## **PHASE PORTRAITS**

Now I want to introduce you to a nice online tool which might
help you plot some of
these slope fields. (if you get tired of doing this by hand and
your calculator can't plot these for you) It is called **PPLANE**
(named for the term *phase plane*, of course). It is being
developed by
**John C. Polking** and
some of his colleagues
at Rice University. The software is really just a set of m-files that
MatLab executes. MatLab is a Matrix Laboratory software package that is
very popular with many mathematicians and engineers.
Before you actually visit the site, you should
remember that there are a few syntax notes that you need to understand.
After all, this is a computing tool, much like a calculator,
and it requires a certain syntax in order to function properly.
You may want to keep this window open while you are using the
tool so that you can read through the following notes as you plug
along for the first time. Just minimize it and pull it up whenever
you need it. The link to **PPLANE** is at the bottom of the
page, but read (at least skim it, please) through the following
before you click. As you may have guessed by now, the directions
are essentially the same as those for the DFIELD tool.
- There will be several windows that pop up as you get started.
As the software is cranking up, you will have to click OK
on the copyright box in order to
get started. Managing all of these windows will require some
ingenuity on your part.
- Two Basic Windows:
- The two basic windows that you will work with are the
**Equation Window** and the
**Phase Plane Window**.You
may have to adjust the size of the Equation Window in order to
see the contents of all the boxes. The Equation Window is where
you enter the differential equation. Notice that there is already
an equation entered for you. There is a box which tells PPLANE
which variable is the independent variable.
There are also some parameters in the equation-- a and b. The
values of those parameters can be adjusted in the boxes under the
heading **Parameter Expressions**.
One can also adjust the maximum and minimum values of the
graph of the slope field by using
**The Display Window** boxes.

- Plotting a phase plane:
- Click on the button in the right hand
corner of the Equation Window. It should say something like
**Graph Phase Plane**. You
can view the phase plane in the Phase Plane window. By
using your mouse to click on a point in the direction field, you
can generate a sketch of the solution to your first order system of
ODEs which passes through that point.
- If you are interested in a couple of examples, try
looking in the Gallery menu at the top of the Equation Window.

- Basic Syntax and Operations:
- addition, +
- subtraction, -
- multiplication, *, (don't forget
this one: 3x is written as 3*x)
- powers, ^, (3^2 means
*three
squared*)

- Functions:
- trigonometric, sin(3*x),
(the parentheses must be there, `sin3x'
will only give you syntax errors and frustrate you) and it
works the same for cosine (cos(3*x)) and tangent (tan(3*x))
functions
- the exponential function e, exp(x) or exp(t^2),
(again the
parenthese are the big deal)

- Quitting: You may quit the application by clicking the
Quit option under the File menu
in the Phase Plane Window. There may also be a ``Quit'' button
at the bottom of the Equation Window.

Click Here
in order to use **PPLANE**. Note: There is
some introductory stuff at the top of the page. In order to crank up
the tool, you must click on the grey button
labelled PPLANE 2005.10

If you are interested in some of the more advanced options available with PPlane, please ask me--or just hunt around on your own and experiment with some things.
#### Last Updated: 08/22/07