The Lorenz attractor, named for Edward N. Lorenz, is a fractal structure corresponding to the longterm behavior of the Lorenz oscillator. The Lorenz oscillator is a 3dimensional dynamical system that exhibits chaotic flow, noted for its lemniscate shape. The map shows how the state of a dynamical system (the three variables of a threedimensional system) evolves over time in a complex, nonrepeating pattern.
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Overview
The attractor itself, and the equations from which it is derived, were introduced by Edward Lorenz in 1963, who derived it from the simplified equations of convection rolls arising in the equations of the atmosphere.
In addition to its interest to the field of nonlinear mathematics, the Lorenz model has important implications for climate and weather prediction. The model is an explicit statement that planetary and stellar atmospheres may exhibit a variety of quasiperiodic regimes that are, although fully deterministic, subject to abrupt and seemingly random change.
From a technical standpoint, the Lorenz oscillator is nonlinear, threedimensional and deterministic. In 2001 it was proven by Warwick Tucker that for a certain set of parameters the system exhibits chaotic behavior and displays what is today called a strange attractor. The strange attractor in this case is a fractal of Hausdorff dimension between 2 and 3. Grassberger (1983) has estimated the Hausdorff dimension to be 2.06 ± 0.01 and the correlation dimension to be 2.05 ± 0.01.
The system also arises in simplified models for lasers (Haken 1975) and dynamos (Knobloch 1981).
Equations
The equations that govern the Lorenz oscillator are:
where σ is called the Prandtl number and ρ is called the Rayleigh number. All σ, ρ, β > 0, but usually σ = 10, β = 8/3 and ρ is varied. The system exhibits chaotic behavior for ρ = 28 but displays knotted periodic orbits for other values of ρ. For example, with ρ = 99.96 it becomes a T(3,2) torus knot.
Rayleigh number
Source code
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