## 7 Conclusions  
This study deals with the dynamic instability analysis for the DTID in transition through PPRR. To achieve the goals, an existent model for FTV of AD was modified by designing an MPCM for FTV of a DTID that included the nonuniformity of AGMI, the nonuniformity of AMMI (_new aspects of the nonuniformities of AGMI and AMMI for DTID's elements that were considered till now linked through CVI_, see Appendix A _Equations (A1)-(A12))_, the quasi-homokinetic TJs of the DTID (_new aspects, see Equations (1) and (5)), the external harmonic excitations induced by the asynchronous electric motor (_new aspects considering higher-order components of the nonuniformity of the torque load, see Equations (6) and (7)_) and the nonuniformity of the reactive torque of the milling system(_new aspects considering higherorder components of the nonuniformity of the reactive torque, see Equations (8) and (9))_. For the investigation of nonstationary dynamic instability of FTV for DTID's elements in the PPRR, the THA was designed for the detection of chaotic dynamic behavior based on the Equations (25) and (26), generating the phase portraits that represent the detection of possible chaos manifestation through PPRR of the DTID's elements. Based on THA, the chaos manifestation for DIE1 was detected through the phase portraits when \(\beta_{1}=9^{0}\), the damping ratio was in the vicinity range \(\xi\in[1.6,7.3]\cdot 10^{-3}\) in the PPRR the excitation frequency being in the vicinity of the value \(\eta_{1}\approx 3066\) Hz (see Figures 11d and 12d), while for \(\beta_{1}=9^{0}\), and the damping ratio \(\xi)/7.3\cdot 10^{-3}\) the phase portrait indicated a limit cycle (see Figure 13d). Using THA, the chaos manifestation for DOE3 was detected through the phase portraits when \(\beta_{2}=9^{0}\), the damping ratio was in the vicinity range \(\xi\in[1.6,6.4]\cdot 10^{-3}\) in the PPRR the excitation frequency being in the vicinity of the value \(\eta_{3}\approx 2792\) Hz (see Figures 14d and 15d), while for \(\beta_{2}=9^{0}\), and the damping ratio \(\xi)/6.4\cdot 10^{-3}\) the phase portrait indicated a limit cycle (see Figure 16d). _A crucial contribution was the computation of the ranges of the principal parametric resonances that vary with the misalignment angles \(\beta_{1}\), \(\beta_{2}\) and with the nonuniformities of DTID's AMMIs_ (see Equations (22) and (23)). _Another significant contribution of this study is the development of a new robust method to certify and confirm the deterministic chaos, namely LEA-PM_ (see the DSOE (29) and (30)). The Lyapunov exponents were computed using Equations (31)-(33) for DIE1 (see Figures 17-20) and Equations (34)-(36) for DOE3 (see Figures 21-24) and were verified by the conditions (37)-(39) of LEA. Using LEA for DIE1 when \(\beta_{1}=9^{0}\), the damping ratio is in the vicinity range \(\xi\in[1.6,11.0]\cdot 10^{-3}\), and the frequency's excitation is in the range \([2950,3200]\) Hz the nonstationary FTV is chaotic (see Figures 17-19). Using, for DOE3, LEA when \(\beta_{2}=9^{0}\), the damping ratio is in the vicinity range \(\xi\in[1.6,6.4]\cdot 10^{-3}\) and the frequency's excitation is in the range \([2650,2950]\) Hz the nonstationary FTV is chaotic (see Figures 21 and 22). The same manifestation is obtained for DOE3 when \(\beta_{2}=9^{0}\), the damping ratio is in the vicinity range \(\xi\in[6.4,10.4]\cdot 10^{-3}\) but the frequency's excitation is in the range \([2700,2925]\) Hz, and therefore, the nonstationary FTV is chaotic (see Figure 23). The images determined using the Poincare Map method for NFTV of DIE1 (see Figures 25-27) and DOE3 (see Figures 28-30) reconfirmed the chaos manifestation of the DTID's elements. From the literature survey [30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42], it was noted that all the research data are based on one method (THA, MLEM, PMM) or two methods (THA-MLEM, THA-PMM) for analyzing the chaotic dynamic behavior of the systems.