Simple Properties of the Lorenz Equations

In this section we ll follow in Lorenz s footsteps. He took the analysis as far as possible using standard techniques, but at a certain stage he found himself confronted with what seemed like a paradox. One by one he had eliminated all the known possibilities for the long-term behavior of his system he showed that in a certain range of parameters, there could be no stable fixed points and no stable limit cycles, yet he also proved that all trajectories remain confined to a bounded region and are eventually attracted to a set of zero volume. What could that set be And how do the trajectories move on it As we ll see in the next section, that set is the strange attractor, and the motion on it is chaotic. 

But first we want to see how Lorenz ruled out the more traditional possibilities. As Sherlock Holmes said inTheSign of Four, When you have eliminated the impossible, whatever remains, however improbable, must be the truth.  

Here T, r, b 0 are parameters, a is the Prandtl number, r is the Rayleigh number, and b has no name. (In the convection problem it is related to the aspect ratio of the rolls.) 

The system (1) has only two nonlinearities, the quadratic terms xy and xz. This should remind you of the waterwheel equations (9.1.9), which had two nonlinearities, and mb I - See Exercise 9.1.3 for the change of variables that transforms the waterwheel equations into the Lorenz equations. 

There is an important symmetry in the Lorenz equations. If we replace 

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Since polarisability is a tensor quantity, the resulting optical properties may also be directionally dependent unless this tensor is isotropic. A simple example is illustrated by Fig. 1 where one can envisage that the interaction of the bond electrons will be greater for the imposed field that is oscillating in a plane parallel to the direction of the bond than for a field oscillating in a plane perpendicular to the bond, i.e. where the polarisability is highest. This interaction leads to a decrease in the velocity of the incident wave by an amount defined by the refractive index, n. For a non-absorbing system, the polarisability is related to the refractive index by the Lorenz-Lorentz equation ... 

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