## 4 Dynamics of NLC Functions
These phenomena of the sudden death and sudden birth of the entanglement [42; 43] are observed only for the log-negativity entanglement for several finite periods (see the nest figure). Moreover, the phase damping \(\gamma=0.4\) leads to increase of the partial SI correlation intervals and to the completely vanishing of the maximal violation of the Bell's inequality. For \(p:0\to p_{c}\), the 3D plots of the Figure 4a-d show that the purity indicator accelerates the effect of the the phase damping. Figure 3: The 3D-evolution of \(L(t)\) in **(a)**, \(U(t)\) in **(b)**, \(N(t)\) in **(c)** and \(B(t)\) in **(d)** when \(J_{x}=J_{z}=2\), \(\chi_{z}=0.5\) and \(\gamma=0\) for the cases: \(p\in[0,1]\) with \(\Delta=0.5\) in (a,b), and \(\Delta\in[0,1.5]\) with \(p=0\) in **(g–k)**. The 2D-evolution of \(L(t)\) (dashed plots), \(U(t)\) (dashed dotted plots), \(B(t)\) (upper solid plots) and \(N(t)\) (solid plots) for \(p=0\) in **(e)** and \(p=0.3\) in **(f)**. In Figure 5, the dynamics for the NLC functions \(L(t)\), \(U(t)\), \(B(t)\) and \(N(t)\) are shown for the case \(p=0.5\). The maximal violation of the Bell's inequality (\(B(t)>2\)) disappears completely. The \(B(t)\) has oscillometry behavior depending on the parameters of the anisotropy, the coupling to the of DM interaction, and the coupling environment. Figure 5b confirms that the DM interaction coupling leads to more regular oscillations, while Figure 5c shows that the anisotropy parameter control the period of the growth of the correlation functions. For the case \(p=0.5\), the deterioration of the NLCs, and the sudden death and sudden birth phenomena are more pronounced than the others cases \(p=0\) and \(p=0.3\). **Case B: \(p\in[p_{c},1]\)**. In this subsection, the non-classical correlations are explored for the purity indicator between \(p_{c}\) and \(1\), \(p\in[p_{c},1]\). From Figures 1, 4a-d, 6 and 7, we conclude that the NLCs grow with the increasing of the purity indicator \(p\). This behavior is opposite to what has been observed for \(p\) between \(0\) and \(p_{c}\)  
Figure 4: The 3D-evolution of \(L(t)\) in (**a**), \(U(t)\) in (**b**), \(N(t)\) in (**c**) and \(B(t)\) in (**d**) as functions of the time and the purity indicator \(p\in[0,1]\) for \(\chi_{2}=0.5\), \(\Delta=1.5\), \(J_{x}=J_{z}=2\) and \(\gamma=0.4\). The 2D-evolution of \(L(t)\) (dashed plots), \(U(t)\) (dashed dotted plots), \(B(t)\) (upper solid plots) and \(N(t)\) (solid plots) for \(p=0\) in (**e**) and \(p=0.3\) in (**f**). From 3D plots of Figure 1 and 2D plots of Figure 6a,b, we notice that: (1) For \(p=1\) and \(\gamma=0\), the effect of the unitary interaction on the NLC evolutions is very weak compared to the case \(p=0.7\). As the purity indicator decreases, \(p:1\to p_{c}\), the unitary interaction effect raise up, and the initial NLCs is deteriorated, except the maximal correlation of the UIN. (2) The maximal violation of the Bell's inequality and SI correlations appear for \(p=1\). While for \(p=0.7\), the SI correlation disappears, the violation of \(B(t)\) is reduced, and the phenomenon of sudden changes manifests only in the dynamics of the UIN. Figure 6c,d shows the effects of the phase damping \(\gamma=0.4\) on the the initial maximal (\(p=1\)) and partial (\(p=0.7\)) NLCs. The NLC amplitudes are reduced when the time evolves. As observed from Figure 6a-d, the deterioration of the NLCs increases as \(p:1\to p_{c}\). The NLC quantifiers are more sensitive to the phase damping, and the log-negativity entanglement is more robust against the phase damping. From Figures 2a-d and 7a,b we observe that a lager value of the DM interaction \(\chi_{z}=3\) leads to the enhancement of the effect of the unitary interaction. In the range \(p\in[p_{c},1]\), the DM interaction has strong effect unlike the case \(p\in[0,p_{c}]\). The NLCs for \(p\) ranging from \(p_{c}\) to \(1\) is plotted in the Figure 3a-d with large value \(\Delta=1.5\). While Figure 7c,d represents the NLCs for a small value of the anisotropy, \(\Delta=0.5\), with \(p=1\) and \(p=0.7\).