The Lorenz system contains three parameters,
denoted ,
and
.
The default values for the parameters are
,
,
and
.
These values appear in panel text items located on
the main panel. The default values correspond
to the values originally studied by Lorenz.
For the default value of
,
the Lorenz
system possesses a global attractor, but for other
parameter values different initial conditions
may end up in different attractors. This exercise
will help you to learn how to alter parameter values
and to control the initial conditions with a high
degree of accuracy.
One method of changing parameters is by editing text
fields in the Selected Point panel. If it is not
already open, open it with
from the menu entry Selected...
under the Settings menu button of the
command window. (Figure
)
Change the text item rho to the value 19.
Clear the view window by
and make sure
the xz plane is displayed. Restore the range of x values
to the original -30 to 30 and the z to -5 to
50. On the Selected Point panel, set the value of the
items x and
z to 0.1, and set the value of
y to -0.1.
Now select the button item Forward in the Orbits panel. The orbit starting at the point (0.1, -0.1, 0.1) is drawn in towards an attracting fixed point. Select Continue and watch the trajectory spiral in to the sink.
Now change the value of y to -0.099,
and select Forward again. The orbit of the
point
(0.1, -0.099, 0.1) is attracted to a
second (different) attracting point! (Figure )
You can also use the mouse to change parameters. By clicking
on the View menu button of the command
window,
open a second 2-D Image window. Change the horizontal and
vertical variables being displayed to the parameters
and
.
Note that by clicking
on different locations
in this window, the values of
and
in the Selected
Point panel change accordingly. You can mark the selected point on
this window by selecting Display... from the Options menu
button on the view window. The popup box allows you to select various
symbols and sizes to mark locations in the view window. Click
on Add Point to actually mark a location with the
symbol you have selected.
Being able to control the initial conditions of a trajectory with a high degree of accuracy is often crucial in the exploration of dynamical systems, and in these cases it is often preferable to input initial conditions as text items rather than via a mouse.