Linearizations in PPLANE5

A major new feature in PPLANE5 is the ability to display the linearization of a system at an equilibrium point. To see how it might be used consider the following situation in the Display Window.

The default system in PPLANE5 with the saddle point at the origin and the stable and unstable orbits plotted.

The linearization is displayed by clicking the appropriate button in the Equilibrium point data Window. Thus there is a two step process involved. First the option in the Solutions menu to Find an equilibrium point must be executed and then the button must be pushed in the Equilibrium point data Window.

The new Linearization Window is the same as the Display Window except that the system used is the linearization of the system in the Display Window. Solution curves for the linearization can be plotted by clicking the mouse at the initial point.

The linearization of the default system at the equilibrium point at the origin.

The Linearization Window is a square window, with the units in the two axes scaled the same. If this window is resized, this feature should be maintained in order to facilitate comparison with the Display Window.

By opening the Linearization Window, and then using the Zoom in square option in the Display Window, the user can compare the two, and observe whether the nonlinear system approaches the linearization in small neighborhoods of the equilibrium point.

For example, after zooming in on the origin in Display Window for the default system we get to the following:

The effect of using the Zoom in Square option to focus on the saddle point at the origin of the default system.

The similarity with the linearization in the previous figure is obvious.


John C. Polking <polking@rice.edu>
Last modified: Fri Jan 28 14:52:49 CST 2000