where * f(x,y) * and * g(x,y) * are user specified
functions. This file has 4 parameters labeled * a,b,c,d * which
you can include in your right-hand sides. Here are some important
commands:

- To quit click on (File) (Quit) [ WinPP: File Exit]
- To turn off the bell so your neighbor won't kill you, click on (File) (Bell off) [NOT in WinPP]
- To change the right-hand side functions
**f,g**click on the (File) (Edit) (Functions) menu items and edit the two functions. Click (Ok) when you are done. [WinPP: Edit RHS] - To use the mouse to enter initial conditions click on (Initialconds) (Mouse) or (mIce) and then click in the phase-plane [WinPP: Run Mouse or Run Mice]
- To manually enter initial data, either
- Click on (Initialconds) (New) and then type them in in the command window or,
- Find the initial condition window, (iconified and labeled ICs) and open it up. Then type in initial data in the window and click on OK. Then click on (Initialconds) (Go) in the main menu [WinPP: Run Go or click on Go in the Initial data window]

- To find fixed points, click on (Sing pts) (Mouse) and click nearby where you think there is a fixed point. Answer the prompts (No) and (No) for now. [WinPP: Run Equilibria Go]
- To find nullclines, click on (Nullclines) (New) [WinPP: PhasePlane Nullclines]
- To draw direction fields, click on (Dir.field/flow) (Direct Field) and choose 10 or so as the grid. [WinPP: Phaseplane DirectField]
- Click on (Restore) to redraw trajectories and click on (Nullcline) (Restore) to redraw nullclines [WinPP: Ctrl-R, Phasplane Redraw ncline.]
- To print out click on (Graphics) (Postscript) and type in the name of a file. Then at the console prompt, print your file remembering the name. [WinPP: File Print Postscript]
- To
**permanently keep**up to 26 trajectories click on (Graphics) (Freeze) (On freeze) and all trajectories that you compute will be stored in succession. When you want to delete all these for a new problem, click on (Graphics) (Freeze) (Remove all). To turn off the permanent freeze option, click on (Graphics) (Freeze) (Off freeze). [WinPP: Graphics Keep curve] - To change the graphics window click on (Window) (Window) and enter the ranges for the axes. [WinPP: Graphics View ]
- To change the view, i.e. the variables on the axes, click on (View axes) (2D) and set the parameters accordingly. [WinPP: Graphics View]

The first example is from another book and is

Draw the nullclines and the direction fields. What is the fixed point?
(It is where the nullclines intersect.) What kind is it? The line **
y=0 ** is invariant; that is, any initial conditions that start with
* y=0 * stay there. This is the unstable manifold for the fixed
point. The stable manifold is not a straight line which is one
instance where linear and nonlinear systems differ. XPP can usually
compute stable and unstable manifolds for nonlinear systems (although
not for this problem!) To compute manifolds, click on (Sing pts)
(Mouse) [WinPP: Run Equilibria Mouse]
and make a guess in the phaseplane. Then answer (yes) to the
question about invariant manifolds. For this problem, it fails to
compute the appropriate eigenvector, but it usually doesn't. ** NOTE
** If XPP fails to compute the ** stable manifold ** here is how
to do it manually. Click on (nUmerics) (Dt) and change the timestep
to its negative so that you can integrate backward in time.[WinPP:
Numerics Int. Pars.] Now click
(Esc) to return to the main menu. Compute two trajectories on either
side of the saddle point, close to it. These will approximate the
stable manifold pretty well. Make sure you return (Dt) to its
positive value in the (nUmerics) menu. If XPP fails to compute the
** unstable manifold ** then just choose a few initial conditions
near the fixed point and integrate forward in time.

The next example is a little bit more complicated. There are 4 fixed points and many interesting features of the phaseportrait. Here are the equations:

with * a=0.25 * as the parameter.
Type them in making sure that you use the "*" symbol for
multiplication. Set the window to be ** -5 < x < 5 ** and
** -5 < y < 5 ** by clicking on (Window) (Window).
Verify that there are 4
fixed points. Find the values of the fixed points and determine their
type (node,saddle, spiral, etc). (Answer No on any questions asked!)
Since XPP has a bit of a problem finding the stable manifolds for
this, lets do it manually. First we will compute the unstable
manifolds by integrating forward. As you have found the origin (0,0)
is a saddle as is the fixed point near (-4.05,1.33). Clear all the
graphs you may have frozen and turn the automatic freeze on.
Integrate with the following sets of initial conditions:

- x=.03, y=.01
- x=-.03,y=-.01
- x=-4.1, y=1.37
- x=-4,y=1.3

- x=-.03, y=.01
- x=.03,y=-.01
- x=-4.2, y=1.32
- x=-3.9,y=1.32

- solutions go off to infinity (A)
- solutions tend to the stable node in the positive quadrant (B)
- solutions tend to the stable spiral in the lower left quadrant (C)

The moral of this is that the stable manifolds of saddle points delineate the regions of behavior for many planar differential equations.

Thus the domain of asymptotic stability (DAS) for the node in the upper right quadrant is all of region C and the DAS for the lower left fixed point is region B.

If there are no saddle points or periodic solutions, often the domain of asymptotic stability for the stable fixed point is the whole plane!

- Draw nullclines
- Find all fixed points and their stability
- Find any stable periodic solutions
- Determine the domain of asymptotic stability for the stable fixed points.
- Compute a few representative trajectories

- dx/dt=x-x^3-y, dy/dt=x-a*y (choose a=0 and a=2)
- dx/dt=-x + tanh(4*x-y), dy/dt = -y + tanh(4*y-x)
- dx/dt=y, dy/dt=x*(1-x^2)+a*y-y^3 (choose a=-.25, a=.25, a=1)