Standard Tier Module
Lorenz Attractor (Chaos Demonstrator)
The Lorenz butterfly with a measured Lyapunov exponent and honest numerical trust.
See it run - a worked example, 100% in this browser tab
The problem
Chaos demos quote the Lyapunov exponent as a constant and offer no signal of whether their own integration is numerically sound.
The local-first solution
This plugin integrates the exact Lorenz 1963 equations with RK4 in the browser, estimates the largest Lyapunov exponent in-run by Benettin renormalization, and reports a mechanism-true numerical trust signal against the published value.
What it does
Exact Lorenz 1963 vector field integrated with classical RK4
Closed-form fixed points sqrt(beta(rho-1)) marked on the attractor
In-run Benettin estimate of the largest Lyapunov exponent vs literature
GeoNum conditioning signal on the cancellation-prone RHS subtraction
Honest Richardson self-check fallback when the GeoNum kernel is unreachable
Honest scope
Exact RHS, RK4 update, and cited fixed points; the Lyapunov exponent is computed and compared to the published ~0.9056, not quoted. The trust verdict is about numerical fidelity only - long-term state past the Lyapunov horizon is intrinsically unpredictable, and the full exponent spectrum is out of scope. A demonstrator, not engineering advice.
Authorities cited
- Lorenz, E. N. (1963). Deterministic Nonperiodic Flow. Journal of the Atmospheric Sciences 20(2), 130-141. DOI 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 - the original equations dx=sigma(y-x), dy=x(rho-z)-y, dz=xy-beta z.
- Strogatz, S. H. (2015). Nonlinear Dynamics and Chaos, 2nd ed., Ch. 9 (eq. 9.2) - the canonical parameters sigma=10, beta=8/3, rho=28, the fixed points C+/- at x=y=+/-sqrt(beta(rho-1)), z=rho-1, and sensitive dependence on initial conditions.
- Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.-M. (1980). Lyapunov Characteristic Exponents for smooth dynamical systems. Meccanica 15, 9-30 - the renormalization method used to estimate the largest Lyapunov exponent in-run.
- Sprott, J. C. (2003). Chaos and Time-Series Analysis. Oxford Univ. Press - published largest Lyapunov exponent lambda ~ 0.9056 for the canonical Lorenz attractor (the reference value compared against).
- Hairer, E., Norsett, S. P., Wanner, G. (1993). Solving Ordinary Differential Equations I, 2nd ed., Springer - the classical 4th-order Runge-Kutta method and its O(dt^4) global error (the basis of the Richardson dt vs dt/2 self-convergence check).
- Goldberg, D. (1991). What Every Computer Scientist Should Know About Floating-Point Arithmetic. ACM Computing Surveys 23(1), 5-48. DOI 10.1145/103162.103163 - conditioning of subtraction (the (y - x) GeoNum drift probe).
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Run the attractor
Run the Lorenz system in the browser and save the structured result to Sandbox, attach it to a Worklog case, or route it into a Gate client portal. Nothing leaves your machine to anyone's cloud.