Dynamic system of Lorenz

In 1963, the meteorologist Edward Lorenz has the first highlighted the probably chaotic character of meteorology.

Model of Lorenz

Mathematically, the coupling of the atmosphere with the Océan is described by the system of coupled partial derivative equations Navier-Stokes of the Mécanique of the fluids. This Système of equations was too much complicated with to numerically solve for the first computers existing at the time of Lorenz. This one had thus the idea to seek a very simplified model of these equations to study a particular physical situation: the phenomenon of Convection of Rayleigh - Bénard. It leads then to a dynamic Système differential having only three degrees of freedom, much simpler to integrate numerically than the starting equations.

Differential dynamic system of Lorenz

This differential connection is written:

$\backslash \; begin\; \{boxes\}\; \backslash \; frac\; \{\backslash \; mathrm\; \{D\}\; X\; (T)\}\{\backslash \; mathrm\; \{D\}\; T\}\; =\; \backslash \; sigma\; \backslash \; bigl\; (there\; (T)\; -\; X\; (T)\; \backslash \; bigr)\; \backslash \; \backslash \; \backslash \; frac\; \{\backslash \; mathrm\; \{D\}\; there\; (T)\}\{\backslash \; mathrm\; \{D\}\; T\}\; =\; \backslash \; rho\; \backslash ,\; X\; (T)\; -\; there\; (T)\; -\; X\; (T)\; \backslash ,\; Z\; (T)\; \backslash \; \backslash \; \backslash \; frac\; \{\backslash \; mathrm\; \{D\}\; Z\; (T)\}\{\backslash \; mathrm\; \{D\}\; T\}\; =x\; (T)\; \backslash ,\; there\; (T)\; -\; \backslash \; beta\; \backslash ,\; Z\; (T)\; \backslash \; end\; \{boxes\}$

In these equations, $\backslash \; sigma,\; \backslash \; rho$ - respectively the Number of Prandtl and the Number of Rayleigh - and β is three real parameters.

X (T) is proportional to the intensity of the movement of convection, there (T) is proportional to the difference in temperature between the ascending and downward currents, and Z (T) is proportional to the variation of the vertical profile of temperature compared to a linear profile.

Fixed points

The fixed points of the system are the solutions (X, there, Z) constant of the differential connection. There are three:

• the point fixes (0, 0,0) , which exists whatever the values of the real parameters $\backslash \; sigma,\; \backslash \; rho$ and β .

• two symmetrical fixed points: $\backslash \; left\; (-\; \backslash \; sqrt\; \{\backslash \; beta\; (\backslash \; rho\; -\; 1)\},\; -\; \backslash \; sqrt\; \{\backslash \; beta\; (\backslash \; rho\; -\; 1)\},\; \backslash \; rho\; -\; 1\; \backslash \; right)$ and: $\backslash \; left\; (\backslash \; sqrt\; \{\backslash \; beta\; (\backslash \; rho\; -\; 1)\},\; \backslash \; sqrt\; \{\backslash \; beta\; (\backslash \; rho\; -\; 1)\},\; \backslash \; rho\; -\; 1\; \backslash \; right)$, which exists only when $\backslash \; rho\; >\; 1$.

Attractile strange

See also: Attractile of Lorenz

When the parameters $\backslash \; sigma,\; \backslash \; rho$ and β take the following values: $\backslash \; sigma\; =\; 10$, $\backslash \; rho\; =\; 28$ and $\backslash \; beta=8/3$, the differential dynamic system of Lorenz presents superb a strange Attracteur in the shape of wings of butterfly, represented on the figure opposite.

For almost all the initial conditions (different from those of the fixed points), the orbit of the system walks on the attractile one, the trajectory starting by being rolled up on a wing, then jumping from one wing to another to start to be rolled up on the other wing, and so on, in an apparently erratic way.

Related articles

• Attractile
• dynamic System
• Theory of chaos
• Meteorology

Random links:

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