Math 312 - Applied Mathematics: Social Sciences

Math 312 - Applied Mathematics: Social Sciences

Spring 2006

Some Notes on Using the PPLANE Java Applet

The PPLANE Java applet was written by John C. Polking, a professor at Rice University. This is a nice phase plane plotter that anyone with a web browser and Java can use.

This page is contains a brief set of notes about using the applet.

The PPLANE Java applet can be found at the dfield and pplane web page at http://math.rice.edu/~dfield/dfpp.html.

Below the title that says "dfield and pplane", you should see two buttons, one labeled DFIELD 2005.10 and the other labeled PPLANE 2005.10. If you don't see these labels on the button, then your web browser probably does not have Java enabled. You will have to install Java or use a different computer. (The computers in the Math Department Computer Lab have Java installed.)

Click on PPLANE to start the applet. This will cause several windows to appear.

One of the windows has a short description of PPLANE in green text. Click on "Ok" in this window to close it.

There are three remaining windows.

The PPLANE Phase Plane window shows the vector field of the current set of equations.
The PPLANE Equation Window lets you enter your equations.
The PPLANE Messages window shows occasional output based on calculations done with the equations.

Let's start with a linear system.
x' = 0.5*x - y
y' = x - y
(This is the same system dicussed in Example 2.12.3 in the notes on Planar Linear Systems.)

In the equation window, enter this system into the first set of text fields in the window. Put the x in the upper field to the left of the =, and don't type the prime ('). Put the right side of the equation to the right of the =. As you type, the fields may be displayed with a red background if the text contains any errors. Don't worry about the red background until you finished typing your equation. If the display is still red after you have finished entering the system, check your equation for typing errors.

Modify the Display Window ranges as follows:
   Minimum x = -2
   Maximim x = 2
   Minimum y = -2
   Maximum y = 2

Now click on the button labeled "Graph Phase Plane". The Phase Plane window will then show the vector field for the system. If you use the mouse to click in the phase plane, the trajectory through the point that you clicked will be computed and plotted.

There are several options available in the Phase Plane window. Here are a few worth noting:

Solution->Show Nullclines Not surprisingly, this will show the nullclines.

Solution->Show Nullcline+arrows This will erase the vector field, and show just the vector field on the nullclines.

Graph->x vs. t
Graph->y vs. t
Graph->both x-t & y-t
These will ask you to click on a previously computed curve in the Phase Plane window, and then it will open up a new window containing the corresponding graphs of x vs. t, y vs. t, or both. I highly recommend trying this! It can help you understand the relationship between phase plane plots and plots of x or y vs. t.

The opening dfield and pplane web page has some instructions for printing.

You can quit the applet by selecting File->Quit in the Equation Window. The applet will automatically quit if you leave the web page.

Solution->Show Nullclines	Not surprisingly, this will show the nullclines.
Solution->Show Nullcline+arrows	This will erase the vector field, and show just the vector field on the nullclines.
Graph->x vs. t Graph->y vs. t Graph->both x-t & y-t	These will ask you to click on a previously computed curve in the Phase Plane window, and then it will open up a new window containing the corresponding graphs of x vs. t, y vs. t, or both. I highly recommend trying this! It can help you understand the relationship between phase plane plots and plots of x or y vs. t.