Solving Linear Matrix ODEs in the Complex Plane

Here are the links to 3 applets I developed for solving linear matrix ODEs of the Fuchsian type. The applets can solve an ODE numerically and find the monodromy matrices. You have some options that affect how the applets normalize the monodromy matrices:

Applet 1 (fks1): http://www.wiu.edu/users/fa101/java/FKS/fks1/. It simply solves the ODE.

Applet 2 (fks2): http://www.wiu.edu/users/fa101/java/FKS/fks2/. It finds the monodromy matrices.

Applet 3 (fks3): http://www.wiu.edu/users/fa101/java/FKS/fks3/. It finds the monodromy matrices for a so-called Frobenius case of the sixth Painleve equation. Using the applet I was able to establish that the monodromy matrices are actually in SL(2,Z) - quite an unexpected result (at least for me). See more on this in the article you can find at http://www.wiu.edu/users/fa101/research.

Instructions

After you specify the parameters, you enter the main phase of the applet. Button Solve! actually solves the equation. But before you do that, you may want to change the settings.

To set up singular points, you click Singularities button. To specify residues, you need to click Residues. Finally, to enter the initial value you click Init button. Usage of those panels is straightforward, except, perhaps, the fact that values are actually set when you click OK button in the panels. Working with paths is different: you can drag points of a path. I really wanted to implement this feature, to give the applets more geometric and “hands-on” feeling, unlike command line-like interface of a CAS. You can, of course, click Path button and enter the points of a path manually. You can also add a point to a path — click + button. To remove a point, click the
corresponding - button. Added points are generated with random values.

If when you drag, the applet is very unresponsive and the dragging is slow, sometimes it’s because you use a Java Virtual Machine (JVM) from Sun. Go to Control Panel, enter the dialog with properties of the Java Plug-in and disable its usage with your browser. Surprisingly, all operations of this kind are fast with Swing Java in Sun JVM and with old Java in Microsoft JVM,but using old Java (AWT based code) in Sun JVM is often inexplicably slow.

In the Monodromy Applet, you click Monodromy button to start the actual numerical procedure. Use Normalization String (Init panel) to set up the order of the monodromy matrices in the cyclic identity. For example, if you want your matrices to satisfy: M1M2M0 = M_infinity^{-1} use 120 as the normalization string.

Sometime, the monodromy panel (with the monodromy matrices) takes time to appear on the screen. Simply wait for about 5 seconds (issues with updating?) after the panel on the right disappears - it should pop up again, with the monodromy matrices.