Since our solar system has literally thousands of bodies, and since it has lasted for many billions of years, I suppose that we can surmise that relatively stable periodic systems of n-bodies can exist.
When I last checked, the best long-term simulations of the solar system were those of Gerald Sussman and Jack Wisdom of the computer science department at MIT. According to their calculations, the solar system is afflicted with chaos, and its motion can only be predicted with any certainty for about 4 million years... though their simulation went out to 100 million.
Here are some fun facts:
They need to take general relativity into account even for the orbit of Jupiter, which precesses about one radian per billion years.
They take the asteroid belt into account only as modification of the sun's quadrupole moment (which they also use to model its oblateness).
The most worrisome thing about the whole simulation - the most complicated and unpredictable aspect of the whole solar system in terms of its gravitational effects on everything else - is the Earth-Moon system, with its big tidal effects.
The sun loses one Earth mass per 100 million years due to radiation, and another quarter Earth mass due to solar wind.
The first planet to go is Mercury! In their simulations, it eventually picks up energy through a resonance and drifts away.
In the bowels of Usenet, John Baez wrote:
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