## 6 Results and Discussions
The similarity of the phase portrait pictures is shown when comparing the results obtained in Figures 14d-16d with data in the literature [33]. It is noted that \"pictures\" in Figures 14d and 15d are like those specific to a _strange attractor_. In this way, the aim of using THA for NFTV of DOE3 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. As can be seen from Figures 11-16, the THA (phases portraits, all the d. figures) was used as _a detection method for possible chaotic dynamic manifestations_ of the DTID's elements (DIE1 and DOE3). [PERSON] et al. [34], had used time-history analysis to study the steady-state nonlinear motion of a damped spring pendulum with an attached linear damped transverse absorber near resonance (primary external and internal resonance) and to investigate the stability of fixed points. Based on this kind of investigation, the above-mentioned dynamic system found that the motion is free of chaos motion [34] (p. 32). [PERSON] et al. [35] use the THA to investigate the stability of nonlinear behavior of a spur gear pair transmission system with backlash, finding the trends in the unstable conditions of these systems. [PERSON] [36], after employing the Pitchfork and Hops Bifurcations to investigate the geared systems in permanent contact, used THA coupled with the Poincare Map Method to detect the creation of limit cycles (new equilibria) in the case of nonlinear softening behavior of geared systems transmission. [PERSON] et al. [37] investigated the mechanism and the characteristics of the varying global compliance (VC) parametric resonances in ball bearings using time-history analysis, phases orbits analysis, Poincare Map Method, and frequency spectra. Thus, the chaotic motions were detected and certified. [PERSON] and [PERSON] [38], calibrated the order of integration of the differential transform method (DTM) using THA for the fractional chaotic Rossler system. Figure 15: Cont. Figure 16: Time-history graphs of NFTV for DOE3, \(\beta_{2}=9^{0}\), \(\xi=11.2\cdot 10^{-3}\) (a) \((\varphi_{3}(t),t)\) (b) \((\frac{\mathrm{d}\varphi_{3}}{\mathrm{d}t}(t),t)\) (c) \(\left(\varphi_{3}(t),\frac{\mathrm{d}\varphi_{1}}{\mathrm{d}t}(t),t\right)\) (d) \(\left(\varphi_{3}(t),\frac{\mathrm{d}\varphi_{3}}{\mathrm{d}t}(t)\right)\). The new method presented in paragraph 5 uses LEA-PM to certify the deterministic chaos. Based on Equations (31)-(33), developing specific MATLAB software, the Lyapunov exponents for the DSOE (29) of NFTV were computed for DIE1 in transition through PPRR. The results are illustrated in Figures 17-20 for the characteristic values of DIE1, such as excitation frequency in the vicinity of \(\eta_{1}=3066\) Hz (see Equation (27), Figure 9, and Table 1), \(\beta_{1}=9^{0}\), \(\Omega_{1}=9632\) rad/s (see Equation (22), Table 1), \(\chi_{\text{nDIE1}}=0.0026\), and the damping ratio in the range \(\xi=0.0016\ldots 0.0114\). Figure 17: Lyapunov exponents of NFTV for DIE1, \(\beta_{1}=9^{0}\), \(\xi=1.6\cdot 10^{-3}\). As can be seen from the Figures 17-20 in (a) is presented the graph variation of the Lyapunov exponent \(\mathrm{L}_{11}\) (see Equation (31), to verify the criteria (37)), in (b) is illustrated the graph variation of the Lyapunov exponent \(\mathrm{L}_{21}\) (see Equation (32), to verify the criteria (37)), in (c) is shown the graph variation of the Lyapunov exponent \(\mathrm{L}_{31}\) ( see Equation (33), to verify the criteria (37)), and in (d) is presented the graph variation of _the sum of Lyapunov exponents_\(\mathrm{L}_{11}+\mathrm{L}_{21}+\mathrm{L}_{31}\) (in order to apply the criteria (39)), for the nonstationary FTV of DIE1 in transition through the PPRR, namely for excitation frequency in the range [2950,3200] Hz. Figure 18: Lyapunov exponents of NFTV for DIE1, \(\beta_{1}=9^{0}\), \(\xi=7.3\cdot 10^{-3}\).