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Next: Homework Up: No Title Previous: Preparing for AUTO.

Running AUTO.

It is best if you just follow along since AUTO is pretty sensitive. All the following commands are done in the AUTO window.

1.
Click on Axes Hi-lo. This will plot the max and min of the solutions; useful for periodic orbits. Fill them in as follows: Click on OK. You have set up the view and also the parameter that will vary in the bifurcation analysis. Main Parm tells the main parameter to vary. It must be one of 5 parameters that are input in the Parameter list. The Y-axis determines what variable to plot. The 2nd Parm will not be used here. The others are self-explanatory.
2.
Click on Numerics. Change the following: The first two items are technical and will be described later. The last two tell AUTO the range of the parameter.
3.
Click on Run, Steady State. If you have prepared AUTO correctly and done what I have said above, you should see a line across the screen. It will have some numbers on it and will be thick up to about I=9.5 and then thin. NOTE. If all you get is a single cross, then you probably screwed something up. Here is how to get out of this mess: (DON'T DO THIS IF IT WORKED!!)
(a)
Click on Grab and tap Enter.
(b)
Click on File, Reset Diagram, Yes.
(c)
Go back to the previous section and make sure you have a real nice fixed point by clicking i l over and over again.
(d)
Repeat all of this section.

4.
If you are here, then congrats, you have successfully computed a curve of fixed points. Thick lines mean stability and thin mean instability, so at about I=9.4 there is a loss of stability of the fixed point. Click on Grab and use the left and right arrow keys to move along the curve of fixed points. Look at the bottom of the AUTO window and watch how the values change. Look to the left at the little circle and watch the eigenvalues leave it. These are the eigenvalues of the linearized system, exponentiated so that stable ones are mapped into the unit disk. Move the arrows until the letters HB appear under the letters Ty at the bottom of the AUTO window. This says that this point is a point of Hopf bifurcation. We expect a branch of periodic orbits to emerge. Click Enter when you have ``Grabbed'' this point.

5.
Now lets try to track this branch. Click on Run, Periodic. If all goes well, a branch of periodic orbits will be computed. It should look like the figure below. Open circle are unstable periodic orbits and solid dots are stable.
6.
Create hardcopy - click on File Postscript, give the file a name and you will have a nice picture.

7.
Now that you are done admiring your nice bifurcation diagram, you should get rid of it and all the AUTO files associated with it. Click on File, Reset Diagram, Yes

Now quit XPP. Click on the Main Window. Click File, Quit, Yes or tap f q y.


\resizebox*{!}{4in}{\rotatebox{270}{\includegraphics{hhh_bd.ps}}}


next up previous
Next: Homework Up: No Title Previous: Preparing for AUTO.
G. Bard Ermentrout
1999-09-15