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That's not really the issue. The three body problem does have an analytical solution in the form of a power series, but the problem is that it converges so slowly to be of any practical use.

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It is though, if you don’t have a closed form solution you need to use an iterative process to calculate the positions, meaning errors will accrue over time. For a closed form solution that wouldn’t be the case.

Thanks for mentioning the existence of an analytical solution at all though, I wasn't aware of that.

There are iterative methods/systems that stabilize over time. For example, symplectic integrators on tame problems oscillate lightly around the true energy of the system over time. The issue here is the properties of the underlying problem, not the set of solution methods.

> meaning errors will accrue over time

This is not universally true. Error behavior is a function of the particular problem, the algorithm used to approximate its solution, and the properties of input data. A large subtopic of numerical analysis is concerned with this kind of stuff. See [1] or [2] to get a flavor.

[1]: https://en.wikipedia.org/wiki/Numerical_stability

[2]: https://en.wikipedia.org/wiki/Lax_equivalence_theorem

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