iPhone connects and disconnects from Macs repeatedly

This is a new development, and it impacts both USB ports on my MacBook Pro, so I doubt it is a hardware issue. It is not the cable, either. Both OSs are up to date. Basically, if I connect my phone to my Mac, whether to charge it or to sync it, it won't stay connected for more than a few seconds -- unless it is fully charged. That last bit isn't foolproof, but sometimes it works. All the other suggestions I've seen (reset network and location settings, etc.) seem to be voodoo. As usual with weird Mac issues, I don't seem to be alone. However, this did start abruptly -- I haven't had this issue with this combo of phone and computer before. Any ideas?