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The orbit of the earth seems to be very predictable. But as it is a many-body problem having sun, earth, moon, jupiter and so on, is it really that stable or will it start making strange movements sooner or later?

Are there asteroids or moons in our solar system with obviously chaotic orbits?

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I think that this Wikipedia article pretty much summarizes all of this.

The stability of the Solar System is a subject of much inquiry in astronomy. Though the planets have been stable when historically observed, and will be in the short term, their weak gravitational effects on one another can add up in unpredictable ways. For this reason (among others) the Solar System is stated to be chaotic, and even the most precise long-term models for the orbital motion of the Solar System are not valid over more than a few tens of millions of years.

The Solar System is stable in human terms, and far beyond, given that none of the planets will probably collide with each other or be ejected from the system in the next few billion years, and the Earth's orbit will be relatively stable.

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