## 4 Dynamics of NLC Functions
While in the time intervals out sides these points, the LQU and UIN have the same correlation that is called the skew-information correlation (SI correlation) [12], it occurs at \(L(t)=U(t))\) meaning that the minimal and maximal skew information correlations are equal. * We have observed that, for \(p=0.3\) and at some particular times, LQU and UIN present quantum correlations, whereas the system is disentangled and does not violates the Bell's inequality. It is worth noting that such similar observations were reported in [40, 41]. The SI correlation depends on the purity indicator, where in the case \(p=0\), it occurs during all the time, whereas in the case \(p=0.3\), the SI correlation occurs in some time windows. Figure 2 shows the effect of the DM interaction coupling \(\chi_{z}=3.0\) on the dynamics behavior of the NLC functions \(L(t)\), \(U(t)\), \(B(t)\) and \(N(t)\). From 3D and 2D plots of Figure 2, we observe that the increase of the coupling of the DM interaction leads to more regular fluctuations, which increase as the purity indicator tends to the critical point \(p_{c}\) (\(p:0\to p_{c}\)). The maxima of the \(L(t)\) and \(B(t)\) are points in their curves of the case \(\chi_{z}=0.5\). The UIN correlation is enhanced by the DM interaction coupling. We note that the entanglement negativity does not change notably in this interval \(p\in[0,p_{c}]\). In Figure 3, the dependence of the NLC functions \(L(t)\), \(U(t)\), \(B(t)\), and \(N(t)\) on the anisotropy parameter \(\Delta\), is plotted for the different cases: (1) In first case, the dynamics of the NLCs are shown for different values of the purity indicator \(p\in[0,1]\) combined with the small value \(\Delta=0.5\) in Figure 3a,b. (2) The NLCs are displayed for the anisotropy interval \(\Delta\in[0,1.5]\) when the two qubits start with pure state in Figure 3g-k, \(p=0\). The 2D-NLC functions are plotted with the same data of Figure 3a,b, but for \(p=0\) in Figure 3e and \(p=0.3\) in Figure 3f. From Figure 3a,b, we note that the generated correlations, via the log-negativity, the maximal Bell function, and LQU, decrease with increasing \(p:0\to p_{c}\). Meanwhile, the opposite behavior is observed for the generated UIN correlation, in other words, the UIN correlations is enhanced by the increase of the purity. The 2D plots show that, for small value of the anisotropy parameter, the period of the regular growth of the log-negativity, Bell function, LQU and UIN correlations grow with the same dynamical behavior. Figure 2: The \(L(t)\) 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 \(\chi_{z}=3\), \(\Delta=1.5\), \(J_{x}=J_{z}=2\) and \(\gamma=0.0\). The \(L(t)\) (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**). Figure 3g-k reveals the effect of the anisotropy for the general choice \(\Delta\in[0,1.5]\). As the anisotropy parameter increases, \(\Delta:0\to 1.5\), the log-negativity, Bell function, and skew information quantifiers raise up (with respect of \(\Delta\)-axis) to their maximal correlations with oscillatory dynamics. The oscillations of the NLC functions along \(\Delta\)-axis increase by increasing time. The log-negativity reaches quickly its maximum than the others quantifiers. In Figure 4, the effect of the phase damping parameter, \(\gamma=0.4\), on the generated correlations is displayed with the same data in Figure 1. We observed that the phase damping leads to only reducing the amplitudes of the NLC functions with the same dynamical behavior. The entanglement log-negativity is more robust against the phase damping than the other NLCs. The green solid curves of Figure 4f, with the phase damping, show that the log-negativity \(N(t)\) suddenly vanishes completely for a finite time and then it undergoes rebirth again.