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.

Goals:

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.

Schedule:

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.

Thanks,

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.