Hi. FGMRES is basically GMRES plus some adaptivity ideas. Whereas GMRES is theoretically guaranteed to eventually converge, FGMRES is a bit more complicated. This was an attempt to show that chaos theory might have a bit to show about what's really at play in FGMRES.
If you are interested in more of what this method even is, please see the FGMRES section of this dissertation.
I did try to publish this a long time ago, but it's rather sloppily written, and now I don't know what to do with it.