## 7 Conclusions
For the first time, three methods are used to study the chaotic behavior of the DTID through the PPRR: THA for detection of the chaos, LEA (a modified MLEM) to certify the existence of deterministic chaos (_strange attractors_), and PM to reconfirm deterministic chaos. The _innovations_ and _contributions_ of the present work can be summarized as follows:  
1. The design of a modified physically consistent model (MPCM) of the FTV for a double tripod industrial driveshaft (DTID) considering the following phenomena: * the nonuniformity of axial mass moments of inertia (AMMI) and axial geometric moments of inertia (AGMI) of the DTID's elements: driving input element 1 (DIE1) and driven output element 3 (DOE3), * the geometric inertia nonuniformity of the tripod joints (TJ1 and TJ2) of the DTID, * the geometric and kinematic nonuniformity of the quasi-constant velocity joints (quasi-CV)), * the nonuniformity of the torque load's higher order received by DTID from the asynchronous electric motor, * the nonuniformity of the reactive torque's higher order of the milling system; * The computation of the free natural frequencies of the DTID's elements as functions of the misalignment angles of the DTID's elements \(\beta_{1}\), \(\beta_{2}\), the AMMI and the AGMI of the DTID's elements; * The use 4. The design of a quantitative method to certify the occurrence of deterministic chaos (strange attractors) LEA (Lyapunov exponents approach), that is a modified MLEM (maximal Lyapunov exponent method) coupled with a new criterion, the sum of all Lyapunov's exponents is negative, for the nonstationary FTV for DTID's elements in transition through principal parametric resonance region (PPRR);
5. The use of Poincare Maps (PM) to reconfirm the occurrence of deterministic chaos (strange attractors) for the nonstationary FTV for DTID's elements in transition through principal parametric resonance region (PPRR);
6. The determination of the range for damping ratio, using THA-LEA-PM, where the deterministic chaos occurs (strange attractor) for the nonstationary FTV for DTID's DIE1 in transition through principal parametric resonance region (PPRR) \(\eta_{1}\in[2950,3200]\) Hz and worst misalignment \(\beta_{1}=9^{0}\), respectively \(\xi\in[1.6,11]\cdot 10^{-3}\);
7. The determination of the range for damping ratio, using THA-LEA-PM, where the deterministic chaos occurs for the nonstationary FTV for DTID's DOE3 in transition through principal parametric resonance region (PPRR) \(\eta_{3}\in[2700,2925]\) Hz and worst misalignment \(\beta_{2}=9^{0}\), \(\xi\in[1.6,10.4]\cdot 10^{-3}\);
8. For the first time, through the chaotic dynamic behavior for the nonstationary FTV for DTID's elements in transition through the principal parametric resonance region (PPRR), it was revealed theoretically the manifestation of pitting, microcracks, and cracks and the similarity between the phenomena of homokinetic transmission with the phenomena of geared systems transmission. In conclusion, the robust method THA-LEA-PM, along with the advantages highlighted earlier, also has the considerable advantage of being \"mathematical affordable\" for designers. This advantage results from the fact that the investigation is possible in the simplest Riemannian space (the space the Euclidian metric), while the newest method of investigating the chaos KCC method, as mentioned in [24], allows this investigation in Finsler spaces (involving the problem of curvature properties of homogeneous Finsler spaces with \((\alpha,\beta)\)-metrics which is one of the central problems in Riemann-Finsler geometry) as well as the management of Jacobi stability analysis for Berwald curvature. In the DTID's design stage, this new robust method THA-LEA-PM prevents the excessive pitting, microcracks, or even failure by cracks for DTID's elements DIE1, DOE3, and trununion tripods axes of the DTID's joints.