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Lorenz attractor
The Lorenz attractor 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.
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].
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.
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].
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.
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.
Tucker, W. (2002). "A Rigorous ODE Solver and Smale's 14th Problem". Found. Comp. Math. 2: 53-117.
Strogatz, Steven H. (1994). Nonlinear Systems and Chaos. Perseus publishing.
Jonas Bergman, Knots in the Lorentz system, Undergraduate thesis, Uppsala University 2004.
P. Grassberger and I. Procaccia (1983). "Measuring the strangeness of strange attractors". Physica D 9: 189-208. doi:10.1016/0167-2789(83)90298-1.
External links
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.
Tucker, W. (2002). "A Rigorous ODE Solver and Smale's 14th Problem". Found. Comp. Math. 2: 53-117.
Strogatz, Steven H. (1994). Nonlinear Systems and Chaos. Perseus publishing.
Jonas Bergman, Knots in the Lorentz system, Undergraduate thesis, Uppsala University 2004.
P. Grassberger and I. Procaccia (1983). "Measuring the strangeness of strange attractors". Physica D 9: 189-208. doi:10.1016/0167-2789(83)90298-1.
External links
http://en.wikipedia.org/wiki/Lorenz_attractor
Eric W. Weisstein, Lorenz Attractor at MathWorld.
Lorenz Equation on planetmath.org
Lorenz Attractor Interactive Animation (you need the Adobe Shockwave plugin)
Levitated.net: computational art and design
3D VRML Lorenz Attractor (you need a VRML viewer plugin)
JAVA Applet - butterfly effect, Lorenz and Rossler attractors
Essay on Lorenz Attractors in J - see J programming language
Eric W. Weisstein, Lorenz Attractor at MathWorld.
Lorenz Equation on planetmath.org
Lorenz Attractor Interactive Animation (you need the Adobe Shockwave plugin)
Levitated.net: computational art and design
3D VRML Lorenz Attractor (you need a VRML viewer plugin)
JAVA Applet - butterfly effect, Lorenz and Rossler attractors
Essay on Lorenz Attractors in J - see J programming language
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