## 1 Introduction  
The present work investigates the ADFTV, taking into consideration the following physical characteristics and phenomena: the fact that the links of the ADE do not manifest the property of isometry due to the ADKN and ADGN [1, 2], and the ADE's links are even considered CVJs (constant velocity joints); the effect of the torque loads, such as the entry gearbox torques [3] (p. 360); and the effects of ADGN of the ICSM, and the effects of the elasticity and damping in torsion for the ADE's link-joints, which are functions of the ADE twisting angle. The ADKN and ADGN that generate the isometry nonuniformity properties of AD (INP-AD) are mentioned in [2, 4] and experimentally studied by [PERSON] in his Ph.D. thesis [5] (pp. 77-94). This phenomenon of INP-AD induces a nonlinear behavior of driveshafts generating forced parametric vibrations in the frequency range of 0.1-12 kHz, as shown by published experiments and data in [5]. [PERSON] and [PERSON] were the first to study the dynamic behavior of driveshafts, and they discovered through experiments the nonlinear parametric vibrations (NPV) of a universal AD joint [6]. Their research revealed and illustrated experiments concerning the INP-AD of quasi-CVJ driveshaft transmission [7] by [PERSON] and [PERSON]. [PERSON] et al. studied the amplitude variations as functions of frequency for the ADFTV considering the complete powertrain [8], developing a dynamic model with 29 DOF (degrees of freedom). [PERSON] et al. developed a model for the nonlinear ADFTV 3 DOF [9]. The rigidity and the damping of ADFTV were determined for calibration. Unfortunately, the DMADFTV developed in [9] does not have a nonlinear characteristic involving ICSM, ADKN, and ADGN for the ADE, respectively. [PERSON] et al. studied and analyzed in [10] the dynamics of a driveshaft system with an interval of uncertainty, while [PERSON] and [PERSON] realized a survey regarding the theoretical modelization of the geometrical structure and its influence on the kinematics of a universal joint [11]. In [12], [PERSON] et al. developed an analysis of the NVH for a truck driveshaft using an SSA (Six Sigma Approach), while in his master thesis, [PERSON] modeled and simulated the kinematics and dynamics of the drive train [13]. The design and analysis of AD were investigated by [PERSON] et al. [14], and [PERSON] et al. developed a model control for vehicle rollover prevention with time-varying speed [15]. Previously, [PERSON], [PERSON], et al. [16] studied and implemented a hierarchical synchronization control for the active rear-axle steering system. The investigation of high-frequency torsional vibration of automotive drive lines (ADL) was done by [PERSON] et al. using genetic algorithms [17]. At the same time, [PERSON] et al. researched diagnosing defects for ADL by analyzing induced vibration signals generated by such defect phenomena [18]. In [19], [PERSON] studied the dynamic parametric behavior of the Cardan shaft, also performing a sensitivity analysis. This study aimed to investigate the dynamic instability of the ADFTV for the ADE by developing a detection method for chaotic manifestation. The detection method represents the construction of a phase portrait of the NSVA as a function of the NSA in transition through APPR for the ADFTV. Such a picture's phase portrait analysis shall be conducted to confirm the chaotic manifestation of the ADFTV, such as Jacobi stability analysis or Lyapunov stability analysis. Many researches have used the Jacobi stability analysis of dynamical systems in the last two decades based on the Kosambi-Cartan-Chern (KCC) theory. [PERSON] et al. studied the Jacobi stability analysis in gravitation and cosmology using the KCC formalism in Riemannian and Finslerian spaces [20], while [PERSON] et al. using the same KCC formalism determined the geometric invariants and the Berwald's connection for the Lorenz system to make the Jacobi stability analysis [21]. In recent years, research concerning Jacobi stability analysis was carried out by [PERSON] et al.