Dependence on Parameters
The Lorenz model shows a rich variety of behavior as the
parameters a,b,c are changed. Usually b and c
are held fixed and a is varied. This corresponds to increasing
the nonlinearity, driving the system further away from equilibrium,
or in the convection context increasing the temperature difference
across the layer of fluid.
Reset the parameters to their default
values by quitting and restarting the applet. You might want to set
tr an=0 for this investigation.
Some examples of the behavior:
- For a < 1 the solution rapidly decays to the origin
X=Y=Z=0. This corresponds to no motion in the fluid
- For a > 1 (e.g. a=5) the orbit approaches one
of two fixed points (depending on the initial values) away
from the origin. The fixed points are at X 2=Y
2=Z=a-1. In the convection context this corresponds to
nonzero but steady fluid flow (in a circulating "roll"
- At larger values of a, for example a=24.1, the
long time dynamics may either approach one of the fixed points or
a strange attractor (depending on the choice of initial values),
which coexist at these values of a. (Choose nearby initial
values to find solutions that converge to the fixed points.)
- For a>24.74 the strange attractor collides with the
fixed points, which become unstable so that practically all
initial values lead to the familiar butterfly dynamics.
- a=28 gives the usual picture.
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Last modified 18 August, 2009