## 1 Introduction
[15] investigated the vibration characteristics of the shaft systems for dredging pumps using the FEM to determine the upper and the lower limits for critical speeds of torsional vibration. It was observed that the rated speed of the dredging pump was far less than the predicted critical speed. Therefore, FEM offers only some reference value on the stability analysis of the shaft system. Experimental investigations and finite element analysis were carried out in the literature [16] to determine the natural frequencies and the mode shapes for steel and CFRP (carbon fiber reinforced plastic) driveshafts. This interest in CFRP is due to the advantages afforded to the industry when using CFRP instead of steel. [PERSON] and [PERSON] demonstrated through [17] that a method of avoiding parametric resonances of the universal joint driveline is accelerating through these regions. As identified in the literature [13, 14, 15, 16, 17], it is necessary to investigate the PPRR locally and globally in close detail. [PERSON], in his Ph.D. thesis [18] (pp. 88-94), experimentally found the pitting and the microcracks on the tripod of the driveshaft. He explained that the driveshaft joints have similar dynamic behavior as the geared system transmission [18] (p. 117). Additionally, he stated that the bending and torsion vibrations manifest a \"strange beating behavior\" [18] (pp. 138-139) that is generated by the engine excitation, geometric and mass nonuniformities, and kinematic nonuniformity of joints. Inthe literature, the investigation of the chaotic behavior of the geared system transmission was performed in [19] (pp. 102-103, 127-131), and it was found that geared systems may have a deterministic chaos vibration that conducts in time to phenomena as pitting and microcracks. These aspects mentioned before must be investigated. One of the present paper's goals is to investigate possible chaotic dynamic behavior for the NFTV in transition through PPRR. [PERSON] and [PERSON] [20] investigated the chaotic motion of an auto-parametric system by simulations using the maximal Lyapunov exponent method (MLEM) and laboratory experiments. In [21], [PERSON] studied the control over chaotic behavior of an active magnetic bearing system by MLEM, Poincare Maps, and phase portraits. [PERSON] et al. investigated the simultaneous resonance and stability analysis of unbalanced asymmetric thin-walled composite shafts in [22] using the multiple-scale approach (MSA) and the phase portrait graphs. As identified in [20; 21; 22], the methods of investigating the chaotic dynamic behavior are the MLEM (quantitative method) and the Poincare Maps method (qualitative method), and for the computation of dynamic instability frontiers regions the perturbation approach MSA is used. The present paper will use time-history analysis to detect the dynamic chaotic behavior using the phase portrait graphs of the NFTV for DTID. The chaos will be investigated using a robust designed method given by the modified Lyapunov exponent's method (MLEM) coupled with Poincare Maps (PM), that is, Lyapunov Exponents Approach-Poincare Map (LEA-PM), of the NFTV for DTID in transition through PPRR. These are the novel aspects of the new method: detection of chaos by analysis of time-history and investigation of chaos by employing LEA-PM. Why is this new and \"mathematical affordable\" method used by the designers of DTID? As mentioned before, most researchers, such as in [21], use MLEM directly without detecting chaos before. The new robust method detects possible chaotic behavior. Additionally, the MLEM is not sufficient for deterministic chaotic behavior, as mentioned in [23], and for this reason, MLEM was modified in a more complex new method, LEA. All the research on chaos manifestations needed a second confirmation for the deterministic chaos (the manifestation of _strange attractors_) due to the complexity of the dynamic phenomena when chaos occurs. Accordingly, the new method involves using a qualitative second method, the PM.