As I understand your system, nothing enters and nothing leaves which makes it a closed system. Imagine that this system is placed on a four wheel cart with low friction bearings and the cart is on a level surface. When the system is off the cart doesn’t move. When the system is turned on the cart will move back and forth but stay in the same average position. This is the result of Newton’s Third Law of Motion that for every action there is an equal and opposite reaction.
This system is equivalent to throwing a ball back and forth between your right hand and your left hand. The ball moves but on average you don’t move from where you are standing.
You may be interested in spacecraft propulsion.
