## 4 Dynamics of NLC Functions  
Our investigations are based on the effect of the anisotropy parameter \(\Delta=J_{x}-J_{y}\), the coupling to the of DM interaction \(\chi_{z}\), and the coupling environment \(\gamma\). The anisotropy parameter is the difference between the two spin-spin coupling constants of the \(x\) and \(y\) directions. In Figure 1, we show the 3D-evolution of the NLC measures \(L(t)\), \(U(t)\), \(B(t)\), and \(N(t)\), where \(p\in[0,1]\) and \(t\in[0,3]\) for the DM interaction coupling \(\chi_{z}=0.5\), the anisotropic parameter \(\Delta=1.5\), and \(J_{x}=2\) in the absence of the the phase damping. At points \((p,t)=(0,0)\), the NLC measures have the values: \(L(0)=U(0)=N(0)=0\) and \(B(0)=1\) as we expected from the initial state of Equation (13). From Figure 1a-d, we observe that the evolutions of the NLC measures strongly depends on the purity indicator. The generated maximal and partial correlations of the LQU, the maximal Bell function and the log-negativity decrease for \(p\) ranging from 0 to \(p_{c}\) (critical points). After these critical points, they are enhanced with the increase of the purity indicator \(p\). While the generated correlations of the UIN is enhanced monotonically with the increase of \(p\) (\(p:0\to 1\)), without a critical point. Moreover, the generated maximal and partial correlations can be delayed/accelerated by particular choices for the purity of the initial states. The two cases \(p\in[0,p_{c}]\) and \(p\in[p_{c},1]\) present opposite dynamical behavior for the NLCs. In the following, we investigate separately these two cases. **Case A:**\(p\in[0,p_{c}]\) In this interval \(p\in[0,p_{c}]\), the generated partial and maximal correlations of the log-negativity, Bell function, and LQU decrease as the purity indicator tends to the critical point \(p_{c}\) (\(p:0\to p_{c}\)). Here, we discuss the dynamics of the NLC for two cases: Pure state \(p=0\) and partially corrected state \(p=0.3\). Therefore, the 2D dynamics for the NLC functions \(L(t)\), \(U(t)\), \(B(t)\), and \(N(t)\) are shown in Figure 1e,f for these cases with the same parameters of the Figure 1a-d. From the 3D and 2D plots of Figure 1, we noted the following. * The unitary DM interaction between the two qubits leads to the fact that the NLC functions present different generated correlations. Where the log-negativity, Bell function, LQU, and UIN correlations present different oscillatory behaviors. The generated maximal correlations depend on the purity indicator. In the case \(p=0\), the NLC functions present partial and maximal correlations, whereas for the case \(p=0.3\), the LQU generates a correlation greater than the other measures. The maximal violation of the Bell's inequality (\(B(t)>2\)) depend on the purity indicator. In the case \(p=0\), it occurs, whereas in the case \(p=0.3\), it appears during small time intervals, in which the LQU and log-negativity correlations are stronger. As \(p:0\to p_{c}\), the non-local correlation of \(B(t)\) decreases. Figure 1: 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 \(\chi_{z}=0.5\), \(\Delta=1.5\), \(J_{x}=J_{z}=2\) and \(\gamma=0.0\). 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**). * The phenomenon of sudden changes appears only in the dynamics behavior of the log-negativity and the UIN correlations, represented by the abrupt changes in their dynamical behaviors. The phenomenon of the sudden changes was coined in [38] and experimentally observed in [39]. * During each period of the LQU and UIN behaviors, the UIN has two sudden change (SC) points. In the time intervals between the SC points, the dynamics of the LQU and UIN show maximal and minimal skew information correlations, respectively.