Well mom, it looks like its either a rug under the runners or find the center point of the runners and place a screw eyelet into the top of the runners and a wire through the eyelet to the floor where you want to be. A small screw will keep the chair from moving and will not damage the wood floor.
if you have a rug under, get two strips of felt, cut them a tiny bit smaller than the bottom of the rocking rails (about 1 inch by 14 inches long) use gorilla glue to glue the strips to the bottom of the rails, let dry for at least 24 hrs, then put the chair back on the rug, in should no longer scoot