## 6 Results and Discussions
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 22, 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 \(\xi=6.4\cdot 10^{-3}\) the deterministic chaos (strange attractor) is manifested. For certification of this manifestation, the last part of the new robust method LEA-PM, the positively, the qualitative PMM, will be used. Figure 22: Lyapunov exponents of NFTV for DOE3, \(\beta_{2}=9^{0}\), \(\xi=6.4\cdot 10^{-3}\). Through the analysis of the graphs illustrated in Figure 23, 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 2700-2925 Hz, the range of PPRR for FTV of DOE3, for \(\xi=11.2\cdot 10^{-3}\) the deterministic chaos (strange attractor) is manifested. For certification of this manifestation, the last part of the new robust method LEA-PM respectively, the qualitative PMM, will be used. Through the analysis of the graphs illustrated in Figure 24, 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 2670-2925 Hz, the range of PPRR for FTV of DOE3, for \(\xi=11.2\cdot 10^{-3}\) 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 \(\xi=(11.2\ldots 16)\cdot 10^{-3}\) the NFTV of DOE3 in transition through PPRR is an ergodic dynamic behavior representing a transition to chaos, as mentioned in [23]. [PERSON] and [PERSON] [39], investigated the nonlinear vibration of a basic torsional system driven by a [PERSON]'s joint using time-history analysis (THA) and the bifurcation analysis, and the quasi-periodic and chaotic motion were revealed. [PERSON] and [PERSON] [40] studied the torsional  
Figure 12:0 Lyapunov exponents of NFTV for DOE3, \(\beta_{2}=9^{0}\), \(\xi=10.4\cdot 10^{-3}\). instabilities in a [PERSON]'s joint-driven system via maximal Lyapunov's exponents (the well-known MLEM). They found a direct link between the maximal Lyapunov exponents and the instability regions for subharmonic and combination resonances. To certify using a qualitative method that the DIE1 and DOE3 have NFTV in the PPRR of chaotic type, MATLAB software was developed based on DSOE (29) for DIE1 and on DSOE (30) for DOE3, to represent the Poincare Maps that represents the intersection of the orbits in the phase portraits with an orthogonal surface at equal periods (N number of points), using the mathematical procedure presented in [32] (p. 194). The Poincare map represents saddle points or saddle separatrices images if the vibration motion is periodic or quasi-periodic. Figure 25 is presented the PM for DIE1 when \(\xi=1.6\cdot 10^{-3}\), \(\eta_{1}=3066.5\,\mathrm{Hz}\) (see Equation (27), Figure 9, and Table 1), \(\beta_{1}=9^{0}\) using 50,000, 15,000, 10,000, and 1000 sections of orthogonal surface to the orbits in the phase portrait \(\left(\varphi_{1},\frac{\mathrm{d}\varphi_{1}}{\mathrm{d}t}\right)\). Figure 12:1 Lyapunov exponents of NFTV for DOE3, \(\beta_{2}=9^{0}\), \(\xi=11.2\cdot 10^{-3}\). From the phase portraits illustrated in Figure 25, the properties of PM can be noted in the case of strange attractor: _diffuse structure of points having a different density of pixels per unit area_ of the surface and _auto-similarity_ of the image. Figures 26 and 27 present the PM for DIE1 when \(\xi=7.3\cdot 10^{-3},\eta_{1}=3066.5\,\mathrm{Hz}\), \(\beta_{1}=9^{0}\) and \(\xi=11\cdot 10^{-3},\eta_{1}=3066.5\,\mathrm{Hz}\), \(\beta_{1}=9^{0}\) using 15,000 to 1000 sections of orthogonal surface to the orbits in the phase portrait \(\left(\varphi_{1},\frac{\mathrm{d}\varphi_{1}}{\mathrm{d}t}\right)\).