  An oscillating sequence of order $d$ is defined by the linearly disjointness
from all $\{e^{2\pi i P(n)} \}_{n=1}^{\infty}$ for all real polynomials $P$ of
degree smaller or equal to $d$. A fully oscillating sequence is defined to be
an oscillating sequence of all orders. In this paper, we give several
equivalent definitions of such sequences in terms of their disjointness from
different dynamical systems on tori.
