###### Abstract  
The present work aims to design a robust method to detect and certify the deterministic chaos or ergodic process for the forced torsional vibrations (FTV) of a double tripod industrial driveshaft (DTID) in transition through the principal parametric resonance region (PPRR). Based on a modified designed model of FTV for an automotive driveshaft (AD), a robust method comprising THA, LEA, and PM (THA-LEA-PM) was created that certifies the existence of deterministic chaos or an ergodic process for the dynamic behavior of DTID's FTV in transition through PPRR. These methods represent a design tool for the DTID's FTV control of the homokinetic driveline transmission in the industry.  
The present work aims to design a robust method to detect and certify the deterministic chaos or ergodic process for the forced torsional vibrations (FTV) of a double tripod industrial driveshaft (DTID) in transition through the principal parametric resonance region (PPRR) which is considered by the researchers in the field as one of the most important resonance regions for the systems having parametric excitations. The DTID's model for FTV considers the following effects: nonuniformities of inertial characteristics of the DTID's elements, the harmonic torque excitation induced by the asynchronous electrical motor used for a heavy-duty grain mill, and the harmonic reaction torque generated by different granulation of the substance needed to be milled. Based on these aspects, a model of the FTV for the DTID was designed which was a modified, physically consistent model already used by the authors to investigate the FTV of automotive driveshafts (homokinetic transmission). For the DTID elements, the dynamic instability for nonstationary FTV in the PPRR using time-history analysis (THA) was analyzed--THA represents the phase portraits. Time-history analysis is a detection method for possible chaotic dynamic behavior for the nonstationary FTV (NFTV) in transition through PPRR. If this dynamic behavior was seen, a new robust method LEA-PM was created to certify and confirm the deterministic chaos for the NFTV of DTID. The new method, LEA-PM, is composed of the Lyapunov exponent's approach (LEA) coupled with the Poincare Map (PM) applied to the global system of differential equations that describe the FTV of DTID in the PPRR. This new robust method, which embeds LEA and PM, LEA-PM, establishes if the mechanical system has a deterministic chaotic dynamic behavior (strange attractor) or an ergodic dynamic process in this resonant region. LEA represents a new method that includes not only the maximal Lyapunov exponent method (MLEM) but also new mathematical criteria that is \"the sum of all Lyapunov exponents has to be negative\" which, coupled with MLEM, indicates the presence of deterministic chaos (strange attractors). THA-LEA-PM had been used for the NFTV of DTID computing the phase portraits, the Lyapunov exponents, and representing the Poincare Maps of the NFTV for the DTID's elements in transition through PPRR, founding deterministic chaos or ergodic dynamic behavior. Based on the obtained results, numerical simulations revealed the pitting manifestations of the DTID's elements, typical for the geared systems transmission, mentioned recently in experimental data research for the homokinetic transmissions. Using the new robust method, THA-LEA-PM (time-history analysis coupled with LEA-PM) can be used in future research for chaotic dynamic analysis of DTID's NFTV transition through superharmonic resonances,subharmonic resonances, combination resonances, and internal resonances. Time-history analysis as a detection method for chaos and LEA-PM as a certifying method for deterministic chaos can be integrated as a design tool for DTID's FTV control of the homokinetic transmission.  
chaotic dynamic behavior; time-history analysis; Lyapunov exponents approach; Poincare Map; principal parametric resonance; multibody dynamics