## 6 Results and Discussions  
Figures 11-13 illustrate the graphs of time-history to analyze the dynamic instability of NFTV for DTID's DIE1 in transition through PPRR \(\eta_{1}\simeq 3066\,\mathrm{Hz}\) (see Equation (27), Figure 9, and Table 1), considering the worst misalignment for DIE1 \(\beta_{1}=9^{0}\), \(\Omega_{1}=9632\,\mathrm{rad}/\mathrm{s}\) (see Equation (22), Table 1), \(\chi_{\mathrm{nDIE1}}=0.0026\), and the damping ratio \(\xi=0.0016\ldots 0.0129\). MATLAB software was developed based on Equation (25) for NFTV of DIE1 to calculate the time-history \((\varphi_{1}(t),\mathrm{t}),(\frac{\mathrm{d}\varphi_{1}}{\mathrm{dt}}(t), \mathrm{t}),\left(\varphi_{1}(t),\frac{\mathrm{d}\varphi_{1}}{\mathrm{dt}}(t), t\right)\)  
Figure 12: Cont. Figure 12: Time–history graphs of NFTV for DIE1, \(\beta_{1}=9^{0}\), \(\xi=7.3\cdot 10^{-3}\) (a) \((\varphi_{1}(t),t)\), (b) \((\frac{\mathrm{d}\varphi_{1}}{\mathrm{d}t}(t),t)\), (c) \(\left(\varphi_{1}(t),\frac{\mathrm{d}\varphi_{1}}{\mathrm{d}t}(t),t\right)\), (d) \(\left(\varphi_{1}(t),\frac{\mathrm{d}\varphi_{1}}{\mathrm{d}t}(t)\right)\). In Figures 11a-c-13a-c, it is remarked that the nonstationary \(\varphi_{1}(t)\), \(\frac{\mathrm{d}\varphi_{1}}{\mathrm{d}t}(t)\) for NFTV of DIE1 manifests the \"beatings\" effects and a rapid decrease of the extreme values with the damping's increases in the range \((1.6\ldots 12.9)10^{-3}\). Figures 11d-13d illustrate that the phase portraits represent the chaotic manifestation of a _strange attractor_ for \(\xi=1.6\cdot 10^{-3}\) and \(\xi=7.3\cdot 10^{-3}\), multiple limit cycles and \"turning points\", respectively, while for \(\xi=12.9\cdot 10^{-3}\) only a limit cycle is manifested meaning that the nonstationary dynamic behavior for DIE1 has an ergodic manifestation starting from the value of the damping ratio. The results obtained in Figures 11d-13d show the similarity of the phase portrait pictures when compared with data in the literature [33]. It is noted that \"pictures\" in Figures 11d and 12d are like those specific to a _strange attractor_. In this way, the aim of using time-history analysis (THA) for NFTV of DIE1 was reached, THA being thus a detection method for chaos manifestation of NFTV for the elements of DTID. For the cases when the damping ratio is in the vicinity range \(\xi=(1.6\ldots 7.3)\cdot 10^{-3}\), the deterministic chaos of NFTV for DIE1 needs to be certified using the new robust method LEA-PM. Figures 14-16 illustrate the graphs of time-history to analyze the dynamic instability of NFTV for DTID's DOE3 in transition through PPR \(\eta_{3}\simeq 2792\) Hz (see Equation (28), Figure 10, and Table 2), considering the worst misalignment for DOE3 \(\beta_{2}=9^{0}\), \(\Omega_{3}=8767\) rad/s (see Equation (23) and Table 2), \(\chi_{\mathrm{nDOE3}}=0.0027\), and the damping ratio \(\xi=0.0016\ldots 0.0112\). MATLAB software was developed based on Equation (26) for NFTV of DOE3 to calculate the time-history \((\varphi_{3}(t),t)\), \((\frac{\mathrm{d}\varphi_{3}}{\mathrm{d}t}(t),t)\), \((\varphi_{3}(t),\frac{\mathrm{d}\varphi_{3}}{\mathrm{d}t}(t),t)\), \((\varphi_{3}(t),\frac{\mathrm{d}\varphi_{3}}{\mathrm{d}t}(t))\). In Figures 14a-c-16a-c, it is remarked that the nonstationary \(\varphi_{3}(t)\), \(\frac{\mathrm{d}\varphi_{3}}{\mathrm{d}t}(t)\) for NFTV of DOE3 manifests the same \"beatings\" effects and a rapid decrease of the extreme values with the damping's increases in the range \((1.6\ldots 11.2)10^{-3}\) as for DIE1 (see Figures 11a-c-13a-c). The graphs illustrated in Figures 14d-16d show that the phase portraits represent the chaotic manifestation of a _strange attractor_ for \(\xi=1.6\cdot 10^{-3}\) and \(\xi=6.4\cdot 10^{-3}\), multiple limit cycles and \"turning points\", respectively, while for \(\xi=11.2\cdot 10^{-3}\) only a limit cycle is manifested meaning that the nonstationary dynamic behavior for DOE3 has an ergodic manifestation starting from this value of the damping ratio.