Lorenz attractor

Graphic representation of a Lorenz attractor

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

{\ displaystyle {\ begin {aligned} {\ dot {X}} & = a (YX) \\ {\ dot {Y}} & = X (bZ) -Y \\ {\ dot {Z}} & = XY-cZ \ end {aligned}}}

The system was formulated around 1963 by the meteorologist Edward N. Lorenz (1917–2008), who developed it as an idealization of a hydrodynamic system. Based on a work by Barry Saltzman (1931–2001), Lorenz was concerned with modeling the conditions in the earth's atmosphere for the purpose of long-term forecasting . However, Lorenz emphasized that the system developed by him only provides realistic results for very limited parameter ranges. ${\ displaystyle a, b, c}$

Closely related to the Lorenz attractor is the catchphrase of the butterfly effect (metaphor from chaos research ). The system of differential equations was repeatedly in the focus of the public, who tried to explain phenomena in the real world with the chaotic behavior of mathematical equations: The Lorenz system was intended to make clear that small causes can have a big effect in atmospheric flow patterns.

Animated Lorenz attractor; click to start the animation

Derivation

To derive the Lorenz equation as a description of convection currents, the following model was considered, which was experimentally investigated by the French physicist Henri Bénard at the turn of the century and theoretically described in 1916 by the British Nobel Prize winner Lord Rayleigh :

Convection model, assigned point in phase space .

A viscous, incompressible fluid (liquid) is located between two plates with a small distance. While small temperature differences between the top and bottom of the layer can still be compensated for by heat conduction, if a critical temperature difference is exceeded, a liquid movement sets in and convection rolls develop , through which more efficient heat transport is realized. Liquid elements heated from below rise due to their lower density and colder liquid volumes sink.

The mathematical description of the model using the Navier-Stokes equations leads to the above-mentioned system of equations via various simplifications, for example finitely broken series representations.

Hermann Haken showed that processes in lasers can also be modeled with the Lorenz system , since the system is equivalent to the Maxwell-Bloch equations .

Chaos theory

Butterfly graphic: A 3D representation of 2900 points of the Lorenz attractor calculated numerically using a Runge-Kutta method with a fixed step size.

The numerical solution of the system shows deterministic chaotic behavior for certain parameter values , the trajectories follow a strange attractor . The Lorenz attractor therefore plays a role in mathematical chaos theory, because the equations represent one of the simplest systems with chaotic behavior.

The typical parameter setting with a chaotic solution is: and , where the Prandtl number and the Rayleigh number can be used for identification. ${\ displaystyle a = 10, b = 28}$ ${\ displaystyle c = 8/3}$ ${\ displaystyle a}$ ${\ displaystyle b}$

After the physicists and meteorologists mentioned clarified the physical and technical principles, numerous well-known mathematicians dealt with the problem in the second half of the 20th century, including the American mathematician John Guckenheimer . The proof that the Lorenz attractor is a so-called strange attractor was only provided in 1999 by the mathematician Warwick Tucker (born 1970, The Lorenz attractor exists , Department of Mathematics, Uppsala University 1998).

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↑ H. Haken: Analogy between higher instabilities in fluids and lasers in Physics Letters 53A (1975)

Web links

[1] H. Haken: Analogy between higher instabilities in fluids and lasers in Physics Letters 53A (1975)