It's a mini-gripper chuck that used to be sold by Craft Supplies.
You drill a small recess the depth of the ribbed section with a forstner bit (30mm or thereabouts I think from memory), insert the ribbed section and tighten. The screw thread expands the 4 ribbed jaws to bite into the wood.
[The body can then be gripped in the jaws of another chuck. - Not sure about this aspect now ]
I have a craft supplies chuck and one have been wondering how to hold small pieces with it. My chuck did not come with that accessory. But it is quite old so maybe they didn't make them then. Does it work the same way as the other expansion jaws?
If it can cope with those, then it shows that it can cope with nearly anything a turner can ask of it, within reason of course.
Until the lowest point of the bark is reached, natural edge bowls tend to induce vibration because the gouge is only cutting the high spots, ie the bowl wings, on each revolution.