This is a tough nut to crack, my friend. The iPhone worked normally on your Mac laptop. It is not working properly on your new desktop. Is there another PC you can use to confirm that the problem is not your iPhone?
I found a thread in the Apple discussions forum where voltage sensitivities due to a failing fan motor in the desktop were enough to cause the iPod (in that case) to disconnect.
http://discussions.apple.com/thread....sageID=6886265
Because it is easy to do, try a reset (home button/sleep button) and re-sync. You need to rule out hardware issues on either device.