We show that for $ η>0 $ and sufficiently large $ n $, every 5-graph on $ n $ vertices with $δ_{2}(H)\ge (91/216+η)\binom{n}{3}$ contains a Hamilton 2-cycle. This minimum 2-degree condition is asymptotically best possible. Moreover, we give some related results on Hamilton $ \ell $-cycles with $ d $-degree for $\ell\le d \le k-1$ and $1\le \ell < k/2$.