iconic butterfly-wing trajectory unfold.
UpdatedMar 2026
ExportsPNG, JPG
Runs onAny browser
PriceFree
Overview
A dynamic math playground.
Lorenz Attractor integrates the Lorenz system — three coupled equations that exhibit sensitive dependence on initial conditions — and renders the path as a fading trail.
The default parameters (σ=10, ρ=28, β=8/3) produce the famous butterfly shape. Small changes in the parameters or the initial state produce wildly different trajectories within seconds.
Features
Built for math-scale experimentation.
01 / Spotlight
Real-time integration of
The three Lorenz equations with adjustable σ, ρ, β.
02
Fading trail; rotate
The 3D view by dragging and zoom with scroll.
03
Export PNG or
JPG of the current frame from the toolbar.
How to use
Workflow
Steps take about a minute.
Use the sidebar to dial in σ, ρ, β, and the initial (x, y, z). Drag to rotate the 3D view, scroll to zoom. Press Reset to clear the trail, and Export PNG/JPG to save a frame.
Use cases
What people use it for
Demonstrating chaos theory and the butterfly effectTeaching dynamical systems and strange attractorsGenerating butterfly-wing trail art for posters or coversExploring sensitive dependence on initial conditions interactively
FAQ
Common questions
What are the 'classic' parameters?+
σ = 10, ρ = 28, β = 8/3. With these values the system is chaotic and produces the iconic butterfly-wing shape.
Why does a small change in the parameters produce a totally different shape?+
That's sensitive dependence on initial conditions — the defining property of chaos. The Lorenz system has a positive Lyapunov exponent, so two trajectories that start arbitrarily close eventually diverge exponentially.
Related