  We investigate both the local and global persistence behaviour in ANNNI
(axial next-nearest neighour Ising) model. We find that when the ratio $\kappa
$ of the second neighbour interaction to the first neighbour interaction is
less than 1, $P(t)$, the probability of a spin to remain in its original state
upto time $t$ shows a stretched exponential decay. For $\kappa > 1$, $P(t)$ has
a algebraic decay but the exponent is different from that of the nearest
neighbour Ising model. The global persistence behaviour shows similar features.
We also conduct some deeper investigations in the dynamics of the ANNNI model
and conclude that it has a different dynamical behaviour compared to the
nearest neighbour Ising model.
