## 6 Results and Discussions
Through the analysis of the graphs illustrated in Figure 17, it can be concluded that the conditions imposed by Equations (37)-(39), as mentioned in [23; 31], are fulfilled and, therefore, in the excitation frequency's range of 2950-3200 Hz, the range of PPRR for FTV of DIE1, for \(\xi=1.6\cdot 10^{-3}\) the deterministic chaos (strange attractor) is manifested. For certification of this manifestation, the last part of the new robust method LEA-PM, the qualitative PMM, will be used. Through the analysis of the graphs illustrated in Figure 18, it can be concluded that the conditions imposed by Equations (37)-(39), as mentioned in [23; 31], are fulfilled and, therefore, in the excitation frequency's range of 2950-3200 Hz, the range of PPRR for FTV of DIE1, for \(\xi=7.3\cdot 10^{-3}\) the deterministic chaos (strange attractor) is manifested. For certification of this manifestation, the last part of the new robust method LEA-PM, the qualitative PMM, will be used. Figure 19: Lyapunov exponents of NFTV for DIE1, \(\beta_{1}=9^{0}\), \(\xi=11\cdot 10^{-3}\). Through the analysis of the graphs illustrated in Figure 19, it can be concluded that the conditions imposed by Equations (37)-(39), as mentioned in [23; 31], are fulfilled and, therefore, in the excitation frequency's range of 2950-3200 Hz, the range of PPRR for FTV of DIE1, for \(\xi=11\cdot 10^{-3}\) the deterministic chaos (strange attractor) is manifested. For certification of this manifestation, the last part of the new robust method LEA-PM, the qualitative PMM, will be used. Through the analysis of the graphs illustrated in Figure 20, it can be concluded that the conditions imposed by Equations (37)-(39), as mentioned in [23; 31], are not fulfilled and, therefore, in the excitation frequency's range of 2950-3105 Hz, the range of PPRR for FTV of DIE1, for \(\xi=11.4\cdot 10^{-3}\) the deterministic chaos (strange attractor) is not possible. Using LEA-PM, it is certified by the conditions imposed (37)-(39) that for damping ratio in the range \(\xi=(11.4\ldots 18.6)\cdot 10^{-3}\) the NFTV of DIE1 in transition through PPRR is an ergodic dynamic behavior representing a transition to chaos, as mentioned in [23]. Figure 20: Lyapunov exponents of Nonstationary FTV for DIE1, \(\beta_{1}=9^{0}\), \(\xi=11.4\cdot 10^{-3}\). Based on Equations (34)-(36), developing specific MATLAB software, the Lyapunov exponents for the DSOE (30) of NFTV were computed for DOE3 in transition through PPRR. The results are illustrated in Figures 21-24 for the characteristic values of DOE3, such as excitation frequency in the vicinity of \(\eta_{3}\simeq 2792\) Hz (see Equation (28), Figure 10, and Table 2), \(\beta_{2}=9^{0}\), \(\Omega_{3}=8767\) rad/s (see Equation (23) and Table 2), \(\chi_{\rm nDOE3}=0.0027\), and the damping ratio in the range \(\xi=0.0016\ldots 0.0114\). As can be seen from the Figures 21-24 in (a) is presented the graph variation of the Lyapunov exponent L\({}_{13}\) (see Equation (34), to verify the criteria (37)), in (b) is illustrated the graph variation of the Lyapunov exponent L\({}_{23}\) ( see Equation (35), to verify the criteria (37)), in (c) is shown the graph variation of the Lyapunov exponent L\({}_{33}\) ( see Equation (36), to verify the criteria (37)), and in (d) is presented the graph variation of _the sum of Lyapunov exponents_ L\({}_{13}+\) L\({}_{23}+\) L\({}_{33}\) (in order to apply the criteria (39)), for the nonstationary FTV of DOE3 in transition through the PPRR, namely for excitation frequency in the range [2650,2950] Hz. Figure 21: Lyapunov exponents of NFTV for DOE3, \(\beta_{2}=9^{0}\), \(\xi=1.6\cdot 10^{-3}\). Through the analysis of the graphs illustrated in Figure 21, it can be concluded that the conditions imposed by Equations (37)-(39), as mentioned in [23; 31], are fulfilled and, therefore, in the excitation frequency's range of 2650-2950 Hz, the range of PPRR for FTV of DOE3, for \(\xi=1.6\cdot 10^{-3}\) the deterministic chaos (strange attractor) is manifested.