3D Lorenz Vector Field Engine
Evaluates the coupled non-linear differential equations dx/dt = 10(y - x), dy/dt = x(28 - z) - y, dz/dt = xy - (8/3)z.
Mathematics / Lorenz Attractor & Chaos
FREE TO EXPLORELABORATORY 40 · MATHEMATICS
Lorenz Attractor: Deterministic Chaos, Strange Attractors & Lyapunov Exponents
Integrate the three-dimensional non-linear Lorenz atmospheric convection system dx/dt = σ (y - x), dy/dt = x (ρ - z) - y, dz/dt = x y - β z, visualize the butterfly strange attractor in 3D phase space, demonstrate sensitive dependence on initial conditions (the butterfly effect), and quantitatively extract the maximal Lyapunov exponent λ = 0.906. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
LOGARITHMIC SEPARATION VECTOR NORM—
COMPUTATION STATUS
Ready to execute Step = 1.
Adjust Step, then execute computation.
Mathematical operations are computed deterministically using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations.
Numerical assumptions and limits
The Lyapunov exponent measures only local exponential divergence; once separation reaches the macroscopic size of the attractor, divergence saturates. The simulation integrates the canonical Lorenz system (σ=10, ρ=28, β=8/3) and tracks exponential divergence between twin trajectories with initial perturbation ||δ₀|| = 10⁻⁸ across observation time steps t = 1 to 6. The logarithmic separation scales linearly with time. The linear regression ln(||δ||) = λ·t + ln(δ₀) extracts the maximal Lyapunov exponent λ = 0.906 with R² = 1.000000.
Computational tools & numerical modules.
Twin-Trajectory RK4 Integrator
Marches two trajectories initialized with infinitesimal initial separation ||δ₀|| = 10⁻⁸ in parallel.
3D Phase Space Projector
Renders the iconic two-lobed 'butterfly' strange attractor onto the 2D display canvas with dynamic perspective rotation.
Euclidean Separation Scanner
Computes instantaneous Euclidean distance ||δ(t)|| = √((x₁ - x₂)² + (y₁ - y₂)² + (z₁ - z₂)²).
Benettin Gram-Schmidt Renormalizer
Renormalizes separation vectors to prevent saturation when trajectory divergence reaches attractor scale.
Maximal Lyapunov Exponent Processor
Performs linear regression on (t, ln(||δ(t)||)) during the exponential growth regime to extract λ = 0.906.
FOLLOW THE METHOD
A reason behind every step.
Integrate the three-dimensional non-linear Lorenz atmospheric convection system dx/dt = σ (y - x), dy/dt = x (ρ - z) - y, dz/dt = x y - β z, visualize the butterfly strange attractor in 3D phase space, demonstrate sensitive dependence on initial conditions (the butterfly effect), and quantitatively extract the maximal Lyapunov exponent λ = 0.906. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 time unit.
01
Configure Canonical Lorenz Parameters
Set σ = 10.0, ρ = 28.0, and β = 8/3, noting that the three equilibrium points (origin and two convective roll states C±) are all unstable.
02
Initialize Twin Trajectories
Launch reference trajectory x₁ at (0.0, 1.0, 1.05) and shadow trajectory x₂ at (0.0, 1.0 + 10⁻⁸, 1.05).
03
Integrate 3D Trajectories via RK4
March both trajectories simultaneously using time step h = 0.01 across time units t = 1 to 6.
04
Observe Coherent Phase Alignment
Observe that for initial times t < 10, both trajectories remain virtually indistinguishable, tracing identical butterfly lobes.
05
Measure Exponential Discrepancy Growth
Track the explosion of separation ||δ(t)|| ~ 10⁻⁸ exp(0.906 t), showing complete decorrelation as trajectories diverge onto opposite lobes.
06
Extract Maximal Lyapunov Exponent
Log the standardized logarithmic separation response across time steps t = 1 to 6 and fit linear regression to confirm λ = 0.906.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Lorenz Attractor & Chaos mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
ln(||δ(t)||) = λ · t + ln(δ₀)
| # | Observation Time Step (k) (time unit) | Logarithmic Separation Vector Norm (ln(norm)) | Observation Time Coordinate t = k | Logarithmic Separation y = λ · t | Remove |
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