Lorenz system

The Lorenz attractor is the strange attractor of a system of three coupled nonlinear ordinary differential equations:

Was formulated to the system in 1963 by the meteorologist Edward N. Lorenz, who developed it as idealization of a hydrodynamic system. Based on a work by Barry Saltzman (1931-2001) it was Lorenz this is a modeling of the conditions in the Earth's atmosphere for the purpose of long-term prediction. However, Lorenz emphasized that the system he developed, c provides at best for very limited ranges of parameters of a, b realistic results.

Closely associated with the Lorenz attractor is the buzzword of the butterfly effect ( metaphor from chaos theory ). The system of differential equations was repeated in the public eye that tried with the chaotic behavior of the mathematical equations to explain phenomena of the real world: For the Lorenz system should make clear that in the atmospheric flow pattern small causes can have a great effect.

Derivation

For the derivation of the Lorenz equation as a description of convection, the following model was considered, experimentally investigated by the turn of the century by the French physicist Henri Bénard (1874-1939) and in 1916 by the British Nobel laureate Lord Rayleigh ( 1842-1919 ) has been described theoretically:

Between two plates at a short distance a viscous incompressible fluid ( liquid) located. While small temperature differences between the top and bottom of the layer can not be balanced by heat conduction, is at a critical temperature difference is exceeded, a liquid movement and leads to the formation of convection rolls, is realized by an efficient heat transport. This rise from the bottom heated fluid elements due to their lower density and colder volumes of liquid from falling. The mathematical description of the model by the Navier -Stokes equations via various simplifications, such as broken finally series representations of the system of equations given above. Hermann hook showed that the Lorenz system also processes in lasers can be modeled as the system is equivalent to the Maxwell Bloch equations.

Chaos theory

The numerical solution of the system shows when certain parameter values ​​deterministically chaotic behavior, the trajectories follow a strange attractor. Thus, the Lorenz attractor for the mathematical chaos theory plays a role, because the equations are arguably one of the simplest systems with chaotic behavior dar.

The typical parameter setting with chaotic solution is: a = 10, b = 28 and c = 8/ 3, where a can be identified with the Prandtl number and b with the Rayleigh number.

After clarification of the physical and technical foundations through said physicists and meteorologists involved in the second half of the 20th century many famous mathematicians with the problem, including the American mathematician John Guckenheimer. The proof that the Lorenz attractor is a strange attractor so-called, has been furnished by the mathematician Warwick Tucker ( born 1970, The Lorenz attractor exists, Department of Mathematics, Uppsala University, 1998) until 1999.

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