  With an underlying common theme of competing length scales, we study the
many-body Schr\"{o}dinger equation in a quasiperiodic potential and discuss its
connection with the Kolmogorov-Arnold-Moser (KAM) problem of classical
mechanics. We propose a possible visualization of such connection in
experimentally accessible many-body observables. Those observables are useful
probes for the three characteristic phases of the problem: the metallic,
Anderson and band insulator phases. In addition, they exhibit fingerprints of
non-linear phenomena such as Arnold tongues, bifurcations and devil's
staircases. Our numerical treatment is complemented with a perturbative
analysis which provides insight on the underlying physics. The perturbation
theory approach is particularly useful in illuminating the distinction between
the Anderson insulator and the band insulator phases in terms of paired sets of
dimerized states.
