Isaac Newton:
https://wiki.tfes.org/Three_Body_Problem
The only significance of that is that there is no analytic solution to the general problem.
But for the Sun, Earth, Moon system we do need an analytic solution. It is an existing system that has been observed for millenia.
Not only that but the Sun's mass is about 333,000 times the Earth's mass and the Earth's mass is about 81 times the Moon's mass.
So in the Sun, Earth, Moon system the Sun is so huge that it can be considered stationary with almost negligible error.
The moon is well inside the Earth's Hill Sphere of the Earth, Sun system and hence:
The Earth, Moon system can then be analysed as a two-bodied system and that two-bodied system and the Sun can be looked on as another two-bodied system.
The approximations mean that the precise orbits are slightly chaotic and so not exactly periodic but history proves that this is small.
The Babylonians were able to predict Solar eclipses using their Saros cycle, which is very good but not a precise prediction of times and locations.
So even the Sun, Earth, Moon system is not perfectly periodic but the rest of the planets, especially Jupiter and Saturn perturb motions still further.
But all this needs a topic of its own.