Clas Blomberg - Physics of life-Elsevier Science (2007)

296 Part VII. Non-linearity
20
10
y
0
-10
-20
-30
-20 -10 0 10 20
x
Figure 28.2 The chaotic trajectories of the Lorenz equation, projected onto the xy-plane. The trajectories
go around the two stationary points (see the text), sometimes around one, sometimes around
the other in an irregular fashion.
The notations here are of a standard type for this equation. We may very briefly say some
words about the situation to which this equation refer. One has a liquid where convection is
coupled to a heat flow by a temperature gradient. Thus, there is a convection pattern established
in the liquid. This is represented by simple trigonometric expressions (usually called
“modes”). Of the three dimensions of the liquid, the convection currents appear around
lines in one direction. The trigonometric mode represents the flows in the two other directions,
of which one is along the temperature gradient. x represents the convection current in
that mode. y represents a temperature contribution that has the same trigonometric behaviour
as the convection current. z is a temperature contribution that only varies in the direction of
the temperature gradient. Its wavelength is half of that of the convection flow. The parameters
represent relevant flow parameters: s is the Prandtl number (the kinematic viscosity
divided by the thermal conductivity), r, the most important parameter is the Rayleigh number
(gah 3 /nk) divided by a certain geometric critical number. g is the ordinary gravity constant,
a the derivative of the density with respect to temperature of the liquid.his the depth
of the liquid in the temperature gradient direction, n the kinematic viscosity, and k the temperature
conduction constant. Finally b is a geometric constant, usually put equal to 8/3.
We start by a characterisation of this model as in the previous schemes. It is primarily of
interest to see what happens for different values of r. Besides choosing the value b 8/3, one
usually puts s 10, while r is the primary parameters that varies. If r 1, the only singular
point is the origin, which then is stable. When r 1, the origin is an unstable stationary
point (saddle point), and there are two new stationary points at x y √(b/r 1);
z r 1 that are stable for r up to the value r c 470/19(s(s b 3)/(s b 1))
24.7368. This value is found by linearising around these points and analysing the eigenvalues