What can be the smallest chaotic system? As I am talking about 'smallest' can I expect that it should be a quantum system? I understand that we use quantum chaos theory instead of perturbation theory when the perturbation is not small. For example when quantum numbers are very high for an EM field interacting with an atom, we cannot use perturbation theory. In this case we will use the theory of quantum chaos.
My question is what is the lower boundary of the size of the system or more specifically of the quantum number in case of a EM-field atom interaction for which we must use quantum chaos theory?
 A: A hydrogen atom can be made to show chaotic behaviour if it is excited to very near it's ionisation energy. The maths is somewhat beyond me, but this paper discusses calculations of the hydrogen atom showing the onset of chaotic behaviour. You can find more by Googling for "Rydberg atom" combined with "chaos" or "chaotic behaviour".
A: Your question is ill posed. On one hand, you cannot use perturbation theory, just when higher order terms cannot be neglected. On the other hand, quantum chaos theory does not provide any magical tool to understand quantum mechanical system non-perturbatively. The character of the question posed by quantum chaos theory forces us to look at system where perturbation theory breaks down. This discipline just happens to use tools developed to understand quantum system in the regime where perturbation theory brakes down.  It's totally up to those methods, when they are valid. From wikipedia I spotted that sometimes they use integrable systems, meaning they can be solved for any value of the coupling constant. Clearly thus obtained solutions are valued also in the perturbative regime, as well as outside.
