I read yesterday the Norton Dome's paper, which shows that some Newtonian systems can be non-causal, based on specific solutions of Newton's laws. The author justifies the solutions in very nice, logically consistent ways, that made me unable to falsify his conclusions.
In brief words, the thought experiment is: If a sphere is on the apex (top) of a dome that can be geometrically described by the equation $h=\frac{2}{3g}r^{\frac{3}{2}}$ (see the Fig. 1a below), we can show with Newton's laws that this sphere can start moving with absolutely no cause (not even probabilistic one). If you find this very bizarre (like I did when I first heard about it), please take a look at the paper before attacking my post.
Getting closer to my question: The author even makes this sound more reasonable by saying that this can be made clearer by considering the reversibility of the system. Consider a sphere at the rim of the dome, and you give it a kick with some initial velocity to reach the apex (see Fig. 1b below). If the force you use is very small, the sphere will not reach the apex. If the force you use is very high, the sphere will go over the apex. If the force is just right, the sphere will exactly stop at the apex. This shows that this system is reversible, because exactly the same way the sphere rested at the apex by a force that reached the apex, if we reverse time, it'll take the same trajectory to go down (ignoring the radial symmetry of the apex).
My question: Following this logic, can't we say that every Newtonian system that reaches a steady state is non-causal, because otherwise it would be non-reversible, time-wise?
Note: Please don't involve Standard Model's CP/T symmetry related topics. I know that this world is CP-violating (and hence T-violating) due to weak interactions. My question is merely about classical mechanics.