Why are solar systems stable and not chaotic? I am not trained in applied mathematics and am asking as a layperson.
It seems that the motions of the planets can be reliably predicted for thousands of years. Yet, if we consider a relatively simple system, the compound pendulum for example, its motion is highly unpredictable.
Chaotic pendulum - guess when it will stop flipping
The pendulum is highly constrained by its pivot points, therefore its components do not escape from the system. However, with a solar system, there is no rigid attachment and it seems intuitive that planets could randomly fly off into space as a result of chaotic orbits.
Question
Is our solar system inherently stable or unstable?
If unstable then how did it hang together long enough to form in the first place?
If stable, how can stability and predictability be consistent with many bodies moving relatively to one another?
The solar system is chaotic, but it is also stable!
The fixed and linkages between the bars of a double pendulum allow for very rapid energy transfer between the arms.  This makes the chaotic motion develop rapidly.
The interactions between planets are gravitational and much much weaker, moreover, the planets are heavier and it takes a lot more energy to deflect them. This means that in the medium to long-term it is possible to predict exactly where a planet will be.
In the very long-term the system is chaotic, but the gravitational linkage between planets is not strong enough to actually cause one of them to deviate very far from its orbit, unless a resonance develops.  So it is chaotic in the sense that we won't know which side of the sun Mars will be on.  But it is stable in the sense that we know that Mars will continue to be the planet between Earth and Jupiter (but perhaps slightly closer to Earth or slighty closer to Jupiter)
There is only one potential resonance: it is possible that in the very distant future, Mercury could develop a resonance with Jupiter, which could (in 3 to 4 billions years) eject it from the solar system.
So it is the very rigidity of the double pendulum that makes chaos develop so quickly.  Whereas the looseness of gravity means the motion of the planets can be predicted far into the future.