I am interested in exploring the concepts of controlling a chaotic system using the OGY method.  This method allows the stabilization of a perodic orbit, by applying a small perturbation that moves the trajectory into a stable direction of the perodic orbit, allowing the trajectory to be pulled into the orbit by the system.  The system I am interested in applying this method to is a driven double pendulum. 
        Minimal:  At least, I want to sucessfully model the system and find periodic orbits of the system.
        Realistic:  Having found a good perodic orbit, I want to apply the above method to it, and stabilize that orbit.
        Lucky:  This method only takes effect when the trajectory is in the neighborhood of the orbit, it does not make any provision for getting the trajectory to the neighborhood.  Eventually, the trajoctory should happen to be close enough, but if there's time, I'd like to look at methods for targeting the neighborhood more directly.
        Oct 23:  Equations of motion, Poincare sections
        Oct 30:  Find periodic orbits
        Nov  6:   Find periodic orbits
        Nov 13:  Analyze stability of chosen orbit
        Nov 20:  Add control parameter to stabilize orbit
        Nov 27:  Look at targeting methods 
        Dec 04:  Add targeting?
    At this stage, I think I have the correct equations of motion, and have it generating a Poincare section (based on the drive frequency) projected into the coordinates of the inner pendulum. 
    I hope this sounds reasonable to you.  I definitely want to try this control method, but if you think a different system would work better, I'm not so attached to the pendulum.   If this seems ok, I will send a more formal version (in a proper format) for Monday.
    I'd like to apologize once more for not getting this to you sooner.
            Matthew Cammack

30 Oct 2003
From: Matthew Cammack


    I wanted to give an update of my project.  I was hoping to send a LaTeX file, but I'm new to the format, so it's taking longer than I expected.  I thought I should go ahead and send this in the mean time.

    Last week I thought I had the equations of motion, but when I went to type them up, they didn't look right.  But I rederived them, and now I think I have them fixed.  I think I've got these equations integrating correctly as well.  I've also added the Jacobian to the program, which was fairly complicated.  I've used that to generate a stroboscopic Poincare section like the example for a single driven pendulum.      This means unfortunately that I'm about where I thought I was last week, but I'm far more confident in these results.  Next on the schedule is to find periodic orbits.