In Sean Carroll's book "The Big Picture," he states (chapter 4, page 35):
Classical mechanics, the system of equations studied by Newton and Laplace, isn't perfectly deterministic. There are examples of cases where a unique outcome cannot be predicted from the current state of the system. This doesn't bother most people, since cases like this are extremely rare—they are essentially infinitely unlikely among the set of all possible things a system could be doing. They are artificial and fun to think about, but not of great import to what happens in the messy world around us.
What are some of these situations he is referring to?
My first guess was something like the conceptual equivalent of a ball sitting exactly atop a hill and asking which direction it will roll. But that doesn't seem right—this would be a case where the "true" solution is that the ball wouldn't roll at all, while in reality there will always be tiny disturbances that cause the ball to roll in some direction. I don't think this is what Carroll is referring to. He seems to suggest that the "true, exact mathematical outcome" is not uniquely determined for certain classical systems.
I've heard that uniqueness theorems for solutions to differential equations don't generally apply to non-linear equations, and wonder if this may have something to do with it. Elucidation and examples would be much appreciated.