What is "magical wand" here? This is simply Newton's equation and a specific dome shape. Trivially no perfect bodies can be built experimentally, and it's specific properties are not robust to perturbation, but there is absolutely nothing magical or strange going on here.
Do you understand the difference between Norton's dome and other domes with balls at rest on the top?
The very point of non-determinism is that the next state doesn't follow uniquely from the current state, causality is broken. Thus there is no "cause" for one or the other trajectory. This is a feature of the Newtonian equations, whether you like it or not.
