  Approximating nonlinear dynamics with a truncated perturbative expan- sion
may be accurate for a while, but it in general breaks down at a long time scale
that is one over the small expansion parameter. There are interesting occasions
in which such breakdown does not happen. We provide a mathematically general
and precise definition of those occasions, in which we prove that the validity
of truncated theory trivially extends to the long time scale. This enables us
to utilize numerical results, which are only obtainable within finite times, to
legitimately predict the dynamic when the expansion parameter goes to zero,
thus the long time scale goes to infinity. In particular, this shows that
existing non-collapsing solutions in the AdS (in)stability problem persist to
the zero-amplitude limit, opposing the conjecture by Dias, Horowitz, Marolf and
Santos that predicts a shrinkage to measure-zero [1]. We also point out why the
persistence of collapsing solutions is harder to prove, and how the recent
interesting progress by Bizon, Maliborski and Rostoworowski is not there yet
[2].
