Whether the universe as defined by modern physics is deterministic or not is an open question, but its substantially deterministic.
The universe would be deterministic if there was an injective function between the state at one time and the state infinitesimally later than that. If that function existed, we could simply apply it repeatedly to determine the state of the system in the future.
For the vast majority of situations (and I mean VAST), this is true for the equations we believe are good predictors of the world. The three body problem, and indeed the n-body problem all fit into this regime. It can be highly chaotic, but the time evolution function is indeed an injective function.
There is one corner case which cuts away at this certainty, metastability. consider the case of a ball on the top of a hill which could roll to the left or the right. It's just perfectly balanced at the top. Rolling one way or the other are both perfectly legitimate results. We would call that state nondeterministic, because the current state does not say which way it falls. It could even stay up there as long as it "pleases."
The more formal version of this argument would be made in Lagrangian Mechanics. In this formulation of mechanics, we define a "Lagrangian" in terms of positions and velocities, and try to find a path through time which minimizes the action, which is the integral of the Lagrangian over time. This path describes the evolution of the system in an injective manner. Well... almost. Actually, in Lagrangian Mechanics one seeks stationary points, which are minima, maxima, or inflection/saddle points, not just minima. It turns out that you can prove that minimizing action is insufficient to describe things we see in the real world, but finding stationary points is sufficient.
So consider the case of a frictionless ball rolling back and forth in a valley, with a hill on each side. If we give the ball a large velocity, the only valid solution to these equations is for the ball to go over one hill and continue on its way, never to return. It had enough energy to go over the hill, and that was that. If we give the ball a small velocity, then the only valid solution to these equations is indeed a minima, and we can watch the ball roll back and forth in the valley forever, never having enough energy to escape.
But what about the perfect velocity, right in the middle? There is a velocity with exactly the same amount of kinetic energy at the bottom as potential energy at the top. When it gets to the top, it has zero velocity -- a metastable state.
As it turns out, this is a saddle point in the Lagrangian formulation. There are three possibilities: it can stay at the top forever, it can roll back into the valley, or it can roll out of the valley. It turns out all are valid stationary points for the action. Indeed, it turns out that it can also fall off of the top at any arbitrary time, and still be a stationary point. We have a truly non-deterministic system there.
These points are infinitely rare in a larger system. You and I know that in practice, something will eventually perturb the ball off of the infinitely perfect metatstable point. Indeed this is a foundation of the concept of entropy. However, in theory, the entire universe could be in one of these saddle points. Such a universe would be nondeterministic.
But if I may bring in a level of empiricism, it would be impossible to deduce the difference between such a nondeterministic universe and a deterministic universe where the key state variables simply have not been measured to sufficient precision yet.
For a concrete example of this, one might look at Norton's Dome. It defines a construct which generates a non-deterministic result using Newtonian Physics, without having to resort to the complexities of Lagrangian Mechanics. It's interesting because there's a lot of disagreement as to whether or not the argument is valid. It violates some principles(Lipschitz continuity), but there is not a consensus as to whether or not that makes the construct "non-physical" or not.