  Mechanisms are elucidated underlying the existence of dynamical systems whose
generic solutions approach asymptotically (at large time) isochronous
evolutions: all their dependent variables tend asymptotically to functions
periodic with the same fixed period. We focus on two such mechanisms,
emphasizing their generality and illustrating each of them via a representative
example. The first example belongs to a recently discovered class of integrable
indeed solvable many-body problems. The second example consists of a broad
class of (generally nonintegrable) models obtained by deforming appropriately
the well-known (integrable and isochronous) many-body problem with inverse-cube
two-body forces and a one-body linear ("harmonic oscillator") force.
