  We study the dynamics of quantum discord and entanglement for two spin qubits
coupled to a spin chain with Dzyaloshinsky-Moriya (DM) interaction. We
numerically and analytically investigate the time evolution of quantum discord
and entanglement for two-qubit initially prepared in a class of $X-$structure
state. In the case of evolution from a pure state, quantum correlations decay
to zero in a very short time at the critical point of the environment. In the
case of evolution from a mixed state, It is found that quantum discord may get
maximized due to the quantum critical behavior of the environment while
entanglement vanishes under the same condition. Moreover, we observed sudden
transition between classical and quantum decoherence when single qubit
interacts with the environment. The effects of DM interaction on quantum
correlations are also considered and revealed in the two cases. It can enhance
the decay of quantum correlations and its effect on quantum correlations can be
strengthened by anisotropy parameter.
