Sounds just like the guy I got yesterday morning, also Gulfstream, altitude unknown. From memory it was something like N34????.
He first appeared over Argentina, jumped to Brazil, then shot off north again to re-appear somewhere near Bermuda before finally pitching up in Warwickshire. He stooged around for a while on a south east course then vanished. (All this from memory of course, but not a mile from the truth)
If this is an experimental US plane, they're really on to something!!! Ho ho.