I'm learning about the classical KAM theorem, and I can't quite infer precisely what the term "quasi-periodic solution" means in the theorem's statement. I'm reading the following introductory note:
math.bu.edu/people/cew/preprints/introkam.pdf
and on the 15th page a discussion of nearly integrable systems starts and the classical KAM theorem is stated on 16 (this is all independent of pages 1-14). Basically, the theorem is stated here as:
If $\omega^*$ is a set of frequencies badly approximable by rational numbers (in the sense of Definition 3.1), and if the integrable part of the Hamiltonian is nondegenerate at action variable $I^*$, then there is an $\epsilon_0$ s.t. as long as the non-integrable part of the Hamiltonian is analytic and bounded (in the sup norm sense) by $\epsilon_0$ then there exists a quasi-periodic solution with frequencies $\omega^*$.
Now, I'm under the impression from other sources that quasi-periodic on a Kronecker torus is basically no different from the frequencies being not rationally dependent, i.e. any set of frequencies that leads to non-periodic trajectories. But when we no longer have the Kronecker tori in the perturbed case, I'm not really sure what the definition of a quasi-periodic trajectory/solution is.
Thanks a lot in advance for any insight!