2D Lorenz Attractor — Butterfly Projection & Lorenz Map

Lorenz parameters

σ (sigma) 10.0

ρ (rho) 28.0

β (beta) 2.67

Integration speed 1.0×

Trail length 2500

Presets

Divergence (butterfly effect)

Trajectory A

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Trajectory B (+1e-5)

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Separation |A−B|

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Time elapsed (model units)

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Two trajectories start 0.00001 apart in x. Watch the separation grow from the FAQ/theory box below — this is the same sensitive dependence the 3D version shows, measured directly as a number here instead of only as a visual fan-out.

Equations & the Lorenz map

The main canvas integrates the real Lorenz system with 4th-order Runge–Kutta and projects the 3D state onto the x–z plane (the classic "butterfly wings" view, seen edge-on).

dx/dt = σ(y − x)
dy/dt = x(ρ − z) − y
dz/dt = xy − βz

The lower chart is the Lorenz map: every time z reaches a local maximum (a peak in the z-coordinate), the simulation records that peak value zₙ, then plots it against the next peak zₙ₊₁. Lorenz discovered in 1963 that these points fall almost exactly on a single curve — proof that a fully three-dimensional, seemingly random-looking attractor is secretly governed by a simple one-dimensional rule from peak to peak.

Trajectory A Trajectory B (perturbed)

Lorenz map — zₙ vs. zₙ₊₁ (successive z-peaks)0 peaks