A plot of the trajectory Lorenz system for values ρ=28, σ = 10, β = 8/3
The Lorenz attractor, named for Edward N. Lorenz, is a 3-dimensional structure corresponding to the long-term behavior of a chaotic flow, noted for its butterfly shape. The map shows how the state of a dynamical system (the three variables of a three-dimensional system) evolves over time in a complex, non-repeating pattern.
Overview
A trajectory of Lorenz's equations, rendered as a metal wire to show direction and 3D structure
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.
From a technical standpoint, the system is nonlinear, three-dimensional 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 arises in lasers, dynamos, and specific waterwheels. [1]
Equations
Trajectory with scales added
The equations that govern the Lorenz attractor 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.
Butterfly effect
Rayleigh number
| The Lorenz attractor for different values of ρ |
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| ρ=14, σ=10, β=8/3 (Enlarge) |
ρ=13, σ=10, β=8/3 (Enlarge) |
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| ρ=15, σ=10, β=8/3 (Enlarge) |
ρ=28, σ=10, β=8/3 (Enlarge) |
| For small values of ρ, the system is stable and evolves to one of two fixed point attractors. When ρ is larger than 24.28, the fixed points become repulsors and the trajectory is repelled by them in a very complex way, evolving without ever crossing itself. |
| Java animation showing evolution for different values of ρ |
See also
References
- Lorenz, E. N. (1963). "Deterministic nonperiodic flow". J. Atmos. Sci. 20: 130-141. doi:10.1175/1520-0469(1963)020%3C0130:DNF%3E2.0.CO;2 .
- Frøyland, J., Alfsen, K. H. (1984). "Lyapunov-exponent spectra for the Lorenz model". Phys. Rev. A 29: 2928–2931. doi:10.1103/PhysRevA.29.2928.
- Jonas Bergman, Knots in the Lorentz system, Undergraduate thesis, Uppsala University 2004.
External links
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