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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 seconddoi:10.1093/acprof:oso/9780199535286.003.0006 fatcat:2rfg5g6i2zh3lnl76dsefde3ly