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How to Change Parameters

The Lorenz system contains three parameters, denoted $\sigma$, $\beta$ and $\rho$. The default values for the parameters are $\sigma = 10$, $\beta = 8/3$, and $\rho = 28$. 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 $\rho = 28$, 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 \framebox{MENU} 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 \framebox{Control-ADJUST} 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 \framebox{SELECT} 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 $\sigma$ and $\rho$. Note that by clicking \framebox{SELECT} on different locations in this window, the values of $\sigma$ and $\rho$ 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 \framebox{SELECT} 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.


next up previous contents
Next: Loading Other Dynamical Systems Up: The Lorenz Equations Previous: Orbits Panel   Contents
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1998-11-02