Stator or Rectifier??

Ok so my battery went completely dead about an hour away from home yesterday. So I got a new one from autozone and hooked it up. Was fine for about 30 miles and then hit a bump and started getting surges and died. It wouldn't start. I had my mother-in-law come and jump me and it started and then when I went to take off it died. Let it charge off the car for a bit and then took off.. Died on me again about 5 miles down the road. Charged it a lil bit longer this time and finally got it the rest of the 8 miles home. Put it on the charger til it said it was fully charged and then disconnected the charger. Went out this morning and it starts just fine but with it running and I take the positive cable off it dies. Would this be the stator or rectifier? And how would I test these? I more than likely don't have the equipment to check them. Any help would be greatly appreciated.

 

WARNING: NEVER run the bike with the battery cables disconnected, you WILL fry the rectifier/regulator!!!
Given that you have a new, fully charged battery in there, I'd say you already have a bad rectifier/regulator to begin with but you need to verify this with a volt meter. You should read 14.5VDC with the bike running about 2,000RPM (this doesn't need to be dead nuts on, it can be a bit lower or higher).
There are more than a few threads on testing the rectifier/regulator out there, hit the search menu listed on the left side of every page.