What is a chaotic orbit?

When many objects interfere (as in a n-body system) it is very difficult to solve the problem. But if the determination of the orbits is hard, it doesn't mean that it is impossible: with computers that are powerful enough, a very complicated n-body system such as the solar system is completely solved, and the orbits of planets are today perfectly known. In other words, the planets orbits are stable: independently on how hard the calculus is, it is possible to predict where the planets will exactly be in the future. In fact, for these objects, the final state of the object (the position and the speed of the planet at a precise moment in the future) depends totally on the initial conditions and on the forces that act on it.
When talking of the orbits of asteroids and comets, an additional concept must be introduced: the concept of chaotic orbits.

Stable and unstable positions

It is much easier to understand the nature of chaotic orbits making a parallelism with stable and unstable positions in the motion of a pendulum. For a pendulum, this concept is very intuitive: a position of stable equilibrium corresponds to a position the body will come back to, when slightly perturbed. For unstable equilibrium, a little deviation from this initial position is sufficient to make the body leave its initial condition.

Forcing this parallelism we can say that stable orbits will be highly predictable orbits (just like the position of stable equilibrium is for the pendulum) while unstable orbits will depend abruptly from the initial conditions (as for the pendulum, starting from a position of unstable equilibrium, the final position depends strongly on the perturbation).

Chaotic orbits and the Lyapounov time

It is important to understand that a chaotic orbit is unpredictable for its nature and therefore, the position of the object can't be perfectly determined in the future. In other words, a chaotic orbit is so sensible to very small changes in the initial conditions (or in the forces that act upon the body) that the prevision of the trajectory over long periods of time is impossible.
This because different orbits are divergent: a very little difference in the initial conditions leads to totally different orbits in the future. Even if this explanation is very intuitive, it is important to understand that chaos is a mathematical concept, that can be formalised and measured by a parameter called the Lyapounov time.

NEOs' chaotic orbits: an example

The most classical example of chaotic orbits is the trajectory of a light body such as an asteroid or a comet. In fact, there are many physical mechanisms (such as the fly-by of a planet, or a resonance phenomena) that act on asteroids and comets, perturbing their initial orbits and making them become NEOs (click here to know more about these mechanisms).
An example of these mechanisms, is the close approach of a planet. As shown in the picture, when the asteroid passes near a planet, even for very similar initial conditions, the orbits diverge.