Lorenz system
1. Introduction to the Lorenz system
This system defines a 3 dimensional trajectory by differential equations with 3 parameters.
Here, \(x\) is proportional to the rate of convection, \(y\) is related to the horizontal temperature variation, and \(z\) is the vertical temperature variation.
We have also three parameters all strictly positive:
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\(\sigma > 0\) relates to the Prandtl number. This number is a dimensionless quantity that puts the viscosity of a fluid in correlation with the thermal conductivity;
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\(r > 0\) relates to the Rayleigh number, it is a control parameter, representing the temperature difference between the top and bottom of the tank;
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\(b > 0\) relates to the physical dimensions of the layer of fluid uniformly heated from below and cooled from above.
We can see that this system is non-linear, because in the second differential equation (\(\frac{dy}{dt}\)) we can see the term \(xz\) and in the third differential equation (\(\frac{dz}{dt}\)) we have \(xy\). The three differential equations form a coupled system.