If you find out what it is, post here for the rest of us.
My car does that, too, just started out of the blue one day. It runs fine when cold, but runs like yours once it warms up.
Your power steering may be dumping some fluid, check the reservoir and hoses for leaks. Are there any cracked wires or does the connector to the power steering switch look damaged?
That switch is the one on the passenger side, it's in the power steering line, has a connector sticking in it right near the top of the engine.
